Principal Stress and Principal Strain

Sign Conventions

• Tensile normal stress is considered positive ans compressive normal stress is considered negative.
• Shear stress acting on a positive face is considered positive if it acts in positive direction and negative if in negative direction.

Transformation of plane stress

The stress system is known in coordinate system xy. We want to find stress in coordinate system x1y1 which is rotated θ degree in anti clockwise direction. 

Transformation Equations are:

\[σ_{x1} = \frac{σ_x + σ_y}{2} + \frac{σ_x - σ_y}{2}cos2θ + τ_{xy}sin2θ\]

\[σ_{y1} = \frac{σ_x + σ_y}{2} - \frac{σ_x - σ_y}{2}cos2θ - τ_{xy}sin2θ\]

\[τ_{x1y1} = - \frac{σ_x - σ_y}{2}sin2θ + τ_{xy}cos2θ\]

Note:

• $σ_x + σ_y  = σ_{x1} + σ_{y1}$

Principal Stresses and Principal Plane

• The maximum or minimum of normal stresses (σ 1 and σ 2) are known as the principal stresses.
• The plane on which principal stresses act is called principal plane.
• The shear stresses are zero on the principal plane.

To find the principal stresses, differentiate the transform equations and we get,  

\[\boxed{tan2θ_p = \frac{2τ_{xy}}{σ_x - σ_y}}\]

$θ_p$ is angle of principal plane.

Principal Stresses are:

\[\boxed{σ_{1,2} = \frac{σ_x + σ_y}{2} ± \sqrt{(\frac{σ_x - σ_y}{2})^2 + (τ_{xy})^2}}\]

Maximum Shear Stress

Say $θ_s$ is angle of plane where shear stress is maximum then,

\[tan2θ_s = - \bigg(\frac{σ_x - σ_y}{2τ_{xy}}\bigg)\]

\[τ_{max} = \sqrt{(\frac{σ_x - σ_y}{2})^2 + (τ_{xy})^2} \quad \quad τ_{min} = -τ_{max}\]

Relation between Maximum shear stress and Principal Stress :

         \[θ_s = θ_p ± 45°\]

 \[τ_{max} = \frac{σ_1 - σ_2}{2}\]

Principal Strains and Principal Angles

All the equations based on stress transformation can be converted to equations of strain by substitute as follows:

\[ ε _x \iff σ_x\]

\[ ε _y \iff σ_y\]

\[\frac{γ_{xy}}{2} \iff τ_{xy}\]

 

• Plane stress doesn't lead to plane strains.