Yield point and yieldstress or strength, y

Offset method finds thisyield stress by assuminga 0.2 % strain (.002)

.002

Big yieldingregion, largeelongation at

constant stress

Point U is the maximum or ultimate stressPoint R is rupture or complete failure

Failure is not an option

Failure Theories

For safety and reliability a structure must be designedand a material selected so that the strength of thestructure exceeds that of the stresses to which itis subjected.

By material or structural failure is meant that the materialeither ruptures so that it can no longer support any loador it undergoes excessive permanent deformation.

Usually expressed in terms of the yield and ultimatestrengths of the material.

Maximum shear stress theory is used for ductile materialsto predict yielding, also known as Coulomb theory orTresca theory.

Assumes that yielding occurs when the maximum shear stress in a material element reaches the value of themaximum shear stress that would be observed at theinstant when yielding occurred if the material weresubject to uniaxial tension.

Assume that a material is subjected to a uniaxialtension until yielding.

Stress at yielding

max = yp/2 this theory

Then says that forany object subjected to a combo of normaland shear stressesfor which the max shear stress is calculated, yieldingwill begin when themax shear = max

Maximum distortion energy theory states that the conditionsfor yielding occurs when the RMS of the differences betweenthe principal stresses is equal to the yield strength of thematerial that is found in a simple tension test. Also known asVon Mises yield theory or the Mises-Hencky theory.

If:

y = SQRT ( max2 - maxmin + min

2 ) Fails!

Maximum normal stress theory is based on the assumptionthat failure by yielding occurs whenever the largestprincipal stress is equal to the yield strength (y) or byrupture whenever the largest principal stress is equal tothe ultimate strength (u). Easiest theory to use.

For comparison of these theories, failure occurs on oroutside the regions shown below.

Most Accurate

Most conservativewrt MDET

Working stress is the maximum possible stress in anobject or structure that must be designed to withstand.

To account for uncertainties in loading dynamics theallowable stress must be set considerably lower thanthe ultimate strength of the material thus providing fora margin of safety while allowing for an efficient use ofwhat could be an expensive material.

Safety factor n is usually defined as the ratio of theultimate strength of a material to the allowable stress.This can also be based on the yield strength too.

allowable = u / n

Other factors to consider in strength of materials include:1. Temperature2. Friction3. Wear and tear4. corrosion

Fatigue and Endurance relative to repeated loadings and unloadings. Failure may occur after a few cycles or aftera million cycles depending on the amplitude of the applied load, the physical propertes of the material, sizeof the structure etc.

Fracture due to repeated loading is called fatigue leading tocomplete structural failure.

u

Periodic stress applied until failure after N cycles. Higher the amplitude the lower the value of N.

Higheramplitude

Fatigue strength for given N

Fatigue life

Endurancelimit

Glass, ceramics

Stress concentration or stress raising

= F / ( A – Ah )

Note that stress arounda hole increases or israised or concentrated!!!

Note that the horizontal edges and the hole are stress free.

A circular hole in a thin plate

rr =

=

r =

ICBS that the above are the stresses for an elastic material.

Elliptical hole in a plate as shown on the left, andthe idea of stress-relieving drilling of holes as shown on the right.

b

aS 21

The above equation provides the tensile stress in thewall at the ends of the major axis of the hole.

Note that is b = a then SCF = 3as before for a circular hole, butif b<<a then SCF >> 3, by addingholes at the end of the crack youcan at least knock the stress downto a SCF = 3 and stop crack fromgrowing.