Stress Transformation

x

y

x′

y′

Consider what we know about Force Transformation

For Stress Transformation, consider a Plane Stress State

Transformation of Plane Stress

2cos2sin2

2sin2cos22

2sin2cos22

xyyx

'y'x

xyyxyx

'y

xyyxyx

'x

Determination of transformed stresses σx′ , σy′ , & τx′y′ at angle θ for a plane stress state where σx , σy , & τxy are known.

Determination of Principal Stresses & Maximum In-Plane Shear Stress

2sin2cos22 xy

yxyx'x

Orientation θp of planes of maximum and minimum normal stresses

The orientation θp of the maximum principal normal stress is found by differentiating the equation for σx′ with respect to θ and setting it equal to zero:

2/2tan

yx

xyp

Determination of Principal Stresses

The solution to this equation has two roots, θp1 and θp2, that are 90o apart.

Using the trigonometric construction above, the in-plane principal stresses σ1 and σ2 can be determined from Eq. 9-1 if the normal stresses σx and σy are known as well as the shear stress τxy.

Orientation θs of maximum in-plane shear stress

Differentiate the equation for τx′y′ with respect to θ and set it equal to zero. The roots of the Eq. are θs1 and θs2.

xy

yxs

2/2tan

Either θs1 or θs2 can be used in Eq. 9-2 to determine the max in-plane shear stress. The average normal stress that acts on the same plane can also be determined using Eq. 9-1.

Maximum in-plane shear stress and the average normal stress

Mohr’s Circle Representation of Normal Stresses, Shear Stresses, Principal Stresses, and Maximum In-Plane Shear Stress

σ

τ

A Hand Crank is subjected to a 1000 N static load.Determine the maximum stresses at the base.

Hand Crank Problem – Example of 3D Mohr’s Circle for Pt. A

σ

τ

Hand Crank Problem – Example of 3D Mohr’s Circle for Pt. B

σ

τ