FEM QUESTIONS

7/22/2019 FEM QUESTIONS

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Hall Ticket No Question Paper Code : A1327

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Affiliated to JNTUH, Hyderabad

B. Tech VI Semester Tutorial Question Bank 2013 - 14

(Regulations: VCE-R11)

FINITE ELEMENT METHODS

(Mechanical Engineering)

PART-A (2 Mark Questions)

Unit-I

1. Define the term finite element.

2. What is meant by discretization of domain?

3. What is meant by stiffness matrix?

4. Define Bar element.

5. Give the advantage and limitation of Ritz vectors.

6. What are the properties of stiffness matrices?

7. What are compatible elements?

8. What is the significance of node numbering?

9. What is the difference between plane stress and plane strain problems?

10. Explain in detail steps involved in analysis of a typical problem using FEM.

11. With Neat sketch derive the Transformation matrix between Element Local Displacements to Global

Displacements.

12. Write down the Constitutive Relationship for Plane Stress Case.

13. Define Singular matrix.

14. What is linear static analysis?

15. Define Stiffness Matrix.

16. Define plane strain analysis.

17. Differentiate between FEM and FEA.

18. List any four popular software FEA packages.

19. Write the stress-strain relations for the plane strain case in matrix form.

20. Write material matrix (D-Matrix) for 3-D problem.

21. Define body force and surface force with an example for each.

22. Give an expression for the total potential energy functional for 3-Dimensional elastic body subjected

to body forces, surface forces and point loads.

23. Give the relationship between natural co-ordinates and Cartesian co-ordinate system in case of 1-D

quadratic element.

24. What are the properties of shape functions?25. Define node, primary node, secondary node and internal node.

26. The shape function assumes _______ value at one node and __________ at the other node.

Unit-II

1. What is Truss?

2. What is local and global coordinate system?

3. Write down the expression of 2D element stiffness matrix for a truss element.

4. Write down the expression of 3D element stiffness matrix for a truss element.

5. Mention the degrees of freedom for a Beam element?

6. Write down the expression for the potential energy of the beam.

7. Explain the Hermit Shape Functions.

8. Write down the expression of element stiffness matrix for a beam element.

9. Write the stiffness matrix for a plane frame element.

10. What is the basic difference between bar and beam elements?

11. What is the basic difference between bar and beam elements?

12. Give stiffness matrix of a simple beam element.


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13. What is meant by stability analysis?

14. What is the difference between explicit and implicit solution of assembled matrix?

15. Write down the expression of element load vector for a beam element.

16. Write down the expression of element shear force for a beam element.

17. Write down the expression of element bending moment for a beam element.

18. Write down the expression of stiffness matrix for one dimensional bar element.

19. Define total potential energy.

20. Write down the expression of shape function N and displacement U for 1-D bar element.

21. Express quadratic shape functions of a line element.

22. Sketch a typical beam element showing degrees of freedom.23. How will you obtain the potential energy of a system?

24. Differentiate between local co-ordinates and global co-ordinates in case of plane truss.

25. Differentiate between plane truss and space truss element.

26. Differentiate between C0and C

1continuity.

27. Write all the Hermite shape functions for 2 noded beam element.

28. Give the expression for a stiffness matrix for a beam element.

29. Write the transformation matrix for 1-D plane truss element.

30 Sketch a space truss element in local and global coordinate system.

Unit-III

1. Write a strain displacement matrix for CST element.

2. Wite down the expression for shape functions for a constant strain triangle element.

3. What is the necessity for Natural coordinate system?

4. What do you mean by C1continuity?

5. What do you mean by axisymmetric problem?

6. How is a quadratic triangular element different from linear triangular element?

7. List any four commonly used axisymmetric elements.

8. When is an element called as isoparametric element?

9. Write down the stiffness matrix for two dimensional constant strain triangle elements.

10. What is the difference between the strong and weak forms of system equations?

11. Explain the important properties of CST element.12. Write down the stiffness matrix for two dimensional CST elements.

13. Write down the displacement function equation for CST element.

14. What are serendipity elements?

15. Define the sub-parametric, iso-parametric and super parametric elements.

16. Write down the stiffness matrix equation for four noded iso-parametric quadrilateral elements.

17. What are shape functions and what are their properties?

18. What are the conditions for a problem to be axisymmetric?

19. Give the strain displacement matrix equation for an axisymmetric triangular element.

20. Write short notes on Axisymmetric problems.

21. What are the convergence requirements for the finite element solutions?

22. Define CST, LST, QST elements.23. Write the shape functions for 2-D CST element.

24. Define iso parametric, sub parametric and super parametric elements.

25. Differentiate between h-convergence and p-convergence.

26. What is meant by axisymmetric solid?

27. Write down the expression for shape functions for axisymmetric triangular element.

28. State the conditions to be satisfied in order to use axisymmetric elements.

29. Sketch ring shaped axisymmetric solid formed by a triangular and quadrilateral element.

30 The sum of all weights of Gauss points in Gaussian quadrature technique must be ____. Explain why?

Unit-IV

1. Sketch a linear and Quadratic hexahedral element.

2. State the purpose of interpolation models.

3. Sketch a cubic threedimensional element.

4. Define the term Static condensation.

5. Explain the analogies between structural, heat transfer and fluid mechanics.

6. Why is a convergence criterion very important in finite element method?


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7. What is the characteristic matrix of 1D potential flow element?

8. What are the two objectives of thermal analysis?

9. What is mode of heat transfer?

10. Write down the finite element equation for one dimensional heat conduction with free end

convection.

11. State the conditions to be satisfied in order to use axisymmetric elements.

12. Write down the expression for stiffness matrix for an axisymmetric triangular element.

13. Write down the general Helmholtz equation.

14. Write down the expression for governing equation of one dimensional heat conduction.

15. Write down the expression for governing differential equation of one dimensional heat transfer inthin fins.

16. Write down the expression for governing differential equation of two dimensional heat transfer in

thin fins.

17. Write down the expression for one point Gaussian quadrature method.

18.

Using two point Gaussian quadrature formula solve

1

1

.dxex

19. Write down the expression for one point Gaussian quadrature method.

20. Write down the expression for two point Gaussian quadrature method.

21. With a sketch, write the boundary conditions for one dimensional heat transfer in thin fins.22. Give the expressions for element matrix and heat rate vector for a thin fin.

23. Write a note on application of FEM in solving scalar field problems.

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

25. Write the finite element equation used to analyse two dimensional heat transfer problem.

26. Write the finite element equation used to analyse two dimensional torsionproblems.

27. Give an expression for B matrix to solve torsion problem.

28. List the different types of elements used to analyse a uniform shaft subjected to torsion.

29. Differentiate between 1-D analysis of a fin and 2-D analysis of a thin plate.

30 Write the boundary conditions for torsion problems.

Unit-V

1. Define the Hamiltons principle.

2. Differentiate consistent and lumped mass matrices.

3. Write down the expression for element mass matrix for one dimensional bar element.

4. Write down the expression for element mass matrix for truss element.

5. Write down the expression for element mass matrix for constant strain triangular element.

6. Write down the expression for element mass matrix for axisymmetric triangular element.

7. Write down the expression for element mass matrix for quadrilateral element.

8. What are the advantages of lumped matrix over consistent matrix?

9. What is meant by a mode in dynamic analysis?

10. Differentiate skeletal and continuum structures.

11. What are the advantages of lumped matrix over consistent matrix?12. Explain the convergence criterion used in FEM.

13. Explain Prefinement & H-refinement in FEM

14. Name any four commercial Finite element packages.

15. Mention the advantages of Finite Element Method.

16. Finite element method is an approximate method (true/false). Also justify your answer.

17. Mention any four FEA popular software packages.

18. Write down applications of FEM.

19. State any two non-linear problems in Finite Element Analysis.

20. Write the common governing differential equation for a field problem.

21. List the type of dynamic analysis problems.

22. What are the types of eigen value problems.23. Give the expression for element mass matrix for 1-D bar element.

24. Give the expression for element mass matrix for truss element.

25. Differentiate between lumped and consistent mass matrices.

26. Give the equation for characteristic polynomial.

27. Differentiate between bar and a beam element.


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28. Write the boundary conditions for eigen value problems.

29. Differentiate between eigen value and eigen vectors.

30 Mention any two method of extracting eigen value and eigen vectors.

PART-B (5 Mark Questions)

Unit-I

1. A bar under axial load is shown below, determine the deection at load applied

by discretizing the bar into two elements, compare the solution with theoretical

solution and comment. E =210GPa L=30cm,Load P =10000N diameter=2cm

2. Determine the nodal displacements, elements stresses for the stepped bar shown in the figure

below.

3. Derive *D+ matrix for the Plane Stress problem using generalized Hookes Law relations.

4. Find the deflection at the free end of the tapered plate shown in the figure below. Calculate the

element stresses.

5. Derive *D+ matrix for plane strain problem using Hookes Law relations.

6. Determine the nodal displacements and element stresses for the stepped bar shown in the figure

below.

7. Explain i) Elimination approach ii) Penalty approach for the treatment of boundary

conditions in an elasticity problem and enumerate the relative advantages of each method.

8. With a suitable example explain the formulation of finite element equations by direct approach.

Assume suitable data for your example.

9. With a help of a neat block diagram, explain the model based simulation process of finite

element method.10. Find displacement of the midpoint of the rod as shown. E=1,A=1,Body force per unit volume

g=1,Length = 2


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11. Explain different approaches of getting the finite element equations and Explain the natural and

geometric boundary conditions.

12. A long rod is subjected to loading and temperature increase of 300C. The totalstrain n at a point is measured to be 1.2 105 If E=200GPa and =12 106/0C. Determine stress at

the point.

13. Derive stresses and strains relations and Derive equivalent nodal force vectors.

14. For the beam and load shown in figure 1 determine slope at 2 and 3 and at the mid

point of the distributed load. E=200GPa,I=5 106mm4.

15. Write notes on the following:

Weighted Residual method.

Initial and Boundary value problems.

16. Type your Question here Define and derive the Hermite shape functions for a two nodded beam

element?

17. Determine the circumference of a circle of radius r using the basic principles of

finite element method.

18. Derive stiffness equations for a bar element from the one dimensional second order

equation by variated approach.

19. Find the deflections and support reactions for the beam shown in figure. Take E = 200 GPa.

20. An axail load P = 300 x 103N is applied to the rod as shown in figure obtain

(a) The assembled stiffness matrix and force vector.

(b) The nodal displacements and element stiffness.

21. List and explain the steps in solving an FEM problem.

22. Differentiate between plane stress and plane strain problems. Also state the stress-strain relations

for both.


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23. What are the advantages of FEM over other methods?

24. Derive strain-displacement matrix [B] and element stiffness matrix [K] for 1-D linear element.

25. Derive the expressions for a nodal load vector in a two noded bar element due to a) Body force b)

Surface force.

26. List all the properties of global stiffness matrix.

27. Derive and sketch quadratic shape functions for 1-D bar element.

28. For the bar shown in the fig.1, determine the nodal displacements and the reactionforces using

direct stiffness method. Take E = 70 GPa; A = 2 X 10-4

m2; K = 2000 KN/m.

P = 8 KN K

2 m 2 m

Fig. 1

29. The displacement field for a 2 noded 1-D bar element is given by U = N1q1+ N2q2 where q1and

q2are nodal displacements and N1and N2are shape functions. Evaluate (1) , N1and N2 at P (x = 750

mm) given x1 = 300 mm and x2= 1200 mm. (2) ifq1 = 0.09 mm andq2 = 0.15 mm, determine the value

of displacement at point P.

30 Obtain displacement at node 2, reactions, stresses and strains in the circular solid stepped bar as

shown in Fig.2. Take E1= 70 GPa, E2= 200 GPa for the element (1)and (2)

0.01m 0.02m

0.2 m 0.3 m

Fig.2

Unit-II

1. Find the defection at the load and the slopes at the ends for the steel shaft

shown in figure consider the shaft to be simply supported at the bearings A and B

I1=1.25x105 mm4 I2=4x104mm4.

2. Find the nodal displacements and element stresses in a plane truss shown in the figure below.

3. Find the vertical deflection and slope at the nodes for the beam shown in the figure below.


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4. Find the displacements at the nodes and element stresses in a plane truss shown in the figure given

below.

5. Find the vertical deflection and slope at the nodes for the beam shown in the figure below.

E = 100 Gpa I = 6 106mm

4


7. Find the slopes at all nodes for the beam shown in the figure below.


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9. Find the vertical deflection and slope at all nodes of the beam shown in the figure below.

E = 200GPa I = 4106mm

4.

10. For the truss structure shown in figure with indicated load, calculate the stress in each element.

11. Consider the truss element with the coordinates 1(10, 10) and 2(50, 40). If the

displacement vector is q = [15 10 21 43 ]Tmm, then determine the vector q, stress in the element

and stiffness matrix if E=70GPa and A=200mm2

12. Compute the support reaction at the other end of continuous beam shown in figure

1. use the concept of boundary element EI=400 units.


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13. The (x, y) coordinates of the nodes i, j and k of a triangular element are (1,

1), (4, 2) and (3, 5) respectively. The shape functions of a point P locatedinside the element are given by N1 = 0.15 and N2 = 0.25. Determine the x

and y coordinates of the point P.

14. For the pin jointed truss shown in figure1. If E = 200 Gpa, determine

(a)Element stiffness matrices .

(b) Global stiffness matrix .

(c) Stress in the element 1.

(d) Strain in the element 2.

15. The (x, y) coordinates of the nodes i, j and k of a triangular element are (1,

1), (3, 2) and (3, 5) respectively. The shape functions of a point P located

inside the element are given by N1 = 0.12 and N2 = 0.25. Determine the x

and y coordinates of the point P.

16. Estimate the displacement vector, stresses and reactions for the truss structure as

shown below Figure

Note: - Area is not given and assumed as A(e) = 1mm2 E is not given. Assumed

as E=2105 N/mm2

17. Derive the element stiffness matrix for the plane truss element.

18. A beam of length 2 m is fixed at both ends. Estimate the deflection at the center

of the beam where load is acting vertically downward of 10 kN. Divide the beam

into two elements. Compare the solution with theoretical calculations. Take E =

2 1011 N/m2, A = 250 mm2.

19. Consider the truss element with the coordinates i(10,10) & q(50,40) If the displace-

ment vector is q=[15 10 21 43]T mm, then determine

(a) The trace vector F

(b) Stress in each element

(c) Stiffness matrix if E= 70 GPA and A= 200 mm2.

20. An elastic bar is having a uniform cross sectional of area `A' mm2 and length `L'

mm. It is _xed at one end and other end is allowed to move along the axis of the

elastic bar. A force `F' KN is acting at the free end and the Youngs Modulus is `E'


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N/mm2. Calculate the displacement at the free end.

21. Calculate the nodal displacements, element stresses and reaction forces for the truss shown

infig.3.Take E =70 GPa. Cross sectional area A = 200 mm2for all the truss members.

Y

800 mm

X

600 mm 300

P = 50 KN

Fig. 3

22. Derive element stiffness matrix for a truss element in global co-ordinate system.

23. Derive and sketch Hermite shape functions for a beam element.

24. A uniform cross-sectional beam of length 1 m is fixed at one end and supported by a roller at the

other end carries a concentrated load of 20000 N at the mid point of the beam. calculate the

deflection under the load. Also find the shear force, bending moment and their directions. Given E =

200 GPa; I = 25X10-6

m4

.25. Derive element stiffness matrix and load vector for a beam element.

26. Determine the deflection and slope under uniformly distributed load and point load for the beam as

shown in Fig.4. Also state the number of elements used, number of degrees of freedom per node

and the size of the global stiffness matrix K. Given E = 200 GPa; I1= 4X10-6

m4; I2= 2X10

-6m

4.

Y 24 KN/m P = 50 KN

X

2 m 1 m 1m

I1Fig.4 Roller support I2

27. Sketch a two-dimensional truss element in a) a local co-ordinate system and b) a global co-ordinate

system.

28. For the two member truss shown in Figure 5. Determine nodal displacements and

stresses in each member.

2 Take E = 70 GPa,

1X10-4

m2

1X10-4

m2

1 3

1m

Fig.5.

29. Sketch a three-dimensional space truss element in a) a local co-ordinate system and b) a global co-

ordinate system.

30 Derive element stiffness matrix for a space truss element in global co-ordinate system.

Unit-III

1. Triangular elements are used for stress analysis of a plate subjected to in plane load. The

components of displacement parallel to (x, y) axes at the nodes i, j and k of an element are found to


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be (- 0.001, 0.01), (-0.002, 0.01) and (- 0.002, 0.02) cm respectively. If the (x, y) coordinates of the

nodes I, j and j are (20, 20), (40, 20) and (40, 40) in cm respectively, find (a) the distribution of the (x,

y) displacement components inside the element and (b) the components of displacement of the

point (xp, yp) = (30, 25) cm.

2. The nodal coordinates and the nodal displacements of a triangular element, under a specific load

condition are given below: Xi = 0, Yi = 0, Xj = 1 mm , Yj = 3 mm, Xk = 4 mm, Yk = 1, u1= 1 mm, u2= -

0.05 mm, u3= 2 mm, v1= 0.5 mm , v2= 1.5 mm and v3= -1 mm. If E = 2 X 105N / mm

2and = 0.3.

Find the stresses in the element.

3. Derive strain displacement [B] matrix for a 3 noded Triangular element?

4. The nodal coordinates of the triangular element are shown in figure. At the interior Point P

The X coordinate is 3.3 and the shape function at node 1 is N1 is 0.3. Determine the shape

functions at nodes 2 and 3 and also the y coordinate of the point P.

5. Calculate the element stiffness matrix [k(e)

], element stress vector {(e)

} and element force vector

{F(e)

} for a plane stress element shown in the figure given below.

6. The nodal coordinates for an axisymmetrical element are given below. Evaluate [B ] matrix for the

element. The coordinates are in centimeters.

Ri= 10.0 Rj= 14.0 Rk=14.0

Zi= 2.0 Zj = 2.0 Zk= 4.0

A body force of 20 N/cm3acts on the above element in the negative R direction. Evaluate

body force vector.

7. A fin of size 2.5 cm X 25 cm X 250 cm extends from a wall. If the wall temperature is maintained at

9000C and the ambient temperature is 30

0C, determine the temperature distribution in the

fin 1-D elements along x-direction. Take k = 35 W/mK and h = 120 W/m2K.

8. List some disadvantages of using 3-D isoparametric elements. Derive the strain displacement matrix

for a tetrahedron element.

9. What do you mean by a aximetric problem? If the displacement functionsu = 3x2+ 60 xy - 20 y

2

= -4 x2- 20xy + 5 y

2

then find the strains Ex, Eyand xyat a point P (-1, 1).


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10. Heat generated in a large plate (0.8 W/m K) at the rate of 4000 W/m3. The

plate is of thickness 25 cm. The outer surface is exposed to an ambient air with

a heat transfer coefficient of 20 W/m2K at 300C. If the inside surface temperature

is 5000C, calculate the temperature at a distance of 10 cm from the inner wall.

Assume cross sectional area is 62.5 mm2.

11. The nodal coordinates of the triangular element are 1 (1,1) , 2 (4,2) , 3 (3,5).

at the interior point P, the x coordinate is 3.3 and N1 is 0.3 . Determine N2,

N3 and y coordinate at point P.

12. Find the deflection at the load of the steel shaft as shown in figure 1: take E = 200

Gpa

13. Consider the truss element with the coordinates i(10,10) & q(50,40) If the displace-




(c) Stiffness matrix if E= 70 GPA and A= 200 mm2.

14. Consider a beam with uniform distributed load as shown in the figure4. Estimate

the deflection at the centre of the beam.

E = 200 Gpa ; A = 25 mm 0 25 mm.

15. The nodal coordinates of the triangular element are shown in figure5. At the

interior Point P the x coordinate is 3.3 and the shape function at node 1 is N1 is 0.3. Determine the

shape functions at nodes 2 and 3 and also the y coordinate of the point P

16. An axisymmetric ring element is shown in _gure 2. Derive the matrices, [B] and

[D]. Take E = 2_105 N/mm2and _ = 0.33.


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17. Explain the convergence criteria in _nite element analysis.

Write about pre-processor, processor, and post-processor in any FEM software.18. Consider the truss element with the coordinates i(10,10) & q(50,40) If the displace-




(c) Sti_ness matrix if E= 70 GPA and A= 200 mm2.

19. Using the general approach of displacement function, derive the force-displacement

relationship and element sti_ness matrix. for a truss bar element.

20. For the two bar truss shown in figure, determine the nodal displacement, element stresses and

support reactions.

21. Show that the element strain-displacement matrix is constant for a 3 noded CST element.

22. Derive element stiffness matrix and load vector for a CST element.

23. 1. The nodal coordinates of a triangular element are shown in fig.1.At the interior point P, thex-coordinate is 3.3 and N1 = 0.3. Determine N2, N3 and the y-coordinate at point P.

y3(4,6)

2(5,3)

1(1,2)

24. Derive weights and gauss points for 2-point methodof Gauss quadrature and hence

evaluate using 2 point guass quadrature formula and check the same with the exact

result.

25. What are the advantages of using the numerical integration? Explain one-point formula.

26. Evaluate using 3 point guass quadrature formula.

27. Using isoparametric representation, derive shape functions for Axisymmetric triangular element.

28. Derive shape functions for four nodedisoparametricelement.

29. Obtain all the shape functions for a nine-noded quadrilateral element.30 Differentiate between Lagranjian and serendipity family of elements.

Unit-IV

1. Explain the procedure for solving transient heat conduction problem using Finite Element Analysis.

2. Determine the nodal temperatures in the multi layered wall shown in the figure below. Using one

.P


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dimensional steady state heat transfer analysis.

k1= 4 w/cm-0C k2 = w/cm-

0C.

3. A metallic fin, with thermal conductivity of 360 W/m K, 0.1 cm thick and 10 cm long extends from a

plane wall whose temperature is 2350C. Determine the temperature distribution and amount of heat

transfer from the fine the air at 200C with a heat transfer coefficient of 9 W/m

2K. Take width of the

fin is 1 m?

4. A Uniform steel fin of length 25 cm, with a rectangular section 5 cm X 2.5 cm. If the heat transfer

takes place by convection from all the sides while the root of the fin is maintained at 3000C,

determine the temperature distribution in the fin. Assume k = 25 W/m K, h = 250 W/m2K and T=

250C.

5. Estimate the temperature distribution in 1-D fin as shown in figure by making into two

elements?

6. Explain the methodology for the treatment of all three boundary conditions in a

1-D heat transfer element?

7. A composite slab consists of 3 materials of different conductivities i.e 20W/m K, 30 W/m K , 50

W/m K of thickness 0.3 m, 0.15 m and 0.15 m respectively. The outer surface is 200C and the

inner surface is exposed to the convective heat transfer coefficient of 25W/m2

K, 8000C.

Determine the temperature distribution with in the wall.

8. A metallic fin with thermal conductivity K=360W/moC, 0.1 cm thick and 10 cm long extends from a

plane wall whose temperature is 235oC. Determine the temperature distribution and the amount of

heat transferred from the fin to the air at 20oC with h = 9W/m2 oC. Take the width of fin to be 2 m.

9. A composite wall consists of three materials. The outer temperature is T=20oC. Convection heat

transfer takes place on the inner surface of the wall with T=800oC and h = 25W/m2 oC .Determine

the temperature distribution in the wall. K1=20W/moC , K2=30W/m

oC ,K3=50W/m

oC ,L1=0.3m

,L2=0.15m ,L3=0.15m .

10. A long bar of rectangular c/s, having the thermal conductivity of 1.5 w/moC is subjected to the

boundary conditions shown in figure. Determine the temperature distribution in the bar.


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11. Heat is generated in a large plate (K=0.8 W/m0C) at the rate 4000 W/m3. The

plate is 25 cm thick. The outside surfaces of the plate are exposed to ambient air

at 300C with a convective heat transfer coefficient of 20W/m2 0C. Determine the

temperature distribution in the wall. Shown in fugure

12. Calculate the temperature distribution and the heat dissipating capacity of a fin

shown in figure 2. The thermal conductivity of the material is 200W/m k. The

surface heat transfer coefficient is 0.5 W/m2k. The ambient temperature is 30oC.

The thickness of the fin is 1 cm the width at the base is 2 cm and at the tip is

1 cm and varies uniformly along the length, what changes takes place in the heat

dissipation capacity of the fin.

13. Derive the element conductivity matrix and load vector for solving 1- D heat con-

duction problems, if one of the surfaces is exposed to a heat transfer coefficient of

h and ambient temperature of T?14. Estimate the temperature distribution in 1-D fin as shown in figure6 by making

into two elements?

15. One side of the brick wall of width 5 m, height 4 m and thickness 0.5 m is exposedto a temperature of - 250C while the other surface is maintained at 320C. If the

thermal conductivity is 0.75 W/m K and the heat transfer coefficient on the colder

side is 50 W/m2 K. Determine

(a) The temperature distribution in the wall and

(b) Heat loss from the wall.

16. What are the basic steps involved in _nite element analysis and explain them briey.

17. For the truss structure shown in _gure 8 with indicated load, calculate displacen-

ments and stress in each element.


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18. For the three stepped bar shown in fig. 2, the fits snugly between the rigid walls at room

temperature. The temperature is then raised by 300C. Determine the displacements at

nodes 2 and 3, stresses in the three sections.

19. a. Explain in detail the applications of isoparametric elements in two and three

dimensional stress analysis.

b. Using Gaussian quadrature evaluate the following integral

20. A long bar of rectangular cross-section, having thermal conductivity of 1.5W/moC

is subjected to the boundary conditions as shown in figure. Determine the temper-

ature distribution in the bar.

21. Write the governing differential equations for one dimensional heat transfer and discuss the various

types of boundary conditions used in solving heat transfer problems.22. Derive element conductivity matrix for one dimensional heat flow element.

23. Comment on the forced and natural boundary conditions for heat transfer problems.

24. Derive element matrices and heat rate vectors for heat transfer through one dimensional thin films.

25. Explain boundary conditions for 2-D heat conduction using a sketch.

26. A metallic fin, with thermal conductivity k =360W/m.0C, 0.1 cm thick and 10 cm long, extends from a


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plane wall whose temperature is 2350C. determine the temperature distribution and amount of heat

transferred from the fin to the air at 200C with h = 9 W/m

2.0C. Take the width of fin to be 1 m.

27. Derive the governing differential equation for 2-D steady state heat conduction.

28. Derive B matrix for torsion problem using triangular element.

29. Using Galerkin approach, derive the stiffness matrix for torsion problems.

30 Discuss scalar field problems with suitable examples.

Unit-V

1. Explain briefly about Eigen value problem. Determine the natural frequencies and mode shapes for

the rod shown in fig. using the characteristic polynomial technique. Assume E = 200 GPa and mass

density7850.kg/m3. L1=L2=0.3m,A1=400mm2,A2=250mm

2

2. Explain Finite Element formulation for obtaining eigen values of a stepped bar in axial vibrations.

3. Distinguish between consistent mass matrix and Lumped mass matrix. Explain Gaussian Quadrative

method.

4. Discuss the methodology to solve the Eigen value problem for the estimation of natural frequencies

of a stepped bar?

5. Consider the axial vibrations of a steel bar shown in the figure:

a) Develop global stiffness and mass matrices,

b) Determine the natural frequencies?

6. Derive the elemental mass matrix for two noded beam element and 2-noded frame element?

7. Evaluate the eigen values, eigen vectors and natural frequencies of a beam of cross section

360 cm2of length 600 mm. Assume youngs modulus as 200 Gpa, density 7850 kg/m3andmoment of Inertia of 3000 mm4. Make into two elements of 300 mm length each

8. Derive the elemental mass matrix for 1-D bar element and 1-D plane truss element?

9. Determine the natural frequencies and mode shapes of a stepped bar as shown in figure

using the characteristic polynomial technique. Assume E = 25 Gpa and density is 7850

kg/m3.

10. Determine the natural frequencies and corresponding mode shapes for the fig. shown below.

= 7850 kg/m3, E = 2 x 1011N/m2.


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11. Derive the expressions for the mass and stiffness matrices of a plate using triangular

elements.

12. Determine the Eigen values and Eigen vectors for the stepped bar shown in figure. E=30times

106 N/m2 ,specific weight = 0.283 Kg/m3 A1= 1 m2 A2= 0.5 m2

L1=10 m. L2 =5 m.

13. Find the natural frequencies and the corresponding mode shapes for the longitudi-nal vibrations for the stepped bar. Assume A1 = 2A and A2 = A ; I1 = I2 = I ;

E1 = E2 = E.

14. Determine the natural frequencies of a simply supported beam of length 800 mm

with the cross sectional area of 75 cm X 25 cm as shown in the figure2.

Take E= 200 Gpa and density of 7850 kg/m3.

15. Discuss the methodology to solve the Eigen value problem for the estimation of

natural frequencies of a stepped bar?

16. Derive the elemental jumped and consistant mass matrices for 1-D bar element and

1-D plane truss element?

17. Evaluate the eigen values, eigen vectors and natural frequencies of a beam of cross

section 360 cm2 of length 600 mm. Assume youngs modulus as 200 Gpa, density

7850 kg/m3 and moment of Inertia of 3000 mm4. Make into two elements of 300

mm length each.

18. Consider the axial vibrations of a steel bar shown in the _gure 1:

(a) Develop global sti_ness and mass matrices,

(b) Determine the natural frequencies?

19. Evaluate the eigen values, eigen vectors and natural frequencies of a beam of cross

section 350 cm2 of length 700 mm. Assume youngs modulus as 200 Gpa, density7850 kg/m3 and moment of Inertia of 3000 mm4. Make into two elements of 300

mm length each.

20. Obtain the eigen values and eigen vectors for the cantilever beam of length 2m using

consistant mass for translation dof with E = 200GPa, = 7500kg/m3.

21. Explain the formulation of finite element model for dynamic analysis.


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22. Derive element mass matrix for 1-D bar element.

23. Derive element mass matrix for 1-D truss element.

24. Derive element mass matrix for 2-D CST element.

25. Derive element mass matrix for 1-D beam element.

26. Explain the different methods to extract eigen values and eigen vectors.

27. List all the properties of eigen vectors.

28. Derive element stiffness matrixmatrix for 1-D bar element.

29. Compute the eigen value and vectors for the cantilever beam using 1-D element. Take E = 200GPa,

breadth, b = 0.01m, height, h = 0.02m, length, l = 1m and density = 7860Kg/m3.

30 Explain consistent mass matrix and lumped mass matrix with examples.

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