Casd Tutorial Sheet

8/3/2019 Casd Tutorial Sheet

1/8

1.

a)If a displacement field is described by

( )( ) 42

422

1063

1062

+=

++=

yyxv

xyyxu

Determine xyyyxx and, at the point x = 1, y = 0. Also, determine the values of

xyyyxx ,, and zz at that particular point if Youngs modulus Eand Poissons ratio

are 6104.2 kN/m2 and 0.3 respectively. Also, judge whether it is a plane stress or plane strain problem. Use the following relationships to compute strains and stress

values

( )ijjiij uu ,,

2

1+=

ijkkijij 2+=

where and are Lames constants given by( ) ( )

211 +=

Eand

( )

+=

12

E.

Also, Kroneckers delta ijhas the following properties

jiij == when1 and jiij = when0 .

b)In a solid body, the six components of stresses at a point are given by x= 40 Mpa,

y = 20 Mpa, z= 30 Mpa, yz = -30 Mpa, zx = -15 Mpa, xy = 10 Mpa. Determine

the normal stress at the point at a point for which the normal is

( )

=

2

1,50.0,50.0,, zyx nnn . Use the following relationships

ijijTn = and the normal stress iin nT= .

9+5=14

2. Consider the functionalI(potential energy) for minimization given by

( ) 21

2

0

8002

1

2

1+

= chdxdxdy

kI

L

With 20=y at x= 60. Given 25,20 == hk and L=60. Use Rayleigh-Ritz method to

determine 21,cc and 3c using the polynomial expression ( )2

321 xcxccxy ++= .14


2/8

3. Solve for the unknown displacements and rotation at pt Cfor beam shown in following

fig. 1 by the member approach of the stiffness method 14

Fig. 1

4. Derive the transformation matrix required to express quantities from member axis

coordinate to the system axis coordinate. Using this transformation matrix, derive the

expression of a stiffness matrix of a truss member in the global coordinate system. 14

5. Consider the differential equation

04002

2

2

=+ xdx

d 10 x

With the boundary conditions: ( ) ( ) 010 ==

Find the solution of the above mentioned differential equation by Galerkins method

using two term approximation function: ( ) ( )xxcxxc += 11~ 2

21

14

6. Find out the deflection and slope at point B of the following beam (Fig. 1) by the

member approach of the stiffness method. Consider AB and BC as member 1 and 2

respectively.

14

L 0.5 L

A

B

C

Wper unit run

L/2 L/2 L

W

2EIEIA

B

C

Fig. 1


3/8

7. Derive Stiffness matrix and the load matrix of a four nodded quadrilateral finite

element using the potential energy approach. The following expression for shape

functions may be used for the derivation purpose:

( )( )iiiN ++= 1141

14

8. Consider a 4 noded rectangular element as shown in figure 2. Assume plane stress

condition with material properties26

/1030 mKNE = , 3.0= and{ }Tq 000032.0006.0003.0002.000= m. Evaluate BJ, and at0= and 0= .

9. Find out the shape function of a nine nodded Lagrangian element by Lagrangian

polynomial expression. The expression is given by

( )( )( ) ( )( ) ( )

( )( ) ( )( ) ( )niiiiiii

niiif

=

+

+

..................

..................

1121

1121forn number

of Gauss/sampling points.

10. Write short notes on any four of the following topics:

a) Equivalent joint loads b) Natural coordinate system and Jacobian matrix, c) The

criterion to be considered for choosing a displacement polynomial to derive the shape

functions, d) The member flexibility of a prismatic member considering six possible

(0,0) (2,0)

(2,1)(0,1)

Fig. 2

C (1, 0.5)


4/8

d.o.f.s at each end, e) The Isoparametric finite element formulation, f) The strain-

displacement matrix.

11. Determine the member stiffness matrix for a beam member with proper diagrammatic

representations.

12. Solve the following pin jointed plane truss to find out the reaction forces at all the

supports by the flexibility method:

13. Find out the value of the following integral by Gauss-Legendre 2 point Gauss

quadrature rules: 6+6=12

a) +

2

2

21 Cosx

dx

b) ( )dxdyxyyx +3

1

2

1

221

14. a) Consider an one dimensional element with one degree of freedom at each end i and

j respectively. Using matrix inversion procedure, prove that the shape functions for these

two nodes are [ ]

=

ij

i

ij

j

xx

xx

xx

xxN wherex represents coordinate of any point inside

the element. 8

b) State the criterion to be considered for choosing a displacement polynomial to derive

the shape functions.

L L

L

W

Fig. 3


5/8

15. a) With proper diagrammatic representations, derive stress equilibrium equation of

linear theory of elasticity which is as follows:

0,

=+ijij f (i)

b) With proper diagrammatic representations, derive boundary conditions corresponding

to eq. (i) which may be represented as follows:

ijijTn = 7+8=15

16. Derive the expression for the Jacobian matrix [ ]J and the strain-displacement matrix

[ ]B for a four noded quadrilateral finite element. The following expression for shape

functions may be used for the derivation purpose:

( )( )iiiN ++= 114

115

17.Consider the following equation governing a variable cross section bar fixed at the

left end and subjected to an axial force at the right end

( ) 02 =

dx

dux

dx

d10


6/8

18. Fig. 4 shows a plane frame subjected to a vertical load Wat the middle of member

BC. Compute the displacements at B. Also find out the member end actions. Given AB

= BC =L having flexural rigidityEI. Ignore the axial effects.

20

The member stiffness matrix for a conventional plane frame member is given below:

W

L, 2EI

L, EI

A

B C

Fig. 4


7/8

[ ]

=

322622

22

22

622322

22

22

12

22

22

22

22

22

22

3

Ll

Lm

LLl

Lm

L

lL

llmlL

llm

mL

lmmmL

lmm

Ll

Lm

LLl

Lm

L

lL

llmlL

llm

mL

lmmmL

lmm

L

EIK

WhereL = The length of plane frame member; l and m are direction cosines given by

Cosl= and Sinm = . The angle is inclination of the member with the horizontal.19. (a) Derive the member flexibility matrix of a beam member using proper diagrams.

(b) Describe the procedure to solve the set of linear algebraic equations given by

[ ]{ } { }fxK = when the unknown vector { }x contains non-zero support settlements.

20. The potential energy of a beam as shown in the following Fig. 4 is given by the

equation 1: 20

Fig. 4

dxqwdx

wdEIL

=

0

2

2

2 .. (1)

Using variational principle, obtain Euler-Lagrange equation. Also find out the kinematic

(essential) and natural boundary conditions.

21. a) A beam has following dimensions:

Length of the beam: 4 m.

Section of the beam: 100x200 mm2

Thickness in the Z direction: 100 mm.


8/8

Youngs modulus: 200 Gpa

Poissons ratio: 0.3

Consider that the beam has been discretized using 8 nos of 4-noded isoparametric

element having equal length; prepare an input data sheet for a basic finite element

program. Assume that the beam is being modeled as a plane strain problem. Show the

mesh division using a proper diagram. Use a 2 x 2 Gauss quadrature integration scheme.

10

b) What do you understand by bandwidth minimization? Why is it helpful for solving

big finite element problems? How the global numbering of nodes should be done to

minimize the bandwidth of a problem (use a proper sketch)? Describe skyline storage

technique. 2+2+2+4=1022. a) With neat sketches, describe the initial strains method of elasto-plastic analysis.

10

b) What role is played by Newton-Raphson method while dealing with elasto-plastic

problems? Describe with proper diagram. 10

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