Elastic Mechanics Assignments

7/21/2019 Elastic Mechanics Assignments

http://slidepdf.com/reader/full/elastic-mechanics-assignments 1/8

Assignments

1. Describe the similarities and the differences between a plane stress problem and a plane strain problem.

Answer

2. The state of stress in a rectangular plate under uniform biaxial loading, as shown in the

following figure, is found to be

=

000

00

00

Y

X

ijσ

Determine the traction vector and the normal and shearing stresses on the oblique plane .

A!"#$

%. &or the case of pure shear, the stress matrix is given b'

0 0

0 0

0 0 0

ij

τ

σ τ

=

where τ is a given constant. Determine the principal stresses and directions.



A!"#$



(. A two)dimensional problem of a rectangular bar stretched b' uniform end loadings results in the

following constant strain field*1 x

C ε = , 2 y C ε = − and 0

xyγ = ,where +1 and +2 are constants.

Assuming the field depends onl' on x and ', integrate the strain)displacement relations to

determine the displacement components and identif' an' rigid)bod' motion terms.

5. Show that the following strain eld: %

x Ayε = ,

%

y Axε = and 2 -

xy Bxy x yγ = +

gives continuous, single-valued displacements in a simply connected region

only if the constants are related by 2%

A B=

6. #xpress all boundar' conditions for each of the problems illustrated in the following figure.



Answer



. aint)/enants principle allows particular boundar' conditions to be replaced b' their

equivalent resultant. &or problems b-,d-, and f- in exercise , the support boundaries that had

fixed displacement conditions can be modified to specif' the staticall' equivalent reactionloadings. Develop the resultant loadings over the fixed boundaries for each of these cases.

Answer



. 3iven the following stress state* 2 2 2 -

x C y x yσ µ = + − ,

2 2 2 - y

C x y xσ µ = + − ,

2 2 - z

C x yσ µ = + , 2 xy

C xyτ µ = − , 0 yz zxτ τ = =

Discuss wh' this stress state ma' not be a solution of a problem in elasticit'.

4. #xplicitl' show that the fourth order pol'nomial Air' stress function ( 2 2 (ax bx y cy+ + will not

satisf' the bi)harmonic equation unless % % 0a b c+ + = .

10. #xamine that the function( ( -a x yφ = − a50- ma' be used as stress function. Derive the

stress components and determine the surface forces acting on the edges of the rectangular plate

shown below 6od' forces are not involved-.

l 72 l 72

x

y

1

h 7 2

h

7 2

O



11. A beam, as shown in the figure, is sub8ected to pure bending in a plane stress state. The

solution to the problem in terms of Air' stress function is given b' %ayϕ = . 1- #xpress the

constant a in terms of the applied bending moment 9. 2- &ind the expressions for the stresses. 2-

&ind the expressions for the strains. 2- &ind the expressions for the displacements.

12 A thin square plate has in it stresses x Cyσ = and y

Cxσ = , and possibl' some shearing

stresses xyτ . Determine the stresses and displacements.

1%. The cantilever beam shown in the figure is sub8ected to distributed shearing forces with

resultant force F : on the right face. Determine the stresses and displacements of the beam without

considering the self weight. ;sing plane stress theor' and thic<ness of the beam is unit 1.

l

h 7 2

1

h 7 2

x

y

F :

x

y

h

t

b

b

x

y



1(. Through a shrin<)fit process, a rigid solid c'linder of radius1

r δ + is to be inserted into the

hollow c'linder of inner radius1

r and outer radius2

r as shown in the following figure-. This

process creates a displacement boundar' condition1

r -r

u δ = . The outer surface of the hollow

c'linder is to remain stress free. Assuming plane strain conditions, determine the resulting stress

field within the c'linder 1 2

r r r < < -.

1=. A long composite c'linder is sub8ected to the external pressure loading as shown. Assuming

ideali>ed perfect bonding between the materials, the normal stress and displacement will be

continuous across the interface1

r r = . ;nder these conditions, determine the stress and

displacement fields in each material.

Documents

Elastic Mechanics Assignments