FEA Tutorials

PSNA COLLEGE OF ENGINEERING AND TECHNOLOGY, DINDIGUL – 624 622. DEPARTMENT OF MECHANICAL ENGINEERING

TUTORIALS III/VI - ME6603 FINITE ELEMENT ANALYSIS Name

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SYLLABUS UNIT I INTRODUCTION Historical Background – Mathematical Modeling of field problems in Engineering – Governing Equations – Discrete and continuous models – Boundary, Initial and Eigen Value problems– Weighted Residual Methods – Variational Formulation of Boundary Value Problems – RitzTechnique – Basic concepts of the Finite Element Method. 64 UNIT II ONE-DIMENSIONAL PROBLEMS One Dimensional Second Order Equations – Discretization – Element types- Linear and Higher order Elements – Derivation of Shape functions and Stiffness matrices and force vectors- Assembly of Matrices - Solution of problems from solid mechanics and heat transfer. Longitudinal vibration frequencies and mode shapes. Fourth Order Beam Equation –Transverse deflections and Natural frequencies of beams. UNIT III TWO DIMENSIONAL SCALAR VARIABLE PROBLEMS Second Order 2D Equations involving Scalar Variable Functions – Variational formulation –Finite Element formulation – Triangular elements – Shape functions and element matrices and vectors. Application to Field Problems - Thermal problems – Torsion of Non circular shafts – Quadrilateral elements – Higher Order Elements. UNIT IV TWO DIMENSIONAL VECTOR VARIABLE PROBLEMS Equations of elasticity – Plane stress, plane strain and axisymmetric problems – Body forces and temperature effects – Stress calculations Plate and shell elements. UNIT V ISOPARAMETRIC FORMULATION Natural co-ordinate systems – Isoparametric elements – Shape functions for iso parametric elements – One and two dimensions – Serendipity elements – Numerical integration and application to plane stress problems

- Matrix solution techniques – Solutions Techniques to Dynamic problems – Introduction to Analysis Software.

INDEX S.NO 1 2 3 4 5 6 7 8 9 10

DATE

TUTORIAL NUMBER

MARK

SIGNATUR

S

E

Subject Name

FINITE ELEMENT ANALYSIS

Subject Code

ME6603

Year

III Year Mechanical Engg. (A, B & C Section)

Semester

VI

Faculty Name

Mr.S.Balu

Tutorial No.

1

Part – A 1. 2. 3. 4. 5.

Briefly explain the variational approach Briefly explain the weighted residual method What is meant by node or joint? What are ‘h’ and ‘p’ versions of Finite Element method? Define total potential energy.

Part – B 1. Explain in detail Direct stiffness method and potential energy method. W (N/m) 2. Find the maximum deflection and bending moment for a simply supported beam subjected load as shown in fig.1 using Raleigh Ritz method. L

3. Determine the displacement at the load applied node of the spring system as shown in figure.

4. List and briefly describe the general steps of the Finite element method.

W

Subject Name


Subject Code

Year


Semester

VI

Faculty Name

Mr.S.Balu

Tutorial No.

2

ME6603

Part – A 1. Why is variational formulation referred to as weak formulation? What are the advantages of the weak form? 2. Distinguish between 1D bar element and 1D beam element. 3. State the principle of minimum potential energy. 4. What are the factors considered in the discretization process? 5. What is meant by discretization in FEM? Part – B 1. Explain preprocessing and post processing in FEM. 2. Solve the following differential equation of a physical phenomenon using Galerkin method. d2y  500 x 2  0, 0  x  1, 2 dx

Trialfunct ion, y  a1 ( x  x 4 ), Boundary conditions are y (0)  0 y (1)  0 3. solve the differential equation for a physical problem expressed as following equation with boundary conditions as y(0) = 0 and y(10) = 0 using (i) point collocation method (ii) sub domain collocation method (iii) least squares method and (iv) Galarkin’s method. d2 y + 100=0, 0 ≤ x ≤10 dx 2 4. Discuss Rayleigh-Ritz and Galarkin methods of formations by taking an example.

Subject Name


Subject Code

Year


Semester

VI

Faculty Name

Mr.S.Balu

Tutorial No.

3

ME6603

Part – A 1. What are the properties of shape functions? 2. Draw the shape function of a two noded line element with one degrees of freedom at each node. 3. Define a local co-ordinate system. 4. Write down the expression of stiffness matrix for one dimensional bar element . 5. Define shape function. Part – B 1. Derive the displacement function U and shape function N for one dimensional Linear bar element based on Global co-ordinate approach. 2. Derive the Stiffness Matrix for one dimensional linear bar element. 3. For a stepped bar loaded as shown in Fig.2, determine (a) nodal displacements (b) support reactions and (c) element stresses.

Fig.2 Assume: E = 20 x 10 N / cm2 4. The tapered bar of uniform thickness t = 10 mm as shown in Fig-3 has Young’s 6

modulus E = 2 x 105MN/m2 and mass density 7890 kg/m3. In addition to its selfweight, the bar is subjected to a point load P = 1KN at its mid point. i. Model the plate with two finite elements. ii. Write the element stiffness matrices and element force vectors. iii. Using the elimination approach, solve for the global displacement vectors

Fig. 3

Subject Name


Subject Code

ME6603

Year


Semester

VI

Faculty Name

Mr.S.Balu

Tutorial No.

4

Part – A 1. Write the expression of shape function N and displacement for one dimensional bar element. 2. Write down the expression of stiffness matrix for a truss element. 3. What are the differences between boundary value problem and initial value problem. 4. How do you calculate the size of the global stiffness matrix? 5. State the principles of virtual work. Part – B 1. The structure shown in fig.4 is subjected

to

an

increase

in

temperature of 80oC. Determine the displacements, stress and support reactions. Assume the following data.

Fig. 4 2. For the two bar truss shown in fig. 5 determine the displacements of node 1 and the stress in element 1-3. 3. Consider a four bar truss as shown in fig. 6. It is given that E = 2 x 105 N/mm2 and Ae = 625 mm2 for all elements.

Fig. 6

Subject Name


Subject Code

Year


Semester

VI

Faculty Name

Mr.S.Balu

Tutorial No.

5

ME6603

Part – A 1. 2. 3. 4. 5.

What What What Write Write

is CST element? is LST element? is QST element? a displacement function equation for CST element. a strain-displacement matrix for CST element.

Part –B 1. Derive the shape function for the CST element. 2. Derive the strain displacement matrix [B]and stress – strain relationship matrix for two dimensional CST elements. 3. For the CST element coordinate are (100,100), (400,100) and (200,400), assemble strain – displacement matrix. Take t=20mm and E=2 x 105 N/mm2. 4. Calculate the stiffness matrix for the CST element coordinate are (10,7.5),(15,5) and (15,10). The co-ordinate are given in units of ‘mm’. Assume plane stress conditions. Take E = 2.1 x 10 5 N/mm2, ν=0.25, t=10mm. 5. Calculate the element stress and the principle angle for the CST element

co-ordinates

(10,7.5),(15,5)

and

(15,10).

The

nodal

displacements are: u1= 2 mm, v1 =1 mm, u2=0.5mm, v2=0 mm, u3=3 mm, v3=1. Take E = 2.1 x 105 N/mm2, ν=0.25, t=10mm.

Subject Name


Subject Code

Year


Semester

VI

Faculty Name

Mr.S.Balu

Tutorial No.

6

ME6603

Part – A 1. What is meant by plane stress analysis? 2. Define plane strain analysis. 3. Write down the stiffness matrix equation for one dimensional heat conduction element. 4. Write down the stiffness matrix equation for two dimensional CST element. 5. Write down the expression for the shape functions for a CST element. Part –B 1. Calculate the element stiffness matrix and the temperature force vector for the plane stress element co-ordinates are (0,0),(2,0) and (1,3). The element experiences a 20oC increases in temperature. Assume coefficient of thermal expansion is 6 x 10 -6/oC. Take E = 2 x 105 N/mm2, ν=0.25, t=10mm. 2. Derive the temperature function (T) and shape function (N) for one dimensional heat conduction element. 3. Derive the stiffness matrix for one dimensional heat conduction element. 4. Derive the finite element equations for a one dimensional heat conduction problem. 5. A steel rod of diameter d=2 cm, length L=5cm and thermal conductivity k=50 W/m0C is exposed at one end to a constant temperature of 320oC. The other end is in ambient air of temperature 20oC with a convection coefficient of h=100 W/m 2-C. Determine the temperature at the midpoint of the rod.

Subject Name


Subject Code

Year


Semester

VI

Faculty Name

Mr.S.Balu

Tutorial No.

7

ME6603

Part –A 1. What is axisymmetric element? 2. What are the conditions for a problem to be axisymmetric? 3. Write down the displacement equation for an axisymmetric triangular element. 4. Write down the shape functions for an axisymmetric triangular element. 5. Give the strain-displacement matrix equation for an axisymmetric triangular element. Part – B 1. Derive the shape function for axisymmetric element. 2. Derive the strain -displacement matrix [B] for axisymmetric element. 3. Derive the stress-strain relationship matrix [D] for axisymmetric element. 4. The nodal co-ordinates for an axisymmetric element are r 1=10mm, z1=10mm, r2=30mm, z2=10mm, r3=30mm,z3=40mm. evaluate [B] matrix for that element.

Subject Name


Subject Code

Year


Semester

VI

Faculty Name

Mr.S.Balu

Tutorial No.

8

ME6603

Part – B 1. The axisymmetric element co-ordinates are (0,0),(50,0) and (50,50). Determine the stiffness matrix. Take E=200 GPa and v=0.25. the coordinates are in ‘mm’. 2. The axisymmetric elements co-ordinates are (0,0), (60,0) and (30,50). Determine the element stresses. Take E=2.1x105 N/mm2 and v=0.25. The co-ordinates are in ‘mm’. the nodal displacements are: u1=0.05 mm, u2=0.02 mm, u3=0mm, w1=0.03 mm, w2=0.02 mm, w3 = 0mm. 3. Calculate the element stiffness matrix and the thermal force vector for the axisymmetric triangular elements co-ordinates are (6,7),(8,7) and (9,10). The co-ordinates are in ‘mm’. The element experiences a 15oC increase in temperature. Take N/mm2, v=0.25.

α=10x10-6/oC, E=2x105

Subject Name


Subject Code

Year


Semester

VI

Faculty Name

Mr.S.Balu

Tutorial No.

9

ME6603

Part – A 1. What is purpose of Isoparametric elements? 2. Write down the functions for 4 noded rectangular element using natural co-ordinate system. 3. Write the Jacobina matrix for four noded quadrilateral element. 4. Define super parametric element. 5. Define the sub parametric element. Part – B 1. Derive the shape function for noded rectangular parent element by using natural co-ordinate system and co-ordinate transformation. 2. Derive the element stiffness matrix equation for 4 nod3ed isoperimetric quadrilateral element. 3. For the isoparametric four noded quadrilateral element co-ordinates are (1,1),(5,1),(6,6) and (1,4). Determine the Cartesian co-ordinates of point P which has local co-ordinates ε=0.5 and η=0.5

Subject Name


Subject Code

Year


Semester

VI

Faculty Name

Mr.S.Balu

Tutorial No.

10

ME6603

Part – B 1. Evaluate [J] at ε=0.5 and η=0.5 for the linear quadrilateral element as shown in fig.7

Fig. 7 2. Establish the strain-displacement matrix for linear quadrilateral element co-ordinates are (1,1), (5,2), (4,5) and (2,4) at Gauss point r = 0.57735 and s = -0.57735 . 1

3. Evaluate the integral by using Gaussian quadrature

∫ x2 dx −1

.

1

4. Evaluate the integral I =

∫ cos πx2 dx −1

by applying 3 point Gaussion

quadrature and compare exact solution.

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