Elements of Theory of Elasticity (with solved examples)

 

Introduction:

Applications of the finite element method include elasticity problems also. Theory of elasticity deals with the stress and displacements in elastic solids generated by external forces.

Some Important Aspects in theory of elasticity:

Stress Components

There are two types of stresses acting on each face of an element namely,

  1. Axial stress ( – normal): acting perpendicular to the face.
  2. Shear Stress (): acting in two components on each face.

Relation between shear stresses:

1. shear stress

 

So, to analyze the element, mainly 6 stress components needs to be evaluated.

Strain Components

strain components

 

Defines the deformation condition at that point.

Note: All 6 stress components are function of the 6 strain components and the matrix relating them is called as ELASTICITY MATRIX.

 

Theory of plane problems

  1. Plane stress problem
  2. Plane Strain problem

 

Plane stress problem:

  • One dimension (say z) is very small in comparison to the other two dimensions.
  • There is in plane loading that is there is no loading in the direction of thickness.

e.g.: shear wall, thin plate.

plane stress

 

 

 

 

 

 

 

Plane strain problem:

  • One dimension is very large as compared to the other two dimensions.
  • Loading is across the larger dimension and same throughout the larger dimension.
  • Every plane section is identical to the other plane section and having same boundary conditions.

e.g.: water dam, retaining walls.

plane strain

 

 

 

 

 

 

 

Steps in finite element formulation:

  1. Discritize the structure.
  2. Obtain the stiffness matrix and load vector for individual element.
  3. Develop the global stiffness matrix and global load vector by assembling the stiffness matrix and load vector of elements.
  4. Solve the global equations.
  5. Obtain the strain and stress in individual elements.

 

 

The Detailed explanation of the topic is given in the pdf embedded below with solved examples. Stiffness matrix for 2D and 3D elements (axisymmetric) is also calculated.

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