R10 Set No: 1Code No: R31031

III B.Tech. I Semester Regular/Supplementary Examinations, January - 2014 FINITE ELEMENT METHODS

(Common to Mechanical Engineering & Auto Mobile Engineering) Time: 3 Hours Max Marks: 75

Answer any FIVE QuestionsAll Questions carry equal marks

**1. (a) Discuss the advantages and disadvantages of FEM over Classical Finite difference method.

(b) Consider a cantilever beam under uniformly distributed load 0 which can be steadily show4

that the deformation of a body is given by differential equation

2 3

d vEI 4dx

q 0 0 with four

boundary conditions dvv ( 0) 0, (0 ) 0 ,

dx

d V2dx

d0 at x L and

dx

v0 at x=L. Find the

3

approximate solution by using weighted residual method.

2. (a). Derive the stiffness matrix for a quadratic element and explain its properties. (b). Derive the Lagrange shape functions and explain its properties.

3. The members of the truss shown in the accompanying figure have a cross-sectional area of 15cm² and are made of aluminum alloy (E=70 GPa). Determine the deflection of each joint, the stress in each member and the reaction forces.

4. Determine the displacements and rotations of the joints for the portal frame shown in the Fig.

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R10Code No: R31031

5. a) Derive the strain displacem ent matrix for constant strain triangle(b).Find the strain - nodal di splacement matrices Be for the element shownnumbers given at the corners

Set No: 1

in the figure .use local

6. For a rectangular eleme nt as shown in the fig assuming pl ain stress conditionE 2 10

5 2N / mm , 0 3 ,displacement vector [(0,0)

(0.45, 0.075) (0.15, 0.08) (0,0)]

evaluate Jacobian matrix, strain displacement matrix and stress when ξ=0,η=0 .

(0, 1) (2, 1)

(0,0) (2,0)

7. (a) Consider a brick wall of thickness L=0.3 m, k=0.7 . The inner surface is at 28ºC and theouter surface is exposed to cold air at -15ºC .The heat transfer coefficient associated with theoutside surface is h= 40 W/m² ..Determine the steady state temperature distribution within thewall and also the heat flux through the wall. Use a two-element model, assuming one dimensionalflow.

w o(b) A metallic fin, with the thermal conductivity k 360 / m, c 0 1 cm thick and 10

cm long,extends from a plane wall whose temperature is 235ºC. Determine the temperature distribution andamount of heat transferred from the fin to the air at 20ºc with h=9 W/m² ºC. Take the width of thefin to be 1 m. Assume that the tip of the fin is insulated.

8. Determine the Eigen values and Eigen vectors for the bar shown in the figure. Take ρ=7800 kg/m³,E=200 GPa

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R10 Set No: 2Code No: R31031

III B.Tech. I Semester Regular/Supplementary Examinations, January - 2014 FINITE ELEMENT METHODS

(Common to Mechanical Engineering & Auto Mobile Engineering) Time: 3 Hours Max Marks: 75

Answer any FIVE Questions All Questions carry equal marks

**

1. (a) Derive the equations of equilibrium in case of a three dimensional stress system. (b) consider a bar subjected to a uniform axial load as shown in the fig. which can be steadily

show that the deformation of a body is given by differential equation d2u2

q 0 0 with dx

boundary conditions u

d(0 ) 0 ,

2u2 0 at x=L. find the approximate solution by using weighted

dx residual method.

2. Find the stress distribution and the reaction forces in the tapered bar shown in Figure using two finite elements under an axial load of P = 1 N.

3. A small rail road bridge is constructed of steel members, all of which have a cross-sectional area of 3250 cm². A train stops on the bridge, and the load applied to the truss on one side of the bridge are as shown in the fig..Estimate how much the point R moves horizontally because of this loading. Also determine the nodal displacements and element stresses

4. (a) Derive the element stiffness matrix for beam element by potential energy approach.(b) Obtain the expression for the shape functions of a beam element by using global co-ordinatesystem.

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R10 Set No: 2Code No: R31031

5. A constant strain triangle element is shown in the fig. the element is subjected to a body force2 3 3

fx x N / m Determine the nodal force vector f Take element thickness =1m.

6. Establish the Jacobian operator [J] of the two dimensional element shown in figure also find the Jacobian determinant.

7. A long bar of rectangular cross-section having thermal conductivity of 1.5 W/m º C is subjected to the boundary conditions shown in the fig. The two opposite sides are maintained at a uniform temperature of 180 ºC; one side is insulated, and the remaining side is subjected to a convectionprocess with T 25

oc And h= 50 W/m² .º C. Determine the temperature distribution in the bar

8. Determine the Eigen values and Eigen vectors for the stepped bar shown in the fig.

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R10 Set No: 3Code No: R31031

III B.Tech. I Semester Regular/Supplementary Examinations, January - 2014 FINITE ELEMENT METHODS

(Common to Mechanical Engineering & Auto Mobile Engineering) Time: 3 Hours Max Marks: 75

Answer any FIVE Questions All Questions carry equal marks

**

1. (a) Derive the stress strain relations for three dimensional problems of isotropic materials. (b)Explain the concept of FEM briefly and outline the procedure.

2. Consider the bar shown in the fig, determine the nodal displacements, element stresses and support reactions

3. For the pin -jointed configuration shown in the figure, determine the stiffness valuesk11 ,k 12 and k22 of the global stiffness matrix.

4. (a) Derive the load vector for the cantilever beam having a point load. (b) A cantilever beam of 1mt length carries a point load of 10 kN at free end. Determine the deflection at the end of beam; also determine shear force and bending moment. Take E= 70 GPa,

4 area A=500 mm², I 2500 mm

5. Evaluate the shape functions N1 , N2 and N3 at the interior point P for the triangular elementshown in the fig

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R10 Set No: 3Code No: R31031

6. For a rectangular elem ent as shown in the fig assuming pl ain stress conditionE 2 10

5 2N / mm , 0 3 displacement vector [(0,0)

(0.45, 0.075) (0.15, 0.08) (0,0)]

evaluate Jacobian matrix, strain displacement matrix and stress when ξ=0,η=0 .(0, 1) (2, 1)

(0,0) (2,0)

7. Consider the shaft with a rectangular cross section shown in the fig. Determine the angle of twistper unit length in terms of twisting moment (M) and shear modulus of the material (G).

8. Find the Eigen values and Eigen vectors for the beam shown in the fig.

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R10 Set No: 4Code No: R31031

III B.Tech. I Semester Regular/Supplementary Examinations, January - 2014 FINITE ELEMENT METHODS

(Common to Mechanical Engineering & Auto Mobile Engineering) Time: 3 Hours Max Marks: 75

Answer any FIVE QuestionsAll Questions carry equal marks

**

1. (a) Explain the terms ‘Plane stress’ and ‘Plane strain’ problems. Give constitutive laws for thesecases.

(b) If a displacement field is described by u

2 2 4( x2 y6 xy )10

2 4

v (3x 6 y y )10Determine x, y, yxy at the point x=1, y=0

2. Derive a one-dimensional linear interpolation formula for a function u=u(x) that is valid in therange u through u as shown the figure

1 2

3. The members of the truss shown in the accompanying figure have a cross-sectional area of 8 cm² and are made of steel (E=200 GPa). Determine the deflection of each joint, the stress in each member and the reaction forces.

4. For the beam and loading shown in the figure determine i) the slopes at 2 and 3 ii) the vertical deflection at the midpoint of the distributed load

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R10 Set No: 4Code No: R31031

5. (a) Derive the element stiffness matrix for constant strain triangle (b) Derive the shape functions of constant strain triangle in the global Co-ordinate system

6. The following fig shows a four -node quadrilateral. The (x,y) coordinates of each nodes are given in the figure. The element displacement vector q is given asq[0 0 0 20

0 0 15r

0 10 0 0 05 ] , Find the following (a) The x, y co-ordinates of a point P whose location in the master element is given by η=0.5 and ξ=0.5 (b) The u,v displacements of point P

7. 7. A composite wall consists of three materials as shown in the figure. The outer o

temperature T 0 c 20 is Convection heat transfer takes place on the inner surface of theo

wall with T 800o c .and h 25 wwall

m2 c Determine the temperature distribution in the

8. considering the axial vibration of a stepped bar find out the following i) Stiffness matrix and mass matrix ii) natural frequency iii) mode shapes

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