Question Bank With Year -Finite Element Analysis

QUESTION BANK ME6603 – FINITE ELEMENT ANALYSIS UNIT -1 INTRODUCTION PART – A List out the various methods of solving boundary value problems. What is meant by post processing? What are the methods generally associated with the finite element analysis? If a displacement field in x direction is given by u =2x 2+4y2+6xy. Determine the strain in x direction. Write about weighted residual method. State the principle of minimum potential energy theorem. What is Rayleigh-Ritz method? What is meant by node? What is discretization? Distinguish between Error in solution and residual What are the advantages of weak formulation? List the various weighted residual methods Compare the Ritz technique with the nodal approximation method Mention the basic steps of Galerkin method. What are non homogenous boundary conditions? Give an example Compare the Rayleigh Ritz method with Nodal approximation method on what basis, collocation points are selected?

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PART – B 1

A beam AB of span ‘l’ simply, supported at ends and carrying a concentrated load W at the center ‘C’ as shown in fig. Determine the deflection at midspan by using Rayleigh-Ritz method and compare with exact solution

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2

A physical phenomenon is governed by the differential equation (d 2w/dx2)-10x2 =5 for 0 ≤ x ≤ 1. The boundary conditions are given by u(0) = w(1) =0. Assuming a trial solution w(x) = a0+a1x+a2x2+a3x3. Determine using Galerkin method the variation of “w” with respect to x. The following differential equation is available for a physical phenomenon (d2y /dx2 -10x2 = 5) : 0 ≤ x ≤ 1. The boundary conditions are : y(0) = 0 y(1) = 0.

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3

1

M. Saravana Kumar M.E

M/J 16

FMCET/Assistant Professor/Mech

Find an approximate solution of the above differential equation by using Galerkin’s method of weighted residuals and also compare with exact solution. Solve the following differential equation using Galerkin’s method of weighted

4

residuals.

2

2

+

= 4 ,0 ≤

≤ 1 with boundary conditions y(0)=0, y(1) =1.

M/J 16 M/J 14

5

A concentrated load P = 50kN is applied at the center of a fixed beam of length 3 M/J 16 m, depth 200 mm and width 120 mm. Calculate the deflection and slope at the M/J 12 5 2 midpoint. Assume E = 2×10 N/mm (II UNIT)

6

Find the eigen value and the corresponding eigen vector for the matrix

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7

Determine using any Weighted Residual technique the temperature distribution along a circular fin of length 6 cm and radius 1 cm. The fin is attached to a boiler whose wall temperature is 140°C and the free end is insulated. Assume convection coefficient h=10 W/cm2 ° C. Conduction coefficient K = 70 W/cm2 °C and T∞ = 40°C. The governing equation for the heat transfer through the fin is given by

A/M 15

− 8

( )

+ ℎ ( )( −

∞)

=0

Derive the governing equation for a tapered rod fixed at one end and subjected to its own self weight and a force P at the other end as shown in Figure. Let the length of the bar be l and let the cross section vary linearly from A1 at the top

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fixed end to A2 at the free end. E and γ represent the young’s modulus and specific weight of the material of the bar. Convert this equation into its weak form and hence determine the matrices for solving using the Ritz Technique

9 10

Write briefly about weighted residual methods A simply supported beam is subjected to uniformly distributed load over entire

2


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span as shown in fig. Determine the bending moment and deflection at midspan by using Rayleigh Ritz method.

11

List and briefly describe the general steps of the finite element method.

12

Using Collocation method, find the maximum displacement of the tapered rod as shown in fig. E = 2×107 N/cm2, ᵞ = 0.075 N/cm3

13 14

Describe the historical background of fem. Explain the relevance of FEA for solving design problems with the aid of examples.

15

The differential equation for a phenomenon is given by

2

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= 0; 0≤x≤5.

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Derive an equation to find the displacement at node 2 of fixed-fixed beam subjected to axial load P at node 2 using Rayleigh-Ritz method.

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2

− 500

2

The boundary condition are y(0) = 0, y(5)=0. Find the approximate solution using any classical technique. Start with minimal possible approximate solution. 16

3



ME6603 – FINITE ELEMENT ANALYSIS UNIT -2 ONE DIMNESIONAL PROBLEMS PART – A Why polynomials are generally used as shape fuctions? What is dynamics problem? Write down the expression of governing equation free axial vibration of rod and transverse vibration of beam. What is the stationary property of total potential energy Give the Lagrange equation of motion and obtain the shape functions for quadratic coordinate transformation. Write about the effective global nodal forces of beam element What are the boundary conditions in FEA heat transfer problem Consistent mass matrix gives accurate results than lumped mass matrix in dynamics analysis of bar element? Write down the expression of stiffness matrix for a truss element. What is meant by transverse vibration Derive the shape functions for a 1D quadratic bar element What are the properties of the stiffness matrix? Derive the mass matrix for a 1D linear bar element Write down the governing equation and for 1 D longitudinal vibration of a bar fixed at one end and give the boundary conditions

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

Derive the convection matrix for a 1-D linear bar element Write the stiffness matrix for a 1 D two noded linear element Give the properties of shape function Write the analogies between structural, heat transfer and fluid mechanics Define shape function What is a truss? What is meant by transverse vibrations? Write down the governing equation for two dimensional steady state heat conduction Define streamline What is meant by transverse vibrations? What is the need for coordinate transformation in solving truss problems? Illustrate the two Hermite shape functions associated with slope as applicable for beam element. What are force vectors? Give example Define p-refinement What are the factors which govern the selection of nodes?

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PART - B 1

For the bar element shown in fig. Calculate the nodal displacements and element stresses. Assume the material as steel. P = 20 kN.

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2

Determine the eigen values for the stepped bar shown in fig.

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3 4

Derive the shape function for One-dimensional Quadratic Bar element. A steel bar of length 800 mm is subjected to an axial load of 3 kN as shown in fig. Find the nodal displacements of the bar, and load vectors.

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6

Determine the nodal displacements, element stresses and support reactions in the truss element shown in figure. Assume that points 1 and 3 are fixed. Take E= 70 Gpa, and A =200 mm2.

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5



7

For the beam shown in fig, determine the displacements and the slopes at the nodes, the forces in each element and the reactions. E = 200 Gpa, I = 1×10-4 m4

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9

The figure shows a uniform Aluminium fin of diameter 25 mm. The root (left end) of the fin is maintained at a temperature of T∞ =120 oC, convection takes place from the lateral (circular) surface and the right (flat) edge of the fin. Assuming k= 200 W/m oC, h=1000 W/m2 oC and T = 20 oC, determine the temperature distribution in the fin using one dimensional element, considering two elements.

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10

Set up the system of equations governing the free transverse vibrations of a M/J 16 simply supported beam modeled by two finite elements. Determine the natural frequency of the system. Determine the maximum deflection and slope in the beam, loaded as shown in A/M 15 Figure. Determine also the reactions at the supports. E = 200 GPa, I=20 × 10-6 m4, q = 5 kN/m and L= 1m.

11

12

13

Derive using Lagrangian Polynomials the shape functions for a one dimensional A/M 15 three noded bar element. Plot the variation of the same. Hence derive the stiffness matrix and load vector. Determine the first two natural frequencies of transverse vibration of the A/M 15 cantilever beam shown in fig and plot the mode shapes

6



14

Determine the first two natural frequencies of longitudinal vibration of the bar shown in Fig. assuming that the bar is discretized into two elements as shown. E and ρ represent the young’s modulus and the mass density of the material of the bar.

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15

A Composite wall consists of three materials as shown in fig. The inside wall temperature is 200 °C and the outside air temperature is 50 °C with a convection coefficient of 10 W/cm2 °C. Determine the temperature along the composite wall.

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16 17

Derive the stiffness matrix for 2D truss element For the bar shown in fig. with length 2L, modulus of elasticity E, mass density e, and cross sectional area A, determine the first two natural frequencies.

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18

Determine the eigen values and natural frequencies of a system whose stiffness and mass matrices are given below

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7



19 20

Derive an expression for temperature function and shape function for one dimensional heat conduction element. A two noded truss element is shown in Figure. The nodal displacements are u 1 =

N/D 15 M/J 14

5 mm and u2 = 8 mm. Calculate the displacement at x =

16

For the two bar truss shown in fig, determine the displacements of node 1 and the stress in element 1-2

17

Derive the equation of motion based on weak form for transverse vibration of a M/J 14 beam. Derive a finite element equation for one dimensional heat conduction with free M/J 14 end convection. Consider the bar shown in fig. Determine the nodal displacements, element stresses and support reactions. N/D 14

18 19

20 21

Compute the slope, deflection and reaction forces for the cantilever beam of length ‘L’ carrying Uniformly distributed load of intensity ‘fo’. Determine the nodal displacements, stress and strain for the bar shown in fig.

8


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22

Determine the first two natural frequencies of longitudinal vibration of the stepped steel bar shown in fig. and plot the mode shapes. All dimensions are in mm. E= 200 Gpa and density 0.78 kg/cc

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23

Determine the temperature distribution along a circular fin of length 5 cm and radius 1 cm. The fin is attached to a boiler whose wall temperature 140 °C and the free end is open to the atmosphere. Assume T∞ =40 °C , h= 10 W/cm °C, k= 70 W/cm °C. A rod fixed at its end is subjected to a varying body force as shown in Fig 1. Use the Rayleigh – Ritz method with an assumed displacement filed u=a0+a1x+a2x2 to determine displacement u(x) and stress σ(x).

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25

Determine the extension of the bar shown in Fig. due to self weight and a concentrated load of 600N applied at its end. Given b1= 200 mm. b2 = 100 mm and t = 20 mm. Use two spar elements to solve the problem. Take E=2×10 5 N/mm2.

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26

A cantilever beam of length 3.4 m has an elastic spring support of stiffness 230 kN/m at its free end, where a point load of 13 kN acts. Take Young’s modulus as 200 Gpa and area moment of inertia of the cross section as 1×10-4 m4. Determine the displacement and slope at the node and the reactions.

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24

9


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27

Discuss the importance of FEA in assisting design process. Solve the ordinary differential equation

2

2

+ 10x2 = 0 , 0≤ x ≤1 Subject to

boundary conditions y(0) = y(1) = 0 using the Galerkin method with the trial functions NO (x) = 0; N1= x(x-x2). A circular fin 0f 40 mm diameter is fixed to a base maintained at 50°C as shown in figure. The fin is insulated on the surface except the end face which is exposed to air at 25°C. The length of the pin is 1000 mm, the fine is made of metal with thermal conductivity of 37W/m K. If the convection heat coefficient with air is 15 W/m2 K. If the convection heat coefficient with air is 15 W/m2 K. Find the temperature distribution at 250, 500, 750 and 1000 mm from base.

28

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10

ME6603 – FINITE ELEMENT ANALYSIS UNIT –III TWO DIMNESIONAL SCALAR VARIBABLE PROBLEM PART – A State the assumptions in the theory of pure torsion. What is an LST element? Define path line and stream line Write a displacement function equation for CST element. What is CST element? Mention two natural boundary conditions as applied to thermal problems Write down the shape functions associated with the three noded linear triangular elements and plot the variation of the same. Write down the conduction matrix for a three noded linear triangular element Why a CST element so called? How do you define two dimensional elements Write down the governing equation for two dimensional steady state heat conduction. Specify the strain displacement matrix of CST element and comment on it Specify the shape fuctions of four node quadrilateral element Distinguish between CST and LST elements Write the finite element equation used to analyze a two dimensional heat transfer problem


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1

The x,y coordinates 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 of the element. Evaluate the straindisplacement relation matrix B for the above same triangular element and explain how stiffness matrix is obtained assuming scalar variable problem.

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2

Calculate the temperature distribution in the stainless steel fin shown in fig. The region can be discretized into 3 elements of equal size.

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3

Calculate the element stiffness matrix and the temperature force vector for the plane stress element. The element experiences a 20 0C increase in temperature. Assume coefficient of thermal expansion is 6×10-6 C. Take E= 2×105 N/mm2, ʋ = 0.25, t= 5mm.

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4

Derive the shape function for constant strain triangular element.

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5

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, u3 = 0.005 mm, v3 = 0.30 mm. Determine the element stresses and the principle angle. Take E = 70 Gpa and Poisson’s ration = 0.3 and use unit thickness for plane strain. All coordinates are in mm.

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6

Derive the element characteristics of a nine noded quadrilateral element.

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7

A bilinear rectangular element has coordinates as shown in Fig and the nodal

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11


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temperatures are T1 = 100 0C, T2 = 60 0C, T3 = 50 0C, T4 = 90 0C. Compute the temperature at the point whose coordinates are (2.5, 2.5).Also determine the 0 80 C isotherm.

8

A plate of dimensions 15 cm×6cm×1cm is subjected to an axial pull of 5kN. Assuming a typical element is of dimensions as shown in the Fig. Determine the strain displacement matrix and constitutive matrix. E=200Gpa, µ =0.3, t=10 mm.

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9

A two dimensional fin is subjected to heat transfer by conduction and convection. It is discretised as shown in Fig, into two elements using linear triangular elements. Derive the conduction, and thermal load vector. How is convection accounted for in solving the problem using Finite element method?

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10

For the constant strain triangular element shown in fig. Assemble straindisplacement matrix. Take t =20 mm and E= 2×105 N/mm2

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12



11 For the isoparametric four noded quadrilateral elements shown in Fig. Determine the Cartesian co-ordinates of point P which has local coordinates ε= 0.5 η= 0.5

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12

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

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13

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

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14

Derive the Jacobian matrix for triangular element with the (x,y) coordinates of the nodes ar (1.5,2), (7.3, 2), and (4,7) at nodes i,j,k. (8)

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Find the Jacobian transformation for four noded quadrilateral element with the (x,y) coordinates of the nodes are (0,0), (2,0), (2,1) and (0,1) at nodes I,j,k,l. Also find the Jacobian at point whose natural coordinates are (0,0) 15

Determine the stiffness matrix for the triangular elements with the (x,y) coordinates of the nodes are (0,4), (8,0) and (0,4) at nodes i,j,k. Assume plane stress condition E=200 Gpa, Poisson’s ratio = 0.35.

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16

Compute the steady state temperature distributions in the plate shown in Fig. by discretizing the domain of interest using triangular elements. Assume Thermal Conductivity k = 1.5 W/m 0C

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13



17

Derive the characteristic matrix for a two dimensional heat conduction problem using triangular element by Galerkin approach.

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18

Consider a rectangular plate of length 2500 mm and width 2500 mm having a thickness of 300 mm. It is subjected to a uniform heat source of 200 W/m3 acting over the whole body. The temperature of the top side of the body is maintained at 130 oC. The body is insulated on the other edges. Take the thermal conductivity of the material as 35 W/m oC. Determine the temperature distribution using triangular element.

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19

A triangular plate of thickness 9 mm has vertices P(40,40), Q(100,40) and R(70,130). It is fixed at P and supported on rollers at Q. There is a vertical downward load of 5 kN applied at R. Take Youngs modulus as 200 Gpa. Determine the nodal displacements accounting for body weight. Take density of material is 7800 kg/m3.

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20

A four node quadrilateral element is defined by nodal coordinates (in ‘mm’) as 1(3,8), 2(10,5), 3(12,18) and 4(5,16). The nodal displacement vector is given by q = [0,0.2,0.1,1.6,1.2,0,0.6]T.

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Evaluate the stress at the point P(7,12) of the element, assuming plane stress condition. Take young’s modulus and poison’s ratio as 30×10 6 N/m2 and 0.3 respectively. 21

Derive the body force and traction (uniformly distributed) force vectors for four node quadrilateral element.

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22

Derive the element characteristic of a four node quadrilateral element.

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23

Calculate displacements and stress in the given triangular plate, fixed along one edge and subjected to concentrated load at its free end. Take E=70Gpa, thickness of the plate = 10 mm and poisson’s ratio =0.3

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ME6603 – FINITE ELEMENT ANALYSIS UNIT -1V TWO DIMNESIONAL SCALAR VARIBABLE PROBLEM PART – A

1 2 3 4 5 6 7 8 9 10

What is meant by plane stress analysis? Write the strain-displacement matrix for a CST element? Write down the stress-strain relationship matrix for an axissymmetric element. What are the types of shell element? List the application of axisymmetric element Give the applications of plane stress and plane strain problems What are the ways by which a 3D problem can be reduced to a 2D problem? What is QST (Quadratic strain Triangle) element? Define dynamic analysis Sketch a finite element model for a long cylinder subjected to an internal pressure using axisymmetric elements. Distinguish between plane stress and plane strain conditions

11

PART – B For the triangular element as shown in the Fig. Determine the straindisplacement matrix [B] and constitutive matrix [D]. Assume plane stress conditions. Take µ = 0.3, E= 30 × 106 N/m2 and thickness t= 0.1 m. Also calculate the element stiffness matrix for the triangular element.

1

2

For the axisymmetric element shown in the fig. Determine the stiffness matrix. Let E= 2.1×103 N/mm2 and µ = 0.25. The coordinates are in mm.

15


N/D 16 N/D 16 M/J 16 M/J 16 M/J 16 13 M/J 16 15 N/D 14 M/J 14 N/D 13 N/D 13 N/D 13

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3

The nodal co-ordinates for an axisymmetric triangular element are given in figure. Evaluate strain – displacement matrix for that element

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4

Calculate the element stiffness matrix for the axisymmetric triangular element shown in Fig. The element experiences a 15 °C increases in temperature. The coordinates are in mm. Take α = 10×10-6/oC, E= 2×105 N/mm2, ν =0.25. The coordinates are (9,10) (6,7) (8,7).

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5

Establish the shape functions and derive the strain displacement matrix for an axisymmetric triangular element. What are the non-zero strain and stress components of axisymmetric element? Explain. Derive the stiffness matrix of an axisymmetric element using potential energy approach. Establish the traction force vector and estimate the nodal forces corresponding to a uniform radial pressure of 7 bar acting on an axisymmetric element as shown in Fig. Take E=200 Gpa and Poisson’s ratio =0.25.

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6

7

16


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ME6603 – FINITE ELEMENT ANALYSIS UNIT -V ISOPARAMETRIC ELEMENT PART – A What is the purpose of isoparametric elements?

1 2

What are the advantages of Gauss quadrature numerical integration for isoparametric elements What is the difference between natural coordinates and local coordinates? What type of analysis preferred in FEA when the structural member subjected to transient vibrations? What are the advantages of Natural Co-ordiante system? Write the shape functions for 1D quadratic iso parameteric element. Name a few FEA packages What is the principle of skyline solution based on Gaussian elimination? What are superparametric elements? Give an example State the advantages of Gaussian elimination technique What is post processing? Give an example What is meant by natural coordinate system?

3 4 5 6 7 8 9 10 11 12 1

2

N/D 14 N/D 14 N/D 14 N/D 13 N/D 13 M/J 13 M/J 13 N/D 12 N/D 16

Evaluate the integral by two point Gaussian quadrature I =∬ (2

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2

2

+3

. Gauss points are +0.57735 and -0.57735 each of weight 1.0000

+

Evaluate the Cartesian coordinate of the point P which has local coordinates ε= 0.6 η= 0.8

17

M/J 16 M/J 16

PART – B Evaluate the Jacobian matrix for the linear quadrilateral element as shown the fig.

4

3

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4

For the four noded elements shown in Fig. (i) determine Jacobian and evaluate its value at the point (1/2,1/3)

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5

Using Gauss Quadrature evaluate the following integral.

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6

Evaluate the Integral ∫ ( x+x2+

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)dx using Gaussian Integration with one,

two and three integration points. Number of Points Location 1 0

±1/ √3 ± 0.774597

2 3 7 8

9

10

1 0.55555, 0.88888

Solve the following equations using the gauss elimination method. 2x1 +3x2+x3 =9 : x1+2x2+3x3 = 6 : 3x1+x2+2x3 =0. Consider the Isoparametric quadrilateral element with nodes 1-4 at (5,5), (11,7), (12,5) and (4,10) respectively. Computer the Jacobian matrix and its determinant at the element centroid Use Gaussian quadrature with two points to evaluate the integral. The Gaussian points are ± 0.5774 and weights at the two points are equal to unity

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The nodal displacements of a rectangular element having nodal coordinates (0,0), (4,0), (4,2) and (0,2) are u1=0 mm, v1= 0mm, u2=0.1 mm, v2 =0.05 mm, u3=0.05 mm v3=-0.05, u4=0 and v4 = 0 mm respectively. Determine the stress matrix at r=0 and s=0 using the isoparametric formulation. Take E=210 Gpa and Poisson’s ratio =0.25. Evaluate the following

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(i)

18

Weights 2

I = ∫ [ 2+ cos ]


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(ii) I=∫ [3 x –x] dx


Question Bank With Year -Finite Element Analysis

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