Stress and Strain Basics

Stress DefinitionEvery structure must be in equilibrium at all times.This is true if you look only at external loads & reactions, but it also is true if you cut a structure in the middle.What balances the external loads & reactions when you cut a structure?Answer: StressesStress is a measure of the internal forces that balance the external forces.Stress has units of force/area (e.g. pounds per square inch psi; Newtons per square meter Pa).For a simple bar in tension, the stress is the force divided by the cross-sectional area.

Stress On An Arbitrary Element in 2DXYx: Normal stress in X-directionxy: Shear stress on the X-face in the Y-directionyx: Shear stress on the Y-face in the X-directiony: Normal stress in Y-directionABCD

Principal StressesIf we rotate the stress element, at some angle the shear stresses will become zero. The normal stresses on the plane with zero shear stresses are the principal stresses.

Stresses in 3DIn 3D the stress state at a point is completely defined by 6 stress components (which make up the stress tensor):x: Normal stress in X-directiony: Normal stress in Y-directionz: Normal stress in Z-directionxy: Shear stress on X-face in Y-directionxz: Shear stress on X-face in Z-directionyz: Shear stress on Y-face in Z-directionWe also then have 3 principal stresses: P1, P2, and P3.Convention says that P1 > P2 > P3.All 3 can be positive (tension), all 3 can be negative (compression), or you can have a combination.

Summary StressesVon Mises stress (from the distortion energy failure theory).

Tresca stress (from the maximum shear stress failure theory).

Special Stress CasesTorsional stressBending stressMembrane stressHoop stressAxial (longitudinal) stressRadial stressHertzian stressThermal stress

Saint-Venants PrincipleThe localized effects of stress concentrations disappear at some distance from the location of the stress concentration.This has powerful implications for the location of boundary conditions.

StrainStrain is a measure of the amount of deformation in a material.Strain is measured as deformation per unit length, so it is a unitless quantity.There are both normal strains () and shearing strains ().For a bar in tension the normal strain is: