**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,

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

Relation between shear stresses:

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

**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.**

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**Theory of plane problems**

- Plane stress problem
- 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 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.

**Steps in finite element formulation:**

- Discritize the structure.
- Obtain the stiffness matrix and load vector for individual element.
- Develop the global stiffness matrix and global load vector by assembling the stiffness matrix and load vector of elements.
- Solve the global equations.
- 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.**