Strain (normal)

A measure of the deformation of the material that is dimensionless. 

		change in length		L
Strain,	=		=
		original length		L

Modulus of elasticity

Metal deformation is proportional to the imposed loads over a range of loads.

Since stress is proportional to load and strain is proportional to deformation, this implies that stress is proportional to strain. Hooke's Law is the statement of that proportionality.

Stress
	=		=	E
Strain

Proportional limit

The greatest stress at which a material is capable of sustaining the applied load without deviating from the proportionality of stress to strain. Expressed in psi (kg/mm²). 

Ultimate strength (tensile)

The maximum stress a material withstands when subjected to an applied load.  Dividing the load at failure by the original cross sectional area determines the value. 

Elastic limit

The point on the stress-strain curve beyond which the material permanently deforms after removing the load .

Yield strength

Point at which material exceeds the elastic limit and will not return to its origin shape or length if the stress is removed.  This value is determined by evaluating a stress-strain diagram produced during a  tensile test.

Poisson's ratio

The ratio of the lateral to longitudinal strain is Poisson's ratio.

		lateral strain
	=
		longitudinal strain

Bending stress

When bending a piece of metal, one surface of the material stretches in tension while the opposite surface compresses.  It follows that there is a line or region of zero stress between the two surfaces, called the neutral axis. Make the following assumptions in simple bending theory:

1. The beam is initially straight, unstressed and symmetric
2. The material of the beam is linearly elastic, homogeneous and isotropic.
3. The proportional limit is not exceeded.
4. Young's modulus for the material is the same in tension and compression
5. All deflections are small, so that planar cross-sections remain planar before and after bending.

Using classical beam formulas and section properties, the following relationship can be derived:

		3 PL
Bending stress,   b	=
		2 w t 2

		P L3
Bending or flexural modulus,  E  b	=
		4 w t 3 y

Where:	P	=	normal force
	l	=	beam length
	w	=	beam width
	t	=	beam thickness
	y	=	deflection at load point

The reported flexural modulus is usually the initial modulus from the stress-strain curve in tension. 

The maximum stress occurs at the surface of the beam farthest from the neutral surface (axis) and is:

		M c		M
Max surface stress,  max	=		=
		I		Z

Where:	M	=	bending moment
	c	=	distance from neutral axis to outer surface where max stress occurs
	I	=	moment of inertia
	Z	=	I/c = section modulus

For a rectangular cantilever beam with a concentrated load at one end, the maximum surface stress is given by:

		3 d E t
max	=
		2 l 2

Yielding

Yielding occurs when the design stress exceeds the material yield strength. Design stress is typically maximum surface stress (simple loading) or Von Mises stress (complex loading conditions). The Von Mises yield criterion states that yielding occurs when the Von Mises stress,   exceeds the yield strength in tension. Often, Finite Element Analysis stress results use Von Mises stresses. Von Mises stress is:

			( 1- 2 )2 +  ( 2- 3 )2 +  ( 1- 3 )2
	=
			2

where ₁, ₂, ₃ are principal stresses.

Safety factor is a function of design stress and yield strength. The following equation denotes safety factor, f_s.

		Y S
fs	=
		D S