Mtech. Finite Element Analysis

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Code No: D109110402 JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

M.Tech I Semester Regular Examinations March 2010 FINITE ELEMENT ANALYSIS

(CAD/CAM) Time: 3hours Max.Marks:60

Answer any five questions All questions carry equal marks

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1. a) State and explain plane stress and plane strain problems. b) If a displacement field is described as follows,

u = (-x2 + 2y2 + 6xy) and v = (3x + 6y y2) Determine the strain components x, y, and xy at the point x=1; y = 0

2. a)What do you understand by Functional approximation? b) Explain how the Raleigh-Ritz method can be used to approximate the solution of a problem.

3. For the pin jointed truss shown in the figure, determine i) Element stiffness matrix ii) Global stiffness matrix iii) Stress in the element 1 iv) Strain in element 2 v) Strain energy of the system

Assume E = 2x105 N/mm2, L1=750mm, L2=900mm, A1=100mm2, A2 =1250mm2

4. Using two equal length beam elements determine central deflection and slope at supports in simply supported beam of span 3 m carrying a central point load of 20 kN. Take EI=8000 kNm2 and make use of symmetry.

5. a) Using three point Gaussian quadrature find x y dA for a triangular element

whose vertices are (1,1), (3,2), and (2, 3). b) Find the shape functions of a quadrilateral element in natural coordinates.

6. a) List some disadvantages of using 3-D elements. b) Derive the Strain displacement matrix for a Tetrahedron Element.

1

300 2

15KN

20KN

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7. A metallic fin with thermal conductivity 360 W/m K, 0.1 cm thick and 10 cm long extends from a plane wall whose temperature is 235o C. Determine the temperature distribution and amount of heat transfer from the fin to the air at 20oC with heat transfer coefficient of 9W/m2K .Take width of fin as 1 m.

8. a) Explain briefly about Eigen value problem. b) Determine the natural frequencies and mode shapes for the rod shown in fig. using the characteristic polynomial technique. Assume E = 200 GPa and mass density 7850.kg/m3. L1=L2=0.3m, A1=400mm2, A2=250mm2

********

A1 A2

L1 L2

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Code No: C0402

JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD M.Tech I- Semester Supplementary Examinations September, 2010

FINITE ELEMENT ANALYSIS (CAD/CAM)

Time: 3hours Max. Marks: 60 Answer any five questions

All questions carry equal marks - - -

1. a) Discuss the Engineering Applications of Finite element method. b) Derive the strain- Displacement relations for a plane stress problem.

2. a) Discuss the convergence requirements in the selection of displacement model for finite element analysis. b) With suitable example explain Galerkins method. 3. For the stepped bar shown in figure, determine nodal displacements, element stress and support reactions. Assume E = 2 x 105 N/mm2, L1=L2=300mm, A1=250mm2, A2=400mm2, P=300 kN, Q=10 kN.

4. A beam of length 3 m is fixed at both ends. Estimate the deflection at the center of the beam where load is acting vertically downward of 20 kN. Divide the beam into two elements. Compare the solution with theoretical calculations. Take E = 2 x 105 N/mm2, A = 250 mm2, I = 200 mm4. 5. a) Derive the shape functions for a triangular linear element in global Co-ordinate system. b) What is a constant strain triangular element? State its properties and applications. 6. a) Sketch any three 3-D structural elements showing their degrees of freedoms. b) Derive the shape functions of any one of the 3-D structural element . 7. A composite slab consists of 3 materials of different conductivities i.e., 20W/m K, 30 W/m K , 50 W/m K of thickness 0.3 m, 0.15 m and 0.15 m respectively. The outer surface is 200C and the inner surface is exposed to the convective heat transfer coefficient of 25W/m2 K, 8000C. Determine the temperature distribution with in the wall. 8. Derive the stiffness and mass matrices for free vibrations in both lumped mass and consistent mass formulations for a bar element.

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Code No: C0402 JAWAHARLAL NEHRU TECHNOLOGICAL UNIVERSITY HYDERABAD

M.Tech I - Semester Examinations, March/April-2011 FINITE ELEMENT ANALYSIS

(CAD/CAM) Time: 3hours Max. Marks: 60


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1. a. List the advantages and disadvantages of FEM over other traditional variational methods. (5)

b. Derive the finite element equation using the potential energy approach. (7)

2. a. Illustrate the Rayleigh-Ritz method in detail by applying it on an axially loaded bar at one end and fixed at one end as shown on fig.1. (6)

Fig. 1

b. Explain about the Lagrangian constraints used in the principles of elasticity with one example. (6)

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

temperature. The temperature is then raised by 300C. Determine the displacements at nodes 2 and 3, stresses in the three sections. (12)

4. a. Derive the B matrix (strain-displacement) for a Constant Strain Triangle (CST)

element using area coordinates. (6) b. Calculate the surface loads for the triangle element shown in fig. 3. (6)

Fig. 3

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

dimensional stress analysis. (6)

b. Using Gaussian quadrature evaluate the following integral

13

1

(4 )d +

+ . (6) 6. Calculate the conductance matrix [K(e)] and load vector {F(e)} for the triangle element

shown in fig.4 . The thermal conductivities are kx = ky = 4 W/cm-oC and h = 0.3 W/cm2 oC. Thickness of the element is 1cm. All coordinates are given in cms. Convection occurs on the side joining modes i and j (12)

Fig. 4

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

consistant mass for translation dof with E = 200GPa, = 7500kg/m3. (12) 8. a. Discuss about Material and Geometric nonlinearity. (6) b. Explain the solution methods for nonlinear algebraic equations. (6)

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M.Tech I - Semester Examinations, October/November-2011 FINITE ELEMENT ANALYSIS



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1.a) Evaluate the integral by using one and two-point Gaussian quadrature and compare with exact value.

I = 1 1 3 2 2

1 1(x x y xy sin 2x cos 2y)dx dy

+ +

+ + + +

b) Determine shape functions for 4 noded tetrahedran element in absolute and intrinsic co-ordinate system.

c) Obtain the D-matrix for Axis symmetric, 2-D and 3-D problems using Hooks law. [12]

2.a) Explain in detail about Galerkin method with an example. b) Determine the temperature distribution in a straight fin of circular c/s. Use three

one dimensional linear elements and consider the tip is insulated. Diameter of fin is 1 cm, length is 6 cm, h = 0.6 W/cm2 C, =250C and base temperature is 1=800C.

c) Derive one dimensional steady state heat conduction equation. [12] 3.a) Discuss in detail about material and Geometrical Non Linearity and how it is to

be incorporated in FEM Formulation. b) Establish the Hermite shape functions for a beam element Derive the equivalent

nodal point loads for a u.d.l. acting on the beam element in the transverse direction and also determine stiffness matrix. [12]

4.a) Derive the stiffness matrix [K] and the load vector for the two dimensional

simplex element shown in Fig.1, also determine nodal displacements of triangular element, strain and stress of an element. Assume E = 260 MPa, = 0.3 and t = 10 mm.

b) Using the isoparametric element, find the Jacobian and inverse of Jacobian matrix for the element shown in Fig.2 (a) & (b) for the following cases.

i) Determine the coordinate of a point P in x-y coordinate system for the = 0.4 and = 0.6.

ii) Determine the coordinate of the Q in and system for the x = 2.5 and y = 1.0. [12]

Contd.2

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5.a) Differentiate between Consistent Mass matrix and Lumped Mass matrix. b) Consider axial vibration of the steel bar shown in Fig.3, (i) develop the global

stiffness and consistent mass matrices and (ii) determine the natural frequencies and mode shapes using the characteristic polynomial technique with help of linear and quadratic shape function concept and E = 210 GPa. [12]

Fig. 3

6.a) Derive the stiffness matrix [K] and the load vector for the two dimensional six

nodded triangular element, also determine nodal displacements of triangular element, strain and stress of an element.

b) Determine the deflection and stresses at the point of load application on a plate, by considering two CST triangular elements shown in Fig. 4. Assume thickness t = 4 mm, Poissons ratio = 0.25 and Youngs Modulus E= 210 GPa. [12]

Fig. 4

7.a) Derive the necessary equations and solve the following problem: In a solid body the state of stress at a point is given below in the tensor form.

Determine the normal stress at the point on a plane for which the direction cosines

of the normal are

21,

21,

31 . = MPa.

402520253015201550

Contd3

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b) Derive an equation for finding out the potential energy by Rayleigh-Ritz method.

Using Rayleigh-Ritz method, find the displacement of the midpoint of the rod shown in Fig.5. Assume E = 1, A = 1, g = 1 by using linear and quadratic shape functions concept. [12]

Fig. 5

8.a) At 20C an axial load P = 300 103N is applied to the rod as shown in Fig.6. The

temperature is then raised to 60C. Assemble the element stiffness matrix and the element temperature force matrix (F). Again determine the nodal displacements and element stresses by considering linear and Quadratic Shape functions. Also find element strains. Assume E1 = 70 109 N/m2, A1 = 900 mm2, 1 = 23 106/C, E2 = 200 109 N/m2, A2 = 1200 mm2, 2 = 11.7 106/C.

Fig.6

b) Determine the deflection at the free end under its own weight using two elements of bar as shown in Fig. 7. Assume E = 200 GPa, = 7800 kg/m3. [12]

Fig. 7

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M.TECH I - SEMESTER EXAMINATIONS, APRIL/MAY-2012 FINITE ELEMENT ANALYSIS



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1.a) Differentiate between planar frame element and space frame element. b) Use finite element method to calculate displacements and stresses of the bar shown in the fig. 1.

A1 = 50 mm2 A2 = 30 mm2 A3 = 25 mm2 50 N 100 mm 100 mm 100 mm Fig.1 E = 200 Gpa. 2.a) Write about different boundary considerations in beams. b) For a beam and loading shown in fig.2, determine the slopes at 2 and 3 and the

vertical deflection at the midpoint of the distributed load. 10 KN/m

1 2 3 1m 1m

Fig.2

Contd....2

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3. Explain the steps involved in analysis of beams. With the help of a simple example explain how boundary conditions are applied.

4.a) Derive the Equilibrium equations and strain displacement relations for three

dimensional bodies. b) Discuss in detail about Galerkins Method and Principal of Minimum potential

energy and discuss with an example. 5.a) An axisymmetric triangular element is subjected to the loading as shown in fig.3

the load is distributed throughout the circumference and normal to the boundary. Derive all the necessary equations and derive the nodal point loads.

100 Mpa 50 cm 30 cm 50 cm 150 Mpa 40 cm

Fig.3 b) How do you calculate element stresses for 3-Dimensional bodies?

Contd....3

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6.a) Discuss in detail about material and Geometrical Non Linearity and how it is to

be incorporated in FEM Formulation b) Discuss in detail about Tetrahedron element and their properties and advantages.

Derive the [B] matrix, Strian and stresses for four nodded Tetrahedron element. 7.a) Distinguish between consistent mass matrix and Lumped mass matrix.. b) Derive one dimensional steady state heat conduction equation and apply for one

dimensional fin problem. Determine the temperature distribution in a straight fin of circular cross section. Use three one dimensional linear elements and assume that the tip is insulated. Diameter of fin is 1 cm, length is 6 cm, h = 0.6 W/cm2 0C, =250C and base temperature is 1=800C.

8.a) Discuss in detail about the general procedure of FEM formulation. b) Derive the Lagrangian and Hermite shape functions for Bar and Beam two

nodded elements.

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Mtech. Finite Element Analysis