ME 1401 FEA- VII Sem (ME)

MAGNA COLLEGE OF ENGINEERINGJDN EDUCATIONAL TRUST

Redhills – Tiruvallur High Road, Magaral, Chennai – 600 055.

ASSIGNMENT QUESTIONS

R4 - MH 1003 – FINITE ELEMENTS ANALYSIS

IV B.E., VII Sem., ME

June’10 – Nov’10

UNIT - I - FUNDAMENTAL CONCEPTS

PART - A

1.1. What is the finite element method?

1.2. How does the finite element method work?

1.3. What are the methods generally associated with finite element analysis?

1.4. List any four advantages of finite element method. [AU, April / May – 2008]

1.5. Define finite difference method.

1.6. What is the limitation of using a finite difference method? [AU,

April / May – 2010]

1.7. Define finite volume method.

1.8. Differentiate finite element method from finite difference method.

1.9. Differentiate finite element method from finite volume method.

1.10. What do you mean by discretization in finite element method?

1.11. Define degree of freedom.

1.12. State the advantage of finite element method over other numerical analysis methods.

1.13. Briefly explain the stages involved in FEA.

1.14. State the fields to which FEA solving procedure is applicable.

1.15. What are the types of boundary conditions?

1.16. What is structural and non-structural problems?

1.17. Distinguish between 1D bar element and 1D beam element. [AU, Nov / Dec – 2009]

1

MAGNA COLLEGE OF ENGG / AQ / R4-MH1003 / VII / ME / JUNE 2010 - NOV 2010

1.18. Write the equilibrium equation for an elemental volume in 3D including the body force.

1.19. What do you mean by boundary condition and boundary value problem?

1.20. Write the difference between initial value problem and boundary value problem.

1.21. List the various methods of solving boundary value problems. [AU, April / May – 2010]

1.22. Briefly explain force method and stiffness method.

1.23. What is aspect ratio?

1.24. Write a short note on stress – strain relation.

1.25. State the effect of Poisson’s ratio.

1.26. Define total potential energy of an elastic body.

1.27. Write the potential energy for beam of span L simply supported at ends, subjected to a concentrated load P at mid span. Assume EI constant. [AU, April / May – 2008, Nov / Dec – 2008]

1.28. State the principle of minimum potential energy. [AU,

Nov / Dec – 2007]

1.29. Define the principle of virtual work.

1.30. Differentiate Von Mises stress and principle stress .

1.31. Write a brief note on the following.

(a) isotropic material

(b) orthotropic material

(c) anisotropic material

1.32. What do you mean by constitutive law? [AU,

Nov / Dec – 2007]

1.33. What are h and p versions of finite element method?

1.34. What is the difference between static and dynamic analysis?

1.35. What is Galerkin method of approximation? [AU,

Nov / Dec – 2009]

1.36. Distinguish between potential energy and potential energy functional.

1.37. Name any four FEA software

PART - B

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

2


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

1.39. 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,


1.40. 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]

1.41. 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 midspan and compare with exact solution using Rayleigh-Ritz method.

Use y = a1 sin ( x/1) + a2 sin (3 x /1) [AU,

Nov / Dec – 2008]

1.42. 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]

1.43. If a displacement field is described by

,

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

3

L EI


1.44. Explain the Gaussian elimination method for the solving of simultaneous linear algebraic equations with an example. [AU, April / May – 2008]

1.45. In a solid body, the six components of the stress at a point are given by x= 40 MPa , y = 20 MPa, z = 30 MPa, yz = -30 MPa, xz = 15 MPa and xy = 10 MPa. Determine the normal stress at the point, on a plane for which the normal is (nx, ny, nz) = ( ½, ½, )

1.46. In a plane strain problem, we have

x = 20,000 psi y = - 10,000 psi E = 30 x 10 6 psi, = 0.3.

Determine the value of the stress z.

1.47. Consider the rod as shown below, where the strain at any point x is given by . Find the displacement at the tip.

1.48. Determine the displacements of nodes of the spring system as shown below.

1.49. 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]

1.50. Use the Rayleigh – Ritz method to find the displacement of the midpoint of the rod shown below.

4


1.51. A rod fixed at its ends is subjected to a varying body force as shown below. Use the Rayleigh - Ritz method with an assumed displacement field to determine displacement u(x) and stress (x)

1.52. 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]

1.53. Use the Rayleigh – Ritz method to find the displacement field u(x) of the rod as shown below. Element 1 is made of aluminium and element 2 is made of steel. The properties are

Eal = 70 GPa A1 = 900 mm2 L1 = 200 mm

Est = 200 GPa A2 = 1200 mm2 L2 = 300 mm

Load = P = 10,000 N. Assume a piecewise linear displacement.

Field u = a1 + a2x for 0 x 200 mm, and u = a3 + a4 x for 200 x 500

mm.

1.54. A steel rod is attached to rigid walls at each end and is subjected to a distributed load T(x) as shown below.

a) Write the expression for potential energy.

b) Determine the displacement u(x) using the Rayleigh – Ritz method. Assume a displacement field u(x) = a0 + a1 x + a2 x2.

5

12

P x


1.55. Derive the stress – strain relation and strain – displacement relation for an element in space.

1.56. Derive the equation of equilibrium in case of a three dimensional stress system. [AU, Nov / Dec – 2008]

1.57. What is constitutive relationship? Express the constitutive relations for a linear elastic isotropic material including initial stress and strain.

[AU, Nov / Dec – 2009]

1.58. Give a detailed note on the following:

a) Rayleigh Ritz method b) Galerkin method

c) least square method and d) collocation method

1.59. Solve the following equation using a two – parameter trial solution by the

a) collocation method (Rd = 0 at x = 1/3 and x = 2/3)

b) Galerkin method.

Then, compare the two solutions with the exact solution

dy / dx + y = 0. 0 x 1

y (0) = 1

1.60. Determine the Galerkin approximation solution of the differential equation

6

30 30

T(x) = 10 x lb / in T(x) = 300 lb / in

x

Traction T(x)


1.61. 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]

1.62. Give a one – parameter Galerkin solution of the following equation, for the

two domain’s shown below.

1.63. a) Solve the system of simultaneous equation using Gaussian elimination method.

b) Give a detailed note on Cholesky decomposition.

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

[AU, Nov / Dec – 2009]

UNIT – II – ONE DIMENSIONAL PROBLEMS

7

u = 0

u = 0 triangular domain

u = 0u = 0

600 600

600

u = 0

u = 0square domain

u = 0u = 0

(1,0)

(1,1)(0,1)

(0,0)


PART - A

2.1. Write a note on node numbering scheme.

2.2. What do you mean by node and element?

2.3. Highlight at least two rules to guide the placement of the nodes when obtaining approximate solution to a differential equation.

[AU, April / May – 2010]

2.4. Justify that few higher order elements are far superior to several lower order elements.

2.5. Define shape function. [AU, Nov /

Dec – 2007]

2.6. What is a shape function? [AU, Nov /

Dec – 2009]

2.7. Differentiate shape function from displacement model.

2.8. State the properties of stiffness matrix. [AU,

Nov / Dec – 2009]

2.9. List the characteristics of shape functions. [AU,


2.10. State the significance of shape function.

2.11. Write the element stiffness matrix for a two noded linear element subjected to axial loading.

2.12. Write the stiffness matrix for the simple beam element given below. [AU, Nov / Dec – 2008]

2.13. Differentiate global stiffness matrix from elemental stiffness matrix.

2.14. What do you mean by banded matrix?

2.15. How will you find the width of a band?

2.16. How do you calculate the size of the global stiffness matrix?

2.17. List the properties of the global stiffness matrix. [AU,


2.18. Give a brief note on the following

(a) elimination approach (b) penalty approach.

2.19. Name the factors which affect the number element in the given domain.

8


2.20. State the requirements to be fulfilled by the approximate solution for its

convergence towards the actual solution.

2.21. What do you mean by continuity weakening?

2.22. Compare the linear polynomial approximation and quadratic polynomial

approximation.

2.23. Why are polynomials generally used as shape functions?

2.24. What do you mean by error in FEA solution?

2.25. What are the types of load acting on the structure?

2.26. Define traction force (T).

2.27. State the assumptions made while finding the forces in a truss.

2.28. How are thermal loads input in finite element analysis? [AU,

Nov / Dec – 2007]

2.29. Why are polynomial type of interpolation functions preferred over

trigonometric functions?

[AU, Nov / Dec –

2007]

2.30. What is an equivalent nodal force? [AU, April /

May – 2008]

2.31. What are called higher order elements? [AU,


2.32. What do you mean by higher order elements .

[AU, Nov / Dec – 2008]

2.33. What are the characteristics of shape functions?

2.34. Give a brief note on the sources of error in FEA.

2.35. State the significance of post processing the solution in FEA.

2.36. What do you know about radially symmetric problem?

2.37. Write the steps involved in developing finite element model.

2.38. Write the boundary condition for a cantilever beam subjected to point load

at its free end.

9


2.39. For a one dimensional fin problem, what are all the boundary conditions

that can be specified at the free end?

2.40. State the differences between a bar element and a truss element.

PART - B

2.41. Consider the rod (a robot arm) as shown below, which is rotating at constant angular velocity = 30 rad/sec. Determine the axial stress distribution in the rod, using two quadratic elements. Consider only the centrifugal force. Ignore bending of the rod.

2.42. An axial load P = 300 x 103 N is applied at 200C to the rod as shown below. The temperature is then raised to 600C

a) Assemble the stiffness (K) and load (F) matrices.

b) Determine the nodal displacements and element stresses.

2.43. 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]

10


2.44. Consider a bar as shown below having a cross sectional area Ae = 1.2 in2

and Young’s modulus E = 30 x 106 psi If q1 = 0.02 in and q2 = 0.025 in, determine the following:

a) the displacement at the point P b) the strain and stress

c) the element stiffness matrix and d) the strain energy in the element.

2.45. A finite element solution using one – dimensional, two – noded elements has been obtained for a rod as shown below.

Displacement are as follows , E = 1N/mm2, area of

each element = 1 mm2, L1-2 = 50 mm, L2-3 = 80 mm, L3-4 = 100 mm.

i) According to the finite element theory, plot the displacement u(x) versus x.

ii) According to the finite element theory, plot the strain (x) versus x.

iii) Determine the B matrix for element 2-3.

iv) Determine the strain energy in the element 1-2 using

11

1x1 : 15 in

x : 20 in 3x2 : 23 in

x

q2q1

P

1 2 3 4 x


2.46. Consider the bar, loaded as shown below. Determine the nodal displacements, element stresses and support reactions. Solve this problem by adopting elimination method for handling boundary conditions. (value of E = 200 x 109 N/m2).

2.47. Consider the bar as shown below. Determine the nodal displacements, element stresses and support reactions. (E = 200 x 109 N/m2)

2.48. An axial load P = 385 KN is applied to the composite block as shown below. Determine the stress in each material.

12

P Rigid plate

200 mm1 2

a a

1 2

30 mm 30 mm

60 mm

Section a – a

E1 = 70,000 MPa , E2 = 105,000 MPa


2.49. For a vertical rod as shown below, find the deflection at A and the stress distribution. E = 100 MPa and weight per unit volume = 0.06 N/cm3. Comment on the stress distribution.

2.50. 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]

2.51. Find the deflection at the free end under its own weight, using divisions of

a) 1 element b) 2 elementsc) 4 elements d) 8 elements and e) 16 elements

Then plot the number of elements versus deflection.

13

2500 cm2

g

A

B

1.6 m

1 m

C

1500 cm2

100 mm

25 mm

1000 mm g E = 200 GPa f = 77 KN/m3

100

100

25

25

top viewfront view


2.52. The plane wall shown below is 0.5 m thick. The left surface of the wall is maintained at a constant temperature of 2000C and the right surface is insulated. The thermal conductivity K = 25 W/MoC and there is a uniform heat generation inside the wall of Q = 400 W/m3. Determine the temperature distribution through the wall thickness using linear elements.

2.53. a) For the discretisation of beam elements as shown below, number the nodes so as to minimize the bandwidth of the assembled stiffness matrix (K)

b) The elements of a row or column of the stiffness matrix of a bar element sum up to zero, but not so for a beam element. Explain why this is so.

2.54. For the beam problem shown below, determine the tip deflection and the slope at the roller support.

14

Mq = 1000 N/m 1000 N

K = 200 N/m

2 m

3 m


2.55. Find the deflection and slope for the following beam section at which point load is applied.

2.56. Solve the following beam as shown below, clamped at one end and spring support at other end. A linearly varying transverse load of maximum magnitude of 100 N/cm applied over the span of 4 cm to 10 cm. Take EI =

2 x 107 N/cm2, .

2.57. Obtain the deflection at the mid point of the beam shown below and determine the reaction.

15


2.58. 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]

Figure shows uniformly loaded beam

2.59. Determine the displacement of node 1 and the stress in element 3, for the three-bar truss as shown below. Take A = 250 mm2, E = 200 GPa for all elements.

2.60. Find the horizontal and vertical displacements of node 1 for the truss shown below. Take A = 300 mm2, E = 2 x105 N/mm2 for each element.

16


2.61. 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.

(i) Form the equation F = KU where K is the assembled stiffness matrix of the structure.

(ii) Find the forces in all the five members. [AU,


2.62. 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 Elements

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

17


2.63. Derive the shape function for a 2 noded beam element and a 3 noded bar element. [AU, Nov / Dec – 2008]

2.64. Why are higher order elements needed? Determine the shape functions of an eight noded rectangular element.

[AU, Nov / Dec – 2007]

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

(ii) Derive the shape functions for a 2D truss element. [AU,

Nov / Dec – 2007]

2.66. 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”.


Figure shows the one-dimensional linear element

UNIT - III - TWO DIMENSIONAL PROBLEMS - SCALAR VARIABLE PROBLEMS

PART – A

3.1. Name few 2-D elements along with a neat sketch.

3.2. State the differences between 2D element and 1D element.

3.3. Give one example each for plane stress and plane strain problems. [AU, Nov / Dec – 2008]

3.4. Give a brief note on natural co-ordinate system.

3.5. Define Lagrange’s interpolation.

18


3.6. Write the Lagrangean shape functions for a 1D, 2 noded element. [AU, Nov / Dec – 2008]

3.7. Obtain the shape function for a 1D quadratic isoparametric element.

3.8. Write the relation to obtain the size of the stiffness matrix for a linear quadrilateral element having Ux and Uy as dof.

3.9. Why is the 3 noded triangular element called as a CST element?

3.10. Write down the interpolation function of a field variable for three-node triangular element.

[AU, April / May –

2010]

3.11. Explain the important properties of CST elements. [AU,

Nov / Dec – 2008]

3.12. Write briefly about LST and QST elements.

3.13. What are CST and LST elements? [AU, Nov / Dec – 2009]

3.14. Write the displacement function equation for CST element.

3.15. Write the strain – displacement matrix for CST element.

3.16. Differentiate CST and LST elements. [AU, Nov / Dec - 2007]

3.17. What do you mean by the terms : c0,c1 and cn continuity? [AU, April / May – 2010]

3.18. Distinguish between C0, C1 and C2 continuity elements.

3.19. What are the different problems governed by 2D scalar field variables?

3.20. Use various number of triangular elements to mesh the given domain in the order of increasing solution refinement.

3.21. Define Pascal triangle.

3.22. Write the significance of Pascal triangle in developing triangular elements.

3.23. Distinguish one from the other of the following

19


a) linear and quadratic triangular elements. b) linear and quadratic Lagrange elements.

3.24. What do you mean by area co-ordinate method?

3.25. State the advantage of serendipity element.

3.26. What do you mean by wrapping?

3.27. Write the node numbering and element connectivity table for the given domain using suitable discretization.

3.28. Plot the variation of shape function with respect node of a 3 noded triangular element.

3.29. Write down the nodal displacement equations for a two dimensional triangular elasticity element.


3.30. Write down the governing differential equation for a two dimensional steady-state heat transfer problem. [AU, Nov / Dec – 2009]

3.31. Name a few boundary conditions involved in any heat transfer analysis. [AU, April / May – 2010]

PART - B

3.32. Calculate the element stresses and the principle angle for the element shown below.

3.33. The nodal co-ordinates of the triangular element is as shown below. At the interior point P, the x- co-ordinate is 3.3 and N1 = 0.3. Determine N2, N3 and the y – co-ordinate at point P.

20


3.34. The (x,y) co-ordinates of nodes i, j and k of a triangular element are given by (0,0), (3,0) and (1.5,4) mm respectively. Evaluate the shape functions N1, N2 and N3 at an interior point P (2, 2.5) mm for the element.

For the same triangular element, obtain the strain-displacement relation matrix B. [AU, Nov / Dec – 2009]

3.35. For the triangular element shown below, obtain the strain – displacement relation matrix B and determine the strains x ,y and xy.

3.36. Find the stresses at the critical point of 1/8th of 1 cm side square bar subjected to torsion

3.37. Compute the element matrices and vectors for the element shown below, when the edges jk and ik experience convection heat loss.

3.38. Compute element matrices and vectors for the elements shown in figure when the edge kj experiences convection heat loss.

[AU, Nov / Dec – 2009]

21


3.39. 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]

3.40. 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]

3.41. 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]

3.42. A long bar of rectangular cross section having thermal conductivity of 1.5 W/M0C is subjected to the boundary condition as shown below. Two opposite sides are maintained at uniform temperature of 180 0C. One side is insulated and the remaining side is subjected to a convection process with T = 850C and h = 50 W/m2c. Determine the temperature distribution in the bar.

22180 0 C0.4 m

180 0 C

0.6 m K = 1.5 W/m0C

h = 50 W/m2 C

T = 250 C


3.43. Determine three points on the 50o C contour line for the rectangular

element shown in the figure. The nodal values are i= 42o C, j=54o C,

k= 56o C and m= 46o C.


Figure shows Nodal coordinates of the rectangular element

3.44. a) Derive the interpolation function 14 for the quadratic triangular element as shown below.

b) Derive the interpolation function of a corner node in a cubic serendipity element.

3.45. 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

23


1 Px1

3

x

3.46. Consider the quadrilateral element as shown below using the linear interpolation functions of a rectangular element, transform the element to the local co-ordinate system and sketch the transformed element.

3.47. Determine the shape functions for a constant strain triangular (CST) element in terms of natural coordinate system.

[AU, Nov / Dec – 2008

3.48. Find the Jacobian matrix for the nine-node rectangular element as shown below. What is the determinant of the Jacobian matrix?

3.49. Compute the steady state temperature distribution for the plate shown in the figure below. A constant temperature of T0 = 1500 C is maintained along the edge y = w and all other edges have zero temperature. The thermal conductivites are Kx = Ky = 1. Assume w = L = 1 and thickness t = 1.

24


3.50. Obtain the finite element solution to the torsion problem for a rectangular cross – sections as shown below. Compute the torque required to produce a twist of 10.

3.51. 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]

3.52. 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]

25


UNIT - IV – TWO DIMENSIONAL PROBLEMS - VECTOR VARIABLE PROBLEMS

PART – A

4.1. State the condition for plane stress problem.

4.2. State the condition for plane strain problem.

4.3. Distinguish between plane stress and plan strain problems. [AU,

Nov / Dec – 2009]

4.4. State the condition for axi-symmetric problem.

4.5. List the required conditions for a problem assumed to be axisymmetric. [AU, April / May – 2010]

4.6. Give examples for the following cases.

a) plane stress problem b) plane strain problem c) axi-symmetric problem

4.7. Define the following terms with suitable examples [AU,


i) plane stress, plan strain ii) node, element and shape functions iii) axisymmetric analysis

4.8. Write the assumptions used to define the given problem as plane stress problem.

4.9. Write the assumptions used to define the given problem as plane strain problem.

4.10. Using general stress - strain relation, obtain plane stress equation.

4.11. Beginning with general elastic stress-strain relation, derive the plane strain condition.

4.12. What are the differences between 2 Dimensional scalar variable and vector variable elements? [AU, Nov / Dec – 2009]

4.13. What are the ways by which a three dimensional problem can be reduced to a two dimensional problem?

26


4.14. Give the stiffness matrix equation for an axisymmetric triangular element.

4.15. What is axisymmetric element?

4.16. Write short notes on axisymmetric problems. [AU,

Nov / Dec - 2007]

4.17. What is meant by axi-symetric field problem? Given an example. [AU, Nov / Dec – 2009]

4.18. What do you mean by constitutive law and give the constitutive law for axi-symmetric problems? [AU,

April / May – 2008, Nov / Dec - 2008]

4.19. Give one example each for plane stress and plane strain problems. [AU, April / May - 2008]

4.20. Give a brief note on static condensation.

4.21. Prove that 2 0 for plane strain condition.

4.22. Differentiate axi – symmetric and cyclic –symmetric structures.

4.23. Differentiate axi-symmetric load and asymmetric load with examples.

4.24. Define the term initial strain.

4.25. State the effect of Poisson’s ratio in plane strain problem.

4.26. How will the stress field vary linearly?

4.27. Compare the changes in the D matrix evolved out of plane strain, plane stress and axi-symmetric problem.

PART - B

4.28. Model a disk with a hole subjected to a concentrated compressive load as shown below. Take Young’s modulus E = 2 x 107 N/cm2, = 0.3, thickness t = 0.5 cm, outer diameter = 10 cm, inner diameter = 5 cm.

4.29. Solve the plane stress problem using three different mesh divisions Compare your deformation and stress results with values obtained from elementary beam theory.

27


4.30. (i) Explain the terms plane stress and plane strain problems. Give the constitutive laws for these cases.

[AU, Nov / Dec – 2007]

(ii) Derive the equations of equilibrium in the case of a three dimensional system.

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

4.31. 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]

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


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


4.34. For the plate with a hole under plane stress, find the deformed shape of the hole and determine the maximum stress distribution along AB by using stresses in elements adjacent to the line thickness t = 1 inch.

4.35. 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]

28


Figure shows Triangular elastic Element

4.36. 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.


Figure shows Triangular Element

4.37. Determine the deflection of a thin plate subjected to extensional load as shown.

29


4.38. Determine the consistent load factor for the CST element under the action of loading as shown below.

4.39. Obtain the consistent nodal vector for the element as shown below.

4.40. A thin elastic plate subjected to uniformly distributed edge load as shown below. Find the stiffness and force matrix of the element.

4.41. A long cylinder of inside diameter 80 mm and outside diameter 120 mm snugly fits in a hole over its full length. The cylinder is then subjected to an internal pressure of 2 MPa. Using two elements on the 10 mm length shown, find the displacement at the inner radius.

30


4.42. Determine the element stresses for the axisymmetric element as shown below. Take E = 2.1 x 105 N/mm2 and = 0.25.

Use the nodal displacements as

u1 = 0.05 mm w1 = 0.03 mm

u2 = 0.02 mm w2 = 0.02 mm

u3 = 0 mm w3 = 0 mm

4.43. Calculate the element stiffness matrix and the thermal force vector for the axisymmetric triangular element as shown below. The element experiences a 150 C increase in temperature. Take = 10 x 10-6 / 0C, E = 2 x 105 N/mm2 and = 0.25

4.44. The triangular element shown in figure is subjected to a constant pressure 10 N/mm2 along the edge ij. Assume E = 200 Gpa, Poisson’s ratio = 0.3 and thickness of the element = 2 mm. The coefficient of

31


thermal expansion of the material = 2 x10-6/ oC and T = 50o C. Determine the constitutive matrix (stress-strain relationship matrix D) and the nodal force vector for the element. [AU, Nov / Dec - 2009]

4.45. For the CST element given below figure assemble strain displacement matrix. Take t = 20 mm, E = 2 x106 N/mm2. [AU, Nov /

Dec - 2008]

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

[AU, Nov / Dec –

2007]

4.47. 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]

32

(200, 400)

(400, 100)(100, 100)


4.48. The open –ended steel cylinder as shown below is subjected to an internal pressure of 1MPa. Find the deformed shape and the distribution of principal stresses.

4.49. The steel flywheel as shown below rotates at 3000 rpm. Find the deformed shape of the fly wheel and give the stress distribution.

4.50. For axi-symmetric pressure loading as shown below, determine the equivalent point loads F1, F2, F3, F4, F7 and F8.

33


4.51. Solve the axi-symmetric field problem as shown below, for the mesh shown there.

T0 = 1000 C Ro= 0.02 m

go = 107 x 2 W/ m3 = internal heat generation

K = 20 W/mc

UNIT - V - ISOPARAMETRIC ELEMENT FORMULATION

PART - A

5.1. What do you mean by isoparametric formulations.? [AU, Nov / Dec – 2007]

5.2. What do you mean by natural co-ordinate system?

5.3. What are the advantages of natural co-ordinates? [AU,

Nov / Dec – 2007]

34


5.4. What are the advantages of natural coordinates over global co-ordinates? [AU, Nov / Dec – 2008]

5.5. Write the natural co-ordinates for the point “P” of the triangular element. The point ‘P’ is the C.G. of the triangle.

[AU, Nov / Dec – 2008]

5.6. State the basic laws on which isoparametric concept is developed. [AU, April / May – 2008]

5.7. Differentiate: local axis and global axis. [AU, April /

May – 2008]

5.8. Define Isoparametric elements? [AU,

Nov / Dec – 2008]

5.9. Define the following term with suitable examples : [AU,


i) Iso-parametric elements

5.10. What is the purpose of isoparametric elements?

5.11. Differentiate x – y space and - space.

5.12. Write the advantages of co-ordinate transformation from Cartesian co-ordinates to natural co-ordinates.

5.13. Write about Jacobian transformation used in co-ordinate transformation.

5.14. Differentate between sub-parametric, isoparametric and super – parametric elements.

5.15. Represent the variation of shape function with respect to nodes for quadratic elements in terms of natural co-ordinates.

5.16. Compare linear model, quadratic model and cubic model in terms of natural co-ordinate system.

5.17. Write a brief note on continuity and compatibility.

5.18. State the advantage of Gaussian integration.

5.19. State the four-point Gaussian quadrature rule.

5.20. Name the commonly used integration method in natural – co-ordinate system.

5.21. Write the relation between weights and Gauss points in Gauss-Legendre quadrature.

5.22. Write down the element force vector equation for a four noded quadrilateral element.

5.23. Write down the Jacobian matrix for a four noded quadrilateral element 35


5.24. Write the shape function for the quadrilateral element in , space.

5.25. Why is four noded quadrilateral element is preferred for axi-symmetric problem than three noded triangular element?

5.26. Distinguish between essential boundary conditions and natural boundary conditions.

[AU, Nov / Dec – 2009]

5.27. Write the advantages of higher order elements in natural co – ordinate system.

5.28. What are the types of non linearity? [AU, Nov /

Dec – 2007]

PART – B

5.29. For the isoparametric quadrilateral element as shown below, the Cartesian co-ordinates of the point P are (6,4). The loads 10 KN and 12 KN are acting in X and Y directions on that point P. Evaluate the nodal equivalent forces.

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

[AU, April / May

– 2008]

5.31. Distinguish between subparametric and superparametric elements. [AU, Nov / Dec – 2009]

5.32. Determine the Jacobian for the (x, y) – (, ) transformation for the element shown below. Also find the area of triangle using determinant method.

36


5.33. Compute the element and force matrix for the four noded rectangular element as shown below.

5.34. 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]

5.35. Derive the shape function for a eight – noded quadrilateral element in , space.

5.36. Consider a rectangular element as shown below. Assume plane stress condition,

inches Evaluate the Jacobian transformation (J), B matrix, and at = 0 and = 0.

37


5.37. 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]

5.38. For a 4 – noded quadrilateral element as shown below, the element displacement vector q is given as . Find the x, y co-ordinate of a point P whose location in the master element is given by = 0.5 and = 0.5

5.39. Write all the shape function for the elements shown below.

38


5.40. For a six – noded linear strain triangular element used for 2 – D stress analysis,

a) show whether the equilibrium equations are satisfied or not within the element i.e

b) comment on the compatibility across element boundaries as well as within the element.

5.41. Derive locations and weights of an order 2 Gauss rule by requiring that it integrates exactly the polynomial = a1 + a2 +a32 +a43 in the interval –1 1. Assume that sampling points and weights are symmetric with respect to the middle of the interval.

5.42. Write short notes on [AU, Nov /

Dec – 2008]

(i) Uniqueness of mapping of isoparametric elements.

(ii) Jacobian matrix.

(iii) Gaussian Quadrature integration technique.

5.43. Integrate

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

5.44. Evaluate the integral and compare with exact results. [AU,

Nov / Dec – 2009]

5.45. (i) Use Gauss quadrature rule (n=2) to numerically integrate [AU, Nov / Dec – 2008]

(ii) Using natural coordinates derive the shape function for a linear quadrilateral element.

5.46. 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]

39


5.47. The area shown is composed of a trapezoid and a rectangle. Evaluate the area using Gauss rules of order 1,2 and 3 and determine the percentage error of each result.

5.48. Use Gaussian quadrature to obtain an exact value of the integral. [AU, April / May – 2010]

5.49. Use one, two and three Gauss quadrature to integrate the following functions. Determine the percentage error of each result.

a)

b)

c)

d)

5.50. Evaluate [i] and [j] for each of the following four elements shown below. Also determine the ratio of element area to the area of a square two units on a side. How is the ratio related to J and why?

2,44,5

1,15,2

s

40


5.51. Evaluate the Jacobian matrix for the isoparametric quadrilateral element as shown below.

*****

41

ME 1401 FEA- VII Sem (ME)

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