FEAQPUPTO APR2010

8/3/2019 FEAQPUPTO APR2010

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B.E./B.Tech. DEGREE EXAMINATION, APRIL/MAY 2010

Seventh Semester

Mechanical Engineering

ME 1401 INTRODUCTION OF FINITE ELEMENT ANALYSIS(Common to Automobile Engineering and Mechatronics Engineering)

Time: Three hours Maximum : 100 marks

Answer ALL questions

PARTA (10x2 = 20 marks)

1. What is the limitation of using a finite difference method?2. List the various methods of solving boundary value problems.3. Write down the interpolation function of a field variable for three-node triangular element.4. Highlight at least two rules to guide the placement of the nodes when obtaining

approximate solution to a differential equation.

5. List the properties of the global stiffness matrix.6. List the characteristics of shape functions7. What do you mean by the terms : c, c1 and cn continuity?8. Write down the nodal displacement equations for a two dimensional triangular elasticity

element.

9. List the required conditions for a problem assumed to be axisymmetric.10.Name a few boundary conditions involved in any heat transfer analysis.

PART B (5 x l2 =60 marks)

11.(a)Discuss the following methods to solve the given differential equation

With the boundary conditions y(0)=0 und y(H) = 0

(i) Variational method(ii) Collocation method.

Or

(b) For the spring system shown in Figure 1, calculate the global stiffness matrix,

displacements of nodes 2 and 3, the reaction forces at node 1 and 4, Also calculate the forces inthe spring 2. Assume, k1 =k3 =100N/m, k2 = 200 N/m, ul = u4 = 0 and P = 500 N.

. Figure 1 Spring System Assembly

12.(a) Determine the joint displacements, the joint reactions, element forces and element stresses ofthe given truss elements.


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(b) Derive the interpolation function for the one dimensional linear element with a length 'L' andtwo nodes, one at each end, designated as i & j. Assume the origin of the coordinate system is to

the left of node 'i',

13.(a) Determine three points on the 50C contour line for the rectangular element shown in theFigure 4. The nodal values are i= 42C, j =54C, k=56C and m =46C.


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(b) The simply supported beam shown in Figure 5 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.

14. (a) For the plane strain element shown in the Figure 6, the nodal displacements are given

as : u1= 0.005 mm, u2 =0.002 mm, u3 =0.0 mm, u4 =0.0mm, u5 =0.004 mm, u6 = 0.0mm. Determine theelement stresses. Take E = 200 Gpa and = 0.3. Use unit thickness for plane strain.

Or ;

(b) Determine the element stiffness matrix and the thermal load vector for the plane stress element

shown in Figure 7. The element experiences 20C increase in temperature. Take E = 15,e6.N/cm2,

= 0.25 , t = 0.5 cm and a = 6e-6/C.


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Or

(b) Define the following terms with suitable examples :(i) Plane stress, Plane strain

(ii) Node, Element and Shape functions

(iii) iso-parametric element

(iv) Axisymmetric analysis


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B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2009.

Seventh Semester


ME 1401 INTRODUCTION OF FINITE ELEMENT ANALYSIS

(Regulation 2004)

(Common to Automobile Engineering and Mechatronics Engineering)

Time: Three hours Maximum: 100

marks

Answer ALL questions.

PART A (10 x 2 = 20 marks)

1. Distinguish between ID bar element and ID beam element.2. What is Galerkin method of approximation?3. What are CST & LST elements? .4. What is a shape function?5. State the properties of stiffness matrix.6. Write down the governing differential equation for a two dimensional steady-state heat transfer

problem.

7. What is meant by axi-symetric field problem? Give an example.8. Distinguish between plane stress and plane strain problems.9. What are the differences between 2 Dimensional scalar variable and vector variable elements?10.Distinguish between essential boundary conditions and natural boundary conditions.

PART B (5 x 16 = 80 marks)

11.(a) (i) What is constitutive relationship? Express the constitutive relations for a linear elasticisotropic material including initial stress and strain. (6)

(ii) Consider the differential equation (d2

y / dx2) + 400x

2=0 for 0 < x < 1 subject

to boundary conditions y(0) = 0; y(l) = 0 .

The functional corresponding to this problem, to be extremized is given by

I= {

-0.5(dy/dx)

2+400x

2y}

Find the solution of the problem using Rayleigh-Ritz method by considering a two-term solution

as y(x) = C1x(l - x) + C2x2(1 - x). (10)

Or

(b) (i) A physical phenomenon is governed by the differential equation ;

(d2w/dx

2) -10x

2=5 for 0 < x < l.

The boundary conditions are given by w(0) = w(l) = 0. By taking a two-term trial solution

as w(x) = Clf1(x)+C2f2(x) with f1(x) = x{x-l) and f'2{x) = x2(x - 1), find the

solution of the problem using the Galerkin method. (10)


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(ii) Solve the following system of equations using Gauss elimination method. (6)

xl +3x2 +2x3 = 13

-2x1+x2-x3 = -3

-5x1 +x2 +3x3 =6

12. (a) The stepped bar shown in Fig. 1 is subjected to an increase

in temperature, T = 80C. Determine the displacements, element stresses and support reactions.

(b) Consider a two-bar truss supported by a spring shown in Fig. 2. Both bars have E = 210 GPa and

A =5.0xl0-4

m2. Bar one has a length of 5 m 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.

13. (a) (i) 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 Nl,N2 and N3

at an interior point P (2, 2.5)

mm for the element. (4)

(ii) For the same triangular element, obtain the strain-displacement relation matrix B. (12)

Or

(b) Compute element matrices and vectors for the element shown in Fig. 3, when the edge kj

experiences convection heat loss. . '


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14. (a) The triangular element shown in Fig. 4 is subjected to a constant pressure 10 N/mm 2 along the

edge ij. Assume E - 200 GPa, Poisson's ratio = 0.3 and thickness of the element = 2 mm. The

coefficient of thermal expansion of the material is a = 2xl0-6/C and T=50C. Determine theconstitutive matrix (stress-strain relationship matrix D) and the nodal force vector for the element.

Or

(b) 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.

15. (a) (i) The Cartesian (global) coordinates of the corner nodes of a quadrilateral element are given by

(0, -1), (-2, 3), (2, 4) and (5, 3). Find the coordinate transformation between the global and local

(natural) coordinates. Using this, determine the Cartesiancoordinates of the point defined by (r, s) = (0.5, 0.5) in the global coordinate system. (8)

(ii) Evaluate the integral . . . .

Or

(b) (i) The Cartesian (global) coordinates of the corner nodes of an isoparametric quadrilateral element

are given by (1, 0), (2, 0), (2.5, 1.5) and (1.5, 1). Find its Jacobian matrix. (12)

(ii) Distinguish between subparametric and superparametric elements. (4)


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B.E./B.Tech. DEGREE EXAMINATION, MAY/JUNE 2009.

Seventh Semester

Automobile Engineering

ME 1401 -- INTRODUCTION OF FINITE ELEMENT ANALYSIS

(Common to Mechanical Engineering and Mechatronics Engineering)

Time : Three hours Maximum : 100 marks



1. Write down the boundary conditions of a cantilever beam AB of span L fixed at A and free at Bsubjected to a uniformly distributed load of P throughout the span.

2. Write down the potential energy function for a three dimensional deformable body in terms ofstrains and displacements.

3. Draw the shape functions of a two noded line element.4. Draw the shape functions of a one dimensional line element with three nodes.5. Write down the lumped mass matrix for the truss element6. Consider a wall of a tank containing a hot liquid at a temperature To with an air stream of

temperature Tx passed on the outside, maintaining a wall temperature of TL at the boundary.

Specify the boundary conditions.7. Write down the constitutive relationship for. the axi-symmetric problem.8. Distinguish between Lagrange and Hermition interpolation functions.9. Define, superametic element.10.Write down the Gauss Integration formula for triangular domains.


11. (a) A simply supported beam (span L and flexural rigidity El) carries two equal

concentrated loads at each of the quarter span points. Using Raleigh-Ritz methoddetermining the deflections under the two loads and the two end slopes.

Or

(b) Use the Gaussian elimination method to solve the following simultaneousequations :

12. (a) A column of length 500 mm is loaded axially as shown in fig. 12 (a). Analyze the column

and evaluate the stress and strains at salient points. The Young's modulus can be taken as E.

Al=62.5mm2

A2 = 125 mm2


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Or

(b) For the prinmatic bar shown in Fig. 12 (b), generate the stiffness matrix corresponding to the

three coordinates indicated. Use the following shape functions.

. .

13. (a) Develop stiffness coefficients due to torsion for a three dimensional beam element.

Or

(b) Calculate the temperature distribution in stainless steel fin shown in Fig. 13 (b). The

region can be discretized into 5 elements and 6 nodes.

14. (a) Develop shape functions for the. nine noded rectangular element

belonging to the Lagrange family.

Or

(b) Develop the shape function for an eight noded brick element.


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15. (a) For a triangular element shown in figure 15 (a), compute the stiffness matrix by using

isoparametric formulation and numerical integration with one point quadrature rule.

E = 2 X 103

kN/cm2, = O.

Or

(b) Evaluate the following integral using two point Gaussian quadrature :


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B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2008

Seventh Semester


ME 1401 - INTRODUCTION OF FINITE ELEMENT ANALYSIS


(Regulations 2004)

Time: Three hours Maximum: 100 marks


Part A - (10 x 2 = 20 marks)

1. Write the potential energy for beam of span 'L' simply supported at ends, subjected to a concentrated

load 'P' at midspan. Assume EI constant.2. What do you mean by higlter order elements?

3. What is constitutive law and give constitutive law for axi-symmetric problems?4. Explain tlte important properties of CST element.

Ans. 1. The strain components are constant throughout the volume of the element.

2. It has six unknown displacement degrees of freedom.

5. Give one example each for plane stress and plane strain problems.

Ans. Generally, members that are thin (those with a small z dimension compared to the in-plane x

and y dimensions) and whose loads act only in the x-y plane can be considered to be under plane stress.

Plates with holes and plates with fillets are coming under plane stress analysis problems. Dams and

pipes subjected to loads that remain constant over their lengths are coming under plane strain analysisproblems.

6. Write the stiffness matrix for the' simple beam element given below.

7. Write the shape functions for a ID, 2 noded element.

8. What are the advantages of natural coordinates over global co-ordinates?

Ans. A natural co-ordinate system is used to define any point inside the element by a set of

dimensionless numbers whose magnitude never exceeds unity. This system is very useful in assemblingof stiffness matrices. But in the global co-ordinate system, the points in the entire structure are defined

using co-ordinates.

9. Define Isoparametric elements.

10. Write the natural co-ordinates for the point 'P' of the triangular element. The point 'P' is the C.G. of

the triangle.


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11.(a) Determine the expression for deflection and bending moment in a simply supported beam

subjected to uniformly distributed load over entire span. Find the deflection and moment at

midspan and compare with exact solution using Rayleigh-Ritz method

Use y = a1sin(x/l)+a2sin(3x/l)

OR(b) Derive the equation of equilibrium in case of a three dimensional stress system.

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

OR

(b) Write the mathematical formulation for a steady state heat transfer conduction problem and

derive the stiffness and force matrices for the same.

13. (a) Find the expression for nodal force vector in a CST element shown in Fig.(1) subject to pressuresPx1, on side 1.

OR

(b) Determine the shape functions for a constant strain triangular (CST) element in terms of natural

coordinate system.

14. (a) For the CST element given below Fig.(3), assemble strain-displacement matrix. Take t = 20 mm,E = 2 X 105 N/mm2.

OR

(b) Derive the expression for constitutive stress-strain relationship and also reduce it to the problem

of plane stress and plane strain.

15.(a) Write short notes on:

(i) Uniqueness of mapping of isoparametrix elements,

(ii) Jacobian matrix.

(iii) Gaussian Quadrature integration technique.

OR

(b) (i) Use Gauss quadrature rule (n = 2) to numerically integrate -11

-11 xy dx dy,

(ii) Use natural co-ordinates derive the shape function for a linear quadrilateral element.


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B.E./B.Tech. DEGREE EXAMINATION, APRIL/MAY 2008.

Seventh Semester




(Regulations 2004)




1. List any four advantages of finite element method.

2. Write the potential energy for beam of span 'L' simply supported at ends, subjected to aconcentrated load 'P' at mid span. Assume EI constant.

3. What are called higher order elements?

4. Write briefly about CST element.

5. What is the governing differential equation for a one dimensional heat transfer?

6. Differentiate : Local axis and Global axis.

7. What is an equivalent nodal force?

8. Give one example each for plane stress and plane strain problems.

9. What do you mean by Constitutive Law and give Constitutive Law for axi-symmetricproblems?

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


11. (a) Explain the Gaussian elimination method for solving of simultaneous linear algebraic

equations with an example.

Or

(b) 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 thefunction Y = A{l -cos(x/2L)}. Given : EI is constant.

12. (a) (i) Derive the shape functions for a 2D beam element. (8)(ii) Derive the shape functions for 2D truss element. (8)

Or

(b) Each of the five bars of the pin jointed truss shown in Figure 12 (b) has a cross sectional area

20 sq.cm. and E = 200 GPa.


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(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. (6)

13. (a) Find the temperature at a point P (1, 1.5) inside the triangular element shown with the

nodal temperatures given as TI= 40C, T

J= 34C, and T

K= 46C. Also determine the location

of the 42C contour line for the triangular element shown in Figure 13 (a). (16)

(b) Calculate the element stiffness matrix and thermal force vector for the plane stress element

shown in Figure 13 (b). The element experiences a rise of 10C. (16)

14. (a) Derive the constant-strain triangular element's stiffness matrix and equations. (16)

Or

(b) Derive the Linear-Strain triangular element's stiffness matrix and equations. (16)

15. (a) Integrate f(x) = 10 + (20x)-(3X2 /10) + (4X3 /100) - (-5X4/1000) +(6x5/10000)

between 8 and 12. Use Gaussian Quadrature Rule. (16)Or

(b) Derive element stiffness matrix for a Linear Isoparametric Quadrilateral element.(16)


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B.E./B.Tech. DEGREE EXAMINATION, NOVEMBER/DECEMBER 2007.

Seventh Semester Mechanical Engineering



(Regulations 2004)




1. State the principle of minimum potential energy.2. Define shape functions.3. What do you mean by Constitutive Law?4. Differentiate CST and LST elements.5. What are the advantages of natural coordinates?6. How thermal loads are input in finite element analysis?7. Why polynomial type of interpolation functions are preferred over trigonometric functions?8. Write short notes on Axisymmetric problems.9. What do you mean by isoparametric formulation?10.What are the types of non linearity?


11. (a) Compute the value of central deflection (Figure 1) by assuming

y =

The beam is uniform throughout and carries a central point load P.

Or

(b) (i) Write short note on Galerkin's method. (8)

(ii) Write briefly about Gaussian Elimination. (8)

12. (a) (i) Derive the shape functions for a 2D beam element. (8)

(ii) Derive the shape functions for 2D truss element. (8)

Or

(b) Why higher order elements are needed? Determine the shape functions of an eight noded

rectangular element.

13. (a) For the constant Strain Triangular element shown in Figure 2, assemble

Strain-displacement matrix. Take t= 20 mm andE = 2xl05 N/mm2 .


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Or

(b) The temperature at the four corners of a four-noded rectangle are T1, T2, T3

and T4 Determine the consistent load vector for a 2-D analysis, aimed to determinethe thermal stresses.

14. (a) Derive the expression for the element stiffness matrix for anaxisymmetric shell element.

Or

(b) (i) Explain the terms "Plane stress" and "Plane strain" problems. Give constitutivelaws for these cases. (8)

(ii) Derive the equations of equilibrium in case of a three dimensional system. (8)

15. (a) Establish the strain-displacement matrix for the linear quadrilateral element as shown in

figure 3 at Gauss pointr =

0.57735 ands =

-57735 .

Or

(b) Derive element stiffness matrix for a Linear Isoparametric Quadrilateral element.

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FEAQPUPTO APR2010