Research ArticleGeneral Analysis of Timoshenko Beams on Elastic Foundation
S Abohadima M Taha and M A M Abdeen
Department of Engineering Mathematics and Physics Faculty of Engineering Cairo University Giza 12211 Egypt
Correspondence should be addressed to M A M Abdeen mtahtahab47engcuedueg
Received 1 September 2015 Accepted 8 October 2015
Academic Editor Zhen-Lai Han
Copyright copy 2015 S Abohadima et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
General analytical solutions for stability free and forced vibration of an axially loadedTimoshenko beam resting on a two-parameterfoundation subjected to nonuniform lateral excitation are obtained using recursive differentiationmethod (RDM) Elastic restraintsfor rotation and translation are assumed at the beam ends to investigate the effect of support weakening on the beam behaviorHowever the effects of rotational inertia and shear stress induced from the axial load are considered The obtained solutions areverified first and then used to investigate the significance of different parameters on the beam behavior In addition solutions offorced vibration are analyzed to highlight the effects of excitation nonhomogeneity on the beam behavior
1 Introduction
The static and dynamic analysis of Timoshenko beams withdifferent configurations are of great importance for the designof many engineering applications Analytical solutions arelimited to study the behavior of Timoshenko beamswith sim-ple configuration due to the mathematical complexity of theproblem Ruta [1] used Chebyshev polynomials to study non-prismatic Timoshenko beams The invalidation of Bernoulli-Euler theory for the cases of free-free and pinned-free shearbeams has been discussed by Kausel [2] Attarnejad et al [3]applied the differential transform (DT) to investigate the freevibration of Timoshenko beams resting on two-parameterelastic foundations Taha and Nassar [4] studied free andforced vibration of stressed Timoshenko beams resting ontwo-parameter foundations using the Adomian decomposi-tion method (ADM) Free and forced vibrations of Timo-shenko beams described by a single difference equation havebeen studied by Majkut [5]
On the other hand numerical methods are used in theanalysis of Timoshenko beams with complex configurationsCheng and Pantelides [6] studied Euler and Timoshenkobeams using continuous models and the stiffness matrixmethod Geist and Mclaughlin [7] discussed the phenom-enon of double frequencies in Timoshenko beams at certainvalues of beam slenderness ratios Chen [8] used differential
quadrature element method (DQEM) to study the vibrationof nonprismatic shear deformable beams resting on elasticfoundations Monsalve et al [9] presented dynamic analysisof Timoshenko beam-column with generalized end condi-tions on an elastic foundation using finite element method(FEM) Kocaturk and Simsek [10] investigated the vibrationof Timoshenko beams under various boundary conditionsusing Lagrange equations and used Lagrange multipliers toaccount for different cases of boundary conditions Nguyen[11] studied the vibration of prestressed Timoshenko beamseither fully or partially supported on an elastic foundationusing the finite elementmethod (FEM) Auciello [12] used theRayleigh-Ritz approach and boundary characteristic orthog-onal polynomials are chosen as trial functions to investigatethe vibration of Timoshenko beams on two-parameter foun-dations
Analytical solutions for boundary value problems arealways preferable compared to numerical solutions as theyare more general and give a better understanding of themodel behavior On the other hand unfortunately analyt-ical solutions are limited to simple and idealized modelsRecursive differentiation method (RDM) is an efficient ana-lytical method proposed by Taha [13] for tackling boundaryvalue problems governed by linear or nonlinear differen-tial equations The method constructs analytical solutionsbased on Taylor expansion and can deal with complicated
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 182523 11 pageshttpdxdoiorg1011552015182523
2 Mathematical Problems in Engineering
configurations of beam-foundation systems in finite domainTaha and Doha [14] used RDM to study dynamics of beam-foundation systems assuming Euler-Bernoulli hypothesis
In the present paper RDM is implemented to obtainanalytical solutions for the differential equations governingthe static and dynamic behavior of axially loaded Timo-shenko beams resting on two-parameter foundation withelastic end restraints and subjected to nonuniform lateralexcitation Both the influence of rotational inertia and theshear stress induced from the effect of axial load proposed byTimoshenko and Gere [15] will be considered The stabilitybehavior of cantilever Timoshenko beam resting on elasticfoundation will be analyzed In addition the influences of theelastic end restraints weakening will be studied Further thesignificance of different parameters on the maximum lateralresponse amplitude of the beam due to different types oflateral excitation will be investigated
2 Formulation of the Governing Equations
21 Dynamic Equations of Timoshenko Beams The equationsof translational and angular motion of an infinitesimalelement of an axially loaded Timoshenko beam subjected tolateral excitation resting on two-parameter foundation shownin Figure 1 are
120597119881
120597119909+ 119902 (119909 119905) minus 119896
1119910 (119909 119905) + 119896
2
1205972119910
1205971199092= 120588119860
1205972119910
1205971199052 (1)
119881 (119909 119905) + 119901120597119910
120597119909+
120597119872
120597119909= 120588119868
1205972120579
1205971199052 (2)
The force-displacement relations considering additionalshear stress induced from the component of the axial load inthe direction of the deformed section (119901 sin(120572) cong 119901120572(119909 119905))
The relation between the deformation components of thebeam element (shown in Figure 1(c)) is
120597119910
120597119909= 120579 + 120574 (5)
where 119864 is the modulus of elasticity of the beam material119868 is the moment of inertia of the beam cross section 120588 isthe density 119860 is the area of the cross section 119860
119904= 120581119860 is
the effective shear area 120581 is a correction factor to take intoaccount the nonuniform distribution of shear stress (120581 = 23
for rectangular cross section) 119901 is the axial applied load 1198961
and 1198962are the linear and shear foundation stiffness factors per
unit length of the beam 119902(119909 119905) is the lateral excitation actingon the beam 120579(119909 119905) is the rotation of the beam cross section120574 is the deformation angle due to the shear force 120572(119909 119905) isthe angle between the axial load 119901 and the normal to thedeformed cross section 119881(119909 119905) is the shear force 119872(119909 119905) is
the bending moment 119910(119909 119905) is the lateral response of thebeam 119909 is the coordinate along the beam and 119905 is the time
Substituting (3) (4) and (5) into (1) and (2) the equationsof motion may be expressed as
119860119904119866(
1205972119910
1205971199092minus
120597120579
120597119909) minus 119901
120597120572
120597119909minus 1198961119910 + 1198962
1205972119910
1205971199092minus 120588119860
1205972119910
1205971199052
= minus119902 (119909 119905)
119860119904119866(
120597119910
120597119909minus 120579) minus 119901120572 + 119901
120597119910
120597119909+ 119864119868
1205972120579
1205971199092minus 120588119868
1205972120579
1205971199052= 0
(6)
Introducing the dimensionless variables 120585 = 119909119871 and 119908 =
where Ω is the excitation frequency Most of the practicalcases for nonuniform continuous loading function can beclosely simulated by assuming quadratic loading function inthe form
119860119904119866120595 (120585) + 119901120572 (120585) minus (119860
119904119866 + 119901)
119889120593
119889120585minus
119864119868
1198712
1198892120595
1198891205852
minus 1205881198601199032Ω2120595 (120585) = 0
(10)
22 Additional Shear Induced from the Axial Load There arethree approaches in dealing with the shear stress inducedfrom the axial load the first approach neglects the additional
Mathematical Problems in Engineering 3
kR0
p
kT0 kT1
q(x t)
k1 and k2
Ly
p
x
kR1
(a)
p
y
x0q(x t)dx
V
M
p
120588Ay∙∙dx
M+120597M
120597xdx
V +120597V
120597xdx
120588I120579∙∙dx
dx
dx
) )k1y minus k2
1205972y
120597x
(b)
x
O
y
120574
120579
dy
dx
(c)
Figure 1 (a) Timoshenko beam on elastic foundation (b) Element forces (c) Total deformation
shear stress the second approach assumes the inclinationangle between the axial load and the outward normal tothe deformed cross section 120572(119909 119905) = 120579(119909 119905) and the thirdapproach assumes 120572(119909 119905) = 120597119910120597119905
According to the first approach (120572(120585) = 0) (10) may berewritten as
(12)Using the dimensionless parameters defined in (12) (11) maybe expressed as
119889120595
119889120585=
1205902
1198782
1198892120593
1198891205852minus
1205901
1198782120593 (120585) +
119876 (120585)
1198782 (13)
1198892120595
1198891205852+ 1205900
119889120593
119889120585minus 1205903120595 (120585) = 0 (14)
where1205900= 1198782+ 119875
1205901= 1198701minus 1205824
119865
1205902= 1198782+ 1198702
1205903= 1198782minus
1205824
119865
1205782
119876 (120585) =1198713
119864119868119902 (120585)
(15)
4 Mathematical Problems in Engineering
where 119878 is the slenderness parameter 120582119865is the frequency
parameter 120578 is the slenderness ratio 119903 is the radius of gyra-tion 119875 is the axial load parameter119870
1is the foundation linear
stiffness parameter and 1198702is the foundation shear stiffness
parameterInserting (14) into (13) the equation describing the lateral
response amplitude of the Timoshenko beam based on thefirst approach is
1198894120593
1198891205854minus (
1205901+ 12059021205903minus 12059001198782
1205902
)1198892120593
1198891205852+
12059011205903
1205902
120593 (120585)
=1205903119876 minus 119876
10158401015840
1205902
(16)
Similarly for the second approach (120572(120585) = 120595(120585)) (11) may beexpressed as
119889120595
119889120585=
1205902
1205900
1198892120593
1198891205852minus
1205901
1205900
120593 (120585) +119876 (120585)
1205900
1198892120595
1198891205852+ 1205900
119889120593
119889120585minus 1205904120595 (120585) = 0
(17)
Also the equation of the lateral response amplitude of theTimoshenko beam according to the second approach is
1198894120593
1198891205854minus (
1205901+ 12059021205904minus 1205902
0
1205902
)1198892120593
1198891205852+
12059011205904
1205902
120593 (120585)
=1205904119876 minus 119876
10158401015840
1205902
(18)
where 1205904= 119875 + 120590
3
According to the third approach (120572(120585) = 119889120593119889120585) thedynamic equations of lateral response amplitude can beobtained as
119889120595
119889120585=
1205905
1198782
1198892120593
1198891205852minus
1205901
1198782120593 (120585) +
119876 (120585)
1198782 (19)
1198892120595
1198891205852+ 1198782 119889120593
119889120585minus 1205903120595 (120585) = 0 (20)
1198894120593
1198891205854minus (
1205901+ 12059031205905minus 1198784
1205905
)1198892120593
1198891205852+
12059011205903
1205905
120593 (120585)
=1205903119876 (120585) minus 119876
10158401015840
1205905
(21)
where
1205905= 1205902minus 119875 (22)
The dynamic equations of different approaches may beexpressed as
119889120595
119889120585= 1198611
1198892120593
1198891205852+ 1198612120593 (120585) + 119861
3119876 (120585)
120595 (120585) = 1198614
1198893120593
1198891205853+ 1198615
119889120593
119889120585+ 11986161198761015840(120585)
(23)
where parameters 119861119894for the first approach are defined as
1198611=
1205902
1198782
1198612= minus
1205901
1198782
1198613=
1
1198782
1198614=
1205902
11987821205903
1198615=
11987821205900minus 1205901
11987821205903
1198616=
1
11987821205903
(24)
And for the second approach they are defined as
1198611=
1205902
1205900
1198612= minus
1205901
1205900
1198613=
1
1205900
1198614=
1205902
12059001205904
1198615=
1205902
0minus 1205901
12059001205904
1198616=
1
12059001205904
(25)
For the third approach they are defined as
1198611=
1205905
1198782
1198612= minus
1205901
1198782
1198613=
1
1198782
1198614=
1205905
11987821205903
1198615=
1198784minus 1205901
11987821205903
1198616=
1
11987821205903
(26)
23 Boundary Conditions due to Elastic Restraints For thecase of the elastic restraints at both ends the physicalboundary conditions at beam ends are
at 119909 = 0
119881 (0 119905) = minus1198961198790
119910 (0 119905)
119872 (0 119905) = 1198961198770
120579 (0 119905)
Mathematical Problems in Engineering 5
at 119909 = 119871
119881 (119871 119905) = 1198961198791
119910 (119871 119905)
119872 (119871 119905) = minus1198961198771
120579 (119871 119905)
(27)
Using the dimensionless variables 120585 and 120593 and (2) (3)and (23) the boundary conditions in dimensionless formassuming the first approach (120572(120585) = 0) may be obtained as
1198893120593
1198891205853+ (
1198615minus 1
1198614
)119889120593
119889120585minus
1198701198790
11987821198614
120593 (0) =minus1198616
1198614
1198761015840(0) (28a)
1198893120593
1198891205853minus
1198611
11986141198701198770
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585minus
1198612
11986141198701198770
120593 (0)
=1198613119876 (0) minus 119861
61198701198770
1198761015840(0)
11986141198701198770
(28b)
1198893120593
1198891205853+ (
1198615minus 1
1198614
)119889120593
119889120585+
1198701198791
11987821198614
120593 (1) =minus1198616
1198614
1198761015840(1) (28c)
1198893120593
1198891205853+
1198611
11986141198701198771
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585+
1198612
11986141198701198771
120593 (1)
= minus1198613119876 (1) + 119861
61198701198770
1198761015840(1)
11986141198701198771
(28d)
For the second approach (120572(120585) = 120595(120585)) the boundaryconditions can be expressed as
1198893120593
1198891205853+ (
12059001198615minus 1198782
12059001198614
)119889120593
119889120585minus
1198701198790
12059001198614
120593 (0) =minus1198616
1198614
1198761015840(0) (29a)
1198893120593
1198891205853minus
1198611
11986141198701198770
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585minus
1198612
11986141198701198770
120593 (0)
=1198613119876 (0) minus 119861
61198701198770
1198761015840(0)
11986141198701198770
(29b)
1198893120593
1198891205853+ (
12059001198615minus 1198782
12059001198614
)119889120593
119889120585+
1198701198791
12059001198614
120593 (1) =minus1198616
1198614
1198761015840(1) (29c)
1198893120593
1198891205853+
1198611
11986141198701198771
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585+
1198612
11986141198701198771
120593 (1)
= minus1198613119876 (1) + 119861
61198701198771
1198761015840(1)
11986141198701198771
(29d)
On the other hand for the third approach (120572(120585) = 1205931015840(120585)) the
boundary conditions can be expressed as
1198893120593
1198891205853+ (
11987821198615minus 1205906
11987821198614
)119889120593
119889120585+
1198701198790
11987821198614
120593 (0) =minus1198616
1198614
1198761015840(0) (30a)
1198893120593
1198891205853minus
1198611
11986141198701198770
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585minus
1198612
11986141198701198770
120593 (0)
=1198613119876 (0) minus 119861
61198701198770
1198761015840(0)
11986141198701198770
(30b)
1198893120593
1198891205853+ (
11987821198615minus 1205906
11987821198614
)119889120593
119889120585minus
1198701198791
11987821198614
120593 (1) =minus1198616
1198614
1198761015840(1) (30c)
1198893120593
1198891205853+
1198611
11986141198701198771
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585+
1198612
11986141198701198771
120593 (1)
= minus1198613119876 (1) + 119861
61198701198771
1198761015840(1)
11986141198701198771
(30d)
where
1205906= 1198782minus 119875
1198701198790
=11987131198961198790
119864119868
1198701198770
=1198711198961198770
119864119868
1198701198791
=1198713119896119879119871
119864119868
1198701198771
=119871119896119877119871
119864119868
(31)
3 Solution of the Governing Equations
31 Applications of the RDM to the Governing Equations Touse RDM the response amplitude equations (16) (18) or(21) are to be rewritten in the recursive form
120593(4)
(120585) = 11986011
120593(0)
+ 11986012
120593(1)
+ 11986013
120593(2)
+ 11986014
120593(3)
+ 1198651(120585)
(32)
where 120593(119898) is the 119898-derivative of 120593 Coefficients 119860
119894119895for the
first approach are
11986011
= minus12059011205903
1205902
11986012
= 0
11986013
=1205901+ 12059021205903minus 12059001198782
1205902
11986014
= 0
1198651(120585) =
1205903119876 minus 119876
10158401015840
1205902
(33)
For the second approach coefficients 119860119894119895are
11986011
= minus12059011205904
1205902
11986012
= 0
6 Mathematical Problems in Engineering
Table 1 Values of frequency parameters of the first three vibration modes
In addition coefficients 119860119894119895for the third approach are
11986011
= minus12059011205903
1205905
11986012
= 0
11986013
=1205901+ 12059031205905minus 1198784
1205905
11986014
= 0
1198651(120585) =
1205903119876 minus 119876
10158401015840
1205905
(35)
The solution of (32) may be expressed as
120593 (120585) =
4
sum
119898=1
119879119898119877119898(120585) + 119877
119865(120585) (36)
where the recursive functions 119877119898(120585) 119898 = 1 4 and the force
function 119877119865(120585) are given by
119877119898(120585) =
120585119898minus1
(119898 minus 1)+
119873
sum
119894=1
119860119894119898
120585119894+3
(119894 + 3) 119898 = 1 4
119877119865(120585) =
119873
sum
119894=1
119865119894(0)
120585119894+3
(119899 + 3)
(37)
The recurrence formulae for coefficients119860119896+1119894
119896 = 1 119873 and119865119894are given by
119860119896+11
= 1198601198964
11986011
119860119896+1119903
= 119860119896119903minus1
+ 1198601198964
1198601119903
119903 = 2 119899
119865119896+1
(120585) = 1198651015840
119896(120585) + 119860
11989641198651(120585)
(38)
The application of the boundary conditions (see (28a)ndash(28d)(29a)ndash(29d) or (30a)ndash(30d)) at 120585 = 0 1 yields a system
of 4-algebraic equations in 4 unknowns 1198791 1198792 1198793 and
1198794 however calculating these unknowns the response ampli-
tude distribution 120593(120585) can be obtainedThe distribution of bending moment 119872(120585) and shearing
force 119881(120585)may be obtained using (2) and (3) respectively
32 Calculations of Critical Loads and Natural FrequenciesThe critical loads and natural frequencies may be calculatedby assuming zero excitation amplitude (119865(120585) = 0) andreplacing the excitation frequencyΩ by the natural frequency120596 of the system The substitution of 120593(120585) and its derivativesinto the boundary conditions yields a systemof homogeneousalgebraic equations whose nontrivial solution yields a two-parameter eigenvalue problem in 119901 and120596The solution of thetwo-parameter eigenvalue problem yields both critical loads119901cr119899 and buckling modes for static case (120596 = 0) and thenatural frequencies of system120596
119899and themode shapes for free
vibration (119901 lt 119901cr)Also critical loads and natural frequencies may be
obtained from forced vibration by detecting the axial loador the excitation frequency at which the response amplitudebecomes unbounded
33 Verification of the Obtained Solutions The natural fre-quency parameter of the lower three modes for Timo-shenko beams 120582
119899 119894 = 1 3 is obtained using the present
solution assuming the third approach and compared withthose obtained from the Rayleigh-Ritz approach [12] Thecomparison is indicated in Table 1 for different boundaryconditions It is found that the results of the two approachesare compatible
4 Numerical Results and Discussion
Theobtained solutions are used to investigate the influence ofbeam and foundation parameters on the stability parametersanddynamic behavior of Timoshenko beamsThe foundationparameters (119870
1and 119870
2) and beam properties are taken
similar to those given in Taha and Doha [14]However for the investigation of the amplitude of the
lateral response the following normalized parameters areintroduced
120574 =119875
119875cr
Mathematical Problems in Engineering 7
1
2
3
4
5
Approach = 1Approach = 2Approach = 3
120578
120582
105 6 7 8 9 3020
(a) No foundation (119864119904 = 0)
Approach = 1Approach = 2Approach = 3
1
2
3
4
5
120582
120578
105 6 7 8 9 3020
(b) Stiff foundation (119864119904 = 108)
Figure 2 Variation of 120582 with 120578 for C-F beams (119875 = 021205872)
19
2
21
22
23
24
25
Approach = 1Approach = 2Approach = 3
Pcr
120578
105 6 7 8 9 3020
(a) No foundation
0
2
4
6
8
10
Approach = 1Approach = 2Approach = 3
Pcr
120578
105 6 7 8 9 3020
(b) Stiff foundation
Figure 3 Variation of 119875cr with 120578 for C-F beams
119882lowast=
119882max119882119906-max
(39)where 120574 is the loading parameter 119882
lowast is the maximumamplitude parameter119882max is the maximum amplitude of theinvestigated case and 119882
119906-max is the maximum amplitude forthe P-P corresponding case
41 Stability Parameters for Cantilever Beams Althoughmany engineering applications belong to the Timoshenkocantilever beam model a few articles are published con-cerning the stability and dynamic behavior of such model
due to its mathematical complications In the present workthe investigation of cantilever beam behavior is highlightedThe comparison between the different approaches dealingwith the shear induced from the axial load on the frequencyparameter 120582 for clamped-free beams (C-F) is shown inFigure 2 case (a) for beams without foundation (119864
119904= 0) and
case (b) for beams resting on stiff foundation (119864119904= 108)
On the other hand the influences of different approacheson quantifying the critical load of C-F beams are indicated inFigure 3 case (a) for beams without foundation and case (b)for beams on stiff foundations It is obvious that consideringthe shear component induced from the axial load decreases
8 Mathematical Problems in Engineering
15
2
25
3
35
NFWF
MFSF
120582
120578
105 6 7 8 9 3020
(a) 119875 = 0
05
1
15
2
25
3
NFWF
MFSF
120582
120578
105 6 7 8 9 3020
(b) 119875 = 021205872
Figure 4 Variation of 120582 with 120578 for C-F beams
28
3
32
34
36
38
4
NFWF
MFSF
KR
100 104
120582
(a) Slenderness ratio 120578 = 10
3
35
25
4
45
5
KR
100 104
120578 = 5120578 = 10
120578 = 20
120578 = 50
120582
(b) No foundations (119864119904 = 0)
Figure 5 Influence of 119870119877on 120582 for different types of foundation and for different 120578 (119875 = 0)
the beam stiffness and the effect is noticeable for beams withslenderness ratio 120582 lt 20
In the following analysis the parametric study is basedon the second approach as it is the rational one because theangle between the axial load and the outward normal to thedeformed cross section 120572(119909 119905) is actually equal to the angle ofcross section rotation 120595(119909 119905) in the deformed configuration
The influence of foundation type on the frequency param-eter for C-F beams is shown in Figure 4 It is found thatas the slenderness ratio increases the natural frequency ofbeams resting on foundation increases due to the increase offoundation share in the overall stiffness of the system Theincrease in the natural frequency parameter is more notice-able for stiff foundations (SF) The critical load of Timo-shenko C-F beam without foundation 119875cr = 021120587
2
42 Weakening of the End Restraints The effect of rotationalstiffness variations on the natural frequency parameter isshown in Figure 5(a) for different types of foundations (120578 =
10) and in Figure 5(b) for different values of slenderness ratio(119864119904= 0)Further the effect of rotational stiffness variations on the
critical load parameter is shown in Figure 6(a) for differenttypes of foundation (120578 = 10) and in Figure 6(b) for differentvalues of slenderness ratio (119864
119904= 0)
The influence of rotational stiffness variations on thenatural frequency parameter is shown in Figure 7 for differentloading ratios and for the cases of no foundation (119864
119904= 0) and
stiff foundations (119864119904= 108)
It is obvious that the significance of real foundation typesfor Timoshenko beam behavior (120578 lt 20) is negligible due to
Mathematical Problems in Engineering 9
5
10
15
20
25
NFWF
MFSF
KR
Pcr
10minus2 100 102 104
(a) 120578 = 10
0
10
20
30
40
120578 = 5120578 = 10
120578 = 20
120578 = 50
Pcr
KR
10minus2 100 102 104
(b) 119864119904 = 0
Figure 6 Influence of 119870119877on 119875cr for different types of foundation and for different 120578
2
25
3
35
4
KR
100 104
120582
120574 = 0120574 = 02
120574 = 04
120574 = 06
(a) No foundation (119864119904 = 0)
2
25
3
35
4
KR
100 104
120582
120574 = 0120574 = 02
120574 = 04
120574 = 06
(b) Stiff foundation (119864119904 = 108)
Figure 7 Influence of 119870119877on 120582 for different loading ratios (120578 = 10)
the high stiffness of the beam relative to the foundation Alsoit is found that the influence of rotational stiffness increaseswith the increase in the slenderness ratio Also it is found thatthe natural frequency decreases as the axial load increases tillthe axial load approaches its critical values then the naturalfrequency approaches zero
43 Forced Vibration Analysis Theweakening of ends elasticrestraints is represented by the variation of the restraint stiff-nessThe variation of normalized maximum response ampli-tude (119882lowast) with the ends rotational stiffness (119870
119877) is indicated
in Figure 8 for different loading types Case (a) representsthe effects of uniform loading functions case (b) representsthe effect of linear loading functions and case (c) represents
the effect of quadratic loading functions As the case of119870119877=
0 represents the P-P beams it is obvious from the figuresthat the displacement due to dynamic loading is greater thanthose for static loading due to the inertia force interferenceat 119870119877
= 0 Also it is clear that as 119870119877increases the stiffness
of the beam increases and the response amplitude decreasesIt is obvious that at a certain combination of the systemparameters the resonance condition is approached andunbounded response amplitude is detected As the axial loadincreases the variations of the end rotational stiffness maycause the system to reach the resonance conditions and theamplitude becomes unbounded If the damping is taken intoconsideration the amplitude will possess a finite value butsuch condition is to be prevented However it is concluded
Figure 8 Variation of119882lowast with 119870119877for different axial loading ratio (120578 = 10)
Mathematical Problems in Engineering 11
that the weakening of the end restraints may offer a param-eters combination coinciding with the resonance conditionsfor the first mode or one of the higher modes and unboundedamplitudes may be reached
5 Conclusions
The RDM is used to investigate the stability behavior ofaxially loaded cantilever Timoshenko beam resting on two-parameter foundation In addition the analysis includesthe investigation of Timoshenko beams resting on two-parameter foundation and subjected to different types of lat-eral excitationThe supports at the beam ends are assumed ofthe elastic type to investigate the impact of the support weak-ening on the beambehavior Both the effect of rotational iner-tia and the correction of the shear stress due to axial loadingare taken into consideration which decreases the beam stiff-ness It is concluded that the significance of the foundation onthe Timoshenko beams (120578 lt 20) behavior is negligiblethe effect of end restraints is more noticeable for slenderbeams and the natural frequency decreases as the axial loadincreases However for the case of the forced vibration theresonance conditions may result from the weakening of theend supports and unbounded response amplitudes may bereached The analysis depicts the simplicity and accuracy ofthe RDM in tackling the boundary value problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Ruta ldquoThe application of Chebyshev polynomials to the solu-tion of the nonprismatic Timoshenko beam vibration problemrdquoJournal of Sound and Vibration vol 296 no 1-2 pp 243ndash2632006
[2] K Kausel ldquoNonclassical modes of unrestrained shear beamsrdquoJournal of Engineering Mechanics vol 128 no 6 pp 663ndash6672002
[3] R Attarnejad A Shahba and S J Semnani ldquoApplication ofdifferential transform in free vibration analysis of timoshenkobeams resting on two-parameter elastic foundationrdquo ArabianJournal for Science and Engineering vol 35 no 2B pp 125ndash1322010
[4] M Taha and M A Nassar ldquoAnalysis of stressed Timoshenkobeams on two parameter foundationsrdquo KSCE Journal of CivilEngineering vol 19 no 1 pp 173ndash179 2015
[5] L Majkut ldquoFree and forced vibration of Timoshenko beamsdescribed by single difference equationrdquo Journal of Theoreticaland Applied Mechanics vol 47 no 1 pp 193ndash210 2009
[6] F Y Cheng and C P Pantelides ldquoDynamic timoshenko beam-column on elastic mediardquo Journal of Structural Engineering vol114 no 7 pp 1524ndash1550 1988
[7] B Geist and J R Mclaughlin ldquoDouble eigen value for theTimoshenko beamrdquo Applied Mathematics Letters vol 10 pp129ndash134 1997
[8] C-N Chen ldquoDQEM vibration analyses of non-prismatic sheardeformable beams resting on elastic foundationsrdquo Journal ofSound and Vibration vol 255 no 5 pp 989ndash999 2002
[9] A L G Monsalve Z D G Medina and A J D Ochoa ldquoTim-oshenko beam-column with generalized end conditions onelastic foundation dynamic-stiffness matrix and load vectorrdquoJournal of Sound and Vibration vol 310 no 4-5 pp 1057ndash10792008
[10] T Kocaturk and M Simsek ldquoFree vibration analysis of Tim-oshenko beams under various boundary conditionsrdquo SigmaJournal of Engineering and Natural Sciences vol 1 pp 30ndash442005
[11] DKNguyen ldquoFree vibration of prestressedTimoshenkobeamsresting on elastic foundationsrdquo Vietnam Journal of Mechanicsvol 29 no 1 pp 1ndash12 2012
[12] N M Auciello ldquoVibrations of Timoshenko beams on twoparameter elastic soilrdquo Engineering Transactions vol 56 no 3pp 187ndash200 2008
[13] M Taha ldquoRecursive differentiation method for boundary valueproblems application to analysis of a beam-column on an elas-tic foundationrdquo Journal of Theoretical and Applied Mechanicsvol 44 no 2 pp 57ndash70 2014
[14] M Taha and E H Doha ldquoRecursive differentiation methodapplication to the analysis of beams on two parameter founda-tionsrdquo Journal of Theoretical and Applied Mechanics vol 53 no1 pp 15ndash26 2015
[15] S P Timoshenko and J M Gere Theory of Elastic StabilityEngineering Societies Monographs McGraw-Hill Book NewYork NY USA 1961
configurations of beam-foundation systems in finite domainTaha and Doha [14] used RDM to study dynamics of beam-foundation systems assuming Euler-Bernoulli hypothesis
In the present paper RDM is implemented to obtainanalytical solutions for the differential equations governingthe static and dynamic behavior of axially loaded Timo-shenko beams resting on two-parameter foundation withelastic end restraints and subjected to nonuniform lateralexcitation Both the influence of rotational inertia and theshear stress induced from the effect of axial load proposed byTimoshenko and Gere [15] will be considered The stabilitybehavior of cantilever Timoshenko beam resting on elasticfoundation will be analyzed In addition the influences of theelastic end restraints weakening will be studied Further thesignificance of different parameters on the maximum lateralresponse amplitude of the beam due to different types oflateral excitation will be investigated
2 Formulation of the Governing Equations
21 Dynamic Equations of Timoshenko Beams The equationsof translational and angular motion of an infinitesimalelement of an axially loaded Timoshenko beam subjected tolateral excitation resting on two-parameter foundation shownin Figure 1 are
120597119881
120597119909+ 119902 (119909 119905) minus 119896
1119910 (119909 119905) + 119896
2
1205972119910
1205971199092= 120588119860
1205972119910
1205971199052 (1)
119881 (119909 119905) + 119901120597119910
120597119909+
120597119872
120597119909= 120588119868
1205972120579
1205971199052 (2)
The force-displacement relations considering additionalshear stress induced from the component of the axial load inthe direction of the deformed section (119901 sin(120572) cong 119901120572(119909 119905))
The relation between the deformation components of thebeam element (shown in Figure 1(c)) is
120597119910
120597119909= 120579 + 120574 (5)
where 119864 is the modulus of elasticity of the beam material119868 is the moment of inertia of the beam cross section 120588 isthe density 119860 is the area of the cross section 119860
119904= 120581119860 is
the effective shear area 120581 is a correction factor to take intoaccount the nonuniform distribution of shear stress (120581 = 23
for rectangular cross section) 119901 is the axial applied load 1198961
and 1198962are the linear and shear foundation stiffness factors per
unit length of the beam 119902(119909 119905) is the lateral excitation actingon the beam 120579(119909 119905) is the rotation of the beam cross section120574 is the deformation angle due to the shear force 120572(119909 119905) isthe angle between the axial load 119901 and the normal to thedeformed cross section 119881(119909 119905) is the shear force 119872(119909 119905) is
the bending moment 119910(119909 119905) is the lateral response of thebeam 119909 is the coordinate along the beam and 119905 is the time
Substituting (3) (4) and (5) into (1) and (2) the equationsof motion may be expressed as
119860119904119866(
1205972119910
1205971199092minus
120597120579
120597119909) minus 119901
120597120572
120597119909minus 1198961119910 + 1198962
1205972119910
1205971199092minus 120588119860
1205972119910
1205971199052
= minus119902 (119909 119905)
119860119904119866(
120597119910
120597119909minus 120579) minus 119901120572 + 119901
120597119910
120597119909+ 119864119868
1205972120579
1205971199092minus 120588119868
1205972120579
1205971199052= 0
(6)
Introducing the dimensionless variables 120585 = 119909119871 and 119908 =
where Ω is the excitation frequency Most of the practicalcases for nonuniform continuous loading function can beclosely simulated by assuming quadratic loading function inthe form
119860119904119866120595 (120585) + 119901120572 (120585) minus (119860
119904119866 + 119901)
119889120593
119889120585minus
119864119868
1198712
1198892120595
1198891205852
minus 1205881198601199032Ω2120595 (120585) = 0
(10)
22 Additional Shear Induced from the Axial Load There arethree approaches in dealing with the shear stress inducedfrom the axial load the first approach neglects the additional
Mathematical Problems in Engineering 3
kR0
p
kT0 kT1
q(x t)
k1 and k2
Ly
p
x
kR1
(a)
p
y
x0q(x t)dx
V
M
p
120588Ay∙∙dx
M+120597M
120597xdx
V +120597V
120597xdx
120588I120579∙∙dx
dx
dx
) )k1y minus k2
1205972y
120597x
(b)
x
O
y
120574
120579
dy
dx
(c)
Figure 1 (a) Timoshenko beam on elastic foundation (b) Element forces (c) Total deformation
shear stress the second approach assumes the inclinationangle between the axial load and the outward normal tothe deformed cross section 120572(119909 119905) = 120579(119909 119905) and the thirdapproach assumes 120572(119909 119905) = 120597119910120597119905
According to the first approach (120572(120585) = 0) (10) may berewritten as
(12)Using the dimensionless parameters defined in (12) (11) maybe expressed as
119889120595
119889120585=
1205902
1198782
1198892120593
1198891205852minus
1205901
1198782120593 (120585) +
119876 (120585)
1198782 (13)
1198892120595
1198891205852+ 1205900
119889120593
119889120585minus 1205903120595 (120585) = 0 (14)
where1205900= 1198782+ 119875
1205901= 1198701minus 1205824
119865
1205902= 1198782+ 1198702
1205903= 1198782minus
1205824
119865
1205782
119876 (120585) =1198713
119864119868119902 (120585)
(15)
4 Mathematical Problems in Engineering
where 119878 is the slenderness parameter 120582119865is the frequency
parameter 120578 is the slenderness ratio 119903 is the radius of gyra-tion 119875 is the axial load parameter119870
1is the foundation linear
stiffness parameter and 1198702is the foundation shear stiffness
parameterInserting (14) into (13) the equation describing the lateral
response amplitude of the Timoshenko beam based on thefirst approach is
1198894120593
1198891205854minus (
1205901+ 12059021205903minus 12059001198782
1205902
)1198892120593
1198891205852+
12059011205903
1205902
120593 (120585)
=1205903119876 minus 119876
10158401015840
1205902
(16)
Similarly for the second approach (120572(120585) = 120595(120585)) (11) may beexpressed as
119889120595
119889120585=
1205902
1205900
1198892120593
1198891205852minus
1205901
1205900
120593 (120585) +119876 (120585)
1205900
1198892120595
1198891205852+ 1205900
119889120593
119889120585minus 1205904120595 (120585) = 0
(17)
Also the equation of the lateral response amplitude of theTimoshenko beam according to the second approach is
1198894120593
1198891205854minus (
1205901+ 12059021205904minus 1205902
0
1205902
)1198892120593
1198891205852+
12059011205904
1205902
120593 (120585)
=1205904119876 minus 119876
10158401015840
1205902
(18)
where 1205904= 119875 + 120590
3
According to the third approach (120572(120585) = 119889120593119889120585) thedynamic equations of lateral response amplitude can beobtained as
119889120595
119889120585=
1205905
1198782
1198892120593
1198891205852minus
1205901
1198782120593 (120585) +
119876 (120585)
1198782 (19)
1198892120595
1198891205852+ 1198782 119889120593
119889120585minus 1205903120595 (120585) = 0 (20)
1198894120593
1198891205854minus (
1205901+ 12059031205905minus 1198784
1205905
)1198892120593
1198891205852+
12059011205903
1205905
120593 (120585)
=1205903119876 (120585) minus 119876
10158401015840
1205905
(21)
where
1205905= 1205902minus 119875 (22)
The dynamic equations of different approaches may beexpressed as
119889120595
119889120585= 1198611
1198892120593
1198891205852+ 1198612120593 (120585) + 119861
3119876 (120585)
120595 (120585) = 1198614
1198893120593
1198891205853+ 1198615
119889120593
119889120585+ 11986161198761015840(120585)
(23)
where parameters 119861119894for the first approach are defined as
1198611=
1205902
1198782
1198612= minus
1205901
1198782
1198613=
1
1198782
1198614=
1205902
11987821205903
1198615=
11987821205900minus 1205901
11987821205903
1198616=
1
11987821205903
(24)
And for the second approach they are defined as
1198611=
1205902
1205900
1198612= minus
1205901
1205900
1198613=
1
1205900
1198614=
1205902
12059001205904
1198615=
1205902
0minus 1205901
12059001205904
1198616=
1
12059001205904
(25)
For the third approach they are defined as
1198611=
1205905
1198782
1198612= minus
1205901
1198782
1198613=
1
1198782
1198614=
1205905
11987821205903
1198615=
1198784minus 1205901
11987821205903
1198616=
1
11987821205903
(26)
23 Boundary Conditions due to Elastic Restraints For thecase of the elastic restraints at both ends the physicalboundary conditions at beam ends are
at 119909 = 0
119881 (0 119905) = minus1198961198790
119910 (0 119905)
119872 (0 119905) = 1198961198770
120579 (0 119905)
Mathematical Problems in Engineering 5
at 119909 = 119871
119881 (119871 119905) = 1198961198791
119910 (119871 119905)
119872 (119871 119905) = minus1198961198771
120579 (119871 119905)
(27)
Using the dimensionless variables 120585 and 120593 and (2) (3)and (23) the boundary conditions in dimensionless formassuming the first approach (120572(120585) = 0) may be obtained as
1198893120593
1198891205853+ (
1198615minus 1
1198614
)119889120593
119889120585minus
1198701198790
11987821198614
120593 (0) =minus1198616
1198614
1198761015840(0) (28a)
1198893120593
1198891205853minus
1198611
11986141198701198770
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585minus
1198612
11986141198701198770
120593 (0)
=1198613119876 (0) minus 119861
61198701198770
1198761015840(0)
11986141198701198770
(28b)
1198893120593
1198891205853+ (
1198615minus 1
1198614
)119889120593
119889120585+
1198701198791
11987821198614
120593 (1) =minus1198616
1198614
1198761015840(1) (28c)
1198893120593
1198891205853+
1198611
11986141198701198771
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585+
1198612
11986141198701198771
120593 (1)
= minus1198613119876 (1) + 119861
61198701198770
1198761015840(1)
11986141198701198771
(28d)
For the second approach (120572(120585) = 120595(120585)) the boundaryconditions can be expressed as
1198893120593
1198891205853+ (
12059001198615minus 1198782
12059001198614
)119889120593
119889120585minus
1198701198790
12059001198614
120593 (0) =minus1198616
1198614
1198761015840(0) (29a)
1198893120593
1198891205853minus
1198611
11986141198701198770
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585minus
1198612
11986141198701198770
120593 (0)
=1198613119876 (0) minus 119861
61198701198770
1198761015840(0)
11986141198701198770
(29b)
1198893120593
1198891205853+ (
12059001198615minus 1198782
12059001198614
)119889120593
119889120585+
1198701198791
12059001198614
120593 (1) =minus1198616
1198614
1198761015840(1) (29c)
1198893120593
1198891205853+
1198611
11986141198701198771
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585+
1198612
11986141198701198771
120593 (1)
= minus1198613119876 (1) + 119861
61198701198771
1198761015840(1)
11986141198701198771
(29d)
On the other hand for the third approach (120572(120585) = 1205931015840(120585)) the
boundary conditions can be expressed as
1198893120593
1198891205853+ (
11987821198615minus 1205906
11987821198614
)119889120593
119889120585+
1198701198790
11987821198614
120593 (0) =minus1198616
1198614
1198761015840(0) (30a)
1198893120593
1198891205853minus
1198611
11986141198701198770
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585minus
1198612
11986141198701198770
120593 (0)
=1198613119876 (0) minus 119861
61198701198770
1198761015840(0)
11986141198701198770
(30b)
1198893120593
1198891205853+ (
11987821198615minus 1205906
11987821198614
)119889120593
119889120585minus
1198701198791
11987821198614
120593 (1) =minus1198616
1198614
1198761015840(1) (30c)
1198893120593
1198891205853+
1198611
11986141198701198771
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585+
1198612
11986141198701198771
120593 (1)
= minus1198613119876 (1) + 119861
61198701198771
1198761015840(1)
11986141198701198771
(30d)
where
1205906= 1198782minus 119875
1198701198790
=11987131198961198790
119864119868
1198701198770
=1198711198961198770
119864119868
1198701198791
=1198713119896119879119871
119864119868
1198701198771
=119871119896119877119871
119864119868
(31)
3 Solution of the Governing Equations
31 Applications of the RDM to the Governing Equations Touse RDM the response amplitude equations (16) (18) or(21) are to be rewritten in the recursive form
120593(4)
(120585) = 11986011
120593(0)
+ 11986012
120593(1)
+ 11986013
120593(2)
+ 11986014
120593(3)
+ 1198651(120585)
(32)
where 120593(119898) is the 119898-derivative of 120593 Coefficients 119860
119894119895for the
first approach are
11986011
= minus12059011205903
1205902
11986012
= 0
11986013
=1205901+ 12059021205903minus 12059001198782
1205902
11986014
= 0
1198651(120585) =
1205903119876 minus 119876
10158401015840
1205902
(33)
For the second approach coefficients 119860119894119895are
11986011
= minus12059011205904
1205902
11986012
= 0
6 Mathematical Problems in Engineering
Table 1 Values of frequency parameters of the first three vibration modes
In addition coefficients 119860119894119895for the third approach are
11986011
= minus12059011205903
1205905
11986012
= 0
11986013
=1205901+ 12059031205905minus 1198784
1205905
11986014
= 0
1198651(120585) =
1205903119876 minus 119876
10158401015840
1205905
(35)
The solution of (32) may be expressed as
120593 (120585) =
4
sum
119898=1
119879119898119877119898(120585) + 119877
119865(120585) (36)
where the recursive functions 119877119898(120585) 119898 = 1 4 and the force
function 119877119865(120585) are given by
119877119898(120585) =
120585119898minus1
(119898 minus 1)+
119873
sum
119894=1
119860119894119898
120585119894+3
(119894 + 3) 119898 = 1 4
119877119865(120585) =
119873
sum
119894=1
119865119894(0)
120585119894+3
(119899 + 3)
(37)
The recurrence formulae for coefficients119860119896+1119894
119896 = 1 119873 and119865119894are given by
119860119896+11
= 1198601198964
11986011
119860119896+1119903
= 119860119896119903minus1
+ 1198601198964
1198601119903
119903 = 2 119899
119865119896+1
(120585) = 1198651015840
119896(120585) + 119860
11989641198651(120585)
(38)
The application of the boundary conditions (see (28a)ndash(28d)(29a)ndash(29d) or (30a)ndash(30d)) at 120585 = 0 1 yields a system
of 4-algebraic equations in 4 unknowns 1198791 1198792 1198793 and
1198794 however calculating these unknowns the response ampli-
tude distribution 120593(120585) can be obtainedThe distribution of bending moment 119872(120585) and shearing
force 119881(120585)may be obtained using (2) and (3) respectively
32 Calculations of Critical Loads and Natural FrequenciesThe critical loads and natural frequencies may be calculatedby assuming zero excitation amplitude (119865(120585) = 0) andreplacing the excitation frequencyΩ by the natural frequency120596 of the system The substitution of 120593(120585) and its derivativesinto the boundary conditions yields a systemof homogeneousalgebraic equations whose nontrivial solution yields a two-parameter eigenvalue problem in 119901 and120596The solution of thetwo-parameter eigenvalue problem yields both critical loads119901cr119899 and buckling modes for static case (120596 = 0) and thenatural frequencies of system120596
119899and themode shapes for free
vibration (119901 lt 119901cr)Also critical loads and natural frequencies may be
obtained from forced vibration by detecting the axial loador the excitation frequency at which the response amplitudebecomes unbounded
33 Verification of the Obtained Solutions The natural fre-quency parameter of the lower three modes for Timo-shenko beams 120582
119899 119894 = 1 3 is obtained using the present
solution assuming the third approach and compared withthose obtained from the Rayleigh-Ritz approach [12] Thecomparison is indicated in Table 1 for different boundaryconditions It is found that the results of the two approachesare compatible
4 Numerical Results and Discussion
Theobtained solutions are used to investigate the influence ofbeam and foundation parameters on the stability parametersanddynamic behavior of Timoshenko beamsThe foundationparameters (119870
1and 119870
2) and beam properties are taken
similar to those given in Taha and Doha [14]However for the investigation of the amplitude of the
lateral response the following normalized parameters areintroduced
120574 =119875
119875cr
Mathematical Problems in Engineering 7
1
2
3
4
5
Approach = 1Approach = 2Approach = 3
120578
120582
105 6 7 8 9 3020
(a) No foundation (119864119904 = 0)
Approach = 1Approach = 2Approach = 3
1
2
3
4
5
120582
120578
105 6 7 8 9 3020
(b) Stiff foundation (119864119904 = 108)
Figure 2 Variation of 120582 with 120578 for C-F beams (119875 = 021205872)
19
2
21
22
23
24
25
Approach = 1Approach = 2Approach = 3
Pcr
120578
105 6 7 8 9 3020
(a) No foundation
0
2
4
6
8
10
Approach = 1Approach = 2Approach = 3
Pcr
120578
105 6 7 8 9 3020
(b) Stiff foundation
Figure 3 Variation of 119875cr with 120578 for C-F beams
119882lowast=
119882max119882119906-max
(39)where 120574 is the loading parameter 119882
lowast is the maximumamplitude parameter119882max is the maximum amplitude of theinvestigated case and 119882
119906-max is the maximum amplitude forthe P-P corresponding case
41 Stability Parameters for Cantilever Beams Althoughmany engineering applications belong to the Timoshenkocantilever beam model a few articles are published con-cerning the stability and dynamic behavior of such model
due to its mathematical complications In the present workthe investigation of cantilever beam behavior is highlightedThe comparison between the different approaches dealingwith the shear induced from the axial load on the frequencyparameter 120582 for clamped-free beams (C-F) is shown inFigure 2 case (a) for beams without foundation (119864
119904= 0) and
case (b) for beams resting on stiff foundation (119864119904= 108)
On the other hand the influences of different approacheson quantifying the critical load of C-F beams are indicated inFigure 3 case (a) for beams without foundation and case (b)for beams on stiff foundations It is obvious that consideringthe shear component induced from the axial load decreases
8 Mathematical Problems in Engineering
15
2
25
3
35
NFWF
MFSF
120582
120578
105 6 7 8 9 3020
(a) 119875 = 0
05
1
15
2
25
3
NFWF
MFSF
120582
120578
105 6 7 8 9 3020
(b) 119875 = 021205872
Figure 4 Variation of 120582 with 120578 for C-F beams
28
3
32
34
36
38
4
NFWF
MFSF
KR
100 104
120582
(a) Slenderness ratio 120578 = 10
3
35
25
4
45
5
KR
100 104
120578 = 5120578 = 10
120578 = 20
120578 = 50
120582
(b) No foundations (119864119904 = 0)
Figure 5 Influence of 119870119877on 120582 for different types of foundation and for different 120578 (119875 = 0)
the beam stiffness and the effect is noticeable for beams withslenderness ratio 120582 lt 20
In the following analysis the parametric study is basedon the second approach as it is the rational one because theangle between the axial load and the outward normal to thedeformed cross section 120572(119909 119905) is actually equal to the angle ofcross section rotation 120595(119909 119905) in the deformed configuration
The influence of foundation type on the frequency param-eter for C-F beams is shown in Figure 4 It is found thatas the slenderness ratio increases the natural frequency ofbeams resting on foundation increases due to the increase offoundation share in the overall stiffness of the system Theincrease in the natural frequency parameter is more notice-able for stiff foundations (SF) The critical load of Timo-shenko C-F beam without foundation 119875cr = 021120587
2
42 Weakening of the End Restraints The effect of rotationalstiffness variations on the natural frequency parameter isshown in Figure 5(a) for different types of foundations (120578 =
10) and in Figure 5(b) for different values of slenderness ratio(119864119904= 0)Further the effect of rotational stiffness variations on the
critical load parameter is shown in Figure 6(a) for differenttypes of foundation (120578 = 10) and in Figure 6(b) for differentvalues of slenderness ratio (119864
119904= 0)
The influence of rotational stiffness variations on thenatural frequency parameter is shown in Figure 7 for differentloading ratios and for the cases of no foundation (119864
119904= 0) and
stiff foundations (119864119904= 108)
It is obvious that the significance of real foundation typesfor Timoshenko beam behavior (120578 lt 20) is negligible due to
Mathematical Problems in Engineering 9
5
10
15
20
25
NFWF
MFSF
KR
Pcr
10minus2 100 102 104
(a) 120578 = 10
0
10
20
30
40
120578 = 5120578 = 10
120578 = 20
120578 = 50
Pcr
KR
10minus2 100 102 104
(b) 119864119904 = 0
Figure 6 Influence of 119870119877on 119875cr for different types of foundation and for different 120578
2
25
3
35
4
KR
100 104
120582
120574 = 0120574 = 02
120574 = 04
120574 = 06
(a) No foundation (119864119904 = 0)
2
25
3
35
4
KR
100 104
120582
120574 = 0120574 = 02
120574 = 04
120574 = 06
(b) Stiff foundation (119864119904 = 108)
Figure 7 Influence of 119870119877on 120582 for different loading ratios (120578 = 10)
the high stiffness of the beam relative to the foundation Alsoit is found that the influence of rotational stiffness increaseswith the increase in the slenderness ratio Also it is found thatthe natural frequency decreases as the axial load increases tillthe axial load approaches its critical values then the naturalfrequency approaches zero
43 Forced Vibration Analysis Theweakening of ends elasticrestraints is represented by the variation of the restraint stiff-nessThe variation of normalized maximum response ampli-tude (119882lowast) with the ends rotational stiffness (119870
119877) is indicated
in Figure 8 for different loading types Case (a) representsthe effects of uniform loading functions case (b) representsthe effect of linear loading functions and case (c) represents
the effect of quadratic loading functions As the case of119870119877=
0 represents the P-P beams it is obvious from the figuresthat the displacement due to dynamic loading is greater thanthose for static loading due to the inertia force interferenceat 119870119877
= 0 Also it is clear that as 119870119877increases the stiffness
of the beam increases and the response amplitude decreasesIt is obvious that at a certain combination of the systemparameters the resonance condition is approached andunbounded response amplitude is detected As the axial loadincreases the variations of the end rotational stiffness maycause the system to reach the resonance conditions and theamplitude becomes unbounded If the damping is taken intoconsideration the amplitude will possess a finite value butsuch condition is to be prevented However it is concluded
Figure 8 Variation of119882lowast with 119870119877for different axial loading ratio (120578 = 10)
Mathematical Problems in Engineering 11
that the weakening of the end restraints may offer a param-eters combination coinciding with the resonance conditionsfor the first mode or one of the higher modes and unboundedamplitudes may be reached
5 Conclusions
The RDM is used to investigate the stability behavior ofaxially loaded cantilever Timoshenko beam resting on two-parameter foundation In addition the analysis includesthe investigation of Timoshenko beams resting on two-parameter foundation and subjected to different types of lat-eral excitationThe supports at the beam ends are assumed ofthe elastic type to investigate the impact of the support weak-ening on the beambehavior Both the effect of rotational iner-tia and the correction of the shear stress due to axial loadingare taken into consideration which decreases the beam stiff-ness It is concluded that the significance of the foundation onthe Timoshenko beams (120578 lt 20) behavior is negligiblethe effect of end restraints is more noticeable for slenderbeams and the natural frequency decreases as the axial loadincreases However for the case of the forced vibration theresonance conditions may result from the weakening of theend supports and unbounded response amplitudes may bereached The analysis depicts the simplicity and accuracy ofthe RDM in tackling the boundary value problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Ruta ldquoThe application of Chebyshev polynomials to the solu-tion of the nonprismatic Timoshenko beam vibration problemrdquoJournal of Sound and Vibration vol 296 no 1-2 pp 243ndash2632006
[2] K Kausel ldquoNonclassical modes of unrestrained shear beamsrdquoJournal of Engineering Mechanics vol 128 no 6 pp 663ndash6672002
[3] R Attarnejad A Shahba and S J Semnani ldquoApplication ofdifferential transform in free vibration analysis of timoshenkobeams resting on two-parameter elastic foundationrdquo ArabianJournal for Science and Engineering vol 35 no 2B pp 125ndash1322010
[4] M Taha and M A Nassar ldquoAnalysis of stressed Timoshenkobeams on two parameter foundationsrdquo KSCE Journal of CivilEngineering vol 19 no 1 pp 173ndash179 2015
[5] L Majkut ldquoFree and forced vibration of Timoshenko beamsdescribed by single difference equationrdquo Journal of Theoreticaland Applied Mechanics vol 47 no 1 pp 193ndash210 2009
[6] F Y Cheng and C P Pantelides ldquoDynamic timoshenko beam-column on elastic mediardquo Journal of Structural Engineering vol114 no 7 pp 1524ndash1550 1988
[7] B Geist and J R Mclaughlin ldquoDouble eigen value for theTimoshenko beamrdquo Applied Mathematics Letters vol 10 pp129ndash134 1997
[8] C-N Chen ldquoDQEM vibration analyses of non-prismatic sheardeformable beams resting on elastic foundationsrdquo Journal ofSound and Vibration vol 255 no 5 pp 989ndash999 2002
[9] A L G Monsalve Z D G Medina and A J D Ochoa ldquoTim-oshenko beam-column with generalized end conditions onelastic foundation dynamic-stiffness matrix and load vectorrdquoJournal of Sound and Vibration vol 310 no 4-5 pp 1057ndash10792008
[10] T Kocaturk and M Simsek ldquoFree vibration analysis of Tim-oshenko beams under various boundary conditionsrdquo SigmaJournal of Engineering and Natural Sciences vol 1 pp 30ndash442005
[11] DKNguyen ldquoFree vibration of prestressedTimoshenkobeamsresting on elastic foundationsrdquo Vietnam Journal of Mechanicsvol 29 no 1 pp 1ndash12 2012
[12] N M Auciello ldquoVibrations of Timoshenko beams on twoparameter elastic soilrdquo Engineering Transactions vol 56 no 3pp 187ndash200 2008
[13] M Taha ldquoRecursive differentiation method for boundary valueproblems application to analysis of a beam-column on an elas-tic foundationrdquo Journal of Theoretical and Applied Mechanicsvol 44 no 2 pp 57ndash70 2014
[14] M Taha and E H Doha ldquoRecursive differentiation methodapplication to the analysis of beams on two parameter founda-tionsrdquo Journal of Theoretical and Applied Mechanics vol 53 no1 pp 15ndash26 2015
[15] S P Timoshenko and J M Gere Theory of Elastic StabilityEngineering Societies Monographs McGraw-Hill Book NewYork NY USA 1961
Figure 1 (a) Timoshenko beam on elastic foundation (b) Element forces (c) Total deformation
shear stress the second approach assumes the inclinationangle between the axial load and the outward normal tothe deformed cross section 120572(119909 119905) = 120579(119909 119905) and the thirdapproach assumes 120572(119909 119905) = 120597119910120597119905
According to the first approach (120572(120585) = 0) (10) may berewritten as
(12)Using the dimensionless parameters defined in (12) (11) maybe expressed as
119889120595
119889120585=
1205902
1198782
1198892120593
1198891205852minus
1205901
1198782120593 (120585) +
119876 (120585)
1198782 (13)
1198892120595
1198891205852+ 1205900
119889120593
119889120585minus 1205903120595 (120585) = 0 (14)
where1205900= 1198782+ 119875
1205901= 1198701minus 1205824
119865
1205902= 1198782+ 1198702
1205903= 1198782minus
1205824
119865
1205782
119876 (120585) =1198713
119864119868119902 (120585)
(15)
4 Mathematical Problems in Engineering
where 119878 is the slenderness parameter 120582119865is the frequency
parameter 120578 is the slenderness ratio 119903 is the radius of gyra-tion 119875 is the axial load parameter119870
1is the foundation linear
stiffness parameter and 1198702is the foundation shear stiffness
parameterInserting (14) into (13) the equation describing the lateral
response amplitude of the Timoshenko beam based on thefirst approach is
1198894120593
1198891205854minus (
1205901+ 12059021205903minus 12059001198782
1205902
)1198892120593
1198891205852+
12059011205903
1205902
120593 (120585)
=1205903119876 minus 119876
10158401015840
1205902
(16)
Similarly for the second approach (120572(120585) = 120595(120585)) (11) may beexpressed as
119889120595
119889120585=
1205902
1205900
1198892120593
1198891205852minus
1205901
1205900
120593 (120585) +119876 (120585)
1205900
1198892120595
1198891205852+ 1205900
119889120593
119889120585minus 1205904120595 (120585) = 0
(17)
Also the equation of the lateral response amplitude of theTimoshenko beam according to the second approach is
1198894120593
1198891205854minus (
1205901+ 12059021205904minus 1205902
0
1205902
)1198892120593
1198891205852+
12059011205904
1205902
120593 (120585)
=1205904119876 minus 119876
10158401015840
1205902
(18)
where 1205904= 119875 + 120590
3
According to the third approach (120572(120585) = 119889120593119889120585) thedynamic equations of lateral response amplitude can beobtained as
119889120595
119889120585=
1205905
1198782
1198892120593
1198891205852minus
1205901
1198782120593 (120585) +
119876 (120585)
1198782 (19)
1198892120595
1198891205852+ 1198782 119889120593
119889120585minus 1205903120595 (120585) = 0 (20)
1198894120593
1198891205854minus (
1205901+ 12059031205905minus 1198784
1205905
)1198892120593
1198891205852+
12059011205903
1205905
120593 (120585)
=1205903119876 (120585) minus 119876
10158401015840
1205905
(21)
where
1205905= 1205902minus 119875 (22)
The dynamic equations of different approaches may beexpressed as
119889120595
119889120585= 1198611
1198892120593
1198891205852+ 1198612120593 (120585) + 119861
3119876 (120585)
120595 (120585) = 1198614
1198893120593
1198891205853+ 1198615
119889120593
119889120585+ 11986161198761015840(120585)
(23)
where parameters 119861119894for the first approach are defined as
1198611=
1205902
1198782
1198612= minus
1205901
1198782
1198613=
1
1198782
1198614=
1205902
11987821205903
1198615=
11987821205900minus 1205901
11987821205903
1198616=
1
11987821205903
(24)
And for the second approach they are defined as
1198611=
1205902
1205900
1198612= minus
1205901
1205900
1198613=
1
1205900
1198614=
1205902
12059001205904
1198615=
1205902
0minus 1205901
12059001205904
1198616=
1
12059001205904
(25)
For the third approach they are defined as
1198611=
1205905
1198782
1198612= minus
1205901
1198782
1198613=
1
1198782
1198614=
1205905
11987821205903
1198615=
1198784minus 1205901
11987821205903
1198616=
1
11987821205903
(26)
23 Boundary Conditions due to Elastic Restraints For thecase of the elastic restraints at both ends the physicalboundary conditions at beam ends are
at 119909 = 0
119881 (0 119905) = minus1198961198790
119910 (0 119905)
119872 (0 119905) = 1198961198770
120579 (0 119905)
Mathematical Problems in Engineering 5
at 119909 = 119871
119881 (119871 119905) = 1198961198791
119910 (119871 119905)
119872 (119871 119905) = minus1198961198771
120579 (119871 119905)
(27)
Using the dimensionless variables 120585 and 120593 and (2) (3)and (23) the boundary conditions in dimensionless formassuming the first approach (120572(120585) = 0) may be obtained as
1198893120593
1198891205853+ (
1198615minus 1
1198614
)119889120593
119889120585minus
1198701198790
11987821198614
120593 (0) =minus1198616
1198614
1198761015840(0) (28a)
1198893120593
1198891205853minus
1198611
11986141198701198770
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585minus
1198612
11986141198701198770
120593 (0)
=1198613119876 (0) minus 119861
61198701198770
1198761015840(0)
11986141198701198770
(28b)
1198893120593
1198891205853+ (
1198615minus 1
1198614
)119889120593
119889120585+
1198701198791
11987821198614
120593 (1) =minus1198616
1198614
1198761015840(1) (28c)
1198893120593
1198891205853+
1198611
11986141198701198771
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585+
1198612
11986141198701198771
120593 (1)
= minus1198613119876 (1) + 119861
61198701198770
1198761015840(1)
11986141198701198771
(28d)
For the second approach (120572(120585) = 120595(120585)) the boundaryconditions can be expressed as
1198893120593
1198891205853+ (
12059001198615minus 1198782
12059001198614
)119889120593
119889120585minus
1198701198790
12059001198614
120593 (0) =minus1198616
1198614
1198761015840(0) (29a)
1198893120593
1198891205853minus
1198611
11986141198701198770
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585minus
1198612
11986141198701198770
120593 (0)
=1198613119876 (0) minus 119861
61198701198770
1198761015840(0)
11986141198701198770
(29b)
1198893120593
1198891205853+ (
12059001198615minus 1198782
12059001198614
)119889120593
119889120585+
1198701198791
12059001198614
120593 (1) =minus1198616
1198614
1198761015840(1) (29c)
1198893120593
1198891205853+
1198611
11986141198701198771
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585+
1198612
11986141198701198771
120593 (1)
= minus1198613119876 (1) + 119861
61198701198771
1198761015840(1)
11986141198701198771
(29d)
On the other hand for the third approach (120572(120585) = 1205931015840(120585)) the
boundary conditions can be expressed as
1198893120593
1198891205853+ (
11987821198615minus 1205906
11987821198614
)119889120593
119889120585+
1198701198790
11987821198614
120593 (0) =minus1198616
1198614
1198761015840(0) (30a)
1198893120593
1198891205853minus
1198611
11986141198701198770
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585minus
1198612
11986141198701198770
120593 (0)
=1198613119876 (0) minus 119861
61198701198770
1198761015840(0)
11986141198701198770
(30b)
1198893120593
1198891205853+ (
11987821198615minus 1205906
11987821198614
)119889120593
119889120585minus
1198701198791
11987821198614
120593 (1) =minus1198616
1198614
1198761015840(1) (30c)
1198893120593
1198891205853+
1198611
11986141198701198771
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585+
1198612
11986141198701198771
120593 (1)
= minus1198613119876 (1) + 119861
61198701198771
1198761015840(1)
11986141198701198771
(30d)
where
1205906= 1198782minus 119875
1198701198790
=11987131198961198790
119864119868
1198701198770
=1198711198961198770
119864119868
1198701198791
=1198713119896119879119871
119864119868
1198701198771
=119871119896119877119871
119864119868
(31)
3 Solution of the Governing Equations
31 Applications of the RDM to the Governing Equations Touse RDM the response amplitude equations (16) (18) or(21) are to be rewritten in the recursive form
120593(4)
(120585) = 11986011
120593(0)
+ 11986012
120593(1)
+ 11986013
120593(2)
+ 11986014
120593(3)
+ 1198651(120585)
(32)
where 120593(119898) is the 119898-derivative of 120593 Coefficients 119860
119894119895for the
first approach are
11986011
= minus12059011205903
1205902
11986012
= 0
11986013
=1205901+ 12059021205903minus 12059001198782
1205902
11986014
= 0
1198651(120585) =
1205903119876 minus 119876
10158401015840
1205902
(33)
For the second approach coefficients 119860119894119895are
11986011
= minus12059011205904
1205902
11986012
= 0
6 Mathematical Problems in Engineering
Table 1 Values of frequency parameters of the first three vibration modes
In addition coefficients 119860119894119895for the third approach are
11986011
= minus12059011205903
1205905
11986012
= 0
11986013
=1205901+ 12059031205905minus 1198784
1205905
11986014
= 0
1198651(120585) =
1205903119876 minus 119876
10158401015840
1205905
(35)
The solution of (32) may be expressed as
120593 (120585) =
4
sum
119898=1
119879119898119877119898(120585) + 119877
119865(120585) (36)
where the recursive functions 119877119898(120585) 119898 = 1 4 and the force
function 119877119865(120585) are given by
119877119898(120585) =
120585119898minus1
(119898 minus 1)+
119873
sum
119894=1
119860119894119898
120585119894+3
(119894 + 3) 119898 = 1 4
119877119865(120585) =
119873
sum
119894=1
119865119894(0)
120585119894+3
(119899 + 3)
(37)
The recurrence formulae for coefficients119860119896+1119894
119896 = 1 119873 and119865119894are given by
119860119896+11
= 1198601198964
11986011
119860119896+1119903
= 119860119896119903minus1
+ 1198601198964
1198601119903
119903 = 2 119899
119865119896+1
(120585) = 1198651015840
119896(120585) + 119860
11989641198651(120585)
(38)
The application of the boundary conditions (see (28a)ndash(28d)(29a)ndash(29d) or (30a)ndash(30d)) at 120585 = 0 1 yields a system
of 4-algebraic equations in 4 unknowns 1198791 1198792 1198793 and
1198794 however calculating these unknowns the response ampli-
tude distribution 120593(120585) can be obtainedThe distribution of bending moment 119872(120585) and shearing
force 119881(120585)may be obtained using (2) and (3) respectively
32 Calculations of Critical Loads and Natural FrequenciesThe critical loads and natural frequencies may be calculatedby assuming zero excitation amplitude (119865(120585) = 0) andreplacing the excitation frequencyΩ by the natural frequency120596 of the system The substitution of 120593(120585) and its derivativesinto the boundary conditions yields a systemof homogeneousalgebraic equations whose nontrivial solution yields a two-parameter eigenvalue problem in 119901 and120596The solution of thetwo-parameter eigenvalue problem yields both critical loads119901cr119899 and buckling modes for static case (120596 = 0) and thenatural frequencies of system120596
119899and themode shapes for free
vibration (119901 lt 119901cr)Also critical loads and natural frequencies may be
obtained from forced vibration by detecting the axial loador the excitation frequency at which the response amplitudebecomes unbounded
33 Verification of the Obtained Solutions The natural fre-quency parameter of the lower three modes for Timo-shenko beams 120582
119899 119894 = 1 3 is obtained using the present
solution assuming the third approach and compared withthose obtained from the Rayleigh-Ritz approach [12] Thecomparison is indicated in Table 1 for different boundaryconditions It is found that the results of the two approachesare compatible
4 Numerical Results and Discussion
Theobtained solutions are used to investigate the influence ofbeam and foundation parameters on the stability parametersanddynamic behavior of Timoshenko beamsThe foundationparameters (119870
1and 119870
2) and beam properties are taken
similar to those given in Taha and Doha [14]However for the investigation of the amplitude of the
lateral response the following normalized parameters areintroduced
120574 =119875
119875cr
Mathematical Problems in Engineering 7
1
2
3
4
5
Approach = 1Approach = 2Approach = 3
120578
120582
105 6 7 8 9 3020
(a) No foundation (119864119904 = 0)
Approach = 1Approach = 2Approach = 3
1
2
3
4
5
120582
120578
105 6 7 8 9 3020
(b) Stiff foundation (119864119904 = 108)
Figure 2 Variation of 120582 with 120578 for C-F beams (119875 = 021205872)
19
2
21
22
23
24
25
Approach = 1Approach = 2Approach = 3
Pcr
120578
105 6 7 8 9 3020
(a) No foundation
0
2
4
6
8
10
Approach = 1Approach = 2Approach = 3
Pcr
120578
105 6 7 8 9 3020
(b) Stiff foundation
Figure 3 Variation of 119875cr with 120578 for C-F beams
119882lowast=
119882max119882119906-max
(39)where 120574 is the loading parameter 119882
lowast is the maximumamplitude parameter119882max is the maximum amplitude of theinvestigated case and 119882
119906-max is the maximum amplitude forthe P-P corresponding case
41 Stability Parameters for Cantilever Beams Althoughmany engineering applications belong to the Timoshenkocantilever beam model a few articles are published con-cerning the stability and dynamic behavior of such model
due to its mathematical complications In the present workthe investigation of cantilever beam behavior is highlightedThe comparison between the different approaches dealingwith the shear induced from the axial load on the frequencyparameter 120582 for clamped-free beams (C-F) is shown inFigure 2 case (a) for beams without foundation (119864
119904= 0) and
case (b) for beams resting on stiff foundation (119864119904= 108)
On the other hand the influences of different approacheson quantifying the critical load of C-F beams are indicated inFigure 3 case (a) for beams without foundation and case (b)for beams on stiff foundations It is obvious that consideringthe shear component induced from the axial load decreases
8 Mathematical Problems in Engineering
15
2
25
3
35
NFWF
MFSF
120582
120578
105 6 7 8 9 3020
(a) 119875 = 0
05
1
15
2
25
3
NFWF
MFSF
120582
120578
105 6 7 8 9 3020
(b) 119875 = 021205872
Figure 4 Variation of 120582 with 120578 for C-F beams
28
3
32
34
36
38
4
NFWF
MFSF
KR
100 104
120582
(a) Slenderness ratio 120578 = 10
3
35
25
4
45
5
KR
100 104
120578 = 5120578 = 10
120578 = 20
120578 = 50
120582
(b) No foundations (119864119904 = 0)
Figure 5 Influence of 119870119877on 120582 for different types of foundation and for different 120578 (119875 = 0)
the beam stiffness and the effect is noticeable for beams withslenderness ratio 120582 lt 20
In the following analysis the parametric study is basedon the second approach as it is the rational one because theangle between the axial load and the outward normal to thedeformed cross section 120572(119909 119905) is actually equal to the angle ofcross section rotation 120595(119909 119905) in the deformed configuration
The influence of foundation type on the frequency param-eter for C-F beams is shown in Figure 4 It is found thatas the slenderness ratio increases the natural frequency ofbeams resting on foundation increases due to the increase offoundation share in the overall stiffness of the system Theincrease in the natural frequency parameter is more notice-able for stiff foundations (SF) The critical load of Timo-shenko C-F beam without foundation 119875cr = 021120587
2
42 Weakening of the End Restraints The effect of rotationalstiffness variations on the natural frequency parameter isshown in Figure 5(a) for different types of foundations (120578 =
10) and in Figure 5(b) for different values of slenderness ratio(119864119904= 0)Further the effect of rotational stiffness variations on the
critical load parameter is shown in Figure 6(a) for differenttypes of foundation (120578 = 10) and in Figure 6(b) for differentvalues of slenderness ratio (119864
119904= 0)
The influence of rotational stiffness variations on thenatural frequency parameter is shown in Figure 7 for differentloading ratios and for the cases of no foundation (119864
119904= 0) and
stiff foundations (119864119904= 108)
It is obvious that the significance of real foundation typesfor Timoshenko beam behavior (120578 lt 20) is negligible due to
Mathematical Problems in Engineering 9
5
10
15
20
25
NFWF
MFSF
KR
Pcr
10minus2 100 102 104
(a) 120578 = 10
0
10
20
30
40
120578 = 5120578 = 10
120578 = 20
120578 = 50
Pcr
KR
10minus2 100 102 104
(b) 119864119904 = 0
Figure 6 Influence of 119870119877on 119875cr for different types of foundation and for different 120578
2
25
3
35
4
KR
100 104
120582
120574 = 0120574 = 02
120574 = 04
120574 = 06
(a) No foundation (119864119904 = 0)
2
25
3
35
4
KR
100 104
120582
120574 = 0120574 = 02
120574 = 04
120574 = 06
(b) Stiff foundation (119864119904 = 108)
Figure 7 Influence of 119870119877on 120582 for different loading ratios (120578 = 10)
the high stiffness of the beam relative to the foundation Alsoit is found that the influence of rotational stiffness increaseswith the increase in the slenderness ratio Also it is found thatthe natural frequency decreases as the axial load increases tillthe axial load approaches its critical values then the naturalfrequency approaches zero
43 Forced Vibration Analysis Theweakening of ends elasticrestraints is represented by the variation of the restraint stiff-nessThe variation of normalized maximum response ampli-tude (119882lowast) with the ends rotational stiffness (119870
119877) is indicated
in Figure 8 for different loading types Case (a) representsthe effects of uniform loading functions case (b) representsthe effect of linear loading functions and case (c) represents
the effect of quadratic loading functions As the case of119870119877=
0 represents the P-P beams it is obvious from the figuresthat the displacement due to dynamic loading is greater thanthose for static loading due to the inertia force interferenceat 119870119877
= 0 Also it is clear that as 119870119877increases the stiffness
of the beam increases and the response amplitude decreasesIt is obvious that at a certain combination of the systemparameters the resonance condition is approached andunbounded response amplitude is detected As the axial loadincreases the variations of the end rotational stiffness maycause the system to reach the resonance conditions and theamplitude becomes unbounded If the damping is taken intoconsideration the amplitude will possess a finite value butsuch condition is to be prevented However it is concluded
Figure 8 Variation of119882lowast with 119870119877for different axial loading ratio (120578 = 10)
Mathematical Problems in Engineering 11
that the weakening of the end restraints may offer a param-eters combination coinciding with the resonance conditionsfor the first mode or one of the higher modes and unboundedamplitudes may be reached
5 Conclusions
The RDM is used to investigate the stability behavior ofaxially loaded cantilever Timoshenko beam resting on two-parameter foundation In addition the analysis includesthe investigation of Timoshenko beams resting on two-parameter foundation and subjected to different types of lat-eral excitationThe supports at the beam ends are assumed ofthe elastic type to investigate the impact of the support weak-ening on the beambehavior Both the effect of rotational iner-tia and the correction of the shear stress due to axial loadingare taken into consideration which decreases the beam stiff-ness It is concluded that the significance of the foundation onthe Timoshenko beams (120578 lt 20) behavior is negligiblethe effect of end restraints is more noticeable for slenderbeams and the natural frequency decreases as the axial loadincreases However for the case of the forced vibration theresonance conditions may result from the weakening of theend supports and unbounded response amplitudes may bereached The analysis depicts the simplicity and accuracy ofthe RDM in tackling the boundary value problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Ruta ldquoThe application of Chebyshev polynomials to the solu-tion of the nonprismatic Timoshenko beam vibration problemrdquoJournal of Sound and Vibration vol 296 no 1-2 pp 243ndash2632006
[2] K Kausel ldquoNonclassical modes of unrestrained shear beamsrdquoJournal of Engineering Mechanics vol 128 no 6 pp 663ndash6672002
[3] R Attarnejad A Shahba and S J Semnani ldquoApplication ofdifferential transform in free vibration analysis of timoshenkobeams resting on two-parameter elastic foundationrdquo ArabianJournal for Science and Engineering vol 35 no 2B pp 125ndash1322010
[4] M Taha and M A Nassar ldquoAnalysis of stressed Timoshenkobeams on two parameter foundationsrdquo KSCE Journal of CivilEngineering vol 19 no 1 pp 173ndash179 2015
[5] L Majkut ldquoFree and forced vibration of Timoshenko beamsdescribed by single difference equationrdquo Journal of Theoreticaland Applied Mechanics vol 47 no 1 pp 193ndash210 2009
[6] F Y Cheng and C P Pantelides ldquoDynamic timoshenko beam-column on elastic mediardquo Journal of Structural Engineering vol114 no 7 pp 1524ndash1550 1988
[7] B Geist and J R Mclaughlin ldquoDouble eigen value for theTimoshenko beamrdquo Applied Mathematics Letters vol 10 pp129ndash134 1997
[8] C-N Chen ldquoDQEM vibration analyses of non-prismatic sheardeformable beams resting on elastic foundationsrdquo Journal ofSound and Vibration vol 255 no 5 pp 989ndash999 2002
[9] A L G Monsalve Z D G Medina and A J D Ochoa ldquoTim-oshenko beam-column with generalized end conditions onelastic foundation dynamic-stiffness matrix and load vectorrdquoJournal of Sound and Vibration vol 310 no 4-5 pp 1057ndash10792008
[10] T Kocaturk and M Simsek ldquoFree vibration analysis of Tim-oshenko beams under various boundary conditionsrdquo SigmaJournal of Engineering and Natural Sciences vol 1 pp 30ndash442005
[11] DKNguyen ldquoFree vibration of prestressedTimoshenkobeamsresting on elastic foundationsrdquo Vietnam Journal of Mechanicsvol 29 no 1 pp 1ndash12 2012
[12] N M Auciello ldquoVibrations of Timoshenko beams on twoparameter elastic soilrdquo Engineering Transactions vol 56 no 3pp 187ndash200 2008
[13] M Taha ldquoRecursive differentiation method for boundary valueproblems application to analysis of a beam-column on an elas-tic foundationrdquo Journal of Theoretical and Applied Mechanicsvol 44 no 2 pp 57ndash70 2014
[14] M Taha and E H Doha ldquoRecursive differentiation methodapplication to the analysis of beams on two parameter founda-tionsrdquo Journal of Theoretical and Applied Mechanics vol 53 no1 pp 15ndash26 2015
[15] S P Timoshenko and J M Gere Theory of Elastic StabilityEngineering Societies Monographs McGraw-Hill Book NewYork NY USA 1961
where 119878 is the slenderness parameter 120582119865is the frequency
parameter 120578 is the slenderness ratio 119903 is the radius of gyra-tion 119875 is the axial load parameter119870
1is the foundation linear
stiffness parameter and 1198702is the foundation shear stiffness
parameterInserting (14) into (13) the equation describing the lateral
response amplitude of the Timoshenko beam based on thefirst approach is
1198894120593
1198891205854minus (
1205901+ 12059021205903minus 12059001198782
1205902
)1198892120593
1198891205852+
12059011205903
1205902
120593 (120585)
=1205903119876 minus 119876
10158401015840
1205902
(16)
Similarly for the second approach (120572(120585) = 120595(120585)) (11) may beexpressed as
119889120595
119889120585=
1205902
1205900
1198892120593
1198891205852minus
1205901
1205900
120593 (120585) +119876 (120585)
1205900
1198892120595
1198891205852+ 1205900
119889120593
119889120585minus 1205904120595 (120585) = 0
(17)
Also the equation of the lateral response amplitude of theTimoshenko beam according to the second approach is
1198894120593
1198891205854minus (
1205901+ 12059021205904minus 1205902
0
1205902
)1198892120593
1198891205852+
12059011205904
1205902
120593 (120585)
=1205904119876 minus 119876
10158401015840
1205902
(18)
where 1205904= 119875 + 120590
3
According to the third approach (120572(120585) = 119889120593119889120585) thedynamic equations of lateral response amplitude can beobtained as
119889120595
119889120585=
1205905
1198782
1198892120593
1198891205852minus
1205901
1198782120593 (120585) +
119876 (120585)
1198782 (19)
1198892120595
1198891205852+ 1198782 119889120593
119889120585minus 1205903120595 (120585) = 0 (20)
1198894120593
1198891205854minus (
1205901+ 12059031205905minus 1198784
1205905
)1198892120593
1198891205852+
12059011205903
1205905
120593 (120585)
=1205903119876 (120585) minus 119876
10158401015840
1205905
(21)
where
1205905= 1205902minus 119875 (22)
The dynamic equations of different approaches may beexpressed as
119889120595
119889120585= 1198611
1198892120593
1198891205852+ 1198612120593 (120585) + 119861
3119876 (120585)
120595 (120585) = 1198614
1198893120593
1198891205853+ 1198615
119889120593
119889120585+ 11986161198761015840(120585)
(23)
where parameters 119861119894for the first approach are defined as
1198611=
1205902
1198782
1198612= minus
1205901
1198782
1198613=
1
1198782
1198614=
1205902
11987821205903
1198615=
11987821205900minus 1205901
11987821205903
1198616=
1
11987821205903
(24)
And for the second approach they are defined as
1198611=
1205902
1205900
1198612= minus
1205901
1205900
1198613=
1
1205900
1198614=
1205902
12059001205904
1198615=
1205902
0minus 1205901
12059001205904
1198616=
1
12059001205904
(25)
For the third approach they are defined as
1198611=
1205905
1198782
1198612= minus
1205901
1198782
1198613=
1
1198782
1198614=
1205905
11987821205903
1198615=
1198784minus 1205901
11987821205903
1198616=
1
11987821205903
(26)
23 Boundary Conditions due to Elastic Restraints For thecase of the elastic restraints at both ends the physicalboundary conditions at beam ends are
at 119909 = 0
119881 (0 119905) = minus1198961198790
119910 (0 119905)
119872 (0 119905) = 1198961198770
120579 (0 119905)
Mathematical Problems in Engineering 5
at 119909 = 119871
119881 (119871 119905) = 1198961198791
119910 (119871 119905)
119872 (119871 119905) = minus1198961198771
120579 (119871 119905)
(27)
Using the dimensionless variables 120585 and 120593 and (2) (3)and (23) the boundary conditions in dimensionless formassuming the first approach (120572(120585) = 0) may be obtained as
1198893120593
1198891205853+ (
1198615minus 1
1198614
)119889120593
119889120585minus
1198701198790
11987821198614
120593 (0) =minus1198616
1198614
1198761015840(0) (28a)
1198893120593
1198891205853minus
1198611
11986141198701198770
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585minus
1198612
11986141198701198770
120593 (0)
=1198613119876 (0) minus 119861
61198701198770
1198761015840(0)
11986141198701198770
(28b)
1198893120593
1198891205853+ (
1198615minus 1
1198614
)119889120593
119889120585+
1198701198791
11987821198614
120593 (1) =minus1198616
1198614
1198761015840(1) (28c)
1198893120593
1198891205853+
1198611
11986141198701198771
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585+
1198612
11986141198701198771
120593 (1)
= minus1198613119876 (1) + 119861
61198701198770
1198761015840(1)
11986141198701198771
(28d)
For the second approach (120572(120585) = 120595(120585)) the boundaryconditions can be expressed as
1198893120593
1198891205853+ (
12059001198615minus 1198782
12059001198614
)119889120593
119889120585minus
1198701198790
12059001198614
120593 (0) =minus1198616
1198614
1198761015840(0) (29a)
1198893120593
1198891205853minus
1198611
11986141198701198770
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585minus
1198612
11986141198701198770
120593 (0)
=1198613119876 (0) minus 119861
61198701198770
1198761015840(0)
11986141198701198770
(29b)
1198893120593
1198891205853+ (
12059001198615minus 1198782
12059001198614
)119889120593
119889120585+
1198701198791
12059001198614
120593 (1) =minus1198616
1198614
1198761015840(1) (29c)
1198893120593
1198891205853+
1198611
11986141198701198771
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585+
1198612
11986141198701198771
120593 (1)
= minus1198613119876 (1) + 119861
61198701198771
1198761015840(1)
11986141198701198771
(29d)
On the other hand for the third approach (120572(120585) = 1205931015840(120585)) the
boundary conditions can be expressed as
1198893120593
1198891205853+ (
11987821198615minus 1205906
11987821198614
)119889120593
119889120585+
1198701198790
11987821198614
120593 (0) =minus1198616
1198614
1198761015840(0) (30a)
1198893120593
1198891205853minus
1198611
11986141198701198770
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585minus
1198612
11986141198701198770
120593 (0)
=1198613119876 (0) minus 119861
61198701198770
1198761015840(0)
11986141198701198770
(30b)
1198893120593
1198891205853+ (
11987821198615minus 1205906
11987821198614
)119889120593
119889120585minus
1198701198791
11987821198614
120593 (1) =minus1198616
1198614
1198761015840(1) (30c)
1198893120593
1198891205853+
1198611
11986141198701198771
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585+
1198612
11986141198701198771
120593 (1)
= minus1198613119876 (1) + 119861
61198701198771
1198761015840(1)
11986141198701198771
(30d)
where
1205906= 1198782minus 119875
1198701198790
=11987131198961198790
119864119868
1198701198770
=1198711198961198770
119864119868
1198701198791
=1198713119896119879119871
119864119868
1198701198771
=119871119896119877119871
119864119868
(31)
3 Solution of the Governing Equations
31 Applications of the RDM to the Governing Equations Touse RDM the response amplitude equations (16) (18) or(21) are to be rewritten in the recursive form
120593(4)
(120585) = 11986011
120593(0)
+ 11986012
120593(1)
+ 11986013
120593(2)
+ 11986014
120593(3)
+ 1198651(120585)
(32)
where 120593(119898) is the 119898-derivative of 120593 Coefficients 119860
119894119895for the
first approach are
11986011
= minus12059011205903
1205902
11986012
= 0
11986013
=1205901+ 12059021205903minus 12059001198782
1205902
11986014
= 0
1198651(120585) =
1205903119876 minus 119876
10158401015840
1205902
(33)
For the second approach coefficients 119860119894119895are
11986011
= minus12059011205904
1205902
11986012
= 0
6 Mathematical Problems in Engineering
Table 1 Values of frequency parameters of the first three vibration modes
In addition coefficients 119860119894119895for the third approach are
11986011
= minus12059011205903
1205905
11986012
= 0
11986013
=1205901+ 12059031205905minus 1198784
1205905
11986014
= 0
1198651(120585) =
1205903119876 minus 119876
10158401015840
1205905
(35)
The solution of (32) may be expressed as
120593 (120585) =
4
sum
119898=1
119879119898119877119898(120585) + 119877
119865(120585) (36)
where the recursive functions 119877119898(120585) 119898 = 1 4 and the force
function 119877119865(120585) are given by
119877119898(120585) =
120585119898minus1
(119898 minus 1)+
119873
sum
119894=1
119860119894119898
120585119894+3
(119894 + 3) 119898 = 1 4
119877119865(120585) =
119873
sum
119894=1
119865119894(0)
120585119894+3
(119899 + 3)
(37)
The recurrence formulae for coefficients119860119896+1119894
119896 = 1 119873 and119865119894are given by
119860119896+11
= 1198601198964
11986011
119860119896+1119903
= 119860119896119903minus1
+ 1198601198964
1198601119903
119903 = 2 119899
119865119896+1
(120585) = 1198651015840
119896(120585) + 119860
11989641198651(120585)
(38)
The application of the boundary conditions (see (28a)ndash(28d)(29a)ndash(29d) or (30a)ndash(30d)) at 120585 = 0 1 yields a system
of 4-algebraic equations in 4 unknowns 1198791 1198792 1198793 and
1198794 however calculating these unknowns the response ampli-
tude distribution 120593(120585) can be obtainedThe distribution of bending moment 119872(120585) and shearing
force 119881(120585)may be obtained using (2) and (3) respectively
32 Calculations of Critical Loads and Natural FrequenciesThe critical loads and natural frequencies may be calculatedby assuming zero excitation amplitude (119865(120585) = 0) andreplacing the excitation frequencyΩ by the natural frequency120596 of the system The substitution of 120593(120585) and its derivativesinto the boundary conditions yields a systemof homogeneousalgebraic equations whose nontrivial solution yields a two-parameter eigenvalue problem in 119901 and120596The solution of thetwo-parameter eigenvalue problem yields both critical loads119901cr119899 and buckling modes for static case (120596 = 0) and thenatural frequencies of system120596
119899and themode shapes for free
vibration (119901 lt 119901cr)Also critical loads and natural frequencies may be
obtained from forced vibration by detecting the axial loador the excitation frequency at which the response amplitudebecomes unbounded
33 Verification of the Obtained Solutions The natural fre-quency parameter of the lower three modes for Timo-shenko beams 120582
119899 119894 = 1 3 is obtained using the present
solution assuming the third approach and compared withthose obtained from the Rayleigh-Ritz approach [12] Thecomparison is indicated in Table 1 for different boundaryconditions It is found that the results of the two approachesare compatible
4 Numerical Results and Discussion
Theobtained solutions are used to investigate the influence ofbeam and foundation parameters on the stability parametersanddynamic behavior of Timoshenko beamsThe foundationparameters (119870
1and 119870
2) and beam properties are taken
similar to those given in Taha and Doha [14]However for the investigation of the amplitude of the
lateral response the following normalized parameters areintroduced
120574 =119875
119875cr
Mathematical Problems in Engineering 7
1
2
3
4
5
Approach = 1Approach = 2Approach = 3
120578
120582
105 6 7 8 9 3020
(a) No foundation (119864119904 = 0)
Approach = 1Approach = 2Approach = 3
1
2
3
4
5
120582
120578
105 6 7 8 9 3020
(b) Stiff foundation (119864119904 = 108)
Figure 2 Variation of 120582 with 120578 for C-F beams (119875 = 021205872)
19
2
21
22
23
24
25
Approach = 1Approach = 2Approach = 3
Pcr
120578
105 6 7 8 9 3020
(a) No foundation
0
2
4
6
8
10
Approach = 1Approach = 2Approach = 3
Pcr
120578
105 6 7 8 9 3020
(b) Stiff foundation
Figure 3 Variation of 119875cr with 120578 for C-F beams
119882lowast=
119882max119882119906-max
(39)where 120574 is the loading parameter 119882
lowast is the maximumamplitude parameter119882max is the maximum amplitude of theinvestigated case and 119882
119906-max is the maximum amplitude forthe P-P corresponding case
41 Stability Parameters for Cantilever Beams Althoughmany engineering applications belong to the Timoshenkocantilever beam model a few articles are published con-cerning the stability and dynamic behavior of such model
due to its mathematical complications In the present workthe investigation of cantilever beam behavior is highlightedThe comparison between the different approaches dealingwith the shear induced from the axial load on the frequencyparameter 120582 for clamped-free beams (C-F) is shown inFigure 2 case (a) for beams without foundation (119864
119904= 0) and
case (b) for beams resting on stiff foundation (119864119904= 108)
On the other hand the influences of different approacheson quantifying the critical load of C-F beams are indicated inFigure 3 case (a) for beams without foundation and case (b)for beams on stiff foundations It is obvious that consideringthe shear component induced from the axial load decreases
8 Mathematical Problems in Engineering
15
2
25
3
35
NFWF
MFSF
120582
120578
105 6 7 8 9 3020
(a) 119875 = 0
05
1
15
2
25
3
NFWF
MFSF
120582
120578
105 6 7 8 9 3020
(b) 119875 = 021205872
Figure 4 Variation of 120582 with 120578 for C-F beams
28
3
32
34
36
38
4
NFWF
MFSF
KR
100 104
120582
(a) Slenderness ratio 120578 = 10
3
35
25
4
45
5
KR
100 104
120578 = 5120578 = 10
120578 = 20
120578 = 50
120582
(b) No foundations (119864119904 = 0)
Figure 5 Influence of 119870119877on 120582 for different types of foundation and for different 120578 (119875 = 0)
the beam stiffness and the effect is noticeable for beams withslenderness ratio 120582 lt 20
In the following analysis the parametric study is basedon the second approach as it is the rational one because theangle between the axial load and the outward normal to thedeformed cross section 120572(119909 119905) is actually equal to the angle ofcross section rotation 120595(119909 119905) in the deformed configuration
The influence of foundation type on the frequency param-eter for C-F beams is shown in Figure 4 It is found thatas the slenderness ratio increases the natural frequency ofbeams resting on foundation increases due to the increase offoundation share in the overall stiffness of the system Theincrease in the natural frequency parameter is more notice-able for stiff foundations (SF) The critical load of Timo-shenko C-F beam without foundation 119875cr = 021120587
2
42 Weakening of the End Restraints The effect of rotationalstiffness variations on the natural frequency parameter isshown in Figure 5(a) for different types of foundations (120578 =
10) and in Figure 5(b) for different values of slenderness ratio(119864119904= 0)Further the effect of rotational stiffness variations on the
critical load parameter is shown in Figure 6(a) for differenttypes of foundation (120578 = 10) and in Figure 6(b) for differentvalues of slenderness ratio (119864
119904= 0)
The influence of rotational stiffness variations on thenatural frequency parameter is shown in Figure 7 for differentloading ratios and for the cases of no foundation (119864
119904= 0) and
stiff foundations (119864119904= 108)
It is obvious that the significance of real foundation typesfor Timoshenko beam behavior (120578 lt 20) is negligible due to
Mathematical Problems in Engineering 9
5
10
15
20
25
NFWF
MFSF
KR
Pcr
10minus2 100 102 104
(a) 120578 = 10
0
10
20
30
40
120578 = 5120578 = 10
120578 = 20
120578 = 50
Pcr
KR
10minus2 100 102 104
(b) 119864119904 = 0
Figure 6 Influence of 119870119877on 119875cr for different types of foundation and for different 120578
2
25
3
35
4
KR
100 104
120582
120574 = 0120574 = 02
120574 = 04
120574 = 06
(a) No foundation (119864119904 = 0)
2
25
3
35
4
KR
100 104
120582
120574 = 0120574 = 02
120574 = 04
120574 = 06
(b) Stiff foundation (119864119904 = 108)
Figure 7 Influence of 119870119877on 120582 for different loading ratios (120578 = 10)
the high stiffness of the beam relative to the foundation Alsoit is found that the influence of rotational stiffness increaseswith the increase in the slenderness ratio Also it is found thatthe natural frequency decreases as the axial load increases tillthe axial load approaches its critical values then the naturalfrequency approaches zero
43 Forced Vibration Analysis Theweakening of ends elasticrestraints is represented by the variation of the restraint stiff-nessThe variation of normalized maximum response ampli-tude (119882lowast) with the ends rotational stiffness (119870
119877) is indicated
in Figure 8 for different loading types Case (a) representsthe effects of uniform loading functions case (b) representsthe effect of linear loading functions and case (c) represents
the effect of quadratic loading functions As the case of119870119877=
0 represents the P-P beams it is obvious from the figuresthat the displacement due to dynamic loading is greater thanthose for static loading due to the inertia force interferenceat 119870119877
= 0 Also it is clear that as 119870119877increases the stiffness
of the beam increases and the response amplitude decreasesIt is obvious that at a certain combination of the systemparameters the resonance condition is approached andunbounded response amplitude is detected As the axial loadincreases the variations of the end rotational stiffness maycause the system to reach the resonance conditions and theamplitude becomes unbounded If the damping is taken intoconsideration the amplitude will possess a finite value butsuch condition is to be prevented However it is concluded
Figure 8 Variation of119882lowast with 119870119877for different axial loading ratio (120578 = 10)
Mathematical Problems in Engineering 11
that the weakening of the end restraints may offer a param-eters combination coinciding with the resonance conditionsfor the first mode or one of the higher modes and unboundedamplitudes may be reached
5 Conclusions
The RDM is used to investigate the stability behavior ofaxially loaded cantilever Timoshenko beam resting on two-parameter foundation In addition the analysis includesthe investigation of Timoshenko beams resting on two-parameter foundation and subjected to different types of lat-eral excitationThe supports at the beam ends are assumed ofthe elastic type to investigate the impact of the support weak-ening on the beambehavior Both the effect of rotational iner-tia and the correction of the shear stress due to axial loadingare taken into consideration which decreases the beam stiff-ness It is concluded that the significance of the foundation onthe Timoshenko beams (120578 lt 20) behavior is negligiblethe effect of end restraints is more noticeable for slenderbeams and the natural frequency decreases as the axial loadincreases However for the case of the forced vibration theresonance conditions may result from the weakening of theend supports and unbounded response amplitudes may bereached The analysis depicts the simplicity and accuracy ofthe RDM in tackling the boundary value problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Ruta ldquoThe application of Chebyshev polynomials to the solu-tion of the nonprismatic Timoshenko beam vibration problemrdquoJournal of Sound and Vibration vol 296 no 1-2 pp 243ndash2632006
[2] K Kausel ldquoNonclassical modes of unrestrained shear beamsrdquoJournal of Engineering Mechanics vol 128 no 6 pp 663ndash6672002
[3] R Attarnejad A Shahba and S J Semnani ldquoApplication ofdifferential transform in free vibration analysis of timoshenkobeams resting on two-parameter elastic foundationrdquo ArabianJournal for Science and Engineering vol 35 no 2B pp 125ndash1322010
[4] M Taha and M A Nassar ldquoAnalysis of stressed Timoshenkobeams on two parameter foundationsrdquo KSCE Journal of CivilEngineering vol 19 no 1 pp 173ndash179 2015
[5] L Majkut ldquoFree and forced vibration of Timoshenko beamsdescribed by single difference equationrdquo Journal of Theoreticaland Applied Mechanics vol 47 no 1 pp 193ndash210 2009
[6] F Y Cheng and C P Pantelides ldquoDynamic timoshenko beam-column on elastic mediardquo Journal of Structural Engineering vol114 no 7 pp 1524ndash1550 1988
[7] B Geist and J R Mclaughlin ldquoDouble eigen value for theTimoshenko beamrdquo Applied Mathematics Letters vol 10 pp129ndash134 1997
[8] C-N Chen ldquoDQEM vibration analyses of non-prismatic sheardeformable beams resting on elastic foundationsrdquo Journal ofSound and Vibration vol 255 no 5 pp 989ndash999 2002
[9] A L G Monsalve Z D G Medina and A J D Ochoa ldquoTim-oshenko beam-column with generalized end conditions onelastic foundation dynamic-stiffness matrix and load vectorrdquoJournal of Sound and Vibration vol 310 no 4-5 pp 1057ndash10792008
[10] T Kocaturk and M Simsek ldquoFree vibration analysis of Tim-oshenko beams under various boundary conditionsrdquo SigmaJournal of Engineering and Natural Sciences vol 1 pp 30ndash442005
[11] DKNguyen ldquoFree vibration of prestressedTimoshenkobeamsresting on elastic foundationsrdquo Vietnam Journal of Mechanicsvol 29 no 1 pp 1ndash12 2012
[12] N M Auciello ldquoVibrations of Timoshenko beams on twoparameter elastic soilrdquo Engineering Transactions vol 56 no 3pp 187ndash200 2008
[13] M Taha ldquoRecursive differentiation method for boundary valueproblems application to analysis of a beam-column on an elas-tic foundationrdquo Journal of Theoretical and Applied Mechanicsvol 44 no 2 pp 57ndash70 2014
[14] M Taha and E H Doha ldquoRecursive differentiation methodapplication to the analysis of beams on two parameter founda-tionsrdquo Journal of Theoretical and Applied Mechanics vol 53 no1 pp 15ndash26 2015
[15] S P Timoshenko and J M Gere Theory of Elastic StabilityEngineering Societies Monographs McGraw-Hill Book NewYork NY USA 1961
Using the dimensionless variables 120585 and 120593 and (2) (3)and (23) the boundary conditions in dimensionless formassuming the first approach (120572(120585) = 0) may be obtained as
1198893120593
1198891205853+ (
1198615minus 1
1198614
)119889120593
119889120585minus
1198701198790
11987821198614
120593 (0) =minus1198616
1198614
1198761015840(0) (28a)
1198893120593
1198891205853minus
1198611
11986141198701198770
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585minus
1198612
11986141198701198770
120593 (0)
=1198613119876 (0) minus 119861
61198701198770
1198761015840(0)
11986141198701198770
(28b)
1198893120593
1198891205853+ (
1198615minus 1
1198614
)119889120593
119889120585+
1198701198791
11987821198614
120593 (1) =minus1198616
1198614
1198761015840(1) (28c)
1198893120593
1198891205853+
1198611
11986141198701198771
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585+
1198612
11986141198701198771
120593 (1)
= minus1198613119876 (1) + 119861
61198701198770
1198761015840(1)
11986141198701198771
(28d)
For the second approach (120572(120585) = 120595(120585)) the boundaryconditions can be expressed as
1198893120593
1198891205853+ (
12059001198615minus 1198782
12059001198614
)119889120593
119889120585minus
1198701198790
12059001198614
120593 (0) =minus1198616
1198614
1198761015840(0) (29a)
1198893120593
1198891205853minus
1198611
11986141198701198770
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585minus
1198612
11986141198701198770
120593 (0)
=1198613119876 (0) minus 119861
61198701198770
1198761015840(0)
11986141198701198770
(29b)
1198893120593
1198891205853+ (
12059001198615minus 1198782
12059001198614
)119889120593
119889120585+
1198701198791
12059001198614
120593 (1) =minus1198616
1198614
1198761015840(1) (29c)
1198893120593
1198891205853+
1198611
11986141198701198771
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585+
1198612
11986141198701198771
120593 (1)
= minus1198613119876 (1) + 119861
61198701198771
1198761015840(1)
11986141198701198771
(29d)
On the other hand for the third approach (120572(120585) = 1205931015840(120585)) the
boundary conditions can be expressed as
1198893120593
1198891205853+ (
11987821198615minus 1205906
11987821198614
)119889120593
119889120585+
1198701198790
11987821198614
120593 (0) =minus1198616
1198614
1198761015840(0) (30a)
1198893120593
1198891205853minus
1198611
11986141198701198770
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585minus
1198612
11986141198701198770
120593 (0)
=1198613119876 (0) minus 119861
61198701198770
1198761015840(0)
11986141198701198770
(30b)
1198893120593
1198891205853+ (
11987821198615minus 1205906
11987821198614
)119889120593
119889120585minus
1198701198791
11987821198614
120593 (1) =minus1198616
1198614
1198761015840(1) (30c)
1198893120593
1198891205853+
1198611
11986141198701198771
1198892120593
1198891205852+
1198615
1198614
119889120593
119889120585+
1198612
11986141198701198771
120593 (1)
= minus1198613119876 (1) + 119861
61198701198771
1198761015840(1)
11986141198701198771
(30d)
where
1205906= 1198782minus 119875
1198701198790
=11987131198961198790
119864119868
1198701198770
=1198711198961198770
119864119868
1198701198791
=1198713119896119879119871
119864119868
1198701198771
=119871119896119877119871
119864119868
(31)
3 Solution of the Governing Equations
31 Applications of the RDM to the Governing Equations Touse RDM the response amplitude equations (16) (18) or(21) are to be rewritten in the recursive form
120593(4)
(120585) = 11986011
120593(0)
+ 11986012
120593(1)
+ 11986013
120593(2)
+ 11986014
120593(3)
+ 1198651(120585)
(32)
where 120593(119898) is the 119898-derivative of 120593 Coefficients 119860
119894119895for the
first approach are
11986011
= minus12059011205903
1205902
11986012
= 0
11986013
=1205901+ 12059021205903minus 12059001198782
1205902
11986014
= 0
1198651(120585) =
1205903119876 minus 119876
10158401015840
1205902
(33)
For the second approach coefficients 119860119894119895are
11986011
= minus12059011205904
1205902
11986012
= 0
6 Mathematical Problems in Engineering
Table 1 Values of frequency parameters of the first three vibration modes
In addition coefficients 119860119894119895for the third approach are
11986011
= minus12059011205903
1205905
11986012
= 0
11986013
=1205901+ 12059031205905minus 1198784
1205905
11986014
= 0
1198651(120585) =
1205903119876 minus 119876
10158401015840
1205905
(35)
The solution of (32) may be expressed as
120593 (120585) =
4
sum
119898=1
119879119898119877119898(120585) + 119877
119865(120585) (36)
where the recursive functions 119877119898(120585) 119898 = 1 4 and the force
function 119877119865(120585) are given by
119877119898(120585) =
120585119898minus1
(119898 minus 1)+
119873
sum
119894=1
119860119894119898
120585119894+3
(119894 + 3) 119898 = 1 4
119877119865(120585) =
119873
sum
119894=1
119865119894(0)
120585119894+3
(119899 + 3)
(37)
The recurrence formulae for coefficients119860119896+1119894
119896 = 1 119873 and119865119894are given by
119860119896+11
= 1198601198964
11986011
119860119896+1119903
= 119860119896119903minus1
+ 1198601198964
1198601119903
119903 = 2 119899
119865119896+1
(120585) = 1198651015840
119896(120585) + 119860
11989641198651(120585)
(38)
The application of the boundary conditions (see (28a)ndash(28d)(29a)ndash(29d) or (30a)ndash(30d)) at 120585 = 0 1 yields a system
of 4-algebraic equations in 4 unknowns 1198791 1198792 1198793 and
1198794 however calculating these unknowns the response ampli-
tude distribution 120593(120585) can be obtainedThe distribution of bending moment 119872(120585) and shearing
force 119881(120585)may be obtained using (2) and (3) respectively
32 Calculations of Critical Loads and Natural FrequenciesThe critical loads and natural frequencies may be calculatedby assuming zero excitation amplitude (119865(120585) = 0) andreplacing the excitation frequencyΩ by the natural frequency120596 of the system The substitution of 120593(120585) and its derivativesinto the boundary conditions yields a systemof homogeneousalgebraic equations whose nontrivial solution yields a two-parameter eigenvalue problem in 119901 and120596The solution of thetwo-parameter eigenvalue problem yields both critical loads119901cr119899 and buckling modes for static case (120596 = 0) and thenatural frequencies of system120596
119899and themode shapes for free
vibration (119901 lt 119901cr)Also critical loads and natural frequencies may be
obtained from forced vibration by detecting the axial loador the excitation frequency at which the response amplitudebecomes unbounded
33 Verification of the Obtained Solutions The natural fre-quency parameter of the lower three modes for Timo-shenko beams 120582
119899 119894 = 1 3 is obtained using the present
solution assuming the third approach and compared withthose obtained from the Rayleigh-Ritz approach [12] Thecomparison is indicated in Table 1 for different boundaryconditions It is found that the results of the two approachesare compatible
4 Numerical Results and Discussion
Theobtained solutions are used to investigate the influence ofbeam and foundation parameters on the stability parametersanddynamic behavior of Timoshenko beamsThe foundationparameters (119870
1and 119870
2) and beam properties are taken
similar to those given in Taha and Doha [14]However for the investigation of the amplitude of the
lateral response the following normalized parameters areintroduced
120574 =119875
119875cr
Mathematical Problems in Engineering 7
1
2
3
4
5
Approach = 1Approach = 2Approach = 3
120578
120582
105 6 7 8 9 3020
(a) No foundation (119864119904 = 0)
Approach = 1Approach = 2Approach = 3
1
2
3
4
5
120582
120578
105 6 7 8 9 3020
(b) Stiff foundation (119864119904 = 108)
Figure 2 Variation of 120582 with 120578 for C-F beams (119875 = 021205872)
19
2
21
22
23
24
25
Approach = 1Approach = 2Approach = 3
Pcr
120578
105 6 7 8 9 3020
(a) No foundation
0
2
4
6
8
10
Approach = 1Approach = 2Approach = 3
Pcr
120578
105 6 7 8 9 3020
(b) Stiff foundation
Figure 3 Variation of 119875cr with 120578 for C-F beams
119882lowast=
119882max119882119906-max
(39)where 120574 is the loading parameter 119882
lowast is the maximumamplitude parameter119882max is the maximum amplitude of theinvestigated case and 119882
119906-max is the maximum amplitude forthe P-P corresponding case
41 Stability Parameters for Cantilever Beams Althoughmany engineering applications belong to the Timoshenkocantilever beam model a few articles are published con-cerning the stability and dynamic behavior of such model
due to its mathematical complications In the present workthe investigation of cantilever beam behavior is highlightedThe comparison between the different approaches dealingwith the shear induced from the axial load on the frequencyparameter 120582 for clamped-free beams (C-F) is shown inFigure 2 case (a) for beams without foundation (119864
119904= 0) and
case (b) for beams resting on stiff foundation (119864119904= 108)
On the other hand the influences of different approacheson quantifying the critical load of C-F beams are indicated inFigure 3 case (a) for beams without foundation and case (b)for beams on stiff foundations It is obvious that consideringthe shear component induced from the axial load decreases
8 Mathematical Problems in Engineering
15
2
25
3
35
NFWF
MFSF
120582
120578
105 6 7 8 9 3020
(a) 119875 = 0
05
1
15
2
25
3
NFWF
MFSF
120582
120578
105 6 7 8 9 3020
(b) 119875 = 021205872
Figure 4 Variation of 120582 with 120578 for C-F beams
28
3
32
34
36
38
4
NFWF
MFSF
KR
100 104
120582
(a) Slenderness ratio 120578 = 10
3
35
25
4
45
5
KR
100 104
120578 = 5120578 = 10
120578 = 20
120578 = 50
120582
(b) No foundations (119864119904 = 0)
Figure 5 Influence of 119870119877on 120582 for different types of foundation and for different 120578 (119875 = 0)
the beam stiffness and the effect is noticeable for beams withslenderness ratio 120582 lt 20
In the following analysis the parametric study is basedon the second approach as it is the rational one because theangle between the axial load and the outward normal to thedeformed cross section 120572(119909 119905) is actually equal to the angle ofcross section rotation 120595(119909 119905) in the deformed configuration
The influence of foundation type on the frequency param-eter for C-F beams is shown in Figure 4 It is found thatas the slenderness ratio increases the natural frequency ofbeams resting on foundation increases due to the increase offoundation share in the overall stiffness of the system Theincrease in the natural frequency parameter is more notice-able for stiff foundations (SF) The critical load of Timo-shenko C-F beam without foundation 119875cr = 021120587
2
42 Weakening of the End Restraints The effect of rotationalstiffness variations on the natural frequency parameter isshown in Figure 5(a) for different types of foundations (120578 =
10) and in Figure 5(b) for different values of slenderness ratio(119864119904= 0)Further the effect of rotational stiffness variations on the
critical load parameter is shown in Figure 6(a) for differenttypes of foundation (120578 = 10) and in Figure 6(b) for differentvalues of slenderness ratio (119864
119904= 0)
The influence of rotational stiffness variations on thenatural frequency parameter is shown in Figure 7 for differentloading ratios and for the cases of no foundation (119864
119904= 0) and
stiff foundations (119864119904= 108)
It is obvious that the significance of real foundation typesfor Timoshenko beam behavior (120578 lt 20) is negligible due to
Mathematical Problems in Engineering 9
5
10
15
20
25
NFWF
MFSF
KR
Pcr
10minus2 100 102 104
(a) 120578 = 10
0
10
20
30
40
120578 = 5120578 = 10
120578 = 20
120578 = 50
Pcr
KR
10minus2 100 102 104
(b) 119864119904 = 0
Figure 6 Influence of 119870119877on 119875cr for different types of foundation and for different 120578
2
25
3
35
4
KR
100 104
120582
120574 = 0120574 = 02
120574 = 04
120574 = 06
(a) No foundation (119864119904 = 0)
2
25
3
35
4
KR
100 104
120582
120574 = 0120574 = 02
120574 = 04
120574 = 06
(b) Stiff foundation (119864119904 = 108)
Figure 7 Influence of 119870119877on 120582 for different loading ratios (120578 = 10)
the high stiffness of the beam relative to the foundation Alsoit is found that the influence of rotational stiffness increaseswith the increase in the slenderness ratio Also it is found thatthe natural frequency decreases as the axial load increases tillthe axial load approaches its critical values then the naturalfrequency approaches zero
43 Forced Vibration Analysis Theweakening of ends elasticrestraints is represented by the variation of the restraint stiff-nessThe variation of normalized maximum response ampli-tude (119882lowast) with the ends rotational stiffness (119870
119877) is indicated
in Figure 8 for different loading types Case (a) representsthe effects of uniform loading functions case (b) representsthe effect of linear loading functions and case (c) represents
the effect of quadratic loading functions As the case of119870119877=
0 represents the P-P beams it is obvious from the figuresthat the displacement due to dynamic loading is greater thanthose for static loading due to the inertia force interferenceat 119870119877
= 0 Also it is clear that as 119870119877increases the stiffness
of the beam increases and the response amplitude decreasesIt is obvious that at a certain combination of the systemparameters the resonance condition is approached andunbounded response amplitude is detected As the axial loadincreases the variations of the end rotational stiffness maycause the system to reach the resonance conditions and theamplitude becomes unbounded If the damping is taken intoconsideration the amplitude will possess a finite value butsuch condition is to be prevented However it is concluded
Figure 8 Variation of119882lowast with 119870119877for different axial loading ratio (120578 = 10)
Mathematical Problems in Engineering 11
that the weakening of the end restraints may offer a param-eters combination coinciding with the resonance conditionsfor the first mode or one of the higher modes and unboundedamplitudes may be reached
5 Conclusions
The RDM is used to investigate the stability behavior ofaxially loaded cantilever Timoshenko beam resting on two-parameter foundation In addition the analysis includesthe investigation of Timoshenko beams resting on two-parameter foundation and subjected to different types of lat-eral excitationThe supports at the beam ends are assumed ofthe elastic type to investigate the impact of the support weak-ening on the beambehavior Both the effect of rotational iner-tia and the correction of the shear stress due to axial loadingare taken into consideration which decreases the beam stiff-ness It is concluded that the significance of the foundation onthe Timoshenko beams (120578 lt 20) behavior is negligiblethe effect of end restraints is more noticeable for slenderbeams and the natural frequency decreases as the axial loadincreases However for the case of the forced vibration theresonance conditions may result from the weakening of theend supports and unbounded response amplitudes may bereached The analysis depicts the simplicity and accuracy ofthe RDM in tackling the boundary value problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Ruta ldquoThe application of Chebyshev polynomials to the solu-tion of the nonprismatic Timoshenko beam vibration problemrdquoJournal of Sound and Vibration vol 296 no 1-2 pp 243ndash2632006
[2] K Kausel ldquoNonclassical modes of unrestrained shear beamsrdquoJournal of Engineering Mechanics vol 128 no 6 pp 663ndash6672002
[3] R Attarnejad A Shahba and S J Semnani ldquoApplication ofdifferential transform in free vibration analysis of timoshenkobeams resting on two-parameter elastic foundationrdquo ArabianJournal for Science and Engineering vol 35 no 2B pp 125ndash1322010
[4] M Taha and M A Nassar ldquoAnalysis of stressed Timoshenkobeams on two parameter foundationsrdquo KSCE Journal of CivilEngineering vol 19 no 1 pp 173ndash179 2015
[5] L Majkut ldquoFree and forced vibration of Timoshenko beamsdescribed by single difference equationrdquo Journal of Theoreticaland Applied Mechanics vol 47 no 1 pp 193ndash210 2009
[6] F Y Cheng and C P Pantelides ldquoDynamic timoshenko beam-column on elastic mediardquo Journal of Structural Engineering vol114 no 7 pp 1524ndash1550 1988
[7] B Geist and J R Mclaughlin ldquoDouble eigen value for theTimoshenko beamrdquo Applied Mathematics Letters vol 10 pp129ndash134 1997
[8] C-N Chen ldquoDQEM vibration analyses of non-prismatic sheardeformable beams resting on elastic foundationsrdquo Journal ofSound and Vibration vol 255 no 5 pp 989ndash999 2002
[9] A L G Monsalve Z D G Medina and A J D Ochoa ldquoTim-oshenko beam-column with generalized end conditions onelastic foundation dynamic-stiffness matrix and load vectorrdquoJournal of Sound and Vibration vol 310 no 4-5 pp 1057ndash10792008
[10] T Kocaturk and M Simsek ldquoFree vibration analysis of Tim-oshenko beams under various boundary conditionsrdquo SigmaJournal of Engineering and Natural Sciences vol 1 pp 30ndash442005
[11] DKNguyen ldquoFree vibration of prestressedTimoshenkobeamsresting on elastic foundationsrdquo Vietnam Journal of Mechanicsvol 29 no 1 pp 1ndash12 2012
[12] N M Auciello ldquoVibrations of Timoshenko beams on twoparameter elastic soilrdquo Engineering Transactions vol 56 no 3pp 187ndash200 2008
[13] M Taha ldquoRecursive differentiation method for boundary valueproblems application to analysis of a beam-column on an elas-tic foundationrdquo Journal of Theoretical and Applied Mechanicsvol 44 no 2 pp 57ndash70 2014
[14] M Taha and E H Doha ldquoRecursive differentiation methodapplication to the analysis of beams on two parameter founda-tionsrdquo Journal of Theoretical and Applied Mechanics vol 53 no1 pp 15ndash26 2015
[15] S P Timoshenko and J M Gere Theory of Elastic StabilityEngineering Societies Monographs McGraw-Hill Book NewYork NY USA 1961
In addition coefficients 119860119894119895for the third approach are
11986011
= minus12059011205903
1205905
11986012
= 0
11986013
=1205901+ 12059031205905minus 1198784
1205905
11986014
= 0
1198651(120585) =
1205903119876 minus 119876
10158401015840
1205905
(35)
The solution of (32) may be expressed as
120593 (120585) =
4
sum
119898=1
119879119898119877119898(120585) + 119877
119865(120585) (36)
where the recursive functions 119877119898(120585) 119898 = 1 4 and the force
function 119877119865(120585) are given by
119877119898(120585) =
120585119898minus1
(119898 minus 1)+
119873
sum
119894=1
119860119894119898
120585119894+3
(119894 + 3) 119898 = 1 4
119877119865(120585) =
119873
sum
119894=1
119865119894(0)
120585119894+3
(119899 + 3)
(37)
The recurrence formulae for coefficients119860119896+1119894
119896 = 1 119873 and119865119894are given by
119860119896+11
= 1198601198964
11986011
119860119896+1119903
= 119860119896119903minus1
+ 1198601198964
1198601119903
119903 = 2 119899
119865119896+1
(120585) = 1198651015840
119896(120585) + 119860
11989641198651(120585)
(38)
The application of the boundary conditions (see (28a)ndash(28d)(29a)ndash(29d) or (30a)ndash(30d)) at 120585 = 0 1 yields a system
of 4-algebraic equations in 4 unknowns 1198791 1198792 1198793 and
1198794 however calculating these unknowns the response ampli-
tude distribution 120593(120585) can be obtainedThe distribution of bending moment 119872(120585) and shearing
force 119881(120585)may be obtained using (2) and (3) respectively
32 Calculations of Critical Loads and Natural FrequenciesThe critical loads and natural frequencies may be calculatedby assuming zero excitation amplitude (119865(120585) = 0) andreplacing the excitation frequencyΩ by the natural frequency120596 of the system The substitution of 120593(120585) and its derivativesinto the boundary conditions yields a systemof homogeneousalgebraic equations whose nontrivial solution yields a two-parameter eigenvalue problem in 119901 and120596The solution of thetwo-parameter eigenvalue problem yields both critical loads119901cr119899 and buckling modes for static case (120596 = 0) and thenatural frequencies of system120596
119899and themode shapes for free
vibration (119901 lt 119901cr)Also critical loads and natural frequencies may be
obtained from forced vibration by detecting the axial loador the excitation frequency at which the response amplitudebecomes unbounded
33 Verification of the Obtained Solutions The natural fre-quency parameter of the lower three modes for Timo-shenko beams 120582
119899 119894 = 1 3 is obtained using the present
solution assuming the third approach and compared withthose obtained from the Rayleigh-Ritz approach [12] Thecomparison is indicated in Table 1 for different boundaryconditions It is found that the results of the two approachesare compatible
4 Numerical Results and Discussion
Theobtained solutions are used to investigate the influence ofbeam and foundation parameters on the stability parametersanddynamic behavior of Timoshenko beamsThe foundationparameters (119870
1and 119870
2) and beam properties are taken
similar to those given in Taha and Doha [14]However for the investigation of the amplitude of the
lateral response the following normalized parameters areintroduced
120574 =119875
119875cr
Mathematical Problems in Engineering 7
1
2
3
4
5
Approach = 1Approach = 2Approach = 3
120578
120582
105 6 7 8 9 3020
(a) No foundation (119864119904 = 0)
Approach = 1Approach = 2Approach = 3
1
2
3
4
5
120582
120578
105 6 7 8 9 3020
(b) Stiff foundation (119864119904 = 108)
Figure 2 Variation of 120582 with 120578 for C-F beams (119875 = 021205872)
19
2
21
22
23
24
25
Approach = 1Approach = 2Approach = 3
Pcr
120578
105 6 7 8 9 3020
(a) No foundation
0
2
4
6
8
10
Approach = 1Approach = 2Approach = 3
Pcr
120578
105 6 7 8 9 3020
(b) Stiff foundation
Figure 3 Variation of 119875cr with 120578 for C-F beams
119882lowast=
119882max119882119906-max
(39)where 120574 is the loading parameter 119882
lowast is the maximumamplitude parameter119882max is the maximum amplitude of theinvestigated case and 119882
119906-max is the maximum amplitude forthe P-P corresponding case
41 Stability Parameters for Cantilever Beams Althoughmany engineering applications belong to the Timoshenkocantilever beam model a few articles are published con-cerning the stability and dynamic behavior of such model
due to its mathematical complications In the present workthe investigation of cantilever beam behavior is highlightedThe comparison between the different approaches dealingwith the shear induced from the axial load on the frequencyparameter 120582 for clamped-free beams (C-F) is shown inFigure 2 case (a) for beams without foundation (119864
119904= 0) and
case (b) for beams resting on stiff foundation (119864119904= 108)
On the other hand the influences of different approacheson quantifying the critical load of C-F beams are indicated inFigure 3 case (a) for beams without foundation and case (b)for beams on stiff foundations It is obvious that consideringthe shear component induced from the axial load decreases
8 Mathematical Problems in Engineering
15
2
25
3
35
NFWF
MFSF
120582
120578
105 6 7 8 9 3020
(a) 119875 = 0
05
1
15
2
25
3
NFWF
MFSF
120582
120578
105 6 7 8 9 3020
(b) 119875 = 021205872
Figure 4 Variation of 120582 with 120578 for C-F beams
28
3
32
34
36
38
4
NFWF
MFSF
KR
100 104
120582
(a) Slenderness ratio 120578 = 10
3
35
25
4
45
5
KR
100 104
120578 = 5120578 = 10
120578 = 20
120578 = 50
120582
(b) No foundations (119864119904 = 0)
Figure 5 Influence of 119870119877on 120582 for different types of foundation and for different 120578 (119875 = 0)
the beam stiffness and the effect is noticeable for beams withslenderness ratio 120582 lt 20
In the following analysis the parametric study is basedon the second approach as it is the rational one because theangle between the axial load and the outward normal to thedeformed cross section 120572(119909 119905) is actually equal to the angle ofcross section rotation 120595(119909 119905) in the deformed configuration
The influence of foundation type on the frequency param-eter for C-F beams is shown in Figure 4 It is found thatas the slenderness ratio increases the natural frequency ofbeams resting on foundation increases due to the increase offoundation share in the overall stiffness of the system Theincrease in the natural frequency parameter is more notice-able for stiff foundations (SF) The critical load of Timo-shenko C-F beam without foundation 119875cr = 021120587
2
42 Weakening of the End Restraints The effect of rotationalstiffness variations on the natural frequency parameter isshown in Figure 5(a) for different types of foundations (120578 =
10) and in Figure 5(b) for different values of slenderness ratio(119864119904= 0)Further the effect of rotational stiffness variations on the
critical load parameter is shown in Figure 6(a) for differenttypes of foundation (120578 = 10) and in Figure 6(b) for differentvalues of slenderness ratio (119864
119904= 0)
The influence of rotational stiffness variations on thenatural frequency parameter is shown in Figure 7 for differentloading ratios and for the cases of no foundation (119864
119904= 0) and
stiff foundations (119864119904= 108)
It is obvious that the significance of real foundation typesfor Timoshenko beam behavior (120578 lt 20) is negligible due to
Mathematical Problems in Engineering 9
5
10
15
20
25
NFWF
MFSF
KR
Pcr
10minus2 100 102 104
(a) 120578 = 10
0
10
20
30
40
120578 = 5120578 = 10
120578 = 20
120578 = 50
Pcr
KR
10minus2 100 102 104
(b) 119864119904 = 0
Figure 6 Influence of 119870119877on 119875cr for different types of foundation and for different 120578
2
25
3
35
4
KR
100 104
120582
120574 = 0120574 = 02
120574 = 04
120574 = 06
(a) No foundation (119864119904 = 0)
2
25
3
35
4
KR
100 104
120582
120574 = 0120574 = 02
120574 = 04
120574 = 06
(b) Stiff foundation (119864119904 = 108)
Figure 7 Influence of 119870119877on 120582 for different loading ratios (120578 = 10)
the high stiffness of the beam relative to the foundation Alsoit is found that the influence of rotational stiffness increaseswith the increase in the slenderness ratio Also it is found thatthe natural frequency decreases as the axial load increases tillthe axial load approaches its critical values then the naturalfrequency approaches zero
43 Forced Vibration Analysis Theweakening of ends elasticrestraints is represented by the variation of the restraint stiff-nessThe variation of normalized maximum response ampli-tude (119882lowast) with the ends rotational stiffness (119870
119877) is indicated
in Figure 8 for different loading types Case (a) representsthe effects of uniform loading functions case (b) representsthe effect of linear loading functions and case (c) represents
the effect of quadratic loading functions As the case of119870119877=
0 represents the P-P beams it is obvious from the figuresthat the displacement due to dynamic loading is greater thanthose for static loading due to the inertia force interferenceat 119870119877
= 0 Also it is clear that as 119870119877increases the stiffness
of the beam increases and the response amplitude decreasesIt is obvious that at a certain combination of the systemparameters the resonance condition is approached andunbounded response amplitude is detected As the axial loadincreases the variations of the end rotational stiffness maycause the system to reach the resonance conditions and theamplitude becomes unbounded If the damping is taken intoconsideration the amplitude will possess a finite value butsuch condition is to be prevented However it is concluded
Figure 8 Variation of119882lowast with 119870119877for different axial loading ratio (120578 = 10)
Mathematical Problems in Engineering 11
that the weakening of the end restraints may offer a param-eters combination coinciding with the resonance conditionsfor the first mode or one of the higher modes and unboundedamplitudes may be reached
5 Conclusions
The RDM is used to investigate the stability behavior ofaxially loaded cantilever Timoshenko beam resting on two-parameter foundation In addition the analysis includesthe investigation of Timoshenko beams resting on two-parameter foundation and subjected to different types of lat-eral excitationThe supports at the beam ends are assumed ofthe elastic type to investigate the impact of the support weak-ening on the beambehavior Both the effect of rotational iner-tia and the correction of the shear stress due to axial loadingare taken into consideration which decreases the beam stiff-ness It is concluded that the significance of the foundation onthe Timoshenko beams (120578 lt 20) behavior is negligiblethe effect of end restraints is more noticeable for slenderbeams and the natural frequency decreases as the axial loadincreases However for the case of the forced vibration theresonance conditions may result from the weakening of theend supports and unbounded response amplitudes may bereached The analysis depicts the simplicity and accuracy ofthe RDM in tackling the boundary value problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Ruta ldquoThe application of Chebyshev polynomials to the solu-tion of the nonprismatic Timoshenko beam vibration problemrdquoJournal of Sound and Vibration vol 296 no 1-2 pp 243ndash2632006
[2] K Kausel ldquoNonclassical modes of unrestrained shear beamsrdquoJournal of Engineering Mechanics vol 128 no 6 pp 663ndash6672002
[3] R Attarnejad A Shahba and S J Semnani ldquoApplication ofdifferential transform in free vibration analysis of timoshenkobeams resting on two-parameter elastic foundationrdquo ArabianJournal for Science and Engineering vol 35 no 2B pp 125ndash1322010
[4] M Taha and M A Nassar ldquoAnalysis of stressed Timoshenkobeams on two parameter foundationsrdquo KSCE Journal of CivilEngineering vol 19 no 1 pp 173ndash179 2015
[5] L Majkut ldquoFree and forced vibration of Timoshenko beamsdescribed by single difference equationrdquo Journal of Theoreticaland Applied Mechanics vol 47 no 1 pp 193ndash210 2009
[6] F Y Cheng and C P Pantelides ldquoDynamic timoshenko beam-column on elastic mediardquo Journal of Structural Engineering vol114 no 7 pp 1524ndash1550 1988
[7] B Geist and J R Mclaughlin ldquoDouble eigen value for theTimoshenko beamrdquo Applied Mathematics Letters vol 10 pp129ndash134 1997
[8] C-N Chen ldquoDQEM vibration analyses of non-prismatic sheardeformable beams resting on elastic foundationsrdquo Journal ofSound and Vibration vol 255 no 5 pp 989ndash999 2002
[9] A L G Monsalve Z D G Medina and A J D Ochoa ldquoTim-oshenko beam-column with generalized end conditions onelastic foundation dynamic-stiffness matrix and load vectorrdquoJournal of Sound and Vibration vol 310 no 4-5 pp 1057ndash10792008
[10] T Kocaturk and M Simsek ldquoFree vibration analysis of Tim-oshenko beams under various boundary conditionsrdquo SigmaJournal of Engineering and Natural Sciences vol 1 pp 30ndash442005
[11] DKNguyen ldquoFree vibration of prestressedTimoshenkobeamsresting on elastic foundationsrdquo Vietnam Journal of Mechanicsvol 29 no 1 pp 1ndash12 2012
[12] N M Auciello ldquoVibrations of Timoshenko beams on twoparameter elastic soilrdquo Engineering Transactions vol 56 no 3pp 187ndash200 2008
[13] M Taha ldquoRecursive differentiation method for boundary valueproblems application to analysis of a beam-column on an elas-tic foundationrdquo Journal of Theoretical and Applied Mechanicsvol 44 no 2 pp 57ndash70 2014
[14] M Taha and E H Doha ldquoRecursive differentiation methodapplication to the analysis of beams on two parameter founda-tionsrdquo Journal of Theoretical and Applied Mechanics vol 53 no1 pp 15ndash26 2015
[15] S P Timoshenko and J M Gere Theory of Elastic StabilityEngineering Societies Monographs McGraw-Hill Book NewYork NY USA 1961
Figure 2 Variation of 120582 with 120578 for C-F beams (119875 = 021205872)
19
2
21
22
23
24
25
Approach = 1Approach = 2Approach = 3
Pcr
120578
105 6 7 8 9 3020
(a) No foundation
0
2
4
6
8
10
Approach = 1Approach = 2Approach = 3
Pcr
120578
105 6 7 8 9 3020
(b) Stiff foundation
Figure 3 Variation of 119875cr with 120578 for C-F beams
119882lowast=
119882max119882119906-max
(39)where 120574 is the loading parameter 119882
lowast is the maximumamplitude parameter119882max is the maximum amplitude of theinvestigated case and 119882
119906-max is the maximum amplitude forthe P-P corresponding case
41 Stability Parameters for Cantilever Beams Althoughmany engineering applications belong to the Timoshenkocantilever beam model a few articles are published con-cerning the stability and dynamic behavior of such model
due to its mathematical complications In the present workthe investigation of cantilever beam behavior is highlightedThe comparison between the different approaches dealingwith the shear induced from the axial load on the frequencyparameter 120582 for clamped-free beams (C-F) is shown inFigure 2 case (a) for beams without foundation (119864
119904= 0) and
case (b) for beams resting on stiff foundation (119864119904= 108)
On the other hand the influences of different approacheson quantifying the critical load of C-F beams are indicated inFigure 3 case (a) for beams without foundation and case (b)for beams on stiff foundations It is obvious that consideringthe shear component induced from the axial load decreases
8 Mathematical Problems in Engineering
15
2
25
3
35
NFWF
MFSF
120582
120578
105 6 7 8 9 3020
(a) 119875 = 0
05
1
15
2
25
3
NFWF
MFSF
120582
120578
105 6 7 8 9 3020
(b) 119875 = 021205872
Figure 4 Variation of 120582 with 120578 for C-F beams
28
3
32
34
36
38
4
NFWF
MFSF
KR
100 104
120582
(a) Slenderness ratio 120578 = 10
3
35
25
4
45
5
KR
100 104
120578 = 5120578 = 10
120578 = 20
120578 = 50
120582
(b) No foundations (119864119904 = 0)
Figure 5 Influence of 119870119877on 120582 for different types of foundation and for different 120578 (119875 = 0)
the beam stiffness and the effect is noticeable for beams withslenderness ratio 120582 lt 20
In the following analysis the parametric study is basedon the second approach as it is the rational one because theangle between the axial load and the outward normal to thedeformed cross section 120572(119909 119905) is actually equal to the angle ofcross section rotation 120595(119909 119905) in the deformed configuration
The influence of foundation type on the frequency param-eter for C-F beams is shown in Figure 4 It is found thatas the slenderness ratio increases the natural frequency ofbeams resting on foundation increases due to the increase offoundation share in the overall stiffness of the system Theincrease in the natural frequency parameter is more notice-able for stiff foundations (SF) The critical load of Timo-shenko C-F beam without foundation 119875cr = 021120587
2
42 Weakening of the End Restraints The effect of rotationalstiffness variations on the natural frequency parameter isshown in Figure 5(a) for different types of foundations (120578 =
10) and in Figure 5(b) for different values of slenderness ratio(119864119904= 0)Further the effect of rotational stiffness variations on the
critical load parameter is shown in Figure 6(a) for differenttypes of foundation (120578 = 10) and in Figure 6(b) for differentvalues of slenderness ratio (119864
119904= 0)
The influence of rotational stiffness variations on thenatural frequency parameter is shown in Figure 7 for differentloading ratios and for the cases of no foundation (119864
119904= 0) and
stiff foundations (119864119904= 108)
It is obvious that the significance of real foundation typesfor Timoshenko beam behavior (120578 lt 20) is negligible due to
Mathematical Problems in Engineering 9
5
10
15
20
25
NFWF
MFSF
KR
Pcr
10minus2 100 102 104
(a) 120578 = 10
0
10
20
30
40
120578 = 5120578 = 10
120578 = 20
120578 = 50
Pcr
KR
10minus2 100 102 104
(b) 119864119904 = 0
Figure 6 Influence of 119870119877on 119875cr for different types of foundation and for different 120578
2
25
3
35
4
KR
100 104
120582
120574 = 0120574 = 02
120574 = 04
120574 = 06
(a) No foundation (119864119904 = 0)
2
25
3
35
4
KR
100 104
120582
120574 = 0120574 = 02
120574 = 04
120574 = 06
(b) Stiff foundation (119864119904 = 108)
Figure 7 Influence of 119870119877on 120582 for different loading ratios (120578 = 10)
the high stiffness of the beam relative to the foundation Alsoit is found that the influence of rotational stiffness increaseswith the increase in the slenderness ratio Also it is found thatthe natural frequency decreases as the axial load increases tillthe axial load approaches its critical values then the naturalfrequency approaches zero
43 Forced Vibration Analysis Theweakening of ends elasticrestraints is represented by the variation of the restraint stiff-nessThe variation of normalized maximum response ampli-tude (119882lowast) with the ends rotational stiffness (119870
119877) is indicated
in Figure 8 for different loading types Case (a) representsthe effects of uniform loading functions case (b) representsthe effect of linear loading functions and case (c) represents
the effect of quadratic loading functions As the case of119870119877=
0 represents the P-P beams it is obvious from the figuresthat the displacement due to dynamic loading is greater thanthose for static loading due to the inertia force interferenceat 119870119877
= 0 Also it is clear that as 119870119877increases the stiffness
of the beam increases and the response amplitude decreasesIt is obvious that at a certain combination of the systemparameters the resonance condition is approached andunbounded response amplitude is detected As the axial loadincreases the variations of the end rotational stiffness maycause the system to reach the resonance conditions and theamplitude becomes unbounded If the damping is taken intoconsideration the amplitude will possess a finite value butsuch condition is to be prevented However it is concluded
Figure 8 Variation of119882lowast with 119870119877for different axial loading ratio (120578 = 10)
Mathematical Problems in Engineering 11
that the weakening of the end restraints may offer a param-eters combination coinciding with the resonance conditionsfor the first mode or one of the higher modes and unboundedamplitudes may be reached
5 Conclusions
The RDM is used to investigate the stability behavior ofaxially loaded cantilever Timoshenko beam resting on two-parameter foundation In addition the analysis includesthe investigation of Timoshenko beams resting on two-parameter foundation and subjected to different types of lat-eral excitationThe supports at the beam ends are assumed ofthe elastic type to investigate the impact of the support weak-ening on the beambehavior Both the effect of rotational iner-tia and the correction of the shear stress due to axial loadingare taken into consideration which decreases the beam stiff-ness It is concluded that the significance of the foundation onthe Timoshenko beams (120578 lt 20) behavior is negligiblethe effect of end restraints is more noticeable for slenderbeams and the natural frequency decreases as the axial loadincreases However for the case of the forced vibration theresonance conditions may result from the weakening of theend supports and unbounded response amplitudes may bereached The analysis depicts the simplicity and accuracy ofthe RDM in tackling the boundary value problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Ruta ldquoThe application of Chebyshev polynomials to the solu-tion of the nonprismatic Timoshenko beam vibration problemrdquoJournal of Sound and Vibration vol 296 no 1-2 pp 243ndash2632006
[2] K Kausel ldquoNonclassical modes of unrestrained shear beamsrdquoJournal of Engineering Mechanics vol 128 no 6 pp 663ndash6672002
[3] R Attarnejad A Shahba and S J Semnani ldquoApplication ofdifferential transform in free vibration analysis of timoshenkobeams resting on two-parameter elastic foundationrdquo ArabianJournal for Science and Engineering vol 35 no 2B pp 125ndash1322010
[4] M Taha and M A Nassar ldquoAnalysis of stressed Timoshenkobeams on two parameter foundationsrdquo KSCE Journal of CivilEngineering vol 19 no 1 pp 173ndash179 2015
[5] L Majkut ldquoFree and forced vibration of Timoshenko beamsdescribed by single difference equationrdquo Journal of Theoreticaland Applied Mechanics vol 47 no 1 pp 193ndash210 2009
[6] F Y Cheng and C P Pantelides ldquoDynamic timoshenko beam-column on elastic mediardquo Journal of Structural Engineering vol114 no 7 pp 1524ndash1550 1988
[7] B Geist and J R Mclaughlin ldquoDouble eigen value for theTimoshenko beamrdquo Applied Mathematics Letters vol 10 pp129ndash134 1997
[8] C-N Chen ldquoDQEM vibration analyses of non-prismatic sheardeformable beams resting on elastic foundationsrdquo Journal ofSound and Vibration vol 255 no 5 pp 989ndash999 2002
[9] A L G Monsalve Z D G Medina and A J D Ochoa ldquoTim-oshenko beam-column with generalized end conditions onelastic foundation dynamic-stiffness matrix and load vectorrdquoJournal of Sound and Vibration vol 310 no 4-5 pp 1057ndash10792008
[10] T Kocaturk and M Simsek ldquoFree vibration analysis of Tim-oshenko beams under various boundary conditionsrdquo SigmaJournal of Engineering and Natural Sciences vol 1 pp 30ndash442005
[11] DKNguyen ldquoFree vibration of prestressedTimoshenkobeamsresting on elastic foundationsrdquo Vietnam Journal of Mechanicsvol 29 no 1 pp 1ndash12 2012
[12] N M Auciello ldquoVibrations of Timoshenko beams on twoparameter elastic soilrdquo Engineering Transactions vol 56 no 3pp 187ndash200 2008
[13] M Taha ldquoRecursive differentiation method for boundary valueproblems application to analysis of a beam-column on an elas-tic foundationrdquo Journal of Theoretical and Applied Mechanicsvol 44 no 2 pp 57ndash70 2014
[14] M Taha and E H Doha ldquoRecursive differentiation methodapplication to the analysis of beams on two parameter founda-tionsrdquo Journal of Theoretical and Applied Mechanics vol 53 no1 pp 15ndash26 2015
[15] S P Timoshenko and J M Gere Theory of Elastic StabilityEngineering Societies Monographs McGraw-Hill Book NewYork NY USA 1961
Figure 4 Variation of 120582 with 120578 for C-F beams
28
3
32
34
36
38
4
NFWF
MFSF
KR
100 104
120582
(a) Slenderness ratio 120578 = 10
3
35
25
4
45
5
KR
100 104
120578 = 5120578 = 10
120578 = 20
120578 = 50
120582
(b) No foundations (119864119904 = 0)
Figure 5 Influence of 119870119877on 120582 for different types of foundation and for different 120578 (119875 = 0)
the beam stiffness and the effect is noticeable for beams withslenderness ratio 120582 lt 20
In the following analysis the parametric study is basedon the second approach as it is the rational one because theangle between the axial load and the outward normal to thedeformed cross section 120572(119909 119905) is actually equal to the angle ofcross section rotation 120595(119909 119905) in the deformed configuration
The influence of foundation type on the frequency param-eter for C-F beams is shown in Figure 4 It is found thatas the slenderness ratio increases the natural frequency ofbeams resting on foundation increases due to the increase offoundation share in the overall stiffness of the system Theincrease in the natural frequency parameter is more notice-able for stiff foundations (SF) The critical load of Timo-shenko C-F beam without foundation 119875cr = 021120587
2
42 Weakening of the End Restraints The effect of rotationalstiffness variations on the natural frequency parameter isshown in Figure 5(a) for different types of foundations (120578 =
10) and in Figure 5(b) for different values of slenderness ratio(119864119904= 0)Further the effect of rotational stiffness variations on the
critical load parameter is shown in Figure 6(a) for differenttypes of foundation (120578 = 10) and in Figure 6(b) for differentvalues of slenderness ratio (119864
119904= 0)
The influence of rotational stiffness variations on thenatural frequency parameter is shown in Figure 7 for differentloading ratios and for the cases of no foundation (119864
119904= 0) and
stiff foundations (119864119904= 108)
It is obvious that the significance of real foundation typesfor Timoshenko beam behavior (120578 lt 20) is negligible due to
Mathematical Problems in Engineering 9
5
10
15
20
25
NFWF
MFSF
KR
Pcr
10minus2 100 102 104
(a) 120578 = 10
0
10
20
30
40
120578 = 5120578 = 10
120578 = 20
120578 = 50
Pcr
KR
10minus2 100 102 104
(b) 119864119904 = 0
Figure 6 Influence of 119870119877on 119875cr for different types of foundation and for different 120578
2
25
3
35
4
KR
100 104
120582
120574 = 0120574 = 02
120574 = 04
120574 = 06
(a) No foundation (119864119904 = 0)
2
25
3
35
4
KR
100 104
120582
120574 = 0120574 = 02
120574 = 04
120574 = 06
(b) Stiff foundation (119864119904 = 108)
Figure 7 Influence of 119870119877on 120582 for different loading ratios (120578 = 10)
the high stiffness of the beam relative to the foundation Alsoit is found that the influence of rotational stiffness increaseswith the increase in the slenderness ratio Also it is found thatthe natural frequency decreases as the axial load increases tillthe axial load approaches its critical values then the naturalfrequency approaches zero
43 Forced Vibration Analysis Theweakening of ends elasticrestraints is represented by the variation of the restraint stiff-nessThe variation of normalized maximum response ampli-tude (119882lowast) with the ends rotational stiffness (119870
119877) is indicated
in Figure 8 for different loading types Case (a) representsthe effects of uniform loading functions case (b) representsthe effect of linear loading functions and case (c) represents
the effect of quadratic loading functions As the case of119870119877=
0 represents the P-P beams it is obvious from the figuresthat the displacement due to dynamic loading is greater thanthose for static loading due to the inertia force interferenceat 119870119877
= 0 Also it is clear that as 119870119877increases the stiffness
of the beam increases and the response amplitude decreasesIt is obvious that at a certain combination of the systemparameters the resonance condition is approached andunbounded response amplitude is detected As the axial loadincreases the variations of the end rotational stiffness maycause the system to reach the resonance conditions and theamplitude becomes unbounded If the damping is taken intoconsideration the amplitude will possess a finite value butsuch condition is to be prevented However it is concluded
Figure 8 Variation of119882lowast with 119870119877for different axial loading ratio (120578 = 10)
Mathematical Problems in Engineering 11
that the weakening of the end restraints may offer a param-eters combination coinciding with the resonance conditionsfor the first mode or one of the higher modes and unboundedamplitudes may be reached
5 Conclusions
The RDM is used to investigate the stability behavior ofaxially loaded cantilever Timoshenko beam resting on two-parameter foundation In addition the analysis includesthe investigation of Timoshenko beams resting on two-parameter foundation and subjected to different types of lat-eral excitationThe supports at the beam ends are assumed ofthe elastic type to investigate the impact of the support weak-ening on the beambehavior Both the effect of rotational iner-tia and the correction of the shear stress due to axial loadingare taken into consideration which decreases the beam stiff-ness It is concluded that the significance of the foundation onthe Timoshenko beams (120578 lt 20) behavior is negligiblethe effect of end restraints is more noticeable for slenderbeams and the natural frequency decreases as the axial loadincreases However for the case of the forced vibration theresonance conditions may result from the weakening of theend supports and unbounded response amplitudes may bereached The analysis depicts the simplicity and accuracy ofthe RDM in tackling the boundary value problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Ruta ldquoThe application of Chebyshev polynomials to the solu-tion of the nonprismatic Timoshenko beam vibration problemrdquoJournal of Sound and Vibration vol 296 no 1-2 pp 243ndash2632006
[2] K Kausel ldquoNonclassical modes of unrestrained shear beamsrdquoJournal of Engineering Mechanics vol 128 no 6 pp 663ndash6672002
[3] R Attarnejad A Shahba and S J Semnani ldquoApplication ofdifferential transform in free vibration analysis of timoshenkobeams resting on two-parameter elastic foundationrdquo ArabianJournal for Science and Engineering vol 35 no 2B pp 125ndash1322010
[4] M Taha and M A Nassar ldquoAnalysis of stressed Timoshenkobeams on two parameter foundationsrdquo KSCE Journal of CivilEngineering vol 19 no 1 pp 173ndash179 2015
[5] L Majkut ldquoFree and forced vibration of Timoshenko beamsdescribed by single difference equationrdquo Journal of Theoreticaland Applied Mechanics vol 47 no 1 pp 193ndash210 2009
[6] F Y Cheng and C P Pantelides ldquoDynamic timoshenko beam-column on elastic mediardquo Journal of Structural Engineering vol114 no 7 pp 1524ndash1550 1988
[7] B Geist and J R Mclaughlin ldquoDouble eigen value for theTimoshenko beamrdquo Applied Mathematics Letters vol 10 pp129ndash134 1997
[8] C-N Chen ldquoDQEM vibration analyses of non-prismatic sheardeformable beams resting on elastic foundationsrdquo Journal ofSound and Vibration vol 255 no 5 pp 989ndash999 2002
[9] A L G Monsalve Z D G Medina and A J D Ochoa ldquoTim-oshenko beam-column with generalized end conditions onelastic foundation dynamic-stiffness matrix and load vectorrdquoJournal of Sound and Vibration vol 310 no 4-5 pp 1057ndash10792008
[10] T Kocaturk and M Simsek ldquoFree vibration analysis of Tim-oshenko beams under various boundary conditionsrdquo SigmaJournal of Engineering and Natural Sciences vol 1 pp 30ndash442005
[11] DKNguyen ldquoFree vibration of prestressedTimoshenkobeamsresting on elastic foundationsrdquo Vietnam Journal of Mechanicsvol 29 no 1 pp 1ndash12 2012
[12] N M Auciello ldquoVibrations of Timoshenko beams on twoparameter elastic soilrdquo Engineering Transactions vol 56 no 3pp 187ndash200 2008
[13] M Taha ldquoRecursive differentiation method for boundary valueproblems application to analysis of a beam-column on an elas-tic foundationrdquo Journal of Theoretical and Applied Mechanicsvol 44 no 2 pp 57ndash70 2014
[14] M Taha and E H Doha ldquoRecursive differentiation methodapplication to the analysis of beams on two parameter founda-tionsrdquo Journal of Theoretical and Applied Mechanics vol 53 no1 pp 15ndash26 2015
[15] S P Timoshenko and J M Gere Theory of Elastic StabilityEngineering Societies Monographs McGraw-Hill Book NewYork NY USA 1961
Figure 6 Influence of 119870119877on 119875cr for different types of foundation and for different 120578
2
25
3
35
4
KR
100 104
120582
120574 = 0120574 = 02
120574 = 04
120574 = 06
(a) No foundation (119864119904 = 0)
2
25
3
35
4
KR
100 104
120582
120574 = 0120574 = 02
120574 = 04
120574 = 06
(b) Stiff foundation (119864119904 = 108)
Figure 7 Influence of 119870119877on 120582 for different loading ratios (120578 = 10)
the high stiffness of the beam relative to the foundation Alsoit is found that the influence of rotational stiffness increaseswith the increase in the slenderness ratio Also it is found thatthe natural frequency decreases as the axial load increases tillthe axial load approaches its critical values then the naturalfrequency approaches zero
43 Forced Vibration Analysis Theweakening of ends elasticrestraints is represented by the variation of the restraint stiff-nessThe variation of normalized maximum response ampli-tude (119882lowast) with the ends rotational stiffness (119870
119877) is indicated
in Figure 8 for different loading types Case (a) representsthe effects of uniform loading functions case (b) representsthe effect of linear loading functions and case (c) represents
the effect of quadratic loading functions As the case of119870119877=
0 represents the P-P beams it is obvious from the figuresthat the displacement due to dynamic loading is greater thanthose for static loading due to the inertia force interferenceat 119870119877
= 0 Also it is clear that as 119870119877increases the stiffness
of the beam increases and the response amplitude decreasesIt is obvious that at a certain combination of the systemparameters the resonance condition is approached andunbounded response amplitude is detected As the axial loadincreases the variations of the end rotational stiffness maycause the system to reach the resonance conditions and theamplitude becomes unbounded If the damping is taken intoconsideration the amplitude will possess a finite value butsuch condition is to be prevented However it is concluded
Figure 8 Variation of119882lowast with 119870119877for different axial loading ratio (120578 = 10)
Mathematical Problems in Engineering 11
that the weakening of the end restraints may offer a param-eters combination coinciding with the resonance conditionsfor the first mode or one of the higher modes and unboundedamplitudes may be reached
5 Conclusions
The RDM is used to investigate the stability behavior ofaxially loaded cantilever Timoshenko beam resting on two-parameter foundation In addition the analysis includesthe investigation of Timoshenko beams resting on two-parameter foundation and subjected to different types of lat-eral excitationThe supports at the beam ends are assumed ofthe elastic type to investigate the impact of the support weak-ening on the beambehavior Both the effect of rotational iner-tia and the correction of the shear stress due to axial loadingare taken into consideration which decreases the beam stiff-ness It is concluded that the significance of the foundation onthe Timoshenko beams (120578 lt 20) behavior is negligiblethe effect of end restraints is more noticeable for slenderbeams and the natural frequency decreases as the axial loadincreases However for the case of the forced vibration theresonance conditions may result from the weakening of theend supports and unbounded response amplitudes may bereached The analysis depicts the simplicity and accuracy ofthe RDM in tackling the boundary value problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Ruta ldquoThe application of Chebyshev polynomials to the solu-tion of the nonprismatic Timoshenko beam vibration problemrdquoJournal of Sound and Vibration vol 296 no 1-2 pp 243ndash2632006
[2] K Kausel ldquoNonclassical modes of unrestrained shear beamsrdquoJournal of Engineering Mechanics vol 128 no 6 pp 663ndash6672002
[3] R Attarnejad A Shahba and S J Semnani ldquoApplication ofdifferential transform in free vibration analysis of timoshenkobeams resting on two-parameter elastic foundationrdquo ArabianJournal for Science and Engineering vol 35 no 2B pp 125ndash1322010
[4] M Taha and M A Nassar ldquoAnalysis of stressed Timoshenkobeams on two parameter foundationsrdquo KSCE Journal of CivilEngineering vol 19 no 1 pp 173ndash179 2015
[5] L Majkut ldquoFree and forced vibration of Timoshenko beamsdescribed by single difference equationrdquo Journal of Theoreticaland Applied Mechanics vol 47 no 1 pp 193ndash210 2009
[6] F Y Cheng and C P Pantelides ldquoDynamic timoshenko beam-column on elastic mediardquo Journal of Structural Engineering vol114 no 7 pp 1524ndash1550 1988
[7] B Geist and J R Mclaughlin ldquoDouble eigen value for theTimoshenko beamrdquo Applied Mathematics Letters vol 10 pp129ndash134 1997
[8] C-N Chen ldquoDQEM vibration analyses of non-prismatic sheardeformable beams resting on elastic foundationsrdquo Journal ofSound and Vibration vol 255 no 5 pp 989ndash999 2002
[9] A L G Monsalve Z D G Medina and A J D Ochoa ldquoTim-oshenko beam-column with generalized end conditions onelastic foundation dynamic-stiffness matrix and load vectorrdquoJournal of Sound and Vibration vol 310 no 4-5 pp 1057ndash10792008
[10] T Kocaturk and M Simsek ldquoFree vibration analysis of Tim-oshenko beams under various boundary conditionsrdquo SigmaJournal of Engineering and Natural Sciences vol 1 pp 30ndash442005
[11] DKNguyen ldquoFree vibration of prestressedTimoshenkobeamsresting on elastic foundationsrdquo Vietnam Journal of Mechanicsvol 29 no 1 pp 1ndash12 2012
[12] N M Auciello ldquoVibrations of Timoshenko beams on twoparameter elastic soilrdquo Engineering Transactions vol 56 no 3pp 187ndash200 2008
[13] M Taha ldquoRecursive differentiation method for boundary valueproblems application to analysis of a beam-column on an elas-tic foundationrdquo Journal of Theoretical and Applied Mechanicsvol 44 no 2 pp 57ndash70 2014
[14] M Taha and E H Doha ldquoRecursive differentiation methodapplication to the analysis of beams on two parameter founda-tionsrdquo Journal of Theoretical and Applied Mechanics vol 53 no1 pp 15ndash26 2015
[15] S P Timoshenko and J M Gere Theory of Elastic StabilityEngineering Societies Monographs McGraw-Hill Book NewYork NY USA 1961
Figure 8 Variation of119882lowast with 119870119877for different axial loading ratio (120578 = 10)
Mathematical Problems in Engineering 11
that the weakening of the end restraints may offer a param-eters combination coinciding with the resonance conditionsfor the first mode or one of the higher modes and unboundedamplitudes may be reached
5 Conclusions
The RDM is used to investigate the stability behavior ofaxially loaded cantilever Timoshenko beam resting on two-parameter foundation In addition the analysis includesthe investigation of Timoshenko beams resting on two-parameter foundation and subjected to different types of lat-eral excitationThe supports at the beam ends are assumed ofthe elastic type to investigate the impact of the support weak-ening on the beambehavior Both the effect of rotational iner-tia and the correction of the shear stress due to axial loadingare taken into consideration which decreases the beam stiff-ness It is concluded that the significance of the foundation onthe Timoshenko beams (120578 lt 20) behavior is negligiblethe effect of end restraints is more noticeable for slenderbeams and the natural frequency decreases as the axial loadincreases However for the case of the forced vibration theresonance conditions may result from the weakening of theend supports and unbounded response amplitudes may bereached The analysis depicts the simplicity and accuracy ofthe RDM in tackling the boundary value problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Ruta ldquoThe application of Chebyshev polynomials to the solu-tion of the nonprismatic Timoshenko beam vibration problemrdquoJournal of Sound and Vibration vol 296 no 1-2 pp 243ndash2632006
[2] K Kausel ldquoNonclassical modes of unrestrained shear beamsrdquoJournal of Engineering Mechanics vol 128 no 6 pp 663ndash6672002
[3] R Attarnejad A Shahba and S J Semnani ldquoApplication ofdifferential transform in free vibration analysis of timoshenkobeams resting on two-parameter elastic foundationrdquo ArabianJournal for Science and Engineering vol 35 no 2B pp 125ndash1322010
[4] M Taha and M A Nassar ldquoAnalysis of stressed Timoshenkobeams on two parameter foundationsrdquo KSCE Journal of CivilEngineering vol 19 no 1 pp 173ndash179 2015
[5] L Majkut ldquoFree and forced vibration of Timoshenko beamsdescribed by single difference equationrdquo Journal of Theoreticaland Applied Mechanics vol 47 no 1 pp 193ndash210 2009
[6] F Y Cheng and C P Pantelides ldquoDynamic timoshenko beam-column on elastic mediardquo Journal of Structural Engineering vol114 no 7 pp 1524ndash1550 1988
[7] B Geist and J R Mclaughlin ldquoDouble eigen value for theTimoshenko beamrdquo Applied Mathematics Letters vol 10 pp129ndash134 1997
[8] C-N Chen ldquoDQEM vibration analyses of non-prismatic sheardeformable beams resting on elastic foundationsrdquo Journal ofSound and Vibration vol 255 no 5 pp 989ndash999 2002
[9] A L G Monsalve Z D G Medina and A J D Ochoa ldquoTim-oshenko beam-column with generalized end conditions onelastic foundation dynamic-stiffness matrix and load vectorrdquoJournal of Sound and Vibration vol 310 no 4-5 pp 1057ndash10792008
[10] T Kocaturk and M Simsek ldquoFree vibration analysis of Tim-oshenko beams under various boundary conditionsrdquo SigmaJournal of Engineering and Natural Sciences vol 1 pp 30ndash442005
[11] DKNguyen ldquoFree vibration of prestressedTimoshenkobeamsresting on elastic foundationsrdquo Vietnam Journal of Mechanicsvol 29 no 1 pp 1ndash12 2012
[12] N M Auciello ldquoVibrations of Timoshenko beams on twoparameter elastic soilrdquo Engineering Transactions vol 56 no 3pp 187ndash200 2008
[13] M Taha ldquoRecursive differentiation method for boundary valueproblems application to analysis of a beam-column on an elas-tic foundationrdquo Journal of Theoretical and Applied Mechanicsvol 44 no 2 pp 57ndash70 2014
[14] M Taha and E H Doha ldquoRecursive differentiation methodapplication to the analysis of beams on two parameter founda-tionsrdquo Journal of Theoretical and Applied Mechanics vol 53 no1 pp 15ndash26 2015
[15] S P Timoshenko and J M Gere Theory of Elastic StabilityEngineering Societies Monographs McGraw-Hill Book NewYork NY USA 1961
that the weakening of the end restraints may offer a param-eters combination coinciding with the resonance conditionsfor the first mode or one of the higher modes and unboundedamplitudes may be reached
5 Conclusions
The RDM is used to investigate the stability behavior ofaxially loaded cantilever Timoshenko beam resting on two-parameter foundation In addition the analysis includesthe investigation of Timoshenko beams resting on two-parameter foundation and subjected to different types of lat-eral excitationThe supports at the beam ends are assumed ofthe elastic type to investigate the impact of the support weak-ening on the beambehavior Both the effect of rotational iner-tia and the correction of the shear stress due to axial loadingare taken into consideration which decreases the beam stiff-ness It is concluded that the significance of the foundation onthe Timoshenko beams (120578 lt 20) behavior is negligiblethe effect of end restraints is more noticeable for slenderbeams and the natural frequency decreases as the axial loadincreases However for the case of the forced vibration theresonance conditions may result from the weakening of theend supports and unbounded response amplitudes may bereached The analysis depicts the simplicity and accuracy ofthe RDM in tackling the boundary value problems
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] P Ruta ldquoThe application of Chebyshev polynomials to the solu-tion of the nonprismatic Timoshenko beam vibration problemrdquoJournal of Sound and Vibration vol 296 no 1-2 pp 243ndash2632006
[2] K Kausel ldquoNonclassical modes of unrestrained shear beamsrdquoJournal of Engineering Mechanics vol 128 no 6 pp 663ndash6672002
[3] R Attarnejad A Shahba and S J Semnani ldquoApplication ofdifferential transform in free vibration analysis of timoshenkobeams resting on two-parameter elastic foundationrdquo ArabianJournal for Science and Engineering vol 35 no 2B pp 125ndash1322010
[4] M Taha and M A Nassar ldquoAnalysis of stressed Timoshenkobeams on two parameter foundationsrdquo KSCE Journal of CivilEngineering vol 19 no 1 pp 173ndash179 2015
[5] L Majkut ldquoFree and forced vibration of Timoshenko beamsdescribed by single difference equationrdquo Journal of Theoreticaland Applied Mechanics vol 47 no 1 pp 193ndash210 2009
[6] F Y Cheng and C P Pantelides ldquoDynamic timoshenko beam-column on elastic mediardquo Journal of Structural Engineering vol114 no 7 pp 1524ndash1550 1988
[7] B Geist and J R Mclaughlin ldquoDouble eigen value for theTimoshenko beamrdquo Applied Mathematics Letters vol 10 pp129ndash134 1997
[8] C-N Chen ldquoDQEM vibration analyses of non-prismatic sheardeformable beams resting on elastic foundationsrdquo Journal ofSound and Vibration vol 255 no 5 pp 989ndash999 2002
[9] A L G Monsalve Z D G Medina and A J D Ochoa ldquoTim-oshenko beam-column with generalized end conditions onelastic foundation dynamic-stiffness matrix and load vectorrdquoJournal of Sound and Vibration vol 310 no 4-5 pp 1057ndash10792008
[10] T Kocaturk and M Simsek ldquoFree vibration analysis of Tim-oshenko beams under various boundary conditionsrdquo SigmaJournal of Engineering and Natural Sciences vol 1 pp 30ndash442005
[11] DKNguyen ldquoFree vibration of prestressedTimoshenkobeamsresting on elastic foundationsrdquo Vietnam Journal of Mechanicsvol 29 no 1 pp 1ndash12 2012
[12] N M Auciello ldquoVibrations of Timoshenko beams on twoparameter elastic soilrdquo Engineering Transactions vol 56 no 3pp 187ndash200 2008
[13] M Taha ldquoRecursive differentiation method for boundary valueproblems application to analysis of a beam-column on an elas-tic foundationrdquo Journal of Theoretical and Applied Mechanicsvol 44 no 2 pp 57ndash70 2014
[14] M Taha and E H Doha ldquoRecursive differentiation methodapplication to the analysis of beams on two parameter founda-tionsrdquo Journal of Theoretical and Applied Mechanics vol 53 no1 pp 15ndash26 2015
[15] S P Timoshenko and J M Gere Theory of Elastic StabilityEngineering Societies Monographs McGraw-Hill Book NewYork NY USA 1961
