Principal Stresses

BIOE 3200 - Fall 2015

Learning ObjectivesDescribe what principal stresses

and maximum shear stress are and why they are important

Calculate principal stresses and maximum shear stress, and the angles at which they occur

Visualize principal stresses and maximum shear stress on a stress element

BIOE 3200 - Fall 2015

Review: Stresses within a rigid body

BIOE 3200 - Fall 2015

Review: Stress Transformations  

BIOE 3200 - Fall 2015

Principal Stresses: Minimum and maximum normal stressesFind by setting derivative of stress

transformation equation to zeroAlso know that shear stress = 0 on principal

plane◦ Use shear stress transformation equation to

develop equations for principal stresses

BIOE 3200 - Fall 2015

Sketching principal stresses on a stress element

BIOE 3200 - Fall 2015

Maximum Shear StressEven if max normal stress is within

limits of material strength, failure can occur in shear

Find direction of max shear stress plane by setting derivative of shear stress transformation equation to zero

BIOE 3200 - Fall 2015

Maximum Shear StressOccurs on plane at which normal

stresses are equal◦ Can also solve by setting normal stress

transformation equations equal to each other and solving for angle of max shear stress plane

◦ Plug angle into shear stress equation to determine equation for max shear stress

BIOE 3200 - Fall 2015

Sketching Principal Stresses and Maximum Shear Stress

Principal stresses occur at angles 1 and 2 from the x-y axes

Maximum shear stress occurs at an angle s from the x-y axes

BIOE 3200 - Fall 2015

s

s