CIE-1 FEM-HKRV- Department of Mechanical Engineering, BMSCE

Modelling and Finite Element Analysis 16ME6DCMFE
Unit 1

Portions for CIE1

Fundamental concepts: Principles of Elasticity:
Equilibrium equations,
strain displacement relationships in matrix form
Generalised Hooke’s Law, Constitutive relationships(3D)
Constitutive relationships for plane stress, plane strain , Axi-symmetric
Potential energy and equilibrium,PMPE and numericals on Springs in combination subjected to
forces
Rayleigh-Ritz method applied to simple problems on axially loaded members,,with point loads and
distributed loads.
Gauss elimination method and
Gaussian quadrature-1pt,2pt and 3 pt formula.
commercial packages-preprocessor, solver and post processor .




1.Explain the significance of Equilibrium equations. Derive the force equilibrium conditions
for 2D state of stress. Write equilibrium equations for 3D.
2.Sketch the variation of stresses in a 3D differential element subjected to body forces.
Establish the equations of force equilibrium in differential form
3.Write down (do not derive) the equations of equilibrium in differential form in terms of
stresses and body forces for a three dimensional element volume with the help of a neat
sketch
4.Explain principal stress, maximum stress and Von Mises stress and bring out the importance of
these in the stress analysis.
5.The stress components at a point in a body are given by x=3xy2z+2x ,y=5xyz+2y
,z=3xy2z+2x ,xy=0, yz=xz=3xy 2 z+2xy. Determine whether these components of stress
satisfy the equilibrium equations or not at the point (1,-1,2).If not then determine the suitable
body force vector required at this point so that these stress components are in equilibrium.
6.Write strain displacement relations for 2D and for 3D in Cartesian coordinates .
7.Strain in a rod of length “L” fixed at one end and subjected to axial loading is given by
x=1+2x2.Find the tip displacement .
8.The displacement field for a body is given by u=(x2+y2)i+(3+z)j+(x2+2y)k.Find the rectangular
strain components at (3,1,-2)


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, Modelling and Finite Element Analysis 16ME6DCMFE
9. Explain plane stress and plane strain problems. Give examples and write stiffness and
compliance matrices
10. Explain Plane stress ,Plane strain and axisymetric problems with examples.write
stiffness matrix for the same

11. A.Identify the idealization of 2D problems shown. State the geometric and
corresponding stress/strain conditions and write stiffness matrices.




• 1.Thin Plate with Hole 2.Thin cantilever plate

 xy y
 xy
x




11b Stress Analysis of the prototype seatbelt component shown in the figure below is to be carried out
for a tensile load. Identify the idealization of 2D problem shown. State the geometric and corresponding
stress/strain conditions and write stiffness matrices.




Given 2 D member has in plane dimensions(x,y) very large compared to Out of plane dimension(z).Since

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, Modelling and Finite Element Analysis 16ME6DCMFE
the plate is thin in the z-direction, there can be little variation in the stress components through the thickness.
Thus they will be approximately zero throughout the entire domain. Under these assumptions, the stress field
can be taken as



 x   x ( x, y),  y   y ( x, y), xy   xy ( x, y)
 z   xz   yz  0, Also,  xz   yz  0,  z  0




   D 



OR

11c) Stress Analysis of the thick Aluminum tang shown in the figure below is to be carried
out. Identify the idealization of 2D problem. State the geometric and corresponding
stress/strain conditions and write stiffness matrices.




Given 2 D member has in plane dimensions(x,y) relatively less compared to Out of plane
dimension(z). If the body forces and tractions on lateral boundaries are independent of the z-
coordinate and have no z-component, then the deformation field can be taken in the reduced form
u  u ( x, y ) , v  v ( x, y ) , w  0
 z   xz   yz  0
Under these assumptions, the stress field can be taken as
 x   x ( x, y),  y   y ( x, y), xy   xy ( x, y)
 z   xz   yz  0, xz   yz  0, Also,  z  0


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, Modelling and Finite Element Analysis 16ME6DCMFE
 
1   0 
Plane Strain    D , D 3 x 3 
E
  1  0 
1  1  2   1  2 
 0 0 
 2 
12. The state of strain at a point is given by x=0.0015, y=-0.0025, z = yz= xz = 0, xy=
-0.004.Determine the stress tensor at the point . E =208 Gpa. and =0.28.
13. Determine the strain tensor for the stress tensor at a point given by
210 140 00
[] = 140 -240 00 where E=208 GPa and =0.3
00 00 00
14. Explain Axi-symmetric problems. Give examples and write stiffness and compliance
matrices.
15. Explain with examples of bar and beam ,essential and Non essential boundary
conditions
16. Explain boundary value problems and Initial value problems with example.
17. Write admissible functions for
i. Axial bar fixed at one end ii. Axial bar fixed at both ends
iii. Cantilever beam , iv Simply supported beam(Polynomial and trigonometric)
iv.Beam fixed at both ends
v.Propped cantilever beam
18. Using matrix notation, develop an expression for the total potential energy functional
for a 3D elastic solid subjected to body forces, surface forces and point loads.
19. State principle of Minimum Potential Energy.
20. Obtain displacements of the nodes in the spring system problems below using PMPE




21. Explain Rayleigh-Ritz method applied to continuum. What are its disadvantages
22. Obtain expression for displacement for uniform bar fixed at one end and loaded by a
point load at free end in tension using Rayleigh Ritz method .


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