CHENDU COLLEGE OF ENGINEERING & TECHNOLOGY

DEPARTMENT OF MECHANICAL ENGG

QUESTIONS BANK

YEAR/SEM : III/VI

SUB NAME : FINITE ELEMENT ANALYSIS SUB CODE : ME 2353

UNIT I FINITE ELEMENT FORMULATION OF BOUNDARY VALUE PROBLEMS

PART A

1. What is the limitation of using a finite difference method?

2. Write about the Galerkin’s residual method.

3. Give two sketches of structures that have both discrete elements and continuum.

4. List the various methods of solving boundary value problems.

5. State the advantages of Gaussian elimination technique.

6. What are interpolation functions?

7. What is meant by node or joint?

8. What is Rayleigh-Ritz method?

9. Distinguish between 1D bar element and 1D beam element.

10. What is Galerkin method of approximation?

11. What is the principle of skyline solution based on Gaussian elimination?

12. Mention the basic steps of Galerkin method.

13. Compare the Rayleigh Ritz method with Nodal Approximation method.

14. On what basis, collocation points are selected?

15. List the types of nodes.

16. What is the need for adopting penalty approach?

17. Explain an initial value problem with an illustrative example.

18. Differentiate between error in solution and residual in the context of weighted residual method.

19. Explain the weighted residual method.

20. Highlight the equivalence and the difference between the Ritz method and finite element method.

21. What are ‘h’ and ‘p’ versions of Finite Element method?

22.What is 'Rayleigh-Ritz method?

23.What is meant by node or joint?

24. What is the principle of skyline solution based on Gaussian elimination?

25. Mention the basic steps of Galerkin method.

PART B

1. Explain the General procedure of Finite Element Analysis. (MAY/JUN 2013)

2. Explain the weighted residual methods with suitable example.

3. Solve the differential equation for a physical problem expressed as d2y / dx2 + 50 = 0, 0 ≤ x ≤1 with

boundary conditions as y (0) = 0 and y (10) = 0 , Trial function y = a1x (10-x); by all weighted residual

methods. (MAY/JUN 2013)

4. Solve the differential equation for a physical problem expressed as d2y / dx2 + 500x2 = 0, 0 ≤x ≤ 1 with

boundary conditions as y (0) = 0 and y (1) = 0. Trial function y = a1x (1-x3); by Point collocation and

Least squares Method. (MAY/JUN 2012)

5. A physical phenomena is governed by differential equation d2y / dx2 - 10x2 = 5, 0 ≤ x ≤1 with boundary

conditions as y(0) = 0 and y(1) = 0. By taking two-term trial solution as y(x) = C1f1(x) + C2f2(x) with

f1(x) = x(x-1) & f2(x) = x2(x-1).Find the solution of the problem using the Galerkin Method. (MAY/JUN

2012)

6. Consider a uniform rod subjected to a uniform axial load as illustrated in Figure. The deformation of the

bar is governed by the differential equation AE d2u/d2x + q0 =0 with the boundary conditions u(0) = 0,

du/dx | x=L = 0, Determine displacement using weighted residual method.

7. The governing differential equation for the fully developed laminar flow is given by µd2v/d2x + ρ cos θ =

0 with the boundary conditions v(L) = 0, du/dx |x=0 = 0, Find velocity distribution v(x) using weighted

residual method. (NOV /DEC 2012)

8. Derive the characteristic equations for the one dimensional bar element by using piece-wise defined

interpolations and weak form of the weighted residual method? (NOV /DEC 2012)

9. Explain Raleigh Ritz method with suitable example

10. Consider a uniform rod subjected to a uniform axial load as illustrated in Figure. Calculate the

displacement and stress in the bar using Raleigh Ritz method and Compare with exact solutions.

11. A rod fixed at its ends is subjected to a varying body force as shown in Figure. Use the Raleigh Ritz

method with an assumed displacement field u(x) = a0 + a1x + a2x2 to Find the displacement u(x) and

stress (x).Plot the variation of the stress in the rod. (E=1, A=1, L=1) (NOV /DEC 2013)

12. Derive the element level equation for one dimensional bar element based on the stationary of a functional.

13. Consider the differential equation for a physical problem expressed as

d2y / dx2 + 300x2 = 0, 0≤ x ≤10 with boundary conditions as y(0) = 0 and y(1) = 0 function corresponding

to this problem is to be extremised is given by I = {(-1/2) d2y / dx2 + 300x2y } dx. Find the solution of the

problem using Rayleigh-Ritz method using a one term solution (y = ax (1 – x3). (NOV /DEC 2013)

14. Solve using Gauss Elimination method.

x1+3x2 + 2x3 = 13

- 2x1 + x2 – x3 = -3

- 5x1 + x2 + 3x3 = 6.

15. A beam of span ‘l’ simply supported at ends and carrying a concentrated load W at the centre ‘C’.

Determine the deflection at mid span by using Rayleigh-Ritz method and compare with exact solutions.

16.A beam of span ‘l’ simply supported at ends is subjected to UDL over entire span. Determine the

deflection at mid span by using Rayleigh-Ritz method and compare with exact solutions.

17. List and briefly describe the general steps of the finite element method. (16) ( MAY /JUN 2014)

18. The differential equation of a physical phenomenon is given by 0 ≤ x ≤ 1

The boundary conditions are: . Obtain one term approximate solution by using Galerkin's method

of weighted residuals. (16) ( MAY /JUN 2014)

19. Solve the differential equation for a physical problem expressed as

((d^2 y)/(dx^2 ))+100=0,0 ≤x ≤10 with boundary conditions as Y (0) = 0 and y(10)=0 using

(i} point collocation method

(ii) Sub domain collocation method

(iii} least squares method and

(iv} Galarkin's method.

20. A simply Supported .beam subjected to uniformly distributed load over entire span and it is

subjected to a point load at the centre of the span. Calculate the deflection using Rayleigh-Ritz method

and compare with exact solutions.

21. Derive the characteristic equations for the one dimensional bar element by using piece-wise defined

interpolations and weak form of the weighted residual method? (MAY/JUN 2012)

22. (i) Derive the element level equation for one dimensional bar element based on the stationary of a

functional.

(MAY/JUN 2012) (10)

(ii) List out the general procedure for FEA problems.(6) (MAY/JUN 2012)

UNIT II ONE DIMENSIONAL FINITE ELEMENT ANALYSIS

PART A

1. State the principle of minimum potential energy theorem.

2. What is the need for developing the overall stiffness matrix of the entire structure in terms of its global

coordinate system? Give an example.

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

differential equation.

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

5. State the significance of shape function.

6. What is post processing? Give an example.

7. Define shape function.

8. What is a truss?

9. What are CST & LST elements?

10. List out the stiffness matrix properties.

11. What is the need for coordinate transformation in solving truss problems?

12. List the properties of a global stiffness matrix.

13. Define p-refinement.

14. What are the factors which govern the selection of nodes?

15. Evaluate the following area integrals for the three node triangular element

16. List the degree of freedom of a) 1D bar b) beam c) plate element.

17. What is meant by aspect ratio? Why is it significant?

18. Define shape function.

19. What is a truss?

20. What is the need for coordinate transformation in solving truss problems?

21. Illustrate the two Hermite shape functions associated with slope as applicable for beam element.

22. State the significance of shape function.

23. What is post processing? Give an example.

24. State the principle of minimum potential energy theorem.

25. What is the need for developing the overall stiffness matrix of the entire structure in terms of its global

coordinate system? Give an example.

PART B

1. Derive the stiffness matrix and finite element equation for one dimensional bar.

2. Derive the stiffness matrix of 2D Truss element. (NOV /DEC 2013)

3. Derive the shape functions of 2D Beam element.

4. Determine the nodal displacements, element stresses and support reactions for the bar loaded as shown in

the figure. (MAY/JUN 2012)

5. A rod is subjected to an axial load P = 600 KN is applied as shown in the figure.

Determine the fallowing

a) Displacement at each node.

b) Stresses in each element

c) Reactions at each node point.

Take A= 250 mm2, E = 20 x 10

5N/mm

2. (MAY/JUN 2011)

6. For a tapered Plate of uniform thickness t = 10 mm as shown in figure. Find the displacements at the nodes

and the reaction force at the support. (MAY/JUN 2013)

7. Determine the nodal displacements, element stresses and support reactions for the bar loaded as shown in

the figure. (MAY/JUN 2011)

8. For the three bar truss shown in figure.

Determine the displacements of node 1 and the stress in element 3. A = 250 mm2 E = 200 GPa

9. Calculate nodal displacement and elemental stresses for the truss shown in figure E = 70 GPa A = 2 cm2

for all the truss members.

10. Derive an expression for shape function and assemble the stiffness matrix for bending in beam elements.

11. Derive an expression of shape functions and the stiffness matrix for one dimensional bar element based

on global co-ordinate approach.

12. A two noded truss element is shown in the fig. The nodal displacements are u1=5 mm and u2 = 8 mm.

Calculate the displacements at x = L/4, L/3 and L/2.

13. A steel plate of uniform thickness 25 mm is subjected to a point load of 420 N at mid depth as shown in

fig. The plate is also subjected to self-weight. If E = 2*105 N/mm2 and density = 0.8*10-4 N/mm3.

Calculate the displacement at each nodal point and stresses in each element.

14. An axial load of 4*105 N is applied at 30°C to the rod as shown in fig. The temperature is then raised to

60°C. Calculate nodal displacements, stresses in each material and reactions at each nodal point. TakeEal

= 0.7*105 N/mm2; Esteel = 2*105 N/mm2; αal = 23*10-6 /°C ; αsteel = 12*10-6 /°C.

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

element.

16. A two node truss element is shown in figure. The nodal displacements are u1 = 5 mm and

u2 = 8 mm . Calculate the displacement at x = (16) (MAY/JUN 2014)

17. For the two bar truss shown in the figure, determine the displacements of node 1 and the stress in

element 1-3 (MAY/JUN 2014)

18. Derive the shape functions for one dimensional linear element using direct method.

19. The loading and other parameters for a two bar truss element is shown in fig.

Determine

i. The element stiffness matrix for each clement

ii. Global stiffness matrix

iii. Nodal displacements

iv. Reaction forces

v. The stresses induced in the elements. Assume E = 200 GPa

20. Determine the shape function and element matrices for quadratic bar element. (MAY/JUN 2012)

21. Find the nodal displacement developed in the planer truss shown in Figure 1 when a vertically downward

load of 1000 N is applied at node 4. The required data are Eiven in the Table 1.

(MAY/JUN 2012)

Element No. 'e' Cross-sectional

area A cm2

Length

1(e)

cm

Young's

Modulus

E(e)

N/cm2 1 2.0 50 2 x 10

6

2 2.0 50 2 x 106

3 1.0 2 x 106

4 1.0 0 2 x 106

Table 1

UNIT – III

PART A

1. List out the limitations of CST element.

2. State Fourier’s law of heat conduction used in FEA.

3. What is meant by primary and secondary node?

4. Distinguish between CST and LST elements.

5. How do you define two dimensional elements?

6. What is QST (Quadratic strain Triangle) element?

7. State the properties of stiffness matrix.

8. Write down the governing differential equation for a two dimensional steady- state heat transfer problem.

9. Specify the strain displacement matrix of CST element and comment on it.

10. What are non-homogenous boundary conditions? Give an example.

11. Define continuity.

12. When triangular element is preferred over quadrilateral elements?

13. What are the types of non-linearity?

14. What are the four basic sets of elasticity equations?

15. List the properties of the global stiffness matrix.

16. List the characteristics of shape functions.

17. Write down the constitutive relationship for the axi symmetric problem.

18. Write down the gauss integration formula for triangular domains.

19. What is constitutive law?

20. What are called higher order elements?

21. What are important properties of a CST element?

22. Give example for plane stress and plane strain problem.

23. Write the lagrangean shape functions for a 1D and 2noded element.

24. What are the advantages of natural co ordinates over global co ordinates?

25. State the basic laws on which isoparametric concept is developed.

PART B

1. Evaluate the stiffness matrix for the triangular element shown in figure (units in mm) under plane stress

condition E = 210 GPa,μ=025, t =1 mm. (NOV /DEC 2013)

2. For the plane stress element shown in figure the nodal displacements are u1 = 2 mm v1 = 1 mm u2 = 0.5

mm v2 = 0 mm u3 = 3 mm v3 = 1 mm Determine the element Stresses σ , σ 1 ,σ 2 and the principal angle .

Let E = 210 GPa, μ=025, t =10 mm. All coordinates are in mm. (NOV /DEC 2012)

3. A thin plate is subjected to traction as shown in figure. Calculate the global stiffness matrix .

(NOV /DEC 2012)

Take t=25mm, E = 2 x 105N/mm

2, ν =0.30.

4. The Cartesian coordinates of the corner nodes of a quadrilateral element are given by (4, 4), (7, 5), (8, 10)

& (3, 8).ϵ = η = 0.5 Find the Jacobian Matrix.

5. For the two dimensional loaded plate shown in figure determine the nodal displacements and element

stress using plane strain condition considering the body force. Take young’s modulus as 200 GPa, poison’s

ratio and density as 7800 kg/m3

6. For the axisymmetric element shown in the figure determine the stiffness matrix. Let E = 2.1 x 10

5 N/mm2

andμ=025 the coordinates are shown in figure are in mm.

7. Establish the Jacobian operator [J] of the quadrilateral element defined by nodal coordinates P(1,1), Q(5,2),

R(4,5) and S(2,4). r = 0.57735 s= -0.57735. Also find the Jacobian determinant. Establish the strain –

displacement matrix for the above element.

8. Derive the shape functions of eight node quadrilateral elements.

9. A wall of 0.6 m thickness having thermal conductivity of 1.2 W/mK. The wall is to be insulated with a

material of thickness 0.06 m having an average thermal conductivity of 0.3 W/mK. The inner surface

temperature is 1000 °C and outside of the insulation is exposed to atmospheric air at 30 °C with heat

transfer coefficient of 30 W/m2K. Calculate the nodal temperatures.

10. A furnace wall is made up of three layers, inside layer with thermal conductivity 8.5 W/mK, the middle

layer with conductivity 0.25 W/mK , the outer layer with conductivity 0.08 W/mK. The respective

thicknesses of the inner, middle and outer layer are 25 cm, 5 cm and 3 cm resp. The inside temperature of

the wall is 600 C and outside of the wall is exposed to atmospheric air at 30 °C with heat transfer

coefficient of 45 W/m2K. Determine the nodal temperatures. (MAY/JUN 2013)

11. Determine the shape functions N1, N2, and N3 at the interior point p for the triangular element shown in

the figure. (16) (MAY/JUN 2014)

12. Determine the shape functions for a constant strain triangular (CST) element in terms of natural co-

ordinate system. (16) (MAY/JUN 2014)

13. Calculate the value of pressure at the point A which is inside the 3 noded triangular element as shown in

fig . The nodal values are , Point A is located at (2,

1.5) assume pressure is linearly varying in the element. Also determine the location of 42 MPa contour

line.

14. Consider the triangular element show in Figure. The element is extracted from a thin plate of

thickness 0.5 cm. The material is bot rolled low carbon steel. The Nodal co-ordinates are x1 = 0; yi = 0;

xj= 0; yj= -1;xk = 2 and yk= - 1cm. Determine the elemental stiffness matrix. Assuming plane stress

analysis. Take = 0.3and E = 2.1xl07 N/cm

2. (MAY/JUN 2012)

15. A four noded rectangular element is shown in Figure. Determine the

Following:

(i) Jacobian Matrix

(ii) Strain Displacement Matrix

(iii) Element stresses

Take E = 2x105 N/mm2, = 0.25

u = [0, 0, 0.003, 0.004, 0.006, 0.004, 0, 0]T

= 0. Assuming plane stress condition. (MAY/JUN

2012)

.

UNIT – IV

PART A

1. List the importance of two dimensional plane stress and plane strain analysis.

2. Give four examples of practical application of axisymmetric elements.

3. Write the finite element equation used to analyses a two dimensional heat transfer problem.

4. State the applications of axisymmetric elements.

5. What is meant by transverse vibrations?

6. Define dynamic analysis.

7. What is meant by axisymmetric field problem? Give an example.

8. Distinguish between plane stress and plane strain problems.

9. Sketch a finite element model for a long cylinder subjected to an internal pressure using axisymmetric

elements.

10. Distinguish between plane stress and plane strain conditions.

11. Why variational formulation is called as weak formulation?

12. Differentiate between upper bound and lower bound solutions with an example.

13. What are the types of Eigen value problems?

14. List the types of dynamic analysis problems.

15. What do you mean by the terms: c0, c1 and cn continuity?

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

17. Write the finite element equation used to analyses a two-dimensional heattransfer problem.

18. State the applications of axisymmetric elements.

19. List the importance of two dimensional plane stress and plane strain analysis.

20. Give four examples of practical application of axisymmetric elements.

21. Why variational formulation is called as weak formulation?

22. Differentiate between upper bound and lower bound solutions with anexample.

23. What are the types of Eigen value problems?

24. List the types of dynamic analysis problems.

25. Write down the nodal displacement equations for a two dimensional triangularelasticity element.

PART B

1. Using two equal length finite elements, determine the natural frequencies of the solid circular shaft fixed at

one end. Length L.

2. Determine the natural frequencies for the stepped bar shown in figure.

3. Determine the natural frequencies of transverse vibration for a beam fixed at both ends. The beam may be

modeled by two elements, each of length L and cross sectional area A. Consider lumped mass matrix

approach. (MAY/JUN 2013)

4. Determine the natural frequencies for the stepped Beam shown in figure.

5. Derive the equations of motion based on weak form for transverse vibration of beam. (MAY/JUN 2014) &

(MAY/JUN 2012)

6. Consider the undamped 2 degree freedom system as shown in figure. find the response of the system when

the first mass alone is given an initial displacement of unity and released from rest.

7. Determine the eigen values and natural frequencies of a system whose stiffness and mass matrices are

given below. (16) (MAY/JUN 2014)

8. (a) Triangular elements are used for the stress analysis of plate subjected to inplane loads. The (x, y)

coordinates of nodes i, j and k of an element are given by (2, 3), (4, 1), and (4, 5) mm respectively. The nodal

displacements are given as:

ui = 2.0 mm, u2= 0.5 mm, u3= 3.0 mm

vi = 1.0 mm, v2 = 0.0 mm, v3 = 0.5 mm

Determine element stresses:

Let E = 160 GPa, Poisson's ratio = 0.25 and thickness of the element t = 10 mm. (16) (NOV /DEC 2012)

9. (i) What are the non-zero strain and stress components of axisymmetric element? Explain. (4)

(ii) Derive the stiffness matrix of an axisymmetric element using potential approach. (12)

10. For the plane stress element whose coordinates are given by ( 100,100), (400, 100) and (200,

400), the nodal displacements are u1 = 2.0mm: v1 = 1.0 mm, u2= 1.0 mm, v 2= 1.5 mm, u3 = 2.5

mm, v3 = 0.5 mm. Determine the element stresses. Assume E = 200 GN/m2, µ = 0.3 and t = 10 mm.

All coordinates are in mm. (MAY/JUN 2013)

11. Derive the equation of motion based on weak form for transverse vibration of a beam. (NOV /DEC

2013)

12. Use iterative procedures to determine the first and third eigen values for the structure shown in fig.

Hence determine the second eigenvalue and the naturalfrequencies of building. Finally, establish the Eigen

vectors and check therest by applyingthe orthogonality properties of Eigen vectors.

Three storey shear structure. . (MAY/JUN 2013)

13. Consider the undamped 2 degree of freedom system as shown in figure. Find the response of the system

when the first mass alone is given an initial displacement of unity and realized from rest. (MAY/JUN 2012)

Figure 4

The mathematical representation of the system for free, Harmonic vibration is given by

14. Consider a uniform cross section bar, as shown in fig of length L made up of material whose young’s

modulus and density is given by E and . Estimate the natural frequencies of axial vibration of the bar

using both consistent and lumped mass matrices. (NOV /DEC 2012)

15. For the plane strain elements shown in figure, the nodal displacements are given as u1 = 0.005 mm, v1 =

0.002 mm, u2= 0.0, v2= 0.0, us = 0.005 mm, v3 = 0.30 mm. Determine the element stresses and the

principle angle. Take E = 70 GPa and Poisson's ratio = 0.3 and use unit thickness for plane strain. All

coordinates are in mm. (NOV /DEC 2013)

UNIT – V

PART A

1. What is the salient feature of an isoparametric element?

2. Give the Lagrange equation of motion and obtain the equation of motion of a two degree of freedom

system.

3. When are isoparameteric elements used?

4. What are force vectors? Give an example.

5. Write down the governing equation for two-dimensional steady state heat conduction.

6. Define streamline.

7. What are the differences between 2 Dimensional scalar variable and vector variable elements?

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

9. What are super parametric elements? Give an example.

10. Specify the shape functions of four node quadrilateral element.

11. What are the characteristics of shape functions?

12. What is meant by natural coordinate system?

13. Name a few boundary conditions involved in any heat transfer analysis.

14. List the method of describing the motion of fluid.

15. List the required conditions for a problem assumed to be axisymmetric.

16. Specify the shape functions of four node quadrilateral element.

17. When are isoparameteric elements used?

18. What are force vectors? Give an example.

19. What is the salient feature of an isoparametric element?

20. Give the Lagrange equation of motion and obtain the equation of motion of atwo degree of freedom

system.

21. What are the characteristics of shape functions?

22. What is meant by natural coordinate system?

23. Define streamline.

24. List the method of describing the motion of fluid.

25. List the required conditions for a problem assumed to be axisymmetric.

26. Name a few boundary conditions involved in any heat transfer analysis.

PART B

1. A wall of 0.6m thickness having thermal conductivity of 1.2 W/m-K the wall is to be insulated with a

material of thickness 0.06 m having a average thermal conductivity of 0.3 W/m-K. The inner surface

temperature is 1000 0C and outside of the insulation is exposed to atmospheric air at 30

0C with heat

transfer coefficient of 35 N/mm2K. Calculate the nodal temperature using FEA. (NOV /DEC 2012)

2. A Steel rod of diameter d= 2 cm and l= 5 cm and thermal conductivity K = 50 W/m0C is exposed at one

end to a constant temperature of 3200C.The other end is in ambient air of temperature 20

0C with a

convection co-efficient of h = 100 W/mm2 0

C. Determine the temperature at the midpoint of the rod using

FEA. (NOV /DEC 2012)

4. Compute element matrices and vectors for the element shown in figure when edge kj experiences

convection heat loss. (NOV /DEC 2013)

5. Derive the stiffness matrix and load vectors for fluid mechanics in 2D finite element (NOV /DEC 2013)

6. For the smooth pipe of variable cross section determine the potentials at the junctions, the velocities in

each pipe. The potentials at the left end is 10m and that at the right end is 2 m.The permeability coefficient

is 1 m/sec.

7. Evaluate the temperature force vector for the axisymmetric triangular element shown in the fig. The

element experiences a 15°C increases in temperature. Take α = 10*10-6/°C, E = 2*105 N/mm2, m = 0.25.

8. Determine the stiffness matrix for the element shown in fig. The co-ordinates are in mm. Take E = 2*105

N/mm2 and m = 0.25.

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

10. For the axisymmetric elements shown in fig, determine the element stresses. Let E = 210Gpa and m =

0.25. The co-ordinates are in mm, u1 = 0.05 mm; w1 = 0.03 mm;u2 = 0.02 mm; w2 = 0.02 mm;u3 = 0 mm

; w3 = 0

11. Derive the strain – displacement matrix [B] for axisymmetric triangular element

12. Derive the stress-strain relationship matrix [D] for the axisymmetric triangular element. (MAY/JUN

2013)

13. Calculate the element stiffness matrix and the thermal force vector for the axisymmetric triangular

element shown in fig. The element experiences a 15°C increase in temperature. The coordinates are in

mm. Take α = 10*10-6/°C; E = 2*105 N/mm2; m = 0.25 (MAY/JUN 2013)

14. Derive a finite element equation for one dimensional heat conduction with free end convection.

(MAY/JUN 2014)

15.Compute the element matrix and vectors for the element shown in the figure when the edges 2-3 and 3-1

experience heat loss. (MAY/JUN 2012)