1. The following differential equation is available for a physical phenomenon.

Trial function is

Boundary conditions are, y (0) = 0 y (10) = 0

Find the value of the parameter a, by the following methods.

(i) collocation (ii) sub-domain (iii) least squares

(iv) Galerkin

2. Discuss the following methods to solve the given differential equation:

with the boundary condition y(0) = 0 and y(H) = 0

(i) Variant method (ii) Collection method.

[AU, April / May – 2010]

3. A cantilever beam of length L is loaded with a point load at the free end. Find

the maximum deflection and maximum bending moment using Rayleigh-Ritz

method using the function Given: EI is constant.

[AU, April / May – 2008]

4. Determine the expression for deflection and bending moment in a simply

supported beam subjected to uniformly distributed load over entire span. Find the

deflection and moment at mid span and compare with exact solution using

Rayleigh-Ritz method.

Use y = a1 sin ( x/1) + a2 sin (3 x /1) [AU, Nov / Dec – 2008]

5. Compute the value of central deflection in the figure below by assuming

. The beam is uniform throughout and carries a central point load P.

[AU, Nov / Dec – 2007]

6. If a displacement field is described by

,

Determine x, y, xy at the point x = 1, y = 0.

7. Explain the Gaussian elimination method for the solving of simultaneous linear

algebraic equations with an example.

[AU, April / May – 2008]

8. For the spring system shown in figure, calculate the global stiffness matrix,

displacements of nodes 2 and 3, the reaction forces at node 1 and 4. Also calculate

the forces in the spring 2. Assume, k1 = k3 = 100 N/m, k2 = 200 N/m, u1 = u4= 0 and

P=500 N. [AU, April / May – 2010]

9. Consider the differential equation for subject to

boundary conditions . The functional corresponding to this

problem, to be extremized is given by

Find the solution of the problem using Rayleigh-Ritz method by considering a two-

term solution as [AU, Nov / Dec –

2009]

L EI

10. Derive the equation of equilibrium in case of a three dimensional stress system.

[AU, Nov / Dec – 2008]

11. What is constitutive relationship? Express the constitutive relations for a linear

elastic isotropic material including initial stress and strain.

[AU, Nov / Dec –

2009]

12. A physical phenomenon is governed by the differential equation

The boundary conditions are given by

. By taking two-term trial solution as with,

find the solution of the problem using the

Galerkin method. [AU, Nov / Dec – 2009]

13. Analyze a simply supported beam subjected to uniformly distributed load

throughout using Rayliegh Ritz method. Adopt one parameter trigonometric

function. Evaluate the maximum deflection and bending moment and compare

with exact solution [AU, Nov / Dec – 2010]

14. Solve the following system of equations using Gauss elimination method.

[AU, Nov / Dec – 2009]

15. Solve the following system of equations using Gauss elimination method.

28r1+6r2 =1

6r1+ 24r2+6r3 =0

6r2+28r3+8r4 =-1

8r3+16r4 =10 [AU, Nov / Dec – 2010]

16. The stepped bar shown in fig is subjected to an increase in temperature,

T=80o C. Determine the displacements, element stresses and support reactions.

[AU, Nov / Dec –

2009]

17. Consider a two-bar supported by a spring shown in figure. Both bars have E =

210 GPa and A=5.0 x10-4 m2. Bar one has a length of 5m and bar two has a length

of 10 m. The spring stiffness is k= 2 kN/m. Determine the horizontal and vertical

displacements at the joint 1 and stresses in each bar. [AU, Nov / Dec – 2009]

18. The simply supported beam shown in figure is subjected to a uniform

transverse load, as shown. Using two equal-length elements and work-equivalent

nodal loads obtain a finite element solution for the deflection at mid-span and

compare it to the solution given by elementary beam theory.

[AU, April / May - 2010]

19. Each of the five bars of the pin jointed truss shown in figure below has a cross

sectional area 20 sq. cm. and E = 200 GPa.

Form the equation F = KU where K is the assembled stiffness matrix of the

structure.

Find the forces in all the five members. [AU, April / May – 2008]

20. Determine the joint displacements, the joint reactions, element forces and

element stresses of the given truss elements. [AU, April / May - 2010]

Figure Truss with applied load

Table 1: Element property Date

Element

s

A

cm2

E

N/m2

L

m

Global

Node

connection

Degree

1 32.2 6.9e 10 2.54 2 to 3 90

2 38.7 20.7e10 2.54 2 to 1 0

3 25.8 20.7e10 3.59 1 to 3 135

21. Derive the shape function for a 2 noded beam element and a 3 noded bar

element. [AU, Nov / Dec – 2008]

22. Why are higher order elements needed? Determine the shape functions of an

eight noded rectangular element. [AU, Nov / Dec – 2007]

23. Derive the shape functions for a 2D beam element. [AU, Nov / Dec – 2007]

24. Derive the shape functions for a 2D truss element. [AU, Nov / Dec – 2007]

25. Derive the interpolation function for the one dimensional linear element with a

length “L” and two nodes, one at each end, designated as “i” and ” j”. Assume the

origin of the coordinate system is to the left of node “i”. [AU, April / May - 2010]

Figure shows the one-dimensional linear element

26. Generate the stiffness matrix of a 2 noded pristmatic bar with one degree of

freedom at each node. [AU, Nov / Dec – 2010]

27. Derive the consistent load vector for a fixed beam subjected to uniformly

distributed load throughout. Use the shape functions corresponding to two degree

freedom at each of the two nodes of the element. Axial deformations can be

neglected. [AU, Nov / Dec – 2010]

28. The temperature at the four corners of a four – noded rectangle are T1, T2 T3

and T4. Determine the consistent load vector for a 2-D analysis, aimed to

determine the thermal stresses. [AU, Nov / Dec - 2007]

29. Find the temperature at a point P(1,1.5) inside the triangular element shown

with the nodal temperatures given as T1 = 400C, TJ = 340C, and TK = 460C. Also

determine the location of the 420C contour line for the triangular element shown in

figure below. [AU, April / May - 2008]

30. Find the expression for nodal vector in a CST element shown in figure

subjected to pressures Px1 on side 1. [AU, Nov / Dec – 2008]

y

2

1 Px1

3

x

31. Determine the shape functions for a constant strain triangular (CST) element in

terms of natural coordinate system. [AU, Nov / Dec – 2008]

32. Calculate the element stiffness matrix and thermal force vector for the plane

stress element shown in figure below. The element experiences a rise of 100C.

[AU, April / May - 2008]

33. For the constant strain triangular element shown in figure below, assemble the

strain – displacement matrix. Take t = 20 mm and E = 2 x 105 N/mm2.

[AU, Nov / Dec - 2007]

34. Analyze the truss in figure and evaluate the stress resultants in member (2).

Assume area of cross section of all the members is same. E = 2 * 105 N / mm2

All dimensions are in meters [AU, Nov / Dec - 2010]

35. The nodal coordinates of the triangular element are shown in figure. At the

interior point ‘P’ the x coordinate is 3.3 and N1= 0.3. Determine the N2, N3 and y

coordinate of ‘P’

36. Explain the terms plane stress and plane strain problems. Give the constitutive

laws for these cases. [AU, Nov / Dec – 2007]

37. Derive the equations of equilibrium in the case of a three dimensional system.

[AU, Nov / Dec – 2007, Nov / Dec - 2007]

38. Derive the expression for constitutive stress-strain relationship and also reduce

it to the problem of plane stress and plane strain. [AU, Nov / Dec - 2008]

39. Derive the constant-strain triangular element’s stiffness matrix and equations.

[AU, April / May - 2008]

40. Derive the linear – strain triangular element’s stiffness matrix and equations.

[AU, April / May – 2008]

41. Determine the element stiffness matrix and the thermal load vector for the

plane stress element shown in figure. The element experiences 20oC increase in

temperature. Take E = 15e6 N/cm2, = 0.25, t = 0.5 cm and a = 6e - 6/o C.

[AU, April /

May - 2010]

Figure shows Triangular elastic Element

42 .For the plane strain element shown in the figure, the nodal displacements are

given as : u1= 0.005 mm, u2 = 0.002 mm, u3=0.0mm, u4 = 0.0 mm, u5 = 0.004

mm, u6 = 0.0 mm. Determine the element stresses. Take E = 200 Gpa and = 0.3.

Use unit thickness for plane strain. [AU, April / May - 2010]

Figure shows Triangular Element

43. For the CST element given below figure assemble strain displacement matrix.

Take t = 20 mm, E = 2 x106 N/mm2. [AU, Nov / Dec - 2008]

44. Derive the expression for the element stiffness matrix for an axisymmetric shell

element. [AU, Nov / Dec – 2007]

45. The (x,y) co-ordinates of nodes i,j and k of an axisymmetric triangular element

are given by (3,4), (6,5) and (5,8) cm respectively. The element displacement (in

cm) vector is given as q = [0.002, 0.001, 0.001, 0.004, -0.003, 0.007]T.

Determine the element strains. [AU, Nov / Dec - 2009]

46. Derive element stiffness matrix for a linear isoparametric quadrilateral element.

[AU, April / May – 2008]

47. Distinguish between subparametric and superparametric elements.

[AU, Nov / Dec – 2009]

48. The Cartesian (global) coordinates of the corner nodes of a quadrilateral

element are given by (0,-1), (-2, 3), (2, 4) and (5, 3). Find the coordinate

transformation between the global and local (natural) coordinates. Using this,

determine the Cartesian coordinates of the point defined by (r,s) = (0.5, 0.5) in the

global coordinate system. [AU, Nov / Dec – 2009]

(200, 400)

(400, 100)(100, 100)

49. The Cartesian (global) coordinates of the corner nodes of an isoparametric

quadrilateral element are given by (1,0), (2,0), (2.5,1.5) and(1.5,1). Find its

Jacobian matrix. [AU, Nov / Dec – 2009]

50. Write short notes on [AU, Nov / Dec – 2008]

(i) Uniqueness of mapping of isoparametric elements.

(ii)Jacobian matrix.

(iii) Gaussian Quadrature integration technique.

51. Integrate

between 8 and 12. Use Gaussian quadrature rule. [AU, April / May – 2008]

52. Evaluate the integral and compare with exact results.

[AU, Nov / Dec –

2009]

53. Use Gauss quadrature rule (n=2) to numerically integrate

[AU, Nov / Dec – 2008]

55. Using natural coordinates derive the shape function for a linear quadrilateral

element. [AU, Nov

/ Dec – 2008]

56. Establish the strain – displacement matrix for the linear quadrilateral element

as shown in figure below at Gauss point r = 0.57735 and s = -57735.

[AU, Nov / Dec – 2007]

2,44,5

1,15,2

s

57. Use Gaussian quadrature to obtain an exact value of the integral.

[AU, April / May – 2010]

58. Compute element matrices and vectors for the elements shown in figure when

the edge kj experiences convection heat loss. [AU, Nov / Dec – 2009]

59. Write the mathermatical formulation for a steady state heat transfer conduction

problem and derive the stiffness and force matrices for the same.

[AU, Nov / Dec – 2008]

60. The temperature at the four corners of a four – noded rectangle are T1, T2 T3

and T4. Determine the consistent load vector for a 2-D analysis, aimed to

determine the thermal stresses. [AU, Nov / Dec - 2007]