# Plane strain stress relationship equation

Concept of strain: if a bar is subjected to a direct load, and hence a stress the bar i.e. We will have the following relation. This expression is identical in form with the equation defining the direct stress on any inclined plane q with Îx and Îy. So the equations of elasticity reduce to: Equilibrium. ∂σ + 0 (due to stress- strain relations) the case of plane stress in place of engineering notation. CIVL 7/ Chapter 6 - Plane Stress/Plane Strain Stiffness Equations - Part 1 . the stress/strain relationships for plane stress and plane strain are necessary.

Locate the points x' and y' 3. Join x' and y' and draw the Mohr's strain circle 4.

Measure the required parameter from this construction. At a certain point, a material is subjected to the following state of strains: The Mohr's strain circle can be drawn as per the procedure described earlier.

Although we can not measure stresses within a structural member, we can measure strains, and from them the stresses can be computed, Even so, we can only measure strains on the surface.

For example, we can mark points and lines on the surface and measure changes in their spacing angles. In doing this we are of course only measuring average strains over the region concerned. Also in view of the very small changes in dimensions, it is difficult to archive accuracy in the measurements In practice, electrical strain gage provide a more accurate and convenient method of measuring strains.

A typical strain gage is shown below. The gage shown above can measure normal strain in the local plane of the surface in the direction of line PQ, which is parallel to the folds of paper. This strain is an average value of for the region covered by the gage, rather than a value at any particular point. The strain gage is not sensitive to normal strain in the direction perpendicular to PQ, nor does it respond to shear strain.

We therefore need to obtain measurements from three strain gages.

### Hooke's Law for Plane Strain

These three gages must be arranged at different orientations on the surface to from a strain rossett. Typical examples have been shown, where the gages are arranged at either or to each other as shown below: A group of three gages arranged in a particular fashion is called a strain rosette.

The action of the stresses is to produce or being about the deformation in the body consider the distortion produced b shear sheer stress on an element or rectangular block This shear strain or slide is f and can be defined as the change in right angle. So we have two types of strain i. A material is said to be elastic if it returns to its original, unloaded dimensions when load is removed.

Within the elastic limits of materials i. There will also be a strain in all directions at right angles to s.

### NPTEL :: Mechanical Engineering - Strength of Materials

The final shape being shown by the dotted lines. It has been observed that for an elastic materials, the lateral strain is proportional to the longitudinal strain.

The ratio of the lateral strain to longitudinal strain is known as the poison's ratio. Consider an element subjected to three mutually perpendicular tensile stresses sxsyand sz as shown in the figure below. In the absence of shear stresses on the faces of the elements let us say that sxsysz are in fact the principal stress.

The resulting strain in the three directions would be the principal strains. We will have the following relation.

For Two dimensional strain: Hence the set of equation as described earlier reduces to Hence a strain can exist without a stress in that direction Hydrostatic stress: The term Hydrostatic stress is used to describe a state of tensile or compressive stress equal in all directions within or external to a body. So let us determine the expression for the volumetric strain. Consider a rectangle solid of sides x, y and z under the action of principal stresses s1s2s3 respectively.

Let a cuboid of material having initial sides of Length x, y and z. Volumetric strains in terms of principal stresses: Futher -ve sign in the expression occurs so as to keep the consistency of sign convention, because OM' moves clockwise with respect to OM it is considered to be negative strain.

The other relevant expressions are the following: