1998 Investigationo f Stressesa t the Fixed End of Deep Cantilever Beam

8/9/2019 1998 Investigationo f Stressesa t the Fixed End of Deep Cantilever Beam

1/10

Gomputers

& Structures

PERGAMON

Computers nd Structures

9

(1998)

329-338

Investigationof stresses t the fixed end of deep cantilever

beams

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

with

mixed boundary conditions is reported in this

paper.

Specifically, it is on the analysis of stresses at the fixed

end of deep cantilever beams, subjected to

uniformly

distributed

shear at

the

free end. An ideal rnathematical

model, based

on a displacement

potential

function, has been used to formulate the

problem.

The solutions

are

presented

in the form of

graphs.

Results

are compared with the elementary solutions and the discrepancy

appears to be

quite

noticeable, specifically

at

the

fixed end. The

present

solution shows that the fixed end

of

a short cantilever beam is an extremely critical zone

and

the

elementary theory of beams completely fails to

predict

stresses

n this zore.

e

1998

Published by Elsevier Science Ltd. All rights reserved.

1. Notation

x , y

E

1)

o

,t

l,

ox

oy

o,..,

a

b

h

k

R

m

n

0

{t

rectangular coordinates

elastic modulus of the rnaterial

Poisson's atio

stress

displacement component in the x-direction

displacement component in the

y-direction

normal stresscomponent in the x-direction

bending stress

shearing stress

beam length

beam depth

mesh length in the

x-direction

mesh length in the

y.direction

ratio of the mesh lengths klh

number of mesh points in x-direction

number of mesh

points

in

y-direction

Airy's stress unction

displacement

potential

function.

2. Introduction

The elementary theories of strength df materials are

unable

to

predict

the

stresses

n

the critical zones of en-

gineering structures. They are very inadequate to give

information regarding local stressesnear

the

loads

and

near the supports of the beam. They are

only approxi-

mately correct in some casesbut most

of the time, vio-

late conditions which are brought to light by the

more

refined investigation

of

the theory

of elasticity.

Among the existing mathematical

models for two

dimensional boundary-value

stress

problems,

the two

displacement function approach

[1]

and

the

stress

unc-

tion

approach

[9]

are noticeable. The solution of

prac-

tical

problems

started mainly

after

the

introduction of

Airy's

stress

function

[9].

But the

difficulties involved

in trying to solve

practical problems

using the stress

function are

pointed

out by Uddin [1] and also by

Durelli

[2].

The

shortcoming of

@-formulation [9]

is

that it accepts

boundary conditions

in

terms of loading

oniy. Boundary restraints specified n terms

of u and v

can not be satisfactorily imposed

on

the

stress

unction

d.

As most of the

problems

of elasticity are

of

mixed

boundary conditions, this

approach

fails to

provide

any explicit understanding of the stress distribution

in

Corresponding author.

0045-7949198/$see

ront

matter

@

1998Published

y ElsevierScience td.

All

rights eserved.

P I I :

S 0 0 4 5

7

9 4 9 ( 9 8 ) 0 02 7 - 8


2/10

0

S.R.

Ahmed

et

al.l

Computers and Structures 69

(1998)

329-338

restrained boundaries which are, in

gen-

most critical zones in terms

of stress.

Again,

displacement function approach that rs the

u,

from two second order elliptic

partial

differ-

[1].

But

the simultaneousevaluation of

problem

more serious when the boundary

conditions

mixture

of

restraints and

stresses.

As

serious

attempts had hardly been made in the

analysis of elastic bodies using

this formulation

far as

present

iterature is concerned.

Although elasticity

problems

were formulated long

solutions of

practical problems

are hardly

se of the inability of managing the

on them. The age-old

S-

principle

is still applied and its merit is evalu-

problems

of solid mechanics

[3,4]

in

full boundary effects could not be taken into

satisfactorily.

Actually, management of bound-

conditions and boundary shapes are the main ob-

to the solution

of

practical problems.

The

for the birth and dominance of the finite

el-

method is merely its superiority in managing the

problem,

Jones and Harwood

[5]

have introduced a new

modeling

approach

for finite-difference

appli-

of displacement

formulation

of solid

mechanics

solved the

problem

of a uniformly loaded cantile-

In this

connection,

they reported that

the ac-

the finite difference method in reproducing

stresses along the boundary was much

of

finite

element analysis. However,

noted

that the

computational effort of the

analysis, under

the new

boundary mod-

somewhat

greater

than that

of finite el-

photoelastic

studies are

problems

like uniformly

beams

on two

supports

[2,6],

only because

could not be fully taken into account

analytical method

of solutions.

As

stated above, neither of the formulations is

suit-

solving

problems

of mixed boundary con-

and hence an ideal mathematical model is used

approach, the

problem

has

formulated in terms

of a single

potential

function,

U,rc}

defined n terms of displacementcomponents,

as

parallel

to the stress function

@

both of them have to

satisfy

the

same bi-harmo-

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

,):

If

we

replace the

stress

unctions

in

Eqs.

(1)-(3)

by dis-

placement

functions u

(x,

y)

and v

(r, y),

which are re-

lated to stress unctions through the expressions

0o,. 0o",,

; - + - ; - : 0 ,

ox dy

0o

"

Eo,.u

; -+ - ;= :U ,

0y dx

E

fau

Ev l

o , :

r _ v 2 L * * ' d ,

E

fSu

0r1

6 t : 1 - r , z l r r * ' * ) ,

E

f\u

Ev l

o 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 o 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 thi s 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


3/10

S.R. Ahmed et al.I Computers and Structures 69

(1998)

329-338

J J I

n ) t

d-vl

u

: ; - - ; ,

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 n terms

of

ry'

(r, y), as defined above, Eq. (7) is automatically satis-

fied and the only condition

that ,lt has

to satisfy

becomes

1]

A 4 r l t , ^ A o , l ,

, A 4 l /

n

*

f

z-'----:----=

;j?

:

0.

(9)

dx- dx'dv' dv-

Therefore, the whole

problem

has been formulated in

such

a

way that a single function

r/

has

to

be evaluated

from the bi-harmonic

Eq.

(9)

associated with the

boundary conditions that are specified

at the

bounding

edges of the beam.

4. Boundary conditions

The

practical problems

in elasticity are normally of

the

boundary-value

type where the conditions that are

imposed 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 stressesat

the

boundary.

Referring to Fig. 1, which illustrates the

present problem

of

cantilever beam, both the top and

bottom edges are free from

loading, the left lateral

edge

is fixed and the right lateral edge is subjected to

uniformly distributed shear.

For both the top and bottom edges, AB and CD,

the normal and tangential stress components, stated

mathematically, are

given

by

o,(x,

y)

:0,

an d

6*y(x ,

)

:0 ,

fo r

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 E

T H

t ,li

r 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 fo r 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

n terms

of

{/

and thus the corresponding

relations between

known functions on

the boundary and the function

ry'

are,

a2lr

u (x , y ) :

- ,

(10 )

0xov

, \

|

f , , ,A2rl , ̂ A2rlr1

v ( x , y ) : - ; :

l t t - v ) ; + + 2 ; + 1 ,

( 1 1 )

l * u f '

' d Y '

d x ' l

E

f

A3rl/ A3{/1

o , ( x , Y ) : : l

'

- v

' l

( 1 2 )

"r\"" '

' '

( l

+

v)2

ax2ay

'

oy3

'

E

la3l./ .^ ,

a3l/

.^.

o , I x . v \ - -

^ l ^ - + Q * v \ = - ^ : - | .

( 1 3 )

/

'

'J

(l

+

v)'

YaY'

dxzdY)

E

f

a3{r a3{/1

6 " u ( x , y ) :

. I v -

^Y\""

/

(

I

+

v)2

L'

a*6r,

ax3l

As far as numerical computation

is conce.rned, t is evi-

dent from

the

expressions

of above conditions that all

the boundary conditions of interest can

easily

be

dis-

cretized in terms of the displacement function r/ by the

method of finite-difference.

5. Solution

procedure

The essential eature of

the numerical approach here

is that the original

governing

differential equations of

the boundary-value

problems

are replaced by a flnite

set

of simultaneous algebraic

equations and the sol-

utions of this set of si multaneous

algebraic equations

provide

us with

an

approximation

for the displacement

and stress within

the

solid

body. About the solution

through the proposed formulation, attention may be

drawn

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',1t1

a y ' - ' a f

l '

x

Fig. l. Deep

cantilever beam

subjected to uniformly distribu-

ted shear at the free end.


4/10

a

Z

S.R.

Ahmed

et al.lComputers

and Structures69

(1998)

329*338

approach. The finite-difference technique,

one

the

oldest numerical methods extensively

used for

equations, is used here to transform

partial

differential Eq.

(9)

partial

differential Eqs.

(i0)-(14),

associ-

the boundary conditions, into their

corre-

algebraic equations. The

discrete values of

potential function, rL (x, y), at the mesh points of

(Fig.

2)

are

obtained from

a sys-

resulting from the

of the

governing

equation and the

pre-

boundary conditions.

The region in which the dependent function

is to be

a desirable number of mesh

and the values

of the function r are sought

points.

To keep

the order of the error

of

difference equations of the boundary conditions

to

boundary, exterior to the

domain, is introduced. The discretization

for the domain concerned is illustrated

in

points can be done in

considering the

of

the

boundary and also the nature

rectangular

grid

are used

all over the region concerned. The

gov-

bi-harmonic equation which is used to evaluate

only at the internal mesh

points

is

in its corresponding

difference equation

difference operators. When all the

deriva-

present

in the bi-harmonic equation

are

replaced

their respective central difference formulae,

the

bi-harmonic

(9)

becomes

/

i

-

2,

)

+

{/

i

+

2,

)}

-

4R\(r

+

R\llr

(i

-

1,

)

+

{/( i*

1,,r)}

4(I

+

R\U/(i , j+ 1)

+

l /( i , j

l)}

+

(6R4

8R2

qVQ,)

+

2R2{ILQt,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 )

R: k lh .

Considering an interior mesh

point

O

(i,

),

it is seen

the algebraic Eq.

(15)

contains

the discretized

the 13 neighboring

mesh

points,

and when

becomes an immediate

neighbor of the

physical

mesh

points,

this equation will

contain mesh

exterior to the

boundary as well as on

the

(Fig.

2).

Thus, the application

of the

difference expression

of the bi-harmonic

to the

points

in the immediate neighborhood

physical

boundary will cause

no difficulties,

an imaginary false boundary exterior

to the

boundary is introduced.

,

PhYsical oundary

Fig. 2. Rectangular

mesh-network f the domain in relation

to the coordinates

ystemand the finite-difference iscretiza-

tion of the bi-harmonic quationat

an arbitrary nternalmesh

point.

5.2. Management oJ

boundary conditions

Normally, the boundary

conditions are specified

either in

terms of loadings or of restraints

or

of

some

combination

of the two. Each mesh

point

on the

physi-

cal boundary of the domain

always entertains two

boundary conditions

at a time out of four

possible,

namely,

(1)

normal stressand

shear stress;

2)

normal

stressand

tangential displacement;

3)

shear

stress

and

normal displacement; and

(4)

normal displacement and

tangential

displacement. The computer

program

is

organized here in

such a

fashion

that, out of these two

conditions,

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 their

appropriate

difference equations, every mesh

point

of

the 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 choice

of mesh

points

on the boundary.

As the differential

equations associated with the

boundary conditions

contain second and third order

derivatives

of the function

r ,

the application

of the

central difference

expression is not

practical

as, most

of the time, it leads to the inclusion

of the

points

ex-

terior to the false

boundary. The derivatives

of

the

boundary

expressions are thus replaced

by their three

point backward or forward difference formulae, keep-

ing

the order of the local truncation

error the same.

Two different

sets of boundary expressions are

used

for

each boundary, one set for the

first half of the edge

and the other set for the

second half. For example, the

finite-difference

expressions or the normal and

tangen-

tial components

of stresson the top

boundary, AB, at

points

closer o A, are

given

by :


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S.R. Ahmed et al.l Computers and Structures 69

(1998)

329-338

Table 1

Specification of the boundary

conditions in relation to corresponding

mesh-points on the boundary

Correspondence

between

mesh-points and

given

boundary

conditions

Boundary Given

boundary conditions Condition/mesh-point

Condition/mesh-point

Top, AB

Bottom, CD

Left, AC

Right,

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>

+

.sv(z,i-)

(,

-

+)

trQ,i

)

.

(+

-

z)'t'tz'i

2)

o's,'l(2'i3)

-t+t{,(r,i)

v(3,i)l

.+{{/(r, i+

)

+

/(3,i

r)}

-

f itt,o,i

z) l,e,i*,)]],

(16)

6,v(2,

)

O#

oz l*

ve,)

+

.

(+

-+)l

z,it*

t-

+fr*rs,l- r.s{{/(2,- |

+

2{{/(3,j 1)

+

{/(3,j

+

I)

+

l ,(4,

*

1)]] .

The

discretization

scheme using

the

neighboring

grid-

points

as required for expressing

the

above

conditions

on the top boundary of the beam,

AB, is illustrated in

Fig. 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 B

is a common

point

of both

the edges AB and BD and

thus it has four boundary conditions-two from each

edge. In solving the

present

beam

problem,

three

con-

ditions out of the four are used, the remaining one

is

treated as redundant. The three conditions

mentioned

above are organized in such a way that the values of ry'

at three

points,

namely, I,

B,

and

2

are evaluated

from

these equations-points

1 and B from the boundary

conditions

coming from edge AB and

point

2 from the

boundary equation

from

edge BD. Table 2 shows the

choices and the

conditions to be satisfied by

the corner

mesh

points

in relation to the

present

cantilever

beam

problem.

An example of

the finite-difference drscretiza-

tion used to evaluate

the

corner

mesh

point

2 is shown

in Fig. 4 and the corresponding

difference equation

is

as follows:

Q,.il

lmaginary

Boundary

+ i

i = 2

i = 3

a - A

t - a

+ i

i : 3

i = 4

i - 5

i = 6

(b)

Fig. 3. Grid-points

for expressing he 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.

n

v

) ' '

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

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

(17)

j+4

+3

+ i-2 i-t

i i+l

i

(a)

I i-z i-t i i+l

V

i


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S.R. Ahmed

et al.I Computers

and Structures

69

(1998)

329*338

Grid-points used

in expressing he boundary

condition

point

2.

- l )

:

. . , : ; . ; l t . 5V (2 ,n )

{5+

3R2(2

y ) }

'

( l

*

v) 'Rtht

x

V(2,

n

-

I)+

{6

+

4R2Q

v)}rl(2,

-

2)

- { 3 +

R 2 ( 2 + v ) } l / Q , n - 3 )

+

0.5{ /(2,

-

4)

+

1.5R2(2+

X/( l , n

-

l )

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

+

l / (3,

n

-

2) l

+

0.5R2(2 vX/( l , n

-

3)

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

( 18 )

of the computational

molecules

developed

difference

operators are

available in

[8].

of the system

of algebraic equations

There are numerous

existing methods

of solving

a

of algebraic equations.

In the

present problem,

of unknowns in the

system of

equations is

but only a few in

each individual

this condition, the

iterative method

be

preferable.

But the

problem

of solving

the

equations

by the iterative

method has

certain

Although this

method works

very well

f boundaryconditions

at the corner

mesh-points

for

certain boundary

conditions, it fails

to

produce

any

solution for

other complex

boundary conditions.

In

certain cases,

the rate of convergence

of iteration

is

extremely slow, which

makes it impractical.

As

this

iterative

method has the

limitation

of

not

always con-

verging

to a solution

and sometimes

converging but

very

slowly, the authors have

thus used

a triangular

decomposition

method ensuring

better reliability

and

better accuracy

of solution in a

shorter

period

of time.

The matrix decomposition

method,

used here, solves

the

present

system of equations

directly. Finally,

the

same

difference equations

as those of the

boundary

conditions

are organized for the

evaluation of displace-

ment

and stress components

at different

sections of the

beam 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 pose

serious difficulty in their

solutions. This

problem

has

been

satisfactorily tackled

by

present

formulation. All

the solutions

of

interest

obtained

through the ry'-formu-

lation

conform to the

symmetric and

anti-symmetric

characteristics

of the

problem

and

also to the famous

S-Venant's

principle

that the effects

of sharp variation

of

a

parameter

on the boundary

die down

and become

uniform with the increase

of distance

of

points

in the

body from the boundary.

In

obtaining numerical

values for the

present pro-

blem,

the beam as the elastic

body is assurrlbd

to be

made of

ordinary steel

(v

:

0.3, E

:

200

GPa).

Graphs are

plotted

at

different constant

values of x for

varying

y

as well as at

constant

y

for varying

x for the

parameters

of interest.

Moreover, the

effect of alb

ratio

on the relevant

displacement

and stress com-

ponents

is explicitly illustrated

here. In

order to make

the results non-dimensional,

the

displacements

are

Physical Boundary

i : l

i : 1

t = 5

i : 4

V

i

\

(2,

n)

+ i

2

n-5 n-4 n-3

n-2 n-l

l=n

Possible

boundary

conditions

Conditions

used

Corresponding

mesh-points

for

evaluation

of

ry'

A

C

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)


7/10

S.R. Ahmed et al.l

Computersand

Structures69

(1998)

329-338

335

0.05

5

0.04

s o.o3

o.o2

0.01

0

0

0.5

't

t l a

Fig.

5. Distribution

of the displacement

component

u along

the neutral

axis of deeo cantilever

beams.

expressed

as the ratio

of actual displacement

to the

depth of the beam and the stressesas the ratio of the

actual

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 is

observed to be nonlinear

in nature

and identical

with

the elementary

solution having third

order

polynomial-

like

behavior. The

general

trend

of the curve reveals

that the displacement

is zero

at the fixed

end and

maximum

at the free

end of the cantilever

beam

which

is in complete

conformity

with the loading

as well as

with the

end conditions. The

effect of the

afb ratio on

the distribution

of u along the

neutral axis

is also illus-

trated

in the same figure. It conforms to the fact that,

at a lower

afb ratio, the end-effects

become

very

pro-

0 0 . 5 1

y l a

Fig.

6.

Distribution

of the displacement

component

y

at var-

ious longitudinal

sections of the cantilever

beams

(alb

:

2).

0 0 . 5 L

t l a

Fig. 7. Distribution

of the displacement omponent

v along

the top boundary

rlb

:

0) of deepcantilever

eams.

minent

and

provide

restriction to the

deflection of the

beam.

From

the distribution

of

the

displacement

com-

ponent

v with respect

to

y

in Fig. 6, it is

seen that this

displacement at the

lree end for a

particular

alb rutio

is

maximum at the

top and bottom fibers, but zero

over the

whole depth at the fixed

end and all along the

neutral axis,

which is fully in conformity

with the

physical

model of the

problem.

The

distribution is

completely

asymmetric about the

neutral axis of the

beam, which conforms

to the assumption

of the el-

ementary theory

of beam that

plane

sqctions remain

plane

during the bending

of beams.

l'

The distribution

of the displacement component

y

over the span is presented in Fig. 7 describing the

effect

of the afb ratio

on the distribution at the top

0.0015

0.001

5x10-a

0

-5x10-a

-0.001

-0.0015

0 0.5

x l b

Fig.

8.

Distribution

of normal

stress component or at various

transverse

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-

0.10

r'--.F

0.25

-.+

0.50

+y/a=0.00

"-o-

0.05

----r

0.10

+

0.50

0.90

+

1.00


8/10

+a/b=1.0

+

2.0

#

3.0

-#

3.5

36

S.R.

Ahnted

et al.l

Computers

and Structures

69

(1998)

329-338

lrl

\ " 0

l.$

-0.001

-0.002

0 0 . 5

1

xl b

9. Distribution

of the normal stress

omponent

dr at the

of deepcantilever

eams.

of the beam. Displacement

in

the direction

of

/

substantially

towards the free

end as the

onger.

Fig.

8 shows the

distribution of normal

stress

com-

o, with respect

to x at various

transverse

sec-

of the beam. From

the distribution,

it is observed

the variation

is sinusoidal in nature

and the

fixed

most critical

section of the beam

as far

as

stress is concerned.

The effect

of the

alb

on the distribution

of o,

at

the

fixed end

of the

in Fig.

9.

As

appears from

the

stresses ncrease

with an increasing

afb ratio

for

same loading. It

may be concluded

from the

distri-

that the most

critical

point

at the fixed

end

o o, is

around xlb:0.1

and 0.9 in

each

0.012

0.006

-0.006

-0.012

10. Distribution

of bending stress

on at th e flxed

end over

depth of deep cantilever

beams.

0

0.5

x l b

Fig.

I l. Distribution

of bending stress

o' over

verse sections

of a deep cantilever

beam

(alb

:

1

various trans-

2.s).

s

Fig.

10 shows the variation

of bending

stress o, at

the restrained

boundary,

showing the effect

of the alb

ratio on the distribution. Stresses are maximum at

both the top

and bottom fibers

with

zero

value at the

mid-section

which makes

the distribution

asymmetric

about

the longitudinal

mid-section

of the beam. It

should

be noted here that, for

a higher

afb

ratio,

the

magnitude

of o, at the top fiber

is higher than that

at

the

bottom fiber. But, in

cases of elementary

solution,

this

magnitude is exactly

the same for

both the top

and

bottom fibers of the

beam. Again, this variation

of

bending stress along

the depth is analyzed

for a

par-

ticular

beam

(olb

:2.5)

mainly to compare..how

the

elementary

solutions match with that

of exacd

solutions

obtained through

this numerical

approach. In the

el-

ementary solution, the distribution of normal stress

component

varies linearly

with depth

everywhere and

0.004

tq

o.oo3

h

61

0.002

0.001

0

-0.001

0 0.5

x l b

Fig.

12. Distribution

of shearing

stress

over he depth

of deepcantilever

eams.

1

o", al the fixed

end

0.5

x l b

+yla=0.00

.---.o-

0.05

+

0.25

+

0.50

0.7s

0.90

*

1.00


9/10

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 1

x 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 bottom

fibers.

As appears

from Fig.

11, the solutions

differ

from

that of the

elementary

solution in

a sense that

the

distribution is

far from linear,

especially,

at around

the fixed

end, and it

remains linear

for other

sections

of the

beam which,

of course, conforms

to the

famous

S-Venant's

principle.

Distribution

of shearing

stress, oxy at the

fixed end

of

the cantilever

beams

(Fig.

12)

reveals that

shearing

stress s

zero at both

the top

and bottom

edges which

conforms

to the

obvious fact

that both

the top and

bottom

edges of the

physical

model

are free

from

shearing

stresses.The

distribution

describing the

effect

of the alb ratio

on shearing

stress along

the restrained

boundaries,

shows

that the

stress varies

nonlinearly,

having

the maximum

values

near the top

and bottom

corners of the

fixed edge

and minimum

at the mid

sec-

tions

which

disagrees

completely with

the elementary

solutions. Also,

the

beam becomes

more

critical in

terms

of

shearing stresses

when the length

of

the beam

is

increased,

keeping

the loading

constant.

Interestingly,

for

this

particular

type

of loading,

at a

higher

af b ratio, the

upper corner

zone

at the fixed

end

becomes

more critical

in terms

of the stresses

han the

lower

zone.

'Finally,

the variation

of shearing

stresses

over the

depth is investigated

at various transverse

sections

of

the 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

of

this

stress component

over

the depth is

simildr

to that

of elementary

solutions at the

mid-sections

of the

beam. Sufficiently

away from the

boundary,

the distri-

butions are

parabolic

in

nature

and they

are identical

in nature

and magnitude

with that

of elementary

sol-

utions. From the

elementary

solution it is observed

that the

magnitude

of

the

shearing stresses

are maxi-

mum

at the mid-section

of the beam. This

is not

agreed upon

by our numerical

solutions and it

differs

mainly around

the fixed ends as

predicted

by the

el-

ementary theory;

it is maximum

at about xlb

:

0.05

and

0.95. Since, n the elementary

formulas

of strength

of materials, the boundary conditions are satisfied in

an approximate way, it fails

to

provide

the

actual dis-

tribution

of stresses at the

boundaries, especially,

at

the restrained

boundaries.

The

present

ry'-formulation

is free from this

type

of shortcoming and is

thus

capable of

providing

the actual stress

distribution at

any

critical section, either at

or far from the restrained

edges.

7. Conclusions

Earlier mathematical

models

of elasticity were very

deficient in

handling the

practical

problems.

No

appro-

priate approach was available in the literature which

could

provide

the

explicit information about

the actual

distribution

of stresses

at the critical regions

of

restrained

boundaries satisfactorily.

The distinguishing

feature

of the

present

ry'-formulation

over the existing

approaches 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 any

combination

of them

and thus the

solutions obtained

are

promising

and satisfactory

for the entire

region of

interest.

Both

the

qualitative

and

quantitative

*esults of deep

cantilever beams,

obtained through the ry'-formulation,

establish the soundnessand appropriateness of the pre-

sent approach.

The comparative

study with elementary

solutions verifies

that the elementary

solutions are

highly

approximate

as they fail to

provide

the sol-

utions in the neighborhood

of restrained

boundaries.

References

[]

Uddin MW. Finite difference

solution of two-dimen-

sional elastic

problems

with mixed

boundaryconditions.

M.Sc. hesis.

CarletonUniversity,

Canada,1966.

[2]

Durelli AJ,

Ranganayakamma .

Parametric

olution of

stressesn beams.

EngngMech 9B9;115(2):401.

[3] Horgan CO, Knowels JK. Recentdevelopment oncern-

ing

S-Venant's

rinciple.

Adv Appl Mech

1983;23:179-

269.

[4]

Parker DF. The

role of S-Venant's

olutions n rod

and

beam heories. Appl

Mech 1979;46:861-6.

[5]

Dow

JO, JonesMS, Harwood

SA. A new

approach o

boundary modeling

for finite difference

applications n

solid mechanics.

nt J Numer

Meth Engng 1990;30:99-

11 3 .

0.003

0.002

t { :

h

s

0.001


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Structures 9

(1998)

329-338

Durelli AJ, Ranganayakamma B.

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the

use of

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Metrol,

SPIE

1987; 14:1:-8.

,

..

Idris

,ABM.

A

new approagh to solution of mixed

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Engineering

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Ehaka, Bangladesh; 1993.

r

[8]

Ahmed

SR.

Numerical

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University of Engineering and Technology, Dhaka,

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993.

[9]

Timoshenko

SP, Goodier JN.

Theory

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[10]

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.r1

$

*t

. ;

Documents

1998 Investigationo f Stressesa t the Fixed End of Deep Cantilever Beam