Assignment Booklet Me2353 Finite Element Analysis Dec 2013 May 2014

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) 1

R.M.K COLLEGE OF ENGINEERING AND

TECHNOLOGY

RSM NAGAR, PUDUVOYAL-601206

ASSIGNMENT QUESTIONS

ME2353 – FINITE ELEMENT ANALYSIS

III B.E., VI Sem., ME DEC’13 – MAY’14

UNIT – I – FINITE ELEMENT FORMULATION OF BOUNDARY VALUE PROBLEMS

PART – A 1.1) What is the finite element method?

1.2) How does the finite element method work?

1.3) What are the main steps involved in FEA. [AU, April / May – 2011]

1.4) Write the steps involved in developing finite element model.

1.5) What are the basic approaches to improve a finite element model?

[AU, Nov / Dec – 2010]

1.6) Write any two advantages of FEM Analysis. [AU, Nov / Dec – 2012]

1.7) What are the methods generally associated with finite element analysis?

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

1.9) What are the applications of FEA? [AU, April / May – 2011]

1.10) Define finite difference method.

1.11) What is the limitation of using a finite difference method? [AU, April / May – 2010] 1.12) Define finite volume method.

1.13) Differentiate finite element method from finite difference method. 1.14) Differentiate finite element method from finite volume method.

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₂ 1.15) What do you mean by discretization in finite element method?

1.16) What is discretization? [AU, Nov / Dec – 2010]

1.17) List the types of nodes. [AU, May / June – 2012]

1.18) Define degree of freedom.

1.19) What is meant by degrees of freedom? [AU, Nov / Dec – 2012]

1.20) State the advantage of finite element method over other numerical analysis

methods.

1.21) State the fields to which FEA solving procedure is applicable. 1.22) What is a structural and non-structural problem?

1.23) Distinguish between 1D bar element and 1D beam element.

[AU, Nov / Dec – 2009, May / June – 2011]

1.24) Write the equilibrium equation for an elemental volume in 3D including the body

force.

1.25) How to write the equilibrium equation for a finite element? [AU, Nov / Dec – 2012] 1.26) Classify boundary conditions. [AU, Nov / Dec – 2011]

1.27) What are the types of boundary conditions?

1.28) What do you mean by boundary condition and boundary value problem?

1.29) Write the difference between initial value problem and boundary value problem. 1.30) What are the different types of boundary conditions? Give examples.

[AU, May / June – 2012]

1.31) List the various methods of solving boundary value problems.

[AU, April / May – 2010]

1.32) Write down the boundary conditions of a cantilever beam AB of span L fixed at A

and free at B subjected to a uniformly distributed load of P throughout the span. [AU, May / June – 2009, 2011]

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₃ 1.34) What is aspect ratio?

1.35) Write a short note on stress – strain relation.

1.36) Write down the stress strain relationship for a three dimensional stress field.

[AU, April / May – 2011]

1.37) State the effect of Poisson’s ratio.

1.38) Define total potential energy of an elastic body.

1.39) 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, Nov / Dec – 2008]

1.40) State the principle of minimum potential energy.

[AU, Nov / Dec – 2007, April / May – 2009]

1.41) How will you obtain total potential energy of a structural system?

[AU, April / May – 2011, May / June – 2012]

1.42) Write down the potential energy function for a three dimensional deformable body

in terms of strain and displacements. [AU, May / June – 2009]

1.43) What should be considered during piecewise trial functions?

[AU, April / May – 2011]

1.44) Name the weighted residual methods. [AU, Nov / Dec – 2011]

1.45) What is the use of Ritz method? [AU, Nov / Dec – 2011]

1.46) Mention the basic steps of Rayleigh-Ritz method. [AU, April / May – 2011]

1.47) Highlight the equivalence and the difference between Rayleigh Ritz method and the

finite element method. [AU, Nov / Dec – 2012]

1.48) Distinguish between Rayleigh Ritz method and finite element method with regard

to choosing displacement function. [AU, Nov / Dec – 2010]

1.49) Why are polynomial types of interpolation functions preferred over trigonometric

functions? [AU, April / May – 2009, May / June – 2013]

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₄ 1.51) Define the principle of virtual work.

1.52) Differentiate Von Mises stress and principle stress.

1.53) What do you mean by constitutive law?[AU, Nov / Dec – 2007, April / May – 2009] 1.54) What are h and p versions of finite element method?

1.55) What is the difference between static and dynamic analysis?

1.56) What is Galerkin method of approximation? [AU, Nov / Dec – 2009]

1.57) What is a weighted resuidal method? [AU, Nov / Dec – 2010]

1.58) Distinguish between potential energy and potential energy functional. 1.59) Name any four FEA software

PART – B

1.60) Explain the step by step procedure of FEA. [AU, Nov / Dec – 2010]

1.61) Explain the general procedure of finite element analysis. [AU, Nov / Dec – 2011] 1.62) Briefly explain the stages involved in FEA.

1.63) Explain the step by step procedure of FEM. [AU, Nov / Dec – 2011]

1.64) List out the general procedure for FEA problems. [AU, May / June – 2012]

1.65) Compare FEM with other methods of analysis. [AU, Nov / Dec – 2010]

1.66) Define discretization. Explain mesh refinement. [AU, Nov / Dec – 2010]

1.67) Explain the process of discretization of a structure in finite element method in

detail, with suitable illustrations for each aspect being & discussed.

[AU, Nov / Dec – 2012]

1.68) Discuss procedure using the commercial package (P.C. Programs) available today

for solving problems of FEM. Take a structural problem to explain the same.

[AU, Nov / Dec – 2011]

1.69) State the importance of locating nodes in finite element model.

[AU, Nov / Dec – 2011]

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₅ (a) isotropic material

(b) orthotropic material (c) anisotropic material

1.71) What are initial and final boundary value problems? Explain.

[AU, Nov / Dec – 2010]

1.72) Explain the Potential Energy Approach [AU, Nov / Dec – 2010]

1.73) Explain the principle of minimization of potential energy. [AU, Nov / Dec – 2011] 1.74) Explain the four weighted residual methods. [AU, Nov / Dec – 2011]

1.75) Explain Ritz method with an example. [AU, April / May – 2011]

1.76) Explain Rayleigh Ritz and Galerkin formulation with example.

[AU, May / June – 2012]

1.77) Write short notes on Galerkin method? [AU, April / May – 2009]

1.78) Discuss stresses and equilibrium of a three dimensional body.

[AU, May / June – 2012]

1.79) Derive the element level equation for one dimensional bar element based on the

station- of a functional. [AU, May / June – 2012]

1.80) Derive the characteristic equations for the one dimensional bar element by using

piece-wise defined interpolations and weak form of the weighted residual method? [AU, May / June – 2012]

1.81) Explain Gaussian elimination method of solving equations.

[AU, April / May – 2011]

1.82) Write briefly about Gaussian elimination? [AU, April / May – 2009]

1.83) The following differential equation is available for a physical phenomenon.

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₆

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 Square (iv) Galerkin

1.84) 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) Collocation method. [AU, April / May – 2010] 1.85) 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]

1.86) A simply supported beam carries uniformly distributed load over the entire span.

Calculate the bending moment and deflection. Assume EI is constant and compare the results with other solution. [AU, Nov / Dec – 2012]

1.87) 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

(

)

(

)

[AU, Nov / Dec – 2008]

1.88) Compute the value of central deflection in the figure below by assuming

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₇

[AU, Nov / Dec – 2007, April / May – 2009]

1.89) If a displacement field is described by

(

)

(

)

Determine the direct strains in x and y directions as well the shear strain at the point

x = 1, y =0. [AU, April / May – 2011]

1.90) 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 ₂)

1.91) 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.

1.92) 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

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₈ 1.93) Use the Rayleigh – Ritz method to find the displacement of the midpoint of the rod

shown in figure. [AU, April / May – 2011]

1.94) Consider the differential equation

subject

to boundary conditions ( ) ( ) The functional corresponding to this problem, to be extremized is given by

∫ (

)

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

two-term solution as

( )

( )

( )

[AU, Nov / Dec – 2009]

1.96) A bar of uniform cross section is clamped at one end and left free at the other end. It

is subjected to a uniform load axial load P as shown in figure. Calculate the displacement and stress in the bar using three terms polynomial following Ritz method. Compare the results with exact solutions. [AU, May / June – 2011]

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₉ 1.97) 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.

[AU, May / June – 2013]

1.98) A simply supported beam (span L and flexural rigidity EI) carries two equal

concentrated loads at each of the quarter span points. Using Raleigh – Ritz method determine the deflections under the two loads and the two end slopes.

[AU, April / May – 2009]

1.99) Analyze a simply supported beam subjected to a uniformly distributed load

throughout using Rayleigh Ritz method. Adopt one parameter trigonometric function. Evaluate the maximum deflection and bending moment and compare with exact solution. [AU, Nov / Dec – 2010]

1.100) Use the Rayleigh – Ritz method to find the displacement field u(x) of the rod as

shown below. Element 1 is made of aluminum 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.

1.101) A fixed beam length of 2L m carries a uniformly distributed load of a w(in N / m)

which run over a length of ‘L’ m from the fixed end, as shown in Figure. Calculate the rotation at point B using FEA. [AU, Nov / Dec – 2011]

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₁₀ 1.102) A rod fixed at its ends is subjected to a varying body force as shown in Figure. Use

the Rayleigh-Ritz method with an assumed displacement field

( )

to find the displacement u(x) and stress σ(x). Plot the variation of the

stress in the rod. [AU, Nov / Dec – 2012]

1.103) A uniform rod subjected to a uniform axial load is illustrated in Figure. The

deformation of the bar is governed by the differential equation given below. Determine the displacement using weighted residual method.

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₁₁ 1.104) 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.

1.105) Derive the stress – strain relation and strain – displacement relation for an element

in space.

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

[AU, Nov / Dec – 2008]

1.107) What is constitutive relationship? Express the constitutive relations for a linear

elastic isotropic material including initial stress and strain. [AU, Nov / Dec – 2009]

1.108) Give a detailed note on the following:

(a) Rayleigh Ritz method (b) Galerkin method (c) Least square method and (d) Collocation method

1.109) Find the approximate deflection of a simply supported beam under a uniformly

distributed load ‘P‘ throughout its span. Using Galerkin and Least square residual

method. [AU, May / June – 2011]

1.110) Solve the differential equation for a physical problem expressed as

with boundary conditions as y (0) = 0 and y (10) = 0 using

(i) Point collocation method

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₁₂ (iii) Least squares method and

(iv) Galerkin method. [AU, May / June – 2013]

1.111) Solve the differential equation for a physical problem expressed as

with boundary conditions as y (0) = 0 and y (10) = 0 using the trail function

( )

Find the value of the parameters a1 by the following methods.

(i) Point collocation method

(ii) Sub domain collocation method

(iii) Least squares method and

(iv) Galerkin method. [AU, Nov / Dec – 2011]

1.112) Solve the following equation using a two – parameter trial solution by the

(a) Collocation method

(

)

(b) Galerkin method.

Then, compare the two solutions with the exact solution

y (0) = 1

1.113) Determine the Galerkin approximation solution of the differential equation

( ) ( )

1.114) Solve the following differential equation using Galerkin’s method.

( )

( )

[AU, April / May – 2011]

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₁₃

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]

1.116) Determine the two parameter solution of the following using Galerkin method.

( ) ( )

[AU, Nov / Dec – 2012]

1.117) Give a one – parameter Galerkin solution of the following equation, for the two

domain’s shown below. 2₂ 2_2 1.            y u x u

1.118) Describe the Gaussian elimination method of solving equations.

[AU, April / May – 2011]

1.119) Explain the Gaussian elimination method for the solving of simultaneous linear

algebraic equations with an example. [AU, April / May – 2008]

1.120) Solve the following system of equations using Gauss elimination method.

[AU, Nov / Dec – 2010] x1 – x2 + x3 = 1

-3x1 + 2x2 – 3x3 = -6

2x1 – 5x2 + 4x3 = 5

1.121) Solve the following system of equations by Gauss Elimination method.

2x1 – 2x2 – x4 = 1 u = 0

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₁₄

2x2 + x3 + 2x4 = 2

x1 – 2x2 + 3x3 – 2x4 = 3 [AU, May / June – 2012]

x2 + 2x3 + 2x4 = 4

1.122) Solve the following equations by Gauss elimination method.

28r1 + 6r2 = 1 6r1 + 24r2 + 6r3 = 0 6r2 + 28r3 + 8r4 = -1

8r3 + 16r4 = 10 [AU, Nov / Dec – 2010, 2012] 1.123) Use the Gaussian elimination method to solve the following simultaneous

equations:

4x1 + 2x2 – 2x3 – 8x4 = 4

x1 + 2x2 + x3 = 2

0.5x1 – x2 + 4x3 + 4x4 = 10

–4x1 – 2x2 – x4 = 0 [AU, April / May – 2009]

1.124) Solve the following system of equations using Gauss elimination method.

x1 + 3x2 + 2x3 = 13

– 2x1 + x2 – x3 = –3

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₁₅ UNIT – II – ONE DIMENSIONAL FINITE ELEMENT ANALYSIS

PART – A 2.1) Write a note on node numbering scheme.

2.2) What do you mean by node and element?

2.3) What are the types of problems treated as one dimensional problem?

[AU, May / June – 2013]

2.4) Highlight at least two rules to guide the placement of the nodes when obtaining approximate solution to a differential equation. [AU, April / May – 2010]

2.5) Define shape function. [AU, Nov / Dec – 2007, April / May – 2009]

2.6) What is a shape function? [AU, Nov / Dec – 2009]

2.7) Differentiate shape function from displacement model.

2.8) Draw the shape function of a two noded line element. [AU, April / May – 2009]

2.9) Draw the shape function of a two noded line element with one degree of freedom at

each node. [AU, Nov / Dec – 2010]

2.10) Draw the shape function for one dimensional line element with three nodes.

[AU, April / May – 2009]

2.11) State the properties of stiffness matrix. [AU, Nov / Dec – 2009, 2010, 2011] 2.12) List out the stiffness matrix properties. [AU, May / June – 2012]

2.13) State the characteristics of shape function. [AU, May / June – 2011]

2.14) List the characteristics of shape functions. [AU, April / May – 2010]

2.15) When does the stiffness matrix of a structure become singular?

[AU, Nov / Dec – 2012]

2.16) State the significance of shape function.

2.17) Write the element stiffness matrix for a two noded linear element subjected to axial

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₁₆ 2.18) Write the stiffness matrix for the simple beam element given below.

[AU, Nov / Dec – 2008]

2.19) What are the properties of global stiffness matrix? [AU, April / May – 2011]

2.20) Write the properties of Global Stiffness Matrix of a one dimensional element.

[AU, May / June – 2012]

2.21) Differentiate global stiffness matrix from elemental stiffness matrix. 2.22) What do you mean by banded matrix?

2.23) How will you find the width of a band?

2.24) How do you calculate the size of the global stiffness matrix?

2.25) List the properties of the global stiffness matrix. [AU, April / May – 2010]

2.26) Give a brief note on the following

(a) elimination approach (b) penalty approach.

2.27) Name the factors which affect the number element in the given domain.

2.28) State the requirements to be fulfilled by the approximate solution for its

convergence towards the actual solution.

2.29) What do you mean by continuity weakening?

2.30) Compare the linear polynomial approximation and quadratic polynomial

approximation.

2.31) Why polynomials are generally used as shape function? [AU, Nov / Dec – 2011]

2.32) Why are polynomial terms preferred for shape functions in finite element method?

[AU, April / May – 2011]

2.33) What do you mean by error in FEA solution? 2.34) What are the types of load acting on the structure? 2.35) Define traction force (T).

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₁₇ 2.36) State the assumptions are made while finding the forces in a truss.

[AU, Nov / Dec – 2011]

2.37) How are thermal loads input in finite element analysis?

[AU, Nov / Dec – 2007, April / May – 2009]

2.38) What is an interpolation function? [AU, May / June – 2012]

2.39) Why are polynomial types of interpolation functions preferred over trigonometric

functions? [AU, Nov / Dec – 2007, April / May – 2009]

2.40) What is an equivalent nodal force? [AU, April / May – 2008]

2.41) What are called higher order elements?

[AU, April / May – 2008, Nov / Dec – 2010, 2011]

2.42) What is higher order element? [AU, Nov / Dec – 2011]

2.43) What do you mean by higher order elements? [AU, Nov / Dec – 2008]

2.44) Why higher order elements are required for FE analysis? [AU, Nov / Dec – 2012] 2.45) What are higher order elements and why are they preferred?

[AU, April / May – 2011]

2.46) What are the characteristics of shape functions?

2.47) Plot the variations of shape function for 1 – D beam element.

[AU, Nov / Dec – 2010]

2.48) When do we resort to 1 D quadratic spar elements? [AU, April / May – 2011] 2.49) Give a brief note on the sources of error in FEA.

2.50) State the significance of post processing the solution in FEA. 2.51) What do you know about radially symmetric problem?

2.52) Write the boundary condition for a cantilever beam subjected to point load at its

free end.

2.53) For a one dimensional fin problem, what are all the boundary conditions that can be

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₁₈ 2.54) Determine the load vector for the beam element shown in Figure

[AU, Nov / Dec – 2012]

2.55) Write the element stiffness matrix of a truss element. [AU, May / June – 2012] 2.56) Sketch a typical truss element showing local global transformation.

[AU, April / May – 2011]

2.57) Differentiate global and local coordinates. [AU, May / June – 2013]

2.58) State the differences between a bar element and a truss element. PART – B

2.59) What are the different types of elements? Explain the significance of each.

[AU, Nov / Dec – 2010]

2.60) Derive and sketch the quadratic shape function for the bar element.

[AU, May / June – 2011]

2.61) Derive the shape function of a quadratic 1 – D element. [AU, Nov / Dec – 2011] 2.62) Derive the shape functions for one dimensional linear element using direct method.

[AU, May / June – 2013]

2.63) Determine the shape function and element matrices for quadratic bar element.

[AU, May / June – 2012]

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

[AU, Nov / Dec – 2011]

2.65) Obtain an expression for the shape function of a linear bar element.

[AU, April / May – 2011]

2.66) Derive shape functions and stiffness matrix for a 2D rectangular element.

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₁₉ 2.67) 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.

2.68) A steel rod of length 1m is subjected to an axial load of 5 kN as shown in figure.

Area of cross section of the rod is 250 mm2. Using 1 – D element equation solve for the deflection of the bar, E = 2*105 N/mm2. Use four elements.

[AU, Nov / Dec – 2010]

2.69) A column of length 500mm is loaded axially as shown in figure. Analyze the

column and evaluate the stress and strain at salient points. The Young’s modulus can be taken as E. Take A1 = 62.5mm2 and A2 = 125mm2

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₂₀ 2.70) Consider a bar as shown in figure. Young’s Modulus E = 2*105 N/mm2. A1 = 2 cm2,

A2 = 1 cm2 and force of 100 N. Determine the nodal displacement.

[AU, Nov / Dec – 2010]

2.71) Consider the bar shown in Figure Axial force P = 30 kN is applied as shown.

Determine the nodal displacement, stresses in each element and reaction forces [AU, May / June – 2012]

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₂₁ 2.72) Consider the bar as shown in figure. Axial force P1 = 20 kN and P2 = 15 kN is

applied as shown in figure. Determine the nodal displacements, stresses in each element and reaction forces. [AU, April / May – 2011]

2.73) Find the nodal displacement and elemental stresses for the bar shown in Figure.

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₂₂ 2.74) 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.

2.75) 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]

2.76) Axial load of 500N is applied to a stepped shaft, at the interface of two bars. The

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₂₃

subjected to all in temperature of 100˚C. Take E1 = 70*103 N/mm2, E2 = 200*103 N/mm2, A1 = 900mm2, A2 = 1200mm2, α1 = 23*10-6 / ˚C, α2 = 11.7*10-6 / ˚C, L1 = 200mm, L2 = 300mm. [AU, Nov / Dec – 2011] 2.77) 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.

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

Displacement are as follows T

mm

Q [-0.2,0,0.6, -0.1] , 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 . 2 1

kq q U  T

2.78) 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).

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₂₄ 2.79) In the figure shown below load P = 60kN is applied. Determine the displacement

field, stress and support reactions in the body. Take E = 20 kN/mm2

[AU, May / June – 2011]

2.80) Consider the bar as shown below. Determine the nodal displacements, element

stresses and support reactions. (E = 200 x 109 N/m2)

2.81) An axial load P = 385 KN is applied to the composite block as shown below.

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₂₅ 2.82) 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.

2.83) 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]

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₂₆ 2.84) Find the deflection at the free end under its own weight, using divisions of

a) 1 element b) 2 elements c) 4 elements d) 8 elements and e) 16 elements Then plot the number of elements versus deflection.

2.85) For the discretization of beam elements as shown below, number the nodes so as to

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₂₇ 2.86) 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.

2.87) For the beam problem shown below, determine the tip deflection and the slope at

the roller support.

2.88) For the beam and loading as shown in figure. Determine the slopes at the two ends

of the distributed load and the vertical deflection at the mid-point of the distributed load. Take E = 200GPa and I = 4*106 _mm4 _{[AU, May / June – 2011]}

2.89) Find the deflection and slope for the following beam section at which point load is

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₂₈ 2.90) 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, 102

EI K

.

2.91) Obtain the deflection at the midpoint of the beam shown below and determine the

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₂₉ 2.92) 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]

2.93) Determine the displacements and slopes at the nodes for the beam shown in figure.

Take k=200kN / m, E=70GPa and I=2x10-4m4. [AU, Nov / Dec – 2012]

2.94) Determine the nodal displacements and slopes for the beam shown in Figure. Find

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₃₀ 2.95) 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.

2.96) Determine the force in the members of the truss as shown in figure.

Take E = 200 GPa [AU, May / June – 2012]

2.97) Determine the nodal displacements and the element stresses for the two dimensional

loaded plate as shown in figure. Assume plane stress condition. Body force may be neglected in comparison to the external forces. Take E = 210GPa, µ = 0.25, Thickness t = 10mm. [AU, May / June – 2011]

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₃₁ 2.98) The loading and other parameters for a two bar truss element is shown in figure

Determine [AU, May / June – 2013]

(i) The element stiffness matrix for each element

(ii) Global stiffness matrix (iii) Nodal displacements (iv) Reaction forces

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₃₂ 2.99) Calculate nodal displacement and elemental stresses for the truss shown in Figure.

E= 70Gpa.cross-sectional area A = 2cm2 for all truss members.

[AU, April / May – 2011]

2.100) 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.

2.101) Each of the five bars of the pin jointed truss shown in figure below has a cross

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₃₃

(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, April / May – 2008]

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

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

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₃₄ 2.103) Determine the joint displacements, the joint reactions, element forces and element

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

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

2.104) Determine the force in the members of the truss shown in figure.

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₃₅ 2.105) Find the nodal displacement developed in the planer truss shown in Figure when a

vertically downward load of 1000 N is applied at node 4. The required data are given in the Table. [AU, May / June – 2012]

Element No. ‘e’

Cross – Sectional area A

cm2 Length l

(e) _cm Young’s Modulus E(e) N/mm2

1 2 √2 50 2 * 106

2 2 √2 50 2 * 106

3 1 √2.5 100 2 * 106

4 1 √2 100 2 * 106

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

[AU, Nov / Dec – 2008]

2.107) Why is higher order elements needed? Determine the shape functions of an eight

noded rectangular element. [AU, Nov / Dec – 2007, April / May – 2009]

2.108) Derive the shape functions for a 2D beam element.

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₃₆ 2.109) Derive the shape functions for a 2D truss element.

[AU, Nov / Dec – 2007, April / May – 2008, 2009]

2.110) Derive the interpolation function for the one dimensional linear element with a

length “L” and two nodes, one at each end, designated as “i” and ” j”. Assume the origin of the coordinate system is to the left of node “i”.

[AU, April / May - 2010]

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₃₇ UNIT – III – TWO DIMENSIONAL FINITE ELEMENT ANALYSIS

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) Define Lagrange’s interpolation.

3.4) What is geometric Isotropy? [AU, May / June – 2013]

3.5) Write the Lagrangean shape functions for a 1D, 2 noded elements.

[AU, Nov / Dec – 2008]

3.6) Obtain the shape function for a 1D quadratic Isoparametric element.

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

3.8) Why is the 3 noded triangular element called as a CST element?

[AU, Nov / Dec – 2010]

3.9) Write down the interpolation function of a field variable for three-node triangular

element. [AU, April / May – 2010]

3.10) What is a CST element? [AU, April / May – 2011]

3.11) Draw the shape functions of a CST element. [AU, Nov / Dec – 2010]

3.12) Explain the important properties of CST elements. [AU, Nov / Dec – 2008]

3.13) Write a note on CST element. [AU, May / June – 2011]

3.14) Write briefly about LST and QST elements.

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

3.16) Define LST element. [AU, Nov / Dec – 2012]

3.17) Write the displacement function equation for CST element. 3.18) Write the strain – displacement matrix for CST element.

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₃₈ 3.20) Evaluate the following area integrals for the three node triangular element

∫ [AU, May / June – 2012]

3.21) A triangular element is shown in Figure and the nodal coordinates are expressed in

mm. Compute the strain displacement matrix. [AU, Nov / Dec – 2012]

3.22) What do you mean by the terms : c0,c1 and cn continuity?

[AU, April / May – 2010]

3.23) Distinguish between C0, C1 and C2 continuity elements.

3.24) What are the different problems governed by 2D scalar field variables?

3.25) Use various number of triangular elements to mesh the given domain in the order of

increasing solution refinement.

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₃₉ 3.27) Write the significance of Pascal triangle in developing triangular elements.

3.28) Distinguish one from the other of the following

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

elements.

3.29) What do you mean by area co-ordinate method? 3.30) State the advantage of serendipity element. 3.31) What do you mean by wrapping?

3.32) Write the node numbering and element connectivity table for the given domain

using suitable discretization.

3.33) Plot the variation of shape function with respect node of a 3 noded triangular

element.

3.34) Write down the nodal displacement equations for a two dimensional triangular

elasticity element. [AU, April / May – 2010]

3.35) Define a plane stress condition. [AU, Nov / Dec – 2011]

3.36) State the condition for plane stress problem.

3.37) Give one example each for plane stress and plane strain problems.

[AU, Nov / Dec – 2008]

3.38) Distinguish between plane stress and plane strain problems. [AU, Nov / Dec – 2009] 3.39) Distinguish plane stress and plane strain conditions. [AU, Nov / Dec – 2010]

3.40) Define plane strain with suitable example. [AU, Nov / Dec – 2012]

3.41) Define plane strain analysis. [AU, Nov / Dec – 2011]

3.42) Define a plane stress problem with a suitable example. [AU, May / June – 2013] 3.43) Explain plane stress problem with an example. [AU, April / May – 2011]

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₄₀ 3.45) Write down the strain displacement relation. [AU, April / May – 2011]

3.46) State whether plane stress or plane strain elements can be used to model the

following structures. Justify your answer. [AU, Nov / Dec – 2012]

(a) A wall subjected to wind load

(b) A wrench subjected to a force in the plane of the wrench.

3.47) Write the assumptions used to define the given problem as plane stress problem. 3.48) Write the assumptions used to define the given problem as plane strain problem. 3.49) Using general stress - strain relation, obtain plane stress equation.

3.50) Beginning with general elastic stress-strain relation, derive the plane strain

condition.

3.51) What are the differences between 2 Dimensional scalar variable and vector variable

elements? [AU, Nov / Dec – 2009]

3.52) What are the ways by which a three dimensional problem can be reduced to a two

dimensional problem?

3.53) How to reduce a 3D problem into a 2D problem? [AU, Nov / Dec – 2012]

3.54) Give the stiffness matrix equation for an axisymmetric triangular element. 3.55) What is axisymmetric element?

3.56) Give examples of axisymmetric problems. [AU, May / June – 2012]

3.57) What is an axisymmetric problem? [AU, April / May – 2011]

3.58) Write short notes on axisymmetric problems.

[AU, Nov / Dec – 2007, April / May – 2009]

3.59) What is meant by axi-symetric field problem? Given an example.

[AU, Nov / Dec – 2009]

3.60) State the situations where the axisymmetric formulation can be applied.

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₄₁ 3.61) Give four applications where axisymmetric elements can be used.

[AU, April / May – 2011]

3.62) State the applications of axisymmetric elements. [AU, Nov / Dec – 2010]

3.63) Write down the constitutive relationship for axisymmetric problem.

[AU, April / May – 2009]

3.64) Write down the constitutive relationship for the plane stress problem.

[AU, Nov / Dec – 2010]

3.65) What do you mean by constitutive law and give the constitutive law for

axi-symmetric problems? [AU, April / May, Nov / Dec – 2008]

3.66) Give one example each for plane stress and plane strain problems.

[AU, April / May - 2008]

3.67) Explain plane strain problem with an example. [AU, May / June – 2012]

3.68) Give a brief note on static condensation. 3.69) Prove that 2  0 for plane strain condition.

3.70) Differentiate axi – symmetric and cyclic –symmetric structures.

3.71) Differentiate axi-symmetric load and asymmetric load with examples. 3.72) State the condition for axi-symmetric problem.

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

[AU, April / May – 2010]

3.74) What are the four basic sets of elasticity equations? [AU, May / June – 2012] 3.75) Give examples for the following cases.

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

3.76) Define the following terms with suitable examples [AU, April / May – 2010]

i) Plane stress, plane strain ii) Node, element and shape functions iii) Axisymmetric analysis iv) Iso – parametric element

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₄₂ 3.78) State the effect of Poisson’s ratio in plane strain problem.

3.79) How will the stress field vary linearly?

3.80) Compare the changes in the D matrix evolved out of plane strain, plane stress and

axi-symmetric problem.

3.81) What do you mean by Isoparametric formulations?

[AU, Nov / Dec – 2007, April / May – 2009]

3.82) Express the shape functions of four node quadrilateral element.

[AU, May / June – 2012]

3.83) What do you understand by a natural co – ordinate system?

[AU, April / May – 2011]

3.84) What do you mean by natural co-ordinate system? [AU, May / June – 2011]

3.85) What are the advantages of natural co-ordinates?

[AU, Nov / Dec – 2007, April / May – 2009]

3.86) What are the advantages of natural coordinates over global co-ordinates?

[AU, Nov / Dec – 2008]

3.87) Give a brief note on natural co-ordinate system.

3.88) 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]

3.89) Show the transformation for mapping x-coordinate system onto a natural coordinate

system for a linear spar element and for a quadratic spar element.

[AU, Nov / Dec – 2012]

3.90) Define a local co – ordinate system. [AU, Nov / Dec – 2011]

3.91) What is area co – ordinates? [AU, Nov / Dec – 2011]

3.92) What do you understand by area co – ordinates? [AU, April / May – 2011]

3.93) State the basic laws on which Isoparametric concept is developed.

[AU, April / May – 2008]

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₄₃ 3.95) Define super parametric element. [AU, April / May – 2009]

3.96) Explain super parametric element. [AU, Nov / Dec – 2010]

3.97) Define Isoparametric elements? [AU, Nov / Dec – 2008]

3.98) Define Isoparametric elements with suitable examples [AU, April / May – 2010] 3.99) Define Isoparametric element formulations. [AU, Nov / Dec – 2012]

3.100) What do you mean by Isoparametric formulation? [AU, April / May – 2011]

3.101) What is the purpose of Isoparametric elements?

3.102) What are the applications of Isoparametric elements? [AU, April / May – 2011] 3.103) Differentiate x – y space and - space.

3.104) Write the advantages of co-ordinate transformation from Cartesian co-ordinates to

natural co-ordinates.

3.105) What is a Jacobian? [AU, Nov / Dec – 2010]

3.106) What is the need of Jacobian? [AU, April / May – 2011]

3.107) Write down the Jacobian matrix. [AU, Nov / Dec – 2010]

3.108) Write about Jacobian transformation used in co-ordinate transformation.

3.109) What is the significance of Jacobian of transformation? [AU, May / June – 2012] 3.110) Differentiate between sub-parametric, iso- parametric and super – parametric

elements.

3.111) Represent the variation of shape function with respect to nodes for quadratic

elements in terms of natural co-ordinates.

3.112) Compare linear model, quadratic model and cubic model in terms of natural

co-ordinate system.

3.113) Write a brief note on continuity and compatibility.

3.114) Write down the element force vector equation for a four noded quadrilateral

element.

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₄₄ 3.116) Write the shape function for the quadrilateral element in ,  space.

3.117) Why is four noded quadrilateral element is preferred for axi-symmetric problem

than three noded triangular element?

3.118) Sketch a four node quadrilateral element along with nodal degrees of freedom.

[AU, April / May – 2011]

3.119) Write down the stiffness matrix for four noded quadrilateral elements.

[AU, May / June – 2011]

3.120) Distinguish between essential boundary conditions and natural boundary

conditions. [AU, Nov / Dec – 2009]

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

3.122) What are the types of non-linearity?

[AU, Nov / Dec – 2007, April / May – 2009, May / June – 2012]

3.123) State the advantage of Gaussian integration. 3.124) State the four-point Gaussian quadrature rule.

3.125) Briefly explain Gaussian quadrature. [AU, April / May – 2011]

3.126) What are the advantages of Gaussian quadrature? [AU, Nov / Dec – 2012]

3.127) What are the weights and sampling points of two point formula of Gauss

quadrature formula? [AU, May / June – 2012]

3.128) Why numerical integration is required for evaluation of stiffness matrix of an

Isoparametric element? [AU, Nov / Dec – 2011]

3.129) Write the Gauss points and weights for two point formula of numerical integration.

[AU, April / May – 2011]

3.130) Write down the Gauss integration formula for triangular domains.

[AU, April / May – 2009]

3.131) Evaluate the integral

_{∫ (}

) using Gaussian quadrature method.

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₄₅ 3.132) Name the commonly used integration method in natural – co-ordinate system.

3.133) Write the relation between weights and Gauss points in Gauss-Legendre quadrature. PART – B

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

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

3.135) What are shape functions? Derive the shape function for the three noded triangular

elements. [AU, Nov / Dec – 2011]

3.136) Derive the element strain displacement matrix and element stiffness matrix of a

CST element. [AU, April / May – 2011]

3.137) Explain the terms plane stress and plane strain problems. Give the constitutive

laws for these cases. [AU, Nov / Dec – 2007, April / May – 2009]

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

[AU, Nov / Dec – 2007, 2008, April / May – 2009]

3.139) 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]

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

[AU, April / May - 2008]

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

[AU, April / May – 2008]

3.142) Derive the stiffness matrix and equations for a LST element.

[AU, Nov / Dec – 2012]

3.143) Derive the element strain displacement matrix and element stiffness matrix of a

triangular element. [AU, May / June – 2012]

3.144) A two noded line element with one translational degree of freedom is subjected to a

uniformly varying load of intensity P1 at node 1 and P2 at node 2. Evaluate the nodal load vector using numerical integration. [AU, Nov / Dec – 2012]

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₄₆ 3.145) Calculate the element stresses x,y,xy, ₁ and ₂ and the principle angle _p for

the element shown below.

3.146) 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.

3.147) 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]

3.148) For the triangular element shown below, obtain the strain – displacement relation

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₄₇ 3.149) Consider the triangular element show in Figure. The element is extracted from a

thin plate of thickness 0.5 cm. The material is hot rolled low carbon steel. The Nodal co-ordinates are xi =0, yi = 0, xj =0, yj = -1, xk =0, yk = -1 cm,. Determine the

elemental stiffness matrix. Assuming plane stress analysis. Take µ = 0.3 and E =

2.1*107 N/cm2 [AU, May / June – 2012]

3.150) Derive the interpolation function 14 for the quadratic triangular element as shown below.

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₄₈

3.151) Derive the interpolation function of a corner node in a cubic serendipity element. 3.152) Find the expression for nodal vector in a CST element shown in figure subjected to

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

3.153) For the CST element given below, assemble stain displacement matrix. Take t = 20

mm and E = 2*105 N/mm2 [AU, Nov / Dec - 2008]

3.154) Calculate the value of pressure at the point A which is inside the 3 noded triangular

elements as shown in figure. The nodal values are φ1 = 40 MPa, φ2 = 34 MPa and φ3 = 46 MPa, Point A is located at (2, 1.5) Assume pressure is linearly varying in the

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₄₉

element. Also determine the location of 42 MPa contour line.

[AU, May / June – 2013]

3.155) Obtain the global stiffness matrix for the plate shown in figure. Taking two

triangular elements. Assume plane stress condition. [AU, May / June – 2012]

3.156) 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.

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₅₀

3.157) For the plane strain element shown in the figure, the nodal displacements are given

as : u1= 0.005 mm, u2 = 0.002 mm, u3=0.0mm, u4 = 0.0 mm, u5 = 0.004 mm, u6 = 0.0 mm. Determine the element stresses. Take E = 200 Gpa and  = 0.3. Use unit thickness for plane strain. [AU, April / May - 2010]

3.158) For the two-dimensional loaded plate as shown in Figure. Determine the nodal

displacements and element stress using plane strain condition considering body force. Take Young’s modulus as 200 GPa, Poisson’s ration as 0.3 and density as 7800 kg/m3. [AU, April / May – 2011]

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₅₁ 3.159) Derive element force vector when linearly varying pressure acts on the side joining

nodes jk of a triangular element shown in Figure and body force of 25N/mm2 acts downwards. Thickness = 5mm. [AU, April / May – 2011]

3.160) For the plane stress element whose coordinates are given by (100,100), (400, 100)

and (200, 4000, the nodal displacements are u1 = 2.0mm, v1 =l.0mm, u2 =l.0mm, v2

=1.5mm, u3 = 2.5mm, v3 = 0.5mm. Determine the element stresses. Assume E =

200 GN/m2, µ = 0.3 and t = 10 mm. All coordinates are in mm.

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₅₂ 3.161) A thin plate a subjected to surface fraction as shown in Figure. Calculate the global

stiffness matrix. Table t = 25 mm, E = 2 *105 N/mm2 and γ = 0.30. Assume plane

stress condition. [AU, Nov / Dec – 2011]

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

3.163) Calculate nodal displacement and elemental stresses for the truss shown in Figure.

E = 70 GPa cross-sectional area A = 2 cm2 for all truss members.

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₅₃ 3.164) A thin elastic plate subjected to uniformly distributed edge load as shown below.

Find the stiffness and force matrix of the element.

3.165) For the configuration as shown in figure determine the deflection at the point load

applications. Use one model method. Assume plane stress condition.

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₅₄ 3.166) Derive the expression for the element stiffness matrix for an axisymmetric shell

element. [AU, Nov / Dec – 2007, April / May – 2009]

3.167) Describe the step by step procedure of solving axisymmetric problem by finite

element formulation. [AU, May / June – 2012]

3.168) Derive an expression for the stiffness matrix of an axisymmetric element.

[AU, April / May – 2011]

3.169) For an axisymmetric triangular element. Obtain the [B] matrix and constitutive

matrix [AU, Nov / Dec – 2010]

3.170) Derive the stress-strain relationship matrix (D) for the axisymmetric triangular

element. [AU, Nov / Dec – 2012]

3.171) Explain the modeling of cylinders subjected to internal and external pressure using

axisymmetric. [AU, Nov / Dec – 2011]

3.172) For a thick cylinder subjected to internal and external pressure, indicate the steps of

finding the radial stress. [AU, Nov / Dec – 2010]

3.173) Derive the material property matrix for axisymmetric elasticity.

[AU, Nov / Dec – 2011]

3.174) 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]

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₅₅ 3.175) 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.

3.176) Determine the stiffness matrix for the axisymmetric element shown in fig, Take E as

2.1* 106 N/mm2 and Poisson's ratio as 0.3. [AU, Nov / Dec – 2012]

3.177) Determine the element stresses for the axisymmetric element as shown below. Take

FINITE ELEMENT ANALYSIS QUESTION BANK by ASHOK KUMAR.R (AP / Mech) ₅₆

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

3.178) Compute the strain displacement matrix for the following axisymmetric element.

Also calculate the element stress vectors. If [q] = [ 3.484 0 3.321 0 0 0]T * 10-3 cm

[D] = [

] [AU, April / May – 2011]

3.179) An open ended steel cylinder has a length of 200mm and the inner and outer

diameters as 68mm and 100mm respectively. The cylinder is subjected to an internal pressure of 2MPa. Determine the deformed shape and distribution of principle stresses. Take E = 200GPa and Poisson’s ratio = 0.3