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

TUTORIALSIII/VI - ME6603 FINITE ELEMENT ANALYSIS

Name:Register No:Degree/Branch:Year/semester:

SYLLABUSUNIT I INTRODUCTIONHistorical 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 PROBLEMSOne 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 PROBLEMSSecond 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 PROBLEMSEquations of elasticity Plane stress, plane strain and axisymmetric problems Body forces and temperature effects Stress calculations - Plate and shell elements.UNIT V ISOPARAMETRIC FORMULATIONNatural 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.NODATETUTORIAL NUMBERMARKSSIGNATURE

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Subject NameFINITE ELEMENT ANALYSISSubject CodeME6603

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

Faculty NameMr.S.BaluTutorial No.1

Part A1. Briefly explain the variational approach 2. Briefly explain the weighted residual method 3. What is meant by node or joint? 4. What are h and p versions of Finite Element method?5. Define total potential energy. Part B 1. Explain in detail Direct stiffness method and potential energy method.2. Find the maximum deflection and bending moment for a simply supported beam subjected load as shown in fig.1 using Raleigh Ritz method.

W L W (N/m)

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.

Subject NameFINITE ELEMENT ANALYSISSubject CodeME6603

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

Faculty NameMr.S.BaluTutorial No.2

Part A1. 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 B1. Explain preprocessing and post processing in FEM.2. Solve the following differential equation of a physical phenomenon using Galerkin method.

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) Galarkins method.

4. Discuss Rayleigh-Ritz and Galarkin methods of formations by taking an example.

Subject NameFINITE ELEMENT ANALYSISSubject CodeME6603

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

Faculty NameMr.S.BaluTutorial No.3

Part A1. 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 B1. 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.2Assume: E = 20 x 106 N / cm24. The tapered bar of uniform thickness t = 10 mm as shown in Fig-3 has Youngs modulus E = 2 x 105MN/m2 and mass density 7890 kg/m3. In addition to its self-weight, 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 NameFINITE ELEMENT ANALYSISSubject CodeME6603

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

Faculty NameMr.S.BaluTutorial No.4

Part A1. 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 B1. 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. 42. 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 NameFINITE ELEMENT ANALYSISSubject CodeME6603

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

Faculty NameMr.S.BaluTutorial No.5

Part A1. What is CST element?2. What is LST element?3. What is QST element?4. Write a displacement function equation for CST element.5. Write a strain-displacement matrix for CST element.Part B1. 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 105 N/mm2, =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, =0.25, t=10mm.

Subject NameFINITE ELEMENT ANALYSISSubject CodeME6603

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

Faculty NameMr.S.BaluTutorial No.6

Part A1. 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 B1. 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, =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/m2C. Determine the temperature at the midpoint of the rod.

Subject NameFINITE ELEMENT ANALYSISSubject CodeME6603

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

Faculty NameMr.S.BaluTutorial No.7

Part A1. 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 B1. 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 r1=10mm, z1=10mm, r2=30mm, z2=10mm, r3=30mm,z3=40mm. evaluate [B] matrix for that element.

Subject NameFINITE ELEMENT ANALYSISSubject CodeME6603

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

Faculty NameMr.S.BaluTutorial No.8

Part B1. 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 co-ordinates are in mm.2. The axisymmetric