Gomputers& Structures

PERGAMON Computers and Structures 69 (1998) 329-338

Investigation of stresses at the fixed end of deep cantileverbeams

S.R. Ahmed, M.R. Khan, K.M.S. Islam, Md.W. Uddin *

Department of'Mechanical Engineering, Bangladesh University of Engineering and Technology, Dhaka-1000, Bangladesh

Received 22 Janrary 1997; accepted 28 April 1998

Abstract

A numerical investigation for the stresses and displacements of a two-dimensional elastic problem withmixed boundary conditions is reported in this paper. Specifically, it is on the analysis of stresses at the fixedend of deep cantilever beams, subjected to uniformly distributed shear at the free end. An ideal rnathematicalmodel, based on a displacement potential function, has been used to formulate the problem. The solutions arepresented in the form of graphs. Results are compared with the elementary solutions and the discrepancyappears to be quite noticeable, specifically at the fixed end. The present solution shows that the fixed end ofa short cantilever beam is an extremely critical zone and the elementary theory of beams completely fails topredict stresses in this zore. e 1998 Published by Elsevier Science Ltd. All rights reserved.

1. Notation

x , yE1)

o, t

l ,

ox

oy

o,..,

a

bhkRmn0{t

rectangular coordinateselastic modulus of the rnaterialPoisson's ratiostressdisplacement component in the x-directiondisplacement component in the y-directionnormal stress component in the x-directionbending stressshearing stressbeam lengthbeam depthmesh length in the x-directionmesh length in the y.directionratio of the mesh lengths klhnumber of mesh points in x-directionnumber of mesh points in y-directionAiry's stress functiondisplacement potential function.

2. Introduction

The elementary theories of strength df materials areunable to predict the stresses in the critical zones of en-gineering structures. They are very inadequate to giveinformation regarding local stresses near the loads andnear the supports of the beam. They are only approxi-mately correct in some cases but most of the time, vio-late conditions which are brought to light by the morerefined investigation of the theory of elasticity.

Among the existing mathematical models for twodimensional boundary-value stress problems, the twodisplacement function approach [1] and the stress func-tion approach [9] are noticeable. The solution of prac-tical problems started mainly after the introduction ofAiry's stress function [9]. But the difficulties involvedin trying to solve practical problems using the stressfunction are pointed out by Uddin [1] and also byDurelli [2]. The shortcoming of @-formulation [9] isthat it accepts boundary conditions in terms of loadingoniy. Boundary restraints specified in terms of u and vcan not be satisfactorily imposed on the stress function

d. As most of the problems of elasticity are of mixedboundary conditions, this approach fails to provideany explicit understanding of the stress distribution in* Corresponding author.

0045-7949198/$ - see front matter @ 1998 Published by Elsevier Science Ltd. All rights reserved.P I I : S 0 0 4 5 - 7 9 4 9 ( 9 8 ) 0 0 1 2 7 - 8

330 S.R. Ahmed et al.l Computers and Structures 69 (1998) 329-338

the region of restrained boundaries which are, in gen-

eral, the most critical zones in terms of stress. Again,

the two displacement function approach that rs the u,

v-formulation involves flnding two functions simul-

taneously from two second order elliptic partial differ-

ential equations [1]. But the simultaneous evaluation of

two functions, satisfying two simultaneous differential

equations, is extremely difficult and this problem

becomes more serious when the boundary conditions

are specified as a mixture of restraints and stresses. As

a result, serious attempts had hardly been made in the

stress analysis of elastic bodies using this formulation

as far as present literature is concerned.

Although elasticity problems were formulated long

before, exact solutions of practical problems are hardly

available because of the inability of managing the

boundary conditions imposed on them. The age-old S-

Venant's principle is still applied and its merit is evalu-

ated in solving problems of solid mechanics [3,4] in

which full boundary effects could not be taken into

account satisfactorily. Actually, management of bound-

ary conditions and boundary shapes are the main ob-

stacles to the solution of practical problems. The

reason for the birth and dominance of the finite el-

ement method is merely its superiority in managing the

boundary conditions. In circumventing this problem,

Dow, Jones and Harwood [5] have introduced a new

boundary modeling approach for finite-difference appli-

cations of displacement formulation of solid mechanics

and solved the problem of a uniformly loaded cantile-

ver beam. In this connection, they reported that the ac-

curacy of the finite difference method in reproducing

the state of stresses along the boundary was much

higher than that of finite element analysis. However,

they have noted that the computational effort of the

finite difference analysis, under the new boundary mod-

eling, is even somewhat greater than that of finite el-

ement analysis. Even now? photoelastic studies are

being carried out for classical problems like uniformly

loaded beams on two supports [2,6], only because

boundary effects could not be fully taken into account

in their analytical method of solutions.

As stated above, neither of the formulations is suit-

able for solving problems of mixed boundary con-

ditions and hence an ideal mathematical model is used

here. In this numerical approach, the problem has

been formulated in terms of a single potential function,

rll U,rc} defined in terms of displacement components,

and is considered as parallel to the stress function @since both of them have to satisfy the same bi-harmo-

nic equation.

3. Formulation of the protrlem

Fortunately, almost all the practical problems of

stress analysis can easily be resolved into two-dimen-

sional problems. A large number of these practical

problems of elasticity are covered by one of the two

simplifying assumptions, namely, either of plane stress

or of plane strain. With reference to a rcctangular

coordinate system, the three governing equations in

terms of the stress variables o", 6y, and o*, for plane

stress and plane strain problems are given by

G+.#)r.r o,): oIf we replace the stress functions in Eqs. (1)-(3) by dis-

placement functions u (x, y) and v (r, y), which are re-

lated to stress functions through the expressions

0o,. 0o",,; - + - ; - : 0 ,ox dy

0o " Eo,.u; -+ - ;= :U ,0y dx

E fau Evlo , : r _ v 2 L * * ' d ,

E fSu , 0r16 t : 1 - r , z l r r * ' * ) ,

E f\u Evlo t !

4 + v ) L O " * l '

then Eq. (3) is redundant and Eqs.

form to

( 1 )

(2)

(3)

(4)

(5)

' 6 )

(1) and (2) trans-

02u , ( l - v \02u , / l + v \ 02v .a * + ( , / a F + ( , / * u r : o '

( t )

02v l l - v\ 02v / l + v\ E2u_ + t _ t _ + t t _ - 0 ( 8 )0 y 2 ' \ 2 ) a x z ' y 2 ) s x s y

where u and v are the displacement components of a

point in the x and y-directions, respectively. The equili-

brium Eqs. (7) and (8) have to be solved now for the

case of a two-dimensional problem when the body

forces are assumed to be absent.

In the present approach, the problem is reduced to

the determination of a single variable instead of evalu-

ating two functions, u and v, simultaneously, from

equilibrium Eqs. (7) and (8). In this case, as in the case

of Airy's stress function S, a potential function r! (x,

y) is defined in terms of displacement components as

S.R. Ahmed et al.I Computers and Structures 69 (1998) 329-338 J J I

n ) t

d-vlu : ; - - ; ,

oxoy

1 l -v : - l ( 1 - u )r - l - v L

When the displacement components in the Eqs. (l)and (8) are replaced by their expressions in terms of ry'(r, y), as defined above, Eq. (7) is automatically satis-fied and the only condition that ,lt has to satisfybecomes [1]

A 4 r l t , ^ A o , l , , A 4 l / n* f z-'----:----= f ;j? : 0. (9)dx- dx'dv' dv-

Therefore, the whole problem has been formulated insuch a way that a single function r/ has to be evaluatedfrom the bi-harmonic Eq. (9) associated with theboundary conditions that are specified at the boundingedges of the beam.

4. Boundary conditions

The practical problems in elasticity are normally ofthe boundary-value type where the conditions that areimposed on the boundary of the elastic body are visu-alized either in terms of edge-flxity or edge-loading,that is, known values of displacements and stresses atthe boundary. Referring to Fig. 1, which illustrates thepresent problem of cantilever beam, both the top andbottom edges are free from loading, the left lateraledge is fixed and the right lateral edge is subjected touniformly distributed shear.

For both the top and bottom edges, AB and CD,the normal and tangential stress components, statedmathematically, are given by

o,(x, y) :0, and

6*y(x , ! ) :0 , for 0 < y /a < l , x fb :0 and 1.

For the left lateral edge, AC, the normal and tangen-tial displacement components are, respectively,

B

T:,r ET Ht ,lir B

u(x, y) - 0, and

v ( x , y ) - - 0 , f o r 0 < x l b < 1 , y f a = 0 ,

and the corresponding boundary conditions for the

::t$rf"tal edge, BD, the normal and tangential stres-

or(x, Y) :0,

o ,y(x , y ) /E :3 .0 x l0-a for 0 < x /b < I , y f a : l .

In order to solve the problem using Eq. (9), the bound-ary conditions are also needed to be expressed in termsof {/ and thus the corresponding relations betweenknown functions on the boundary and the function ry'are,

a2lru ( x , y ) : - , ( 10 )

0xov

, \ | f , , ,A2rl, , ^A2rlr1v ( x , y ) : - ; : l t t - v ) ; + + 2 ; + 1 , ( 1 1 )l * u f ' ' d Y ' d x ' l

E f A3rl/ A3{/1o , ( x , Y ) : : l ' - v ' l ( 1 2 )"r\""' r ' ' ( l + v)2 lax2ay

' oy3 l '

E la3l./ .^ , a3l/ l .^.o , I x . v \ - - ^ l ^ - + Q * v \ = - ^ : - | . ( 1 3 )/ ' 'J ' ( l + v)' YaY' dxzdY)

E f a3{r a3{/16 " u ( x , y ) : . I v -" ̂ Y\"" '/ '

( I + v)2 L' a*6r, ax3l '

As far as numerical computation is conce.rned, it is evi-dent from the expressions of above conditions that allthe boundary conditions of interest can easily be dis-cretized in terms of the displacement function r/ by themethod of finite-difference.

5. Solution procedure

The essential feature of the numerical approach hereis that the original governing differential equations ofthe boundary-value problems are replaced by a flniteset of simultaneous algebraic equations and the sol-utions of this set of simultaneous algebraic equationsprovide us with an approximation for the displacementand stress within the solid body. About the solutionthrough the proposed formulation, attention may bedrawn to the points described below.

5.1. Method of solution

The limitation and complexity associated with ana-lytical solutions [7] leads to the conclusion that a nu-merical modeling for this class of problem is the only

A',]t , "A',1t1a y ' - ' a f l '

x

Fig. l. Deep cantilever beam subjected to uniformly distribu-ted shear at the free end.

t a a) J Z S.R. Ahmed et al.lComputers and Structures 69 (1998) 329*338

plausible approach. The finite-difference technique, oneof the oldest numerical methods extensively used forsolving differential equations, is used here to transformthe fourth order bi-harmonic partial differential Eq. (9)

and also the partial differential Eqs. (i0)-(14), associ-ated with the boundary conditions, into their corre-sponding algebraic equations. The discrete values ofthe potential function, rL (x, y), at the mesh points ofthe domain concerned (Fig. 2) are obtained from a sys-tem of linear algebraic equations resulting from thediscretization of the governing equation and the pre-

scribed boundary conditions.

The region in which the dependent function is to beevaluated is divided into a desirable number of meshpoints and the values of the function r! are soughtonly at these points. To keep the order of the error ofthe difference equations of the boundary conditions toa minimum, a new false boundary, exterior to thephysical domain, is introduced. The discretizationscheme for the domain concerned is illustrated inFig. 2. The division into mesh points can be done inany regular or irregular manner, but considering therectangular shape of the boundary and also the natureof the differential equations involved, rectangular gridpoints are used all over the region concerned. The gov-

erning bi-harmonic equation which is used to evaluatethe function only at the internal mesh points isexpressed in its corresponding difference equationusing central difference operators. When all the deriva-tives present in the bi-harmonic equation are replacedby their respective central difference formulae, thecomplete finite-difference equation for bi-harmonicEq. (9) becomes

R4 u/ (i - 2, i) + {/ (i + 2, i)} - 4 R\(r + R\llr (i - 1, i)+ {/( i* 1,,r)} - 4(I + R\U/( i , j+ 1) + l /( i , j - l )}

+ (6R4 + 8R2 + qVQ,) + 2R2{ILQ - t,j - l)+ { / ( i - l , j+ 1) + l / ( i + l , j - 1)+ l / ( i+ l , j + i ) ) + { r ( i , j -2 )+ { / ( i , j *2 ) : g (15 )

where R: k lh .

Considering an interior mesh point O (i, j), it is seenthat the algebraic Eq. (15) contains the discretizedvariable of the 13 neighboring mesh points, and whenO becomes an immediate neighbor of the physical

boundary mesh points, this equation will contain meshpoints exterior to the boundary as well as on theboundary itself (Fig. 2). Thus, the application of thecentral difference expression of the bi-harmonicequation to the points in the immediate neighborhoodof the physical boundary will cause no difficulties,provided an imaginary false boundary exterior to thephysical boundary is introduced.

, PhYsical Boundary

Fig. 2. Rectangular mesh-network of the domain in relationto the coordinates system and the finite-difference discretiza-tion of the bi-harmonic equation at an arbitrary internal meshpoint.

5.2. Management oJ boundary conditions

Normally, the boundary conditions are specifiedeither in terms of loadings or of restraints or of somecombination of the two. Each mesh point on the physi-cal boundary of the domain always entertains twoboundary conditions at a time out of four possible,namely, (1) normal stress and shear stress; (2) normalstress and tangential displacement; (3) shear stress andnormal displacement; and (4) normal displacement andtangential displacement. The computer program isorganized here in such a fashion that, out of these twoconditions, one is used for evaluation of r! ay the con-cerned boundary point and the other one for the corre-sponding point on the exterior false boundary. Thus,when the boundary conditions are expressed by theirappropriate difference equations, every mesh point ofthe domain will have a single linear algebraic equation.Table I lists the boundary conditions for each bound-ary of the beam along with the corresponding choiceof mesh points on the boundary.

As the differential equations associated with theboundary conditions contain second and third orderderivatives of the function r!, the application of thecentral difference expression is not practical as, mostof the time, it leads to the inclusion of the points ex-terior to the false boundary. The derivatives of theboundary expressions are thus replaced by their threepoint backward or forward difference formulae, keep-ing the order of the local truncation error the same.Two different sets of boundary expressions are usedfor each boundary, one set for the first half of the edgeand the other set for the second half. For example, thefinite-difference expressions for the normal and tangen-tial components of stress on the top boundary, AB, atpoints closer to A, are given by:

S.R. Ahmed et al.l Computers and Structures 69 (1998) 329-338

Table 1Specification of the boundary conditions in relation to corresponding mesh-points on the boundary

Correspondence betweenmesh-points and given boundary conditions

Boundary Given boundary conditions Condition/mesh-point Condition/mesh-point

Top, ABBottom, CDLeft, ACRight, BD

6x, oxy

6x, oxy

U t l

o , , - 6 - , ,

6-lQi)o,l(m- lj)ul(i,2)o,rl(i,n - I)

6'ylQi)6*yl(mi)vl(i, l)orl(i,n)

o*(Z,i): sfio,,6t(+ - s)r,rz,i>

+ t.sv(z,i- 1) + (, - +) trQ,i + t)

. (+ - z)'t'tz'i + 2) + o's,'l(2'i + 3)

-t+t{,(r,i) + v(3,i)l

.+{{/(r,i+ l) + {/(3,i + r)}

- f itt,o,i +z) +l,e,i*,)]], (16)

6,v(2, i) : O# oz wl* ve, i) +

. (+ -+)l rz,it* (t-

+ fr*rs, il - r.s{{/(2, i - |+ 2{{/(3,j - 1) + {/(3,j + I)

+ l ,(4, j* 1)]] .

The discretization scheme using the neighboring grid-points as required for expressing the above conditionson the top boundary of the beam, AB, is illustrated inFig. 3. Special treatments are also adopted for the cor-ner mesh points which are generally points of 'tran-

sition' in the boundary conditions. Referring to Fig. 4,assuming ,B as the corner mesh point, it is seen that Bis a common point of both the edges AB and BD andthus it has four boundary conditions-two from eachedge. In solving the present beam problem, three con-ditions out of the four are used, the remaining one istreated as redundant. The three conditions mentionedabove are organized in such a way that the values of ry'at three points, namely, I, B, and 2 are evaluated fromthese equations-points 1 and B from the boundaryconditions coming from edge AB and point 2 from the

boundary equation from edge BD. Table 2 shows thechoices and the conditions to be satisfied by the cornermesh points in relation to the present cantilever beamproblem. An example of the finite-difference drscretiza-tion used to evaluate the corner mesh point 2 is shownin Fig. 4 and the corresponding difference equation is

as follows:

Q,.il lmaginary Boundary

+ ii = 2i = 3a - At - a

+ i

i : 3

i = 4i - 5

i = 6

(b)

Fig. 3. Grid-points for expressing the boundary conditions on

the top edge at points closer to A, (a) for normal stress com-ponent, o, (b) for tangential stress component, oxl.

( , - 1 ( \ t e . i \\ v / ' '

" '

1(\r,0. nv ) ' '

) + r l t ( 2 , j + l ) I

I - 0 . s { { / ( 4 , j - I )

(17)

j+4j+3

+ i-2 i-t i i+l

i(a)

I i-z i-t i i+lVi

S.R. Ahmed et al.I Computers and Structures 69 (1998) 329*338

Fig. 4. Grid-points used in expressing the boundary conditiono, for mesh point 2.

o r ( 2 , n - l ) : . . , ! : ; . ; l t . 5 V ( 2 , n ) - { 5 + 3 R 2 ( 2 + y ) }' ( l * v) 'Rtht

x V(2, n - I)+ {6 + 4R2Q * v)}rl(2, n - 2)- { 3 + R 2 ( 2 + v ) } l / Q , n - 3 )

+ 0.5{/(2, n - 4) + 1.5R2(2+ vX/( l , n - l )

+ l / (3, n - l ) j - 2R2Q+ v){( /1, n - 2)

+ l / (3, n - 2) l + 0.5R2(2 + vX/( l , n - 3)

+{ / (3 ,n -3 ) l l . ( 1 8 )

The details of the computational molecules developedfor the finite difference operators are available inRef. [8].

5.3. Solution of the system of algebraic equations

There are numerous existing methods of solving asystem of algebraic equations. In the present problem,the number of unknowns in the system of equations isextremely large but only a few in each individualequation. Under this condition, the iterative methodmay be preferable. But the problem of solving thedifference equations by the iterative method has certainshortcomings. Although this method works very well

Table 2Management of boundary conditions at the corner mesh-points

for certain boundary conditions, it fails to produce anysolution for other complex boundary conditions. Incertain cases, the rate of convergence of iteration isextremely slow, which makes it impractical. As thisiterative method has the limitation of not always con-verging to a solution and sometimes converging butvery slowly, the authors have thus used a triangulardecomposition method ensuring better reliability andbetter accuracy of solution in a shorter period of time.The matrix decomposition method, used here, solvesthe present system of equations directly. Finally, thesame difference equations as those of the boundaryconditions are organized for the evaluation of displace-ment and stress components at different sections of thebeam from the known values of rlr.

6. Results and discussion

Numerical solution with mixed and variable bound-ary conditions has rarely been attempted as the bound-ary conditions of these practical elastic problems poseserious difficulty in their solutions. This problem hasbeen satisfactorily tackled by present formulation. Allthe solutions of interest obtained through the ry'-formu-lation conform to the symmetric and anti-symmetriccharacteristics of the problem and also to the famousS-Venant's principle that the effects of sharp variationof a parameter on the boundary die down and becomeuniform with the increase of distance of points in thebody from the boundary.

In obtaining numerical values for the present pro-blem, the beam as the elastic body is assurrlbd to bemade of ordinary steel (v : 0.3, E : 200 GPa).Graphs are plotted at different constant values of x forvarying y as well as at constant y for varying x for theparameters of interest. Moreover, the effect of albratio on the relevant displacement and stress com-ponents is explicitly illustrated here. In order to makethe results non-dimensional, the displacements are

Physical Boundary

i : li : 1

t = 5

i : 4

Vi

\(2, n)

+ i2

n-5 n-4 n-3 n-2 n-l l=n

Cornerpoint

Possibleboundaryconditions Conditions used

Correspondingmesh-points forevaluation of ry'

Top left, A

Top right, B

Bottom left, C

Bottom right, D

lor, o*y) on AB

[u, v] on AC

lo* o,yf on AB

loy, 6,y] on BD

lo", o*yl on CD

lu, vf on AC

for,orr) on CD

lo, oryT on BD

lo,o*r,v]

fo"ro*rroy)

fo *,o *r,v)

fo ",o ryo r)

(2,2) , (1,2) , (2,1)

(2,n - l ) , ( l ,n - l ) , (2,n)

(m - 1,2) , (m,2), (m - l , l )

(m - l ,n - l ) , (m,n - I ) ,(m - I ,n)

S.R. Ahmed et al.l Computers and Structures 69 (1998) 329-338 335

0.05

5 0.04

s o.o3

o.o2

0.01

00 0.5 't

t l a

Fig. 5. Distribution of the displacement component u alongthe neutral axis of deeo cantilever beams.

expressed as the ratio of actual displacement to thedepth of the beam and the stresses as the ratio of theactual stress to the elastic modulus of the beam ma-terial.

Fig. 5 shows the distribution of displacement com-ponent u along the neutral axis of the beam. It isobserved to be nonlinear in nature and identical withthe elementary solution having third order polynomial-like behavior. The general trend of the curve revealsthat the displacement is zero at the fixed end andmaximum at the free end of the cantilever beam whichis in complete conformity with the loading as well aswith the end conditions. The effect of the af b ratio onthe distribution of u along the neutral axis is also illus-trated in the same figure. It conforms to the fact that,at a lower af b ratio, the end-effects become very pro-

0 0 . 5 1y l a

Fig. 6. Distribution of the displacement component y at var-ious longitudinal sections of the cantilever beams (alb : 2).

0 0 . 5 Lt l a

Fig. 7. Distribution of the displacement component v alongthe top boundary (rlb : 0) of deep cantilever beams.

minent and provide restriction to the deflection of thebeam.

From the distribution of the displacement com-ponent v with respect to y in Fig. 6, it is seen that thisdisplacement at the lree end for a particular alb rutiois maximum at the top and bottom fibers, but zeroover the whole depth at the fixed end and all along theneutral axis, which is fully in conformity with thephysical model of the problem. The distribution iscompletely asymmetric about the neutral axis of thebeam, which conforms to the assumption of the el-ementary theory of beam that plane sqctions remainplane during the bending of beams. l'

The distribution of the displacement component y

over the span is presented in Fig. 7 describing theeffect of the af b ratio on the distribution at the top

0.0015

0.001

5x10-a

0

-5x10-a

-0.001

-0.00150 0.5

x l b

Fig. 8. Distribution of normal stress component or at varioustransverse sections of a deep cantilever beam (alb : 2.5).

tS

\

tq

s

+4lb=1.0"# 2.0+ 3.0---€- 3.5

+xlb=0.00--tr- 0.10r'--.F 0.25-.+ 0.50

+y/a=0.00"-o- 0.05----r 0.10+ 0.50

0.90+ 1.00

+a/b=1.0+ 2.0# 3.0-# 3.5

336 S.R. Ahnted et al.l Computers and Structures 69 (1998) 329-338

lrl\ " 0l.$

-0.001

-0.0020 0 . 5 1

xl b

Fig. 9. Distribution of the normal stress component dr at thefixed end of deep cantilever beams.

edge of the beam. Displacement in the direction of /also increases substantially towards the free end as thebeam becomes longer.

Fig. 8 shows the distribution of normal stress com-ponent o, with respect to x at various transverse sec-tions of the beam. From the distribution, it is observedthat the variation is sinusoidal in nature and the fixededge is the most critical section of the beam as far asthe normal stress is concerned. The effect of the albratio on the distribution of o, at the fixed end of thebeam is illustrated in Fig. 9. As appears from thegraph, stresses increase with an increasing af b ratio forthe same loading. It may be concluded from the distri-bution that the most critical point at the fixed endwith respect to o, is around xlb:0.1 and 0.9 in eachof the beams.

0.012

0.006

-0.006

-0.012

Fig. 10. Distribution of bending stress on at the flxed end overthe depth of deep cantilever beams.

0 0.5x l b

Fig. I l. Distribution of bending stress o' oververse sections of a deep cantilever beam (alb :

1

various trans-2.s).

Fr

s

Fig. 10 shows the variation of bending stress o, atthe restrained boundary, showing the effect of the albratio on the distribution. Stresses are maximum atboth the top and bottom fibers with zero value at themid-section which makes the distribution asymmetricabout the longitudinal mid-section of the beam. Itshould be noted here that, for a higher af b ratio, themagnitude of o, at the top fiber is higher than that atthe bottom fiber. But, in cases of elementary solution,this magnitude is exactly the same for both the topand bottom fibers of the beam. Again, this variation ofbending stress along the depth is analyzed for a par-ticular beam (olb :2.5) mainly to compare..how theelementary solutions match with that of exacd solutionsobtained through this numerical approach. In the el-ementary solution, the distribution of normal stresscomponent varies linearly with depth everywhere and

0.004

tq o.oo3

h

61 0.002

0.001

0

-0.0010 0.5

x l b

Fig. 12. Distribution of shearing stressover the depth of deep cantilever beams.

1

o", al the fixed end

0.5x l b

+yla=0.00.---.o- 0.05+ 0.25+ 0.50

0.7s0.90

* 1.00

S.R. Ahmed et al. I Computers and Structures 69 ( 1998) 329-338 J J I

+yla=0.00* 0.05.-.---Cr- 0.10# 0.50

0.90+ 1.oo

0 0 . 5 1x l b

Fig. 13. Distribution of shearing stress o,, over various trans-verse sections of a deep cantilever beam (alb : 2.5).

the magnitude is maximum at the top and bottomfibers. As appears from Fig. 11, the solutions differfrom that of the elementary solution in a sense thatthe distribution is far from linear, especially, at aroundthe fixed end, and it remains linear for other sectionsof the beam which, of course, conforms to the famousS-Venant's principle.

Distribution of shearing stress, oxy at the fixed endof the cantilever beams (Fig. 12) reveals that shearingstress is zero at both the top and bottom edges whichconforms to the obvious fact that both the top andbottom edges of the physical model are free fromshearing stresses. The distribution describing the effectof the alb ratio on shearing stress along the restrainedboundaries, shows that the stress varies nonlinearly,having the maximum values near the top and bottomcorners of the fixed edge and minimum at the mid sec-tions which disagrees completely with the elementarysolutions. Also, the beam becomes more critical interms of shearing stresses when the length of the beamis increased, keeping the loading constant.Interestingly, for this particular type of loading, at ahigher af b ratio, the upper corner zone at the fixed endbecomes more critical in terms of the stresses than thelower zone.'Finally, the variation of shearing stresses over the

depth is investigated at various transverse sections ofthe beam mainly for comparing its characteristic beha-vior with the elementary solutions. From the distri-bution in Fig. 13, it is observed that the variation ofthis stress component over the depth is simildr to thatof elementary solutions at the mid-sections of thebeam. Sufficiently away from the boundary, the distri-butions are parabolic in nature and they are identicalin nature and magnitude with that of elementary sol-

utions. From the elementary solution it is observedthat the magnitude of the shearing stresses are maxi-mum at the mid-section of the beam. This is notagreed upon by our numerical solutions and it differsmainly around the fixed ends as predicted by the el-ementary theory; it is maximum at about xlb : 0.05and 0.95. Since, in the elementary formulas of strengthof materials, the boundary conditions are satisfied inan approximate way, it fails to provide the actual dis-tribution of stresses at the boundaries, especially, atthe restrained boundaries. The present ry'-formulationis free from this type of shortcoming and is thuscapable of providing the actual stress distribution atany critical section, either at or far from the restrainededges.

7. Conclusions

Earlier mathematical models of elasticity were verydeficient in handling the practical problems. No appro-priate approach was available in the literature whichcould provide the explicit information about the actualdistribution of stresses at the critical regions ofrestrained boundaries satisfactorily. The distinguishingfeature of the present ry'-formulation over the existingapproaches is that, here, all modes of boundary con-ditions can be satisfied exactly, whether they are speci-fied in terms of loading or physical restraints or anycombination of them and thus the solutions obtainedare promising and satisfactory for the entire region ofinterest.

Both the qualitative and quantitative *esults of deepcantilever beams, obtained through the ry'-formulation,establish the soundness and appropriateness of the pre-sent approach. The comparative study with elementarysolutions verifies that the elementary solutions arehighly approximate as they fail to provide the sol-utions in the neighborhood of restrained boundaries.

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