Question

Principal stress and maximum shear stress derivation

Answer 1

Answer:

Rotating Stresses from x-y Coordinate

System to new x'-y' Coordinate System

Rotating the stress state of a stress element can give stresses for any angle. But usually, the maximum normal or shear stresses are the most important. Thus, this section will find the angle which will give the maximum (or minimum) normal stress.

Start with the basic stress transformation equation for the x or y direction.

To maximize (or minimize) the stress, the derivative of σx′ with respective to the rotation angle θ is equated to zero. This gives,

dσx′ / dθ = 0 - (σx - σx) sin2θp + 2τxy cos2θp = 0

where subscript p represents the principal angle that produces the maximum or minimum. Rearranging gives,

Principal Stresses, σ1 and σ2,

at Principal Angle, θp

The angle θp can be substituted back into the rotation stress equation to give the actual maximum and minimum stress values. These stresses are commonly referred to as σ1 (maximum) and σ2 (minimum),

For certain stress configurations, the absolute value of σ2 (minimum) may actually be be larger than σ1 (maximum).

For convenience, the principal stresses, σ1 and σ2, are generally written as,

where the +/- is the only difference between the two stress equations.

It is interesting to note that the shear stress, τx′y′ will go to zero when the stress element is rotated θp.

Answer 2

Answer:

Principal Stress: Maximum and minimum normal stress possible for a specific point on a structural element. Shear stress is 0 at the orientation where principal stresses occur. Principal Angle: The angle of orientation at which principal stresses occur for a specific point.