Sadhana Vol. 33, Part 6, December 2008, pp. 781–801. © Printed in India
Free vibration of semi-rigid connected Reddy–Bickfordpiles embedded in elastic soil
YUSUF YESILCE∗ and HIKMET H CATAL
Dokuz Eylul University, Civil Engineering Department, Engineering Faculty,35160, Buca, Izmir, Turkeye-mail: [email protected]
MS received 14 February 2008; revised 28 April 2008
Abstract. The literature on free vibration analysis of Bernoulli–Euler and Tim-oshenko piles embedded in elastic soil is plenty, but that of Reddy–Bickford pilespartially embedded in elastic soil with/without axial force effect is fewer. The soilthat the pile partially embedded in is idealized by Winkler model and is assumedto be two-layered. The pile part above the soil is called the first region and theparts embedded in the soil are called the second and the third region, respectively.It is assumed that the behaviour of the material is linear-elastic, that axial forcealong the pile length to be constant and the upper end of the pile that is semi-rigidsupported against rotation is modelled by an elastic spring. The governing dif-ferential equations of motion of the rectangular pile in free vibration are derivedusing Hamilton’s principle and Winkler hypothesis. The terms are found directlyfrom the solutions of the differential equations that describe the deformations ofthe cross-section according to the high-order theory. The models have six degreesof freedom at the two ends, one transverse displacement and two rotations, andthe end forces are a shear force and two end moments. Natural frequencies of thepile are calculated using transfer matrix and the secant method for non-trivial solu-tion of linear homogeneous system of equations obtained due to values of axialforces acting on the pile, total and embedded lengths of the pile, the linear-elasticrotational restraining stiffness at the upper end of the pile and to the boundary con-ditions of the pile. Two different boundary conditions are considered in the study.For the first boundary condition, the pile’s end at the first region is semi-rigid con-nected and not restricted for horizontal displacement and the end at the third regionis free and for the second boundary condition, the pile’s end at the first region issemi-rigid connected and restricted for horizontal displacement and the end at thethird region is fixed supported. The calculated natural frequencies of semi-rigidconnected Reddy–Bickford pile embedded in elastic soil are given in tables andcompared with results of Timoshenko pile model.
Keywords. Axial force effect; free vibration; Reddy–Bickford pile; semi-rigidconnected; transfer matrix.
∗For correspondence
781
782 Yusuf Yesilce and Hikmet H Catal
Figure 1. Cross-section displace-ments in different beam theories(Wang et al 2000) (a) Bernoulli–EulerBeam Theory (BET); (b) TimoshenkoBeam Theory (TBT); (c) Reddy–Bickford Beam Theory (RBT).
1. Introduction
The analysis of the piles embedded in elastic soil is similar to the analysis of beams on elasticfoundations.
The analysis of beams has been performed over the years mostly using Bernoulli–Eulerbeam theory (BET). The classical Bernoulli–Euler beam is well studied for slender beams,where the transverse shear deformation can be safely disregarded. This theory is based onthe assumption that plane sections of the cross-section remain plane and perpendicular tothe beam axis. The cross-sectional displacements are shown in (figure 1a), and expressedas
u(x, z, t) = −z · ∂w0(x, t)
∂x(1)
w(x, z, t) = w0(x, t), (2)
where w0(x, t) is the lateral displacement of the beam neutral axis, z is the distance from thebeam neutral axis.
For moderately thick beams Bernoulli–Euler beam theory can be modified in order to takeinto account the transverse shear effect in a simplified way. For example, the well-knownTimoshenko beam theory (TBT) predicts a uniform shear distribution, so necessitating theuse of a so-called shear factor (Cowper 1966, Gruttmann & Wagner 2001, Murthy 1970). Thecross-sectional displacements of Timoshenko beam theory are shown in (figure 1b) and the
Free vibration of semi-rigid connected Reddy–Bickford piles 783
equations for Timoshenko beam theory which relaxes the restriction on the angle of shearingdeformations are;
u(x, z, t) = z · φ(x, t) (3)
w(x, z, t) = w0(x, t), (4)
where φ(x, t) represents the rotation of a normal to the axis of the beam. Han et al (1999)presented a comprehensive study of Bernoulli–Euler, Rayleigh, Shear and Timoshenko beamtheories.
The real shear deformation distribution is not uniform along the depth of the beam, sothat Timoshenko beam theory is not recommended for composite beams, where the accuratedetermination of the shear stresses is required. Especially, it was found that the Timoshenkoshear deformation theory has some major numerical problems such as locking in the numericalanalysis for composite materials. The other problem was the need to supply an artificiallyderived shear correction factor. Although some remedies were devised, as a result, severalhigher-order theories have emerged. These theories, with small variations, are due to severalauthors relax the restriction on the warping of the cross-section and allow variation in thelongitudinal direction of the beam which is cubic (Bickford 1982, Heyliger & Reddy 1988,Levinson 1981, Wang et al 2000).
In this paper, Reddy–Bickford beam theory (RBT) is used, which seems a good compromisebetween accuracy and simplicity (Bickford 1982, Wang et al 2000). The cross-sectionaldisplacements of Reddy–Bickford beam theory are shown in (figure 1c) and according toReddy–Bickford beam theory, the displacements of the rectangular beam can be written as(Wang et al 2000, Reddy 2002, Reddy 2007)
u(x, z, t) = z · φ(x, t) − α · z3 ·[φ(x, t) + ∂w(x, t)
∂x
], (5)
w(x, z, t) = w0(x, t), (6)
where α = 43·h2 ; h is height of the beam.
Bernoulli–Euler beam theory does not consider the shear stress in the cross-section and theassociated strains. Thus, the shear angle is taken as zero through the height of the cross-section.Timoshenko beam theory assumes constant shear stress and shear strain in the cross-section.On the top and bottom edges of the beam the free surface condition is thus violated. The useof a shear correction factor, in various forms including the effect of Poisson’s ratio, does notcorrect this fault of the theory, but rather artificially adjusts the solutions to match the staticor dynamic behaviour of the beam. Reddy–Bickford beam theory and the other high-ordertheories remedy this physical mismatch at the free edges by assuming variable shear strainand shear stress along the height of the cross-section. Then there is no need for the shearcorrection factor. The high-order theory is more exact and represents much better physics ofthe problem. It results in a sixth-order theory compared to the fourth-order of the other less-accurate theories. This yields a six-degree-of-freedom element with six end forces, a shearforce, bending moment and a high-order moment, at the two ends of the beam element.
The determination of the natural frequencies is crucial in the dynamic analysis of piles thatare partially embedded in the soil. Previously, it was widely assumed that the upper ends ofpiles used to support offshore structures, marine and harbor structures or bridges are fullyrigidly connected. In reality, these connections are neither fully rigid nor flexible. They fallbetween fully rigid and flexible connections, depending on the cross-sectional and material
784 Yusuf Yesilce and Hikmet H Catal
properties of the piles. A more reasonable way is to treat them as semi-rigid connections inthe structural analysis.
The analysis of a partially supported pile is similar to that of a beam that is elasticallysupported. Previously, numerous researchers studied the behaviour of beams supported byelastic foundations (Hetenyi 1955). Doyle & Pavlovic (1982) solved the partial differentialequation for free vibration of beams partially attached to elastic foundation using variableseparating method and neglecting axial force and shear effects. Boroomand & Kaynia (1991)studied dynamic analysis of pile–soil–pile interaction for vertical piles in a homogeneoussoil by using a Fourier expansion of variables. Aviles & Sanchez-Sesma (1983) studied theusefulness of a row of rigid piles as an isolating barrier for elastic waves. They formulatedthe problem as one of multiple scattering and diffraction. Aviles & Sanchez-Sesma (1988)presented a theoretical analysis to solve the problem of foundation isolation, using a row ofelastic piles as an isolating barrier for elastic waves. Liao & Sangrey (1978) employed anacoustic model for the use of rows of piles as passive isolation barriers to reduce groundvibrations. West & Mafi (1984) solved the partial differential equation for free vibration of anelastic beam on elastic foundation that is subjected to axial force by using initial value method.Yokoyama (1991) studied the free vibration motion of Timoshenko beam on two-parameterselastic foundation. Esmailzadeh & Ohadi (2000) investigated vibration and stability analysisof non-uniform Timoshenko beams under axial and distributed tangential loads. Catal (2002)calculated natural frequencies and relative stiffness of the pile for non-trivial solution of linearhomogeneous system of equations obtained due to the values of axial forces acting on thepile, the shape factors, and the boundary conditions of the pile with both ends free and bothends simply supported by using Timoshenko beam theory and transfer matrix. Further, heproceeded to determine the natural frequencies of Timoshenko piles partially embedded inthe soil, but semi-rigidly connected at the upper ends, using the method of initial values (Catal2006). Lin & Chang (2005) studied free vibration analysis of multi-span Timoshenko beamwith an arbitrary number of flexible constraints by transfer matrix method. Demirdag & Catal(2007) investigated spectral analysis of semi-rigid supported single storey frames modelledas Timoshenko column with attached mass. Demirdag (2008) studied elastically-supportedTimoshenko column with attached masses is under consideration to obtain its free vibrationnatural frequencies using two different algorithm; transfer matrix method and finite elementmethod. Yesilce & Catal (2008) calculated normalized natural frequencies of Timoshenkopile due to the different values of axial force using transfer matrix and considering rotatoryinertia.
2. The mathematical model and formulation
A rectangular pile partially embedded in the soil whose upper end is semi-rigid connectedagainst rotation is presented in (figure 2). The pile part above the soil is called the first regionand the parts embedded in the soil are called the second and the third region, respectively.
Using Hamilton’s principle and Eqs. (5) and (6); the equations of motion can be writtenfor each region as (Wang et al 2000, Eisenberger 2003):
− 68
105· EIx · ∂2φj (xj , t)
∂x2j
+ 16
105· EIx · ∂3wj(xj , t)
∂x3j
+ 8
15· AG
·[φj (xj , t) + ∂wj (xj , t)
∂xj
]= 0 (0 ≤ xj ≤ Lj) (j = 1, 2, 3) (7)
Free vibration of semi-rigid connected Reddy–Bickford piles 785
− m · ∂2w1(x1, t)
∂t2+ 8
15· AG ·
[∂φ1(x1, t)
∂x1+ ∂2w1(x1, t)
∂x21
]+ 16
105· EIx
· ∂3φ1(x1, t)
∂x31
− 1
21· EIx · ∂4w1(x1, t)
∂x41
−N · ∂2w1(x1, t)
∂x21
= 0 (0 ≤ x1 ≤ L1) (8)
− m · ∂2wk(xk, t)
∂t2+ 8
15· AG ·
[∂φk(xk, t)
∂xk
+ ∂2wk(xk, t)
∂x2k
]+ 16
105· EIx · ∂3φk(xk, t)
∂x3k
− 1
21·EIx · ∂
4wk(xk, t)
∂x4k
−CS(k−1) ·wk(xk, t)−N · ∂2wk(xk, t)
∂x2k
= 0(0 ≤ xk ≤ Lk) (k = 2, 3),
(9)
where wj(xj , t) is displacement function for j th region of the pile, φj (xj , t) represents therotation of a normal to the axis for j th region of the pile, m is mass per unit length of thepile, L1 is pile length above the soil, L2 is pile length embedded in the second region, L3 ispile length embedded in the third region, L is total length of the pile, N is the constant axialcompressive force, A is the cross-section area, Ix is moment of inertia, E, G are Young’smodulus and shear modulus of the pile, CS1 = CR1.b and CS2 = CR2.b in which CR1, CR2
Figure 2. Pile partially embedded in the elas-tic soil.
786 Yusuf Yesilce and Hikmet H Catal
are the modulus of subgrade reaction for the second and the third regions, respectively andb is width of the pile, x1, x2 and x3 are pile positions for the first, the second and the thirdregions, t is time variable.
Assuming that the motion is harmonic we substitute for wj(zj , t) and φj (rj , t) the follow-ing:
wj(zj , t) = wj(zj ) · sin(ω · t) (10)
φj (zj , t) = φj (zj ) · sin(ω · t) (j = 1, 2, 3) (11)
and obtain a system of two coupled ordinary equation for each region as:
− 68
105· EIx
L2· d2φj (zj )
dz2j
+ 16
105· EIx
L3· d3wj(zj )
dz3j
+ 8
15· AG
·[φj (zj ) + 1
L· dwj (zj )
dzj
]= 0 (j = 1, 2, 3) (12)
m · ω2 · w1 (z1) + 8
15· AG
L·[dφ1(z1)
dz1+ 1
L· d2w1(z1)
dz21
]+ 16
105· EIx
L3
· d3φ1(z1)
dz31
− 1
21· EIx
L4· d4w1(z1)
dz41
− Nr · π2 · EIx
L4· d2w1(z1)
dz21
= 0 (13)
m · ω2 · wk(zk) + 8
15· AG
L·[dφk(zk)
dzk
+ 1
L· d2wk(zk)
dz2k
]+ 16
105
· EIx
L3· d3φk(zk)
dz3k
− 1
21· EIx
L4· d4wk(zk)
dz4k
− CS(k−1) · wk(zk) − Nr
· π2 · EIx
L4· d2wk(zk)
dz2k
= 0 (k = 2, 3), (14)
where z = xL
, dimensionless position parameter; Nr = N ·L2
π2·EIx, non-dimensionalized multi-
plication factor for the axial compressive force and ω is natural frequency of the pile.It is assumed that the solution is
wj(zj ) = Cj · ei·sj ·zj (15)
φj (zj ) = Pj · ei·sj ·zj (j = 1, 2, 3) (16)
and substituting Eqs. (15) and (16) into Eqs. (12), (13) and (14) results in(
8
15· AG + 68
105· EIx
L2· s2
j
)· Pj
+(
8
15· AG
L· sj · i − 16
105· EIx
L3· s3
j · i
)· Cj = 0 (j = 1, 2, 3) (17)
Free vibration of semi-rigid connected Reddy–Bickford piles 787
(8
15· AG
L· s1 · i − 16
105· EIx
L3· s3
1 · i
)· P1
+(
m · ω2 − 8
15· AG
L2· s2
1 − 1
21· EIx
L4· s4
1 + Nr · π2 · EIx
L4· s2
1
)· C1 = 0
(18)
(8
15· AG
L· sk · i − 16
105· EIx
L3· s3
k · i
)· Pk
+(
m · ω2 − 8
15· AG
L2· s2
k − 1
21· EIx
L4· s4
k − CS(k−1) + Nr · π2 · EIx
L4· s2
k
)
· Ck = 0 (k = 2, 3), (19)
where i = √−1.Eqs. (17), (18) and (19) can be written in matrix form for the two unknowns Pj and Cj and
the non-trivial solution will be obtained when the determinant of the coefficient matrix willbe zero for each region, i.e.
[− 4
525· (EIx)
2
L6
]· s6
1 +[
68
105· Nr · π2 · (EIx)
2
L6− 8
15· AG · EIx
L4
]· s4
1
+(
68
105· EIx
L2· m · ω2 + 8
15· Nr · π2 · EIx · AG
L4
)· s2
1
+ 8
15· AG · m · ω2 = 0 (20)
[− 4
525· (EIx)
2
L6
]· s6
k +[
68
105· Nr · π2 · (EIx)
2
L6− 8
15· AG · EIx
L4
]· s4
k
+(
68
105· EIx
L2· (m · ω2 − CS(k−1)
)+ 8
15· Nr · π2 · EIx · AG
L4
)· s2
k
+ 8
15· AG · (m · ω2 − CS(k−1)) = 0. (21)
Thus, we have a sixth-order equation with the unknowns for each region, resulting in sixvalues and the general solution for each region can be written as:
wj(zj , t) = [Cj1 · ei·sj1·zj + Cj2 · ei·sj2·zj + Cj3 · ei·sj3·zj + Cj4 · ei·sj4·zj
+Cj5 · ei·sj5·zj + Cj6 · ei·sj6·zj ] · sin(ω · t) (22)
φj (zj , t) = [Pj1 · ei·sj1·zj + Pj2 · ei·sj2·zj + Pj3 · ei·sj3·zj + Pj4 · ei·sj4·zj
+Pj5 · ei·sj5·zj + Pj6 · ei·sj6·zj ] · sin(ω · t) (j = 1, 2, 3). (23)
788 Yusuf Yesilce and Hikmet H Catal
The thirty-six constants, Cj1, . . . , Cj6 and Pj1, . . . , Pj6, will be found from Eqs. (17), (18),(19) and boundary conditions.
For each region, the expression for bending rotation w′j (zj , t) is given by
w′j (zj , t) = 1
L· dwj (zj )
dzj
· sin(ω · t) (j = 1, 2, 3). (24)
For each region, the shear force function Qj(zj , t) can be obtained by using Eqs. (22) and(23) as:
Qj(zj , t) = −8 · AG
15·(
φj (zj ) + 1
L· dwj (zj )
dzj
)· sin(ω · t)
+ Nr · π2 · EIx
L3· dwj (zj )
dzj
· sin(ω · t)
+ EIx
21 · L3· d3wj(zj )
dz3j
· sin(ω · t)
− 16 · EIx
105 · L2· d2φj (zj )
dz2j
· sin(ω · t) (j = 1, 2, 3). (25)
Similarly, the bending moment function Mj(zj , t) can be obtained by using Eqs. (22) and(23) as:
Mj(zj , t) =(
− EIx
21 · L2· d2wj(zj )
dz2j
− Nr · π2 · EIx
L2· wj(zj )
+16 · EIx
105 · L· dφj (zj )
dzj
)· sin(ω · t). (26)
For each region, the higher-order moment function Mhj(zj , t) can be obtained as:
Mhj(zj , t) =(
16 · EIx
105 · L2· d2wj(zj )
dz2j
− 68 · EIx
105 · L· dφj (zj )
dzj
)· sin(ω · t) (j = 1, 2, 3).
(27)
3. Obtaining the transfer matrices of the pile
The position for each region is written due to the values of the transverse displacementwj(zj , t), bending rotation w′
j (zj , t), rotation of normal φj (zj , t), shear force Qj(zj , t),bending moment Mh(zj , t) and higher-order moment function Mhj(zj , t) at the locations ofzj and t for Reddy–Bickford pile, as (j = 1, 2, 3):
〈Sj (zj , t)〉T = 〈wj(zj ) w′j (zj ) φj (zj ) Mj(zj ) Mhj (zj ) Qj (zj )〉T · sin(ω.t), (28)
where {Sj (zj , t)} shows the position vector for each region.
Free vibration of semi-rigid connected Reddy–Bickford piles 789
All terms of the position vector in Eq. (28), is written reducing the sin(ωt) terms as:
{S1(z1)} =
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
w1(z1)
w′1(z1)
φ1(z1)
M1(z1)
Mh1(z1)
Q1(z1)
⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭
=
⎡⎢⎢⎢⎢⎢⎢⎢⎣
A11(z1) A12(z1) A13(z1) A14(z1) A15(z1) A16(z1)
A21(z1) A22(z1) A23(z1) A24(z1) A25(z1) A26(z1)
A31(z1) A32(z1) A33(z1) A34(z1) A35(z1) A36(z1)
A41(z1) A42(z1) A43(z1) A44(z1) A45(z1) A46(z1)
A51(z1) A52(z1) A53(z1) A54(z1) A55(z1) A56(z1)
A61(z1) A62(z1) A63(z1) A64(z1) A65(z1) A66(z1)
⎤⎥⎥⎥⎥⎥⎥⎥⎦
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
C11
C12
C13
C14
C15
C16
⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭
(29)
{S2(z2)} =
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
w2(z2)
w′2(z2)
φ2(z2)
M2(z2)
Mh2(z2)
Q2(z2)
⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭
=
⎡⎢⎢⎢⎢⎢⎢⎢⎣
B11(z2) B12(z2) B13(z2) B14(z2) B15(z2) B16(z2)
B21(z2) B22(z2) B23(z2) B24(z2) B25(z2) B26(z2)
B31(z2) B32(z2) B33(z2) B34(z2) B35(z2) B36(z2)
B41(z2) B42(z2) B43(z2) B44(z2) B45(z2) B46(z2)
B51(z2) B52(z2) B53(z2) B54(z2) B55(z2) B56(z2)
B61(z2) B62(z2) B63(z2) B64(z2) B65(z2) B66(z2)
⎤⎥⎥⎥⎥⎥⎥⎥⎦
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
C21
C22
C23
C24
C25
C26
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭
(30)
{S3(z3)} =
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
w3(z3)
w′3(z3)
φ3(z3)
M3(z3)
Mh3(z3)
Q3(z3)
⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭
=
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣
C11(z3) C12(z3) C13(z3) C14(z3) C15(z3) C16(z3)
C21(z3) C22(z3) C23(z3) C24(z3) C25(z3) C26(z3)
C31(z3) C32(z3) C33(z3) C34(z3) C35(z3) C36(z3)
C41(z3) C42(z3) C43(z3) C44(z3) C45(z3) C46(z3)
C51(z3) C52(z3) C53(z3) C54(z3) C55(z3) C56(z3)
C61(z3) C62(z3) C63(z3) C64(z3) C65(z3) C66(z3)
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
C31
C32
C33
C34
C35
C36
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭
(31)
Eqs. (29), (30) and (31) can be written in closed form, as [(j = 1, . . . , 6), (m =1, . . . , 6), (n = 1, . . . , 6)]:
{S1(z1)} = [Ajm(z1)] · {C1n} (32)
{S2(z2)} = [Bjm(z2)] · {C2n} (33)
{S3(z3)} = [Cjm(z3)] · {C3n}. (34)
The coefficient vectors {C1n}, {C2n}, {C3n} are obtained from Eqs. (32), (33) and (34) forz1 = 0, z2 = 0 and z3 = 0 with the condition that the matrices [Ajm(z1)], [Bjm(z2)] and[Cjm(z3)] are not singular. Substituting these coefficient vectors into the Eqs. (32), (33) and(34) respectively, gives,
{S1(z1)} = [F1(z1)] · {S1(0)} (35)
{S2(z2)} = [F2(z2)] · {S2(0)} (36)
{S3(z3)} = [F3(z3)] · {S3(0)} (37)
790 Yusuf Yesilce and Hikmet H Catal
where;
[F1(z1)] = [Ajm(z1)] · [Ajm(0)]−1 (38)
[F2(z2)] = [Bjm(z2)] · [Bjm(0)]−1 (39)
[F3(z3)] = [Cjm(z3)] · [Cjm(0)]−1 (40)
are the transfer matrices for the first, the second and the third regions of the pile, respectively.The transfer matrices for the three regions can be combined to yield one global transfer
matrix using the characteristics of transfer matrix. Thus, the position vector of the pile end atthe third region can be related to the position vector of the pile end at the first region as
{S1(z1 = 1)} =[F1
(z1 = L1
L
)]·[F2
(z2 = L2
L
)]·[F3
(z3 = L3
L
)]· {S3(z3 = 0)}
(41)
{S1(z1 = 1)} = [F(z = 1)] · {S3(z3 = 0)}, (42)
where [F(z = 1)] shows the global transfer matrix that transfers the values of the positionvector of the pile end at the third region to the values of the position vector of the pile end atthe first region and can be written as:
[F(z = 1)] =
⎡⎢⎢⎢⎢⎢⎢⎢⎣
F11 F12 F13 F14 F15 F16
F21 F22 F23 F24 F25 F26
F31 F32 F33 F34 F35 F36
F41 F42 F43 F44 F45 F46
F51 F52 F53 F54 F55 F56
F61 F62 F63 F64 F65 F66
⎤⎥⎥⎥⎥⎥⎥⎥⎦
, (43)
where the term Fjm shows the terms of the global transfer matrix [F(z = 1)].
4. Obtaining natural frequencies for semi-rigid connected pile
The behaviour of the pile end that is semi-rigid supported against rotation at the first regionis modelled by an elastic spring. The rotational spring rigidities are related with fixity factorthat is defined as below (Monforton & Wu 1963):
f = 1
1 + 3·EIx
Cθ ·L, (44)
where Cθ is the rotational restraining stiffness at the upper end of the pile in the first region.Bending moment function at semi-rigid connected end is written as a linear function of
rotational restraining stiffness and bending rotation as (Wang et al 2000):
M1
(z1 = L1
L
)= −Cθ · φ1
(z1 = L1
L
), (45)
Free vibration of semi-rigid connected Reddy–Bickford piles 791
Figure 3. (a) Pile whose end at the first region is semi-rigid connected and not restricted for horizontaldisplacement and the end at the third region is free (The first model). (b) Pile whose end at the firstregion is semi-rigid connected and restricted for horizontal displacement and the end at the third regionis fixed supported (The second model).
where M1(z1 = L1
L
)and φ1
(z1 = L1
L
)are the bending moment and the rotation of normal for
the first region.Similarly, high-order moment function at semi-rigid connected end is written as a lin-
ear function of rotational restraining stiffness and rotation of normal as (Wang et al2000):
Mh1
(z1 = L1
L
)= 0, (46)
where Mh1(z1 = L1
L
)is the high-order moment for the first region.
Using Eq. (42), the position vectors at free and the fixed supported ends are transferredto the position vectors at the semi-rigid connected end by Eqs. (47) and (48), respec-tively. For the first model, the pile’s end at the first region is semi-rigid connected andnot restricted for horizontal displacement and the end at the third region is free as in fig-ure 3a. For the second model, the pile’s end at the first region is semi-rigid connected andrestricted for horizontal displacement and the end at the third region is fixed supported as infigure 3b.
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
w1(z1 = L1
L
)w′
1
(z1 = L1
L
)φ1(z1 = L1
L
)Cθ · w′
1
(z1 = L1
L
)Cθ · φ1
(z1 = L1
L
)0
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
=
⎡⎢⎢⎢⎢⎢⎢⎢⎣
F11 F12 F13 F14 F15 F16
F21 F22 F23 F24 F25 F26
F31 F32 F33 F34 F35 F36
F41 F42 F43 F44 F45 F46
F51 F52 F53 F54 F55 F56
F61 F62 F63 F64 F65 F66
⎤⎥⎥⎥⎥⎥⎥⎥⎦
·
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
w3(z3 = 0)
w′3(z3 = 0)
φ3(z3 = 0)
000
⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭
(47)
792 Yusuf Yesilce and Hikmet H Catal
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
0
w′1
(z1 = L1
L
)φ1(z1 = L1
L
)Cθ · w′
1
(z1 = L1
L
)Cθ · φ1
(z1 = L1
L
)Q1
(z1 = L1
L
)
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
=
⎡⎢⎢⎢⎢⎢⎢⎢⎣
F11 F12 F13 F14 F15 F16
F21 F22 F23 F24 F25 F26
F31 F32 F33 F34 F35 F36
F41 F42 F43 F44 F45 F46
F51 F52 F53 F54 F55 F56
F61 F62 F63 F64 F65 F66
⎤⎥⎥⎥⎥⎥⎥⎥⎦
·
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
000
M3(z3 = 0)
Mh3(z3 = 0)
Q3(z3 = 0)
⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭
(48)
Eqs. (47) and (48) can be written in matrix form as:
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣
F11 F12 F13 −1 0 0F21 F22 F23 0 −1 0F31 F32 F33 0 0 −1F41 F42 F43 0 −Cθ 0F51 F52 F53 0 0 −Cθ
F61 F62 F63 0 0 0
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦
·
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
w3(z3 = 0)
w′3(z3 = 0)
φ3(z3 = 0)
w1(z1 = L1
L
)w′
1
(z1 = L1
L
)φ1(z1 = L1
L
)
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭
=
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
000000
⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭
(49)
⎡⎢⎢⎢⎢⎢⎢⎢⎣
F14 F15 F16 0 0 0F24 F25 F26 −1 0 0F34 F35 F36 0 −1 0F44 F45 F46 −Cθ 0 0F54 F55 F56 0 −Cθ 0F64 F65 F66 0 0 −1
⎤⎥⎥⎥⎥⎥⎥⎥⎦
·
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
M3(z3 = 0)
Mh3(z3 = 0)
Q3(z3 = 0)
w′1
(z1 = L1
L
)φ1(z1 = L1
L
)Q1
(z1 = L1
L
)
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭
=
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
000000
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭
. (50)
For non-trivial solutions of this problem, following relations are written by using Eqs. (49)and (50), as:
∣∣∣∣∣∣∣∣∣∣∣∣∣
F11 F12 F13 −1 0 0F21 F22 F23 0 −1 0F31 F32 F33 0 0 −1F41 F42 F43 0 −Cθ 0F51 F52 F53 0 0 −Cθ
F61 F62 F63 0 0 0
∣∣∣∣∣∣∣∣∣∣∣∣∣
= 0 (51)
∣∣∣∣∣∣∣∣∣∣∣∣∣
F14 F15 F16 0 0 0F24 F25 F26 −1 0 0F34 F35 F36 0 −1 0F44 F45 F46 −Cθ 0 0F54 F55 F56 0 −Cθ 0F64 F65 F66 0 0 −1
∣∣∣∣∣∣∣∣∣∣∣∣∣
= 0. (52)
Free vibration of semi-rigid connected Reddy–Bickford piles 793
5. Numerical analysis and discussions
In this paper, for numerical analysis, two models that are partially embedded in Winklersoil and whose ends above the soil are semi-rigid connected by an elastic spring having therotational spring rigidities of Cθ , as one being not restricted for horizontal displacement andthe other being restricted for horizontal displacement, respectively, are considered. For twoexamples, natural frequencies of the pile, ωi(i = 1, 2, 3) are calculated by using computerprograms prepared by authors. Natural frequencies are found by determining values for whichthe determinant of the coefficient matrix is equal to zero. There are various methods forcalculating the roots of the frequency equation. One commonly used and simple technique isthe secant method in which a linear interpolation is employed. The eigenvalues, the naturalfrequencies, are determined by a trial and error method based on interpolation and the bisectionapproach. One such procedure consists of evaluating the determinant for a range of frequencyvalues, ωi . When there is a change of sign between successive evaluations, there must be aroot lying in this interval. The iterative computations are determined when the value of thedeterminant changed sign due to a change of 10−4 in the value of ωi .
For each example, CS1 = 15000 kN/m3 and CS2 = 60000 kN/m3. The length of the pile istaken as 15 m and 30 m.
The all numerical results of this paper are obtained based on uniform, rectangular Reddy–Bickford and Timoshenko piles with the following data:
h = 0·50 m; b = 0·30 m; EIx = 6·5625 × 103 kNm2;AG = 121500 kN; m = 0·50 kN.sec2/m;
for the axial force effect Nr = 0·25 and 1·00, fixity factors are taken as f = 0·25 andf = 0·75.
The all numerical results are given for the following three models: Timoshenko model withtwo values for shear correction factor k and Reddy–Bickford model. Many values for theshear correction factor k were suggested, but in this paper, the original values suggested byTimoshenko k = 5
6 and k = 1417 are used (Gruttmann & Wagner 2001).
Natural frequencies of the pile are obtained from the solution of Eqs. (51) and (52) accordingto the boundary conditions, by using the computer program for Nr = 0·25 and Nr = 1·0 withthe values of L, L1/L, L2/L and L3/L taken from table 1.
For L = 15 m, the frequency values obtained for the first three modes of Reddy–Bickfordand Timoshenko piles whose end at the first region is semi-rigid connected and not restrictedfor horizontal displacement and the end at the third region is free are presented in table 2;for L = 30 m, the frequency values obtained for the first three modes of the same piles arepresented in table 3 being compared with the frequency values obtained for Nr = 0·25 and1·00, f = 0·25 and 0·75.
Table 1. Values of L1, L2 and L3 with respect to values of L, L1/L, L2/L and L3/L.
L1/L = 0·50 L2/L = 0·30 L3/L = 0·20 L1/L = 0·50 L2/L = 0·20 L3/L = 0·30
L(m) L1(m) L2(m) L3(m) L(m) L1(m) L2(m) L3(m)
15 7·5 4·5 3·0 15 7·5 3·0 4·530 15·0 9·0 6·0 30 15·0 6·0 9·0
794 Yusuf Yesilce and Hikmet H Catal
Tabl
e2.
The
first
thre
ena
tura
lfre
quen
cies
ofR
eddy
–Bic
kfor
dan
dT
imos
henk
opi
les
who
seen
dat
the
first
regi
onis
sem
i-ri
gid
conn
ecte
dan
dno
tre
stri
cted
for
hori
zont
aldi
spla
cem
enta
ndth
een
dat
the
thir
dre
gion
isfr
ee,L
=15
m.
L=
15m
f=
0·25
L1/L
=0·5
0L
2/L
=0·3
0L
3/L
=0·2
0L
1/L
=0·5
0L
2/L
=0·2
0L
3/L
=0·3
0ωi
(rad/sec)N
r=
0·25
Nr=
1·00
Nr=
0·25
Nr=
1·00
TB
TT
BT
TB
TT
BT
TB
TT
BT
TB
TT
BT
RB
T(k
=5/
6)(k
=14
/17
)R
BT
(k=
5/6)
(k=
14/17
)R
BT
(k=
5/6)
(k=
14/17
)R
BT
(k=
5/6)
(k=
14/17
)
ω1
6·127
46·0
826
6·082
36·2
982
6·198
56·1
981
6·134
06·0
891
6·088
86·3
051
6·205
46·2
050
ω2
34·00
9733
·8781
33·87
3332
·8500
32·69
5432
·6906
34·03
7733
·9054
33·90
0632
·8792
32·72
4032
·7191
ω3
88·89
1488
·9223
88·90
2887
·5489
87·62
4187
·6048
88·94
3688
·9725
88·95
2987
·6051
87·67
8187
·6587
f=
0·75
L1/L
=0·5
0L
2/L
=0·3
0L
3/L
=0·2
0L
1/L
=0·5
0L
2/L
=0·2
0L
3/L
=0·3
0
Nr=
0·25
Nr=
1·00
Nr=
0·25
Nr=
1·00
ωi
(rad/sec)
TB
TT
BT
TB
TT
BT
TB
TT
BT
TB
TT
BT
RB
T(k
=5/
6)(k
=14
/17
)R
BT
(k=
5/6)
(k=
14/17
)R
BT
(k=
5/6)
(k=
14/17
)R
BT
(k=
5/6)
(k=
14/17
)
ω1
7·657
17·3
756
7·375
17·0
486
6·656
56·6
560
7·665
27·3
834
7·382
97·0
574
6·665
06·6
645
ω2
40·41
8239
·1142
39·10
8039
·2134
37·77
7737
·7716
40·45
0239
·1444
39·13
8139
·2468
37·80
9337
·8031
ω3
96·99
0795
·2717
95·24
9295
·6900
93·90
8093
·8857
97·04
1095
·3205
95·29
7995
·7448
93·96
1093
·9386
Free vibration of semi-rigid connected Reddy–Bickford piles 795
Tabl
e3.
The
first
thre
ena
tura
lfre
quen
cies
ofR
eddy
–Bic
kfor
dan
dT
imos
henk
opi
les
who
seen
dat
the
first
regi
onis
sem
i-ri
gid
conn
ecte
dan
dno
tre
stri
cted
for
hori
zont
aldi
spla
cem
enta
ndth
een
dat
the
thir
dre
gion
isfr
ee,L
=30
m.
L=
30m
f=
0·25
L1/L
=0·5
0L
2/L
=0·3
0L
3/L
=0·2
0L
1/L
=0·5
0L
2/L
=0·2
0L
3/L
=0·3
0ωi
(rad/sec)N
r=
0·25
Nr=
1·00
Nr=
0·25
Nr=
1·00
TB
TT
BT
TB
TT
BT
TB
TT
BT
TB
TT
BT
RB
T(k
=5/
6)(k
=14
/17
)R
BT
(k=
5/6)
(k=
14/17
)R
BT
(k=
5/6)
(k=
14/17
)R
BT
(k=
5/6)
(k=
14/17
)
ω1
1·748
91·7
438
1·743
81·7
885
1·776
21·7
761
1·749
01·7
439
1·743
91·7
886
1·776
31·7
762
ω2
9·946
49·9
296
9·929
09·6
504
9·629
59·6
290
9·946
59·9
297
9·929
19·6
505
9·629
69·6
290
ω3
26·99
0326
·9513
26·94
8426
·6369
26·60
0226
·5973
26·99
0426
·9514
26·94
8526
·6370
26·60
0326
·5974
f=
0·75
L1/L
=0·5
0L
2/L
=0·3
0L
3/L
=0·2
0L
1/L
=0·5
0L
2/L
=0·2
0L
3/L
=0·3
0
Nr=
0·25
Nr=
1·00
Nr=
0·25
Nr=
1·00
ωi
(rad/sec)
TB
TT
BT
TB
TT
BT
TB
TT
BT
TB
TT
BT
RB
T(k
=5/
6)(k
=14
/17
)R
BT
(k=
5/6)
(k=
14/17
)R
BT
(k=
5/6)
(k=
14/17
)R
BT
(k=
5/6)
(k=
14/17
)
ω1
2·167
12·1
284
2·128
42·0
088
1·955
11·9
551
2·167
22·1
285
2·128
52·0
089
1·955
21·9
552
ω2
11·65
0911
·4670
11·46
6311
·3302
11·12
7311
·1265
11·65
1011
·4671
11·46
6411
·3303
11·12
7411
·1266
ω3
29·24
9828
·9423
28·93
9028
·8924
28·57
2328
·5690
29·24
9928
·9424
28·93
9128
·8925
28·57
2428
·5691
796 Yusuf Yesilce and Hikmet H Catal
For L = 15 m, the frequency values obtained for the first three modes of Reddy–Bickfordand Timoshenko piles whose end at the first region is semi-rigid connected and restricted forhorizontal displacement and the end at the third region is fixed supported are presented intable 4; for L = 30 m, the frequency values obtained for the first three modes of the same pilesare presented in table 5 being compared with the frequency values obtained for Nr = 0·25and 1·00, f = 0·25 and 0·75.
The first two natural frequency values of Reddy–Bickford pile are higher than the first twonatural frequency values of Timoshenko pile for both models and for all values of Nr , L andf . For both models, the third natural frequency values of Reddy–Bickford pile are lower forf = 0·25 and L = 15 m; are higher for f = 0·75 and L = 30 m, than the third naturalfrequency values of Timoshenko pile. The third natural frequency values of Reddy–Bickfordpile are higher than the third natural frequency values of Timoshenko pile for both models,L = 30 m and all values of f . The differences between Reddy–Bickford beam theory andTimoshenko beam theory are more prominent for the higher frequencies.
For all boundary conditions, the differences between natural frequency values of Timo-shenko pile models with k = 5
6 and k = 1417 are small.
For f = 0·25 and for the condition of the other variables (L, L1/L, L2/L and L3/L ratios)are constant, as the axial compressive force acting to Reddy–Bickford pile and Timoshenkopile whose end at the first region is semi-rigid connected and not restricted for horizontaldisplacement and the end at the third region is free, is increased, the first natural frequencyvalues of Reddy–Bickford pile and Timoshenko pile are increased but the second and the thirdnatural frequency values are decreased. The natural frequency values of Reddy–Bickford pileand Timoshenko pile are decreased as the axial force is increased for f = 0·25 in the secondmodel and for f = 0·75 at all boundary conditions. These results indicate that, the boundaryconditions (especially the end at the third region) of Reddy–Bickford pile and Timoshenkopile are important for the effect of axial force.
As the total length of the pile is increased, a decrease is observed in natural frequencyvalues for both beam theories; for all boundary conditions and the condition of the othervariables (f, Nr , L1/L, L2/L and L3/L ratios) are being constant. This result indicates thatthe increase in the length of the pile leads to a reduction in natural frequency values for bothbeam theories.
A decrease is observed in natural frequency values of the first three modes of the pile forthe condition of total pile length, Nr and L1/L ratio being constant and of L2/L ratio beinggreater than L3/L ratio, for both support conditions and both beam theories. This decrease ismore prominent for the short piles.
Natural frequencies values are different for two combinations of boundary conditions. Forthe condition of all variables are constant, the first three natural frequency values of the pilewhose end at the first region is semi-rigid connected and restricted for horizontal displacementand the end at the third region is fixed supported are higher than the first three natural frequencyvalues of the pile whose end at the first region is semi-rigid connected and not restricted forhorizontal displacement and the end at the third region is free. This result indicates that thetypes of supporting affect the natural frequency values of the pile.
6. Summary and conclusion
In this study, starting from the governing differential equations of motion in free vibration,transfer matrices are developed by using Reddy–Bickford beam theory and the iterative-
Free vibration of semi-rigid connected Reddy–Bickford piles 797
Tabl
e4.
The
first
thre
ena
tura
lfre
quen
cies
ofR
eddy
–Bic
kfor
dan
dT
imos
henk
opi
les
who
seen
dat
the
first
regi
onis
sem
i-ri
gid
conn
ecte
dan
dre
stri
cted
for
hori
zont
aldi
spla
cem
enta
ndth
een
dat
the
thir
dre
gion
isfix
edsu
ppor
ted,
L=
15m
.
L=
15m
f=
0·25
L1/L
=0·5
0L
2/L
=0·3
0L
3/L
=0·2
0L
1/L
=0·5
0L
2/L
=0·2
0L
3/L
=0·3
0ωi
(rad/sec)
Nr=
0·25
Nr=
1·00
Nr=
0·25
Nr=
1·00
TB
TT
BT
TB
TT
BT
TB
TT
BT
TB
TT
BT
RB
T(k
=5/
6)(k
=14
/17
)R
BT
(k=
5/6)
(k=
14/17
)R
BT
(k=
5/6)
(k=
14/17
)R
BT
(k=
5/6)
(k=
14/17
)
ω1
23·50
2723
·4191
23·41
5922
·0924
22·03
0622
·0275
23·52
3923
·4399
23·43
6622
·1163
22·05
2422
·0492
ω2
72·22
6172
·1754
72·15
8570
·8519
70·66
2570
·6458
72·27
5372
·2225
72·20
5570
·9039
70·71
2570
·6957
ω3
139·2
844
139·6
384
139·5
974
137·8
130
138·2
810
138·2
401
139·3
032
139·6
627
139·6
217
137·8
316
138·3
044
138·2
635
f=
0·75
L1/L
=0·5
0L
2/L
=0·3
0L
3/L
=0·2
0L
1/L
=0·5
0L
2/L
=0·2
0L
3/L
=0·3
0
Nr=
0·25
Nr=
1·00
Nr=
0·25
Nr=
1·00
ωi
(rad/sec)
TB
TT
BT
TB
TT
BT
TB
TT
BT
TB
TT
BT
RB
T(k
=5/
6)(k
=14
/17
)R
BT
(k=
5/6)
(k=
14/17
)R
BT
(k=
5/6)
(k=
14/17
)R
BT
(k=
5/6)
(k=
14/17
)
ω1
28·37
5227
·1491
27·14
4627
·2392
25·99
2125
·9877
28·39
9327
·1718
27·16
7327
·2642
26·01
5826
·0114
ω2
78·14
4176
·3510
76·33
1076
·6952
74·93
4574
·9148
78·19
3976
·3984
76·37
8376
·7481
74·98
4874
·9650
ω3
144·8
557
143·3
242
143·2
794
143·4
875
142·0
368
141·9
921
144·8
906
143·3
586
143·3
137
143·5
151
142·0
662
142·0
215
798 Yusuf Yesilce and Hikmet H Catal
Tabl
e5.
The
first
thre
ena
tura
lfr
eque
ncie
sof
Red
dy–B
ickf
ord
and
Tim
oshe
nko
pile
sw
hose
end
atth
efir
stre
gion
isse
mi-
rigi
dco
nnec
ted
and
rest
rict
edfo
rho
rizo
ntal
disp
lace
men
tand
the
end
atth
eth
ird
regi
onis
fixed
supp
orte
d,L
=30
m.
L=
30m
f=
0·25
L1/L
=0·5
0L
2/L
=0·3
0L
3/L
=0·2
0L
1/L
=0·5
0L
2/L
=0·2
0L
3/L
=0·3
0ωi
(rad/sec)N
r=
0·25
Nr=
1·00
Nr=
0·25
Nr=
1·00
TB
TT
BT
TB
TT
BT
TB
TT
BT
TB
TT
BT
RB
T(k
=5/
6)(k
=14
/17
)R
BT
(k=
5/6)
(k=
14/17
)R
BT
(k=
5/6)
(k=
14/17
)R
BT
(k=
5/6)
(k=
14/17
)
ω1
6·848
26·8
402
6·839
86·4
948
6·488
66·4
882
6·848
36·8
403
6·839
96·4
949
6·488
66·4
883
ω2
21·72
8321
·6986
21·69
6321
·3219
21·29
7521
·2952
21·72
8421
·6987
21·69
6421
·3220
21·29
7621
·2953
ω3
44·66
9444
·6216
44·61
4044
·2449
44·20
7544
·2001
44·66
9544
·6217
44·61
4144
·2450
44·20
7644
·2002
f=
0·75
L1/L
=0·5
0L
2/L
=0·3
0L
3/L
=0·2
0L
1/L
=0·5
0L
2/L
=0·2
0L
3/L
=0·3
0
Nr=
0·25
Nr=
1·00
Nr=
0·25
Nr=
1·00
ωi
(rad/sec)
TB
TT
BT
TB
TT
BT
TB
TT
BT
TB
TT
BT
RB
T(k
=5/
6)(k
=14
/17
)R
BT
(k=
5/6)
(k=
14/17
)R
BT
(k=
5/6)
(k=
14/17
)R
BT
(k=
5/6)
(k=
14/17
)
ω1
8·065
77·9
103
7·909
87·7
722
7·613
77·6
132
8·065
87·9
104
7·909
97·7
723
7·613
87·6
133
ω2
23·23
0422
·9722
22·96
9622
·8494
22·59
3522
·5909
23·23
0522
·9723
22·96
9722
·8495
22·59
3622
·5910
ω3
46·32
9145
·9953
45·98
7145
·9199
45·59
4545
·5864
46·32
9245
·9954
45·98
7245
·9200
45·59
4645
·5865
Free vibration of semi-rigid connected Reddy–Bickford piles 799
based computer programs are developed for solution of linear-homogeneous frequency equa-tion set relating to free vibration of different supported two piles partially embedded inelastic soil. Variation in free vibration natural frequencies for the first three modes of thepile is investigated for Nr = 0·25 and Nr = 1·0, f = 0·25 and f = 0·75, due to sup-porting conditions of pile ends and different lengths of the pile. Natural frequency val-ues obtained from Reddy–Bickford beam theory are compared with the results of Tim-oshenko beam theory. As shown, the differences between Reddy–Bickford beam theoryand Timoshenko beam theory become more prominent for free vibration of piles embed-ded in elastic soil. So that, to be on safer side it is recommended to use the higher-ordertheory.
Notation
The following symbols are used in this paper.
A cross-section area of the pileb width of the pileCR1 modulus of subgrade reaction for the second regionCR2 modulus of subgrade reaction for the third regionCθ rotational restraining stiffness at the upper end of the pile in the first regionE Young’s modulusf fixity factor[F(z = 1)] global transfer matrixFjm terms of the global transfer matrixG shear modulush height of the pileIx moment of inertiak shape factor due to cross-section geometryL total length of the pileL1 pile length above the soilL2 pile length embedded in the second regionL3 pile length embedded in the third regionm mass per unit length of the pileMh(zj , t) bending moment function for j th regionMhj(zj , t) higher-order moment function for j th regionN axial compressive forceNr non-dimensionalized multiplication factor for the axial compressive forceQj(zj , t) shear force function for j th region{Sj (zj , t)} position vector for j th regionz dimensionless position parametert time variablewj(zj , t) transverse displacement function for j th regionw0(x, t) lateral displacement of the beam neutral axisw′
j (zj , t) bending rotation function for j th regionxj position for j th regionφj (zj , t) rotation of a normal to the axis of the pile for j th regionω natural frequency
800 Yusuf Yesilce and Hikmet H Catal
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