Yield point and yield stress or strength,  y Offset method finds this yield stress by assuming a 0.2 % strain (.002).002 Big yielding region, large elongation at constant stress Point U is the maximum or ultimate stress Point R is rupture or complete failure Failure is not an option

Failure Theories For safety and reliability a structure must be designed and a material selected so that the strength of the structure exceeds that of the stresses to which it is subjected. By material or structural failure is meant that the material either ruptures so that it can no longer support any load or it undergoes excessive permanent deformation. Usually expressed in terms of the yield and ultimate strengths of the material.

Maximum shear stress theory is used for ductile materials to predict yielding, also known as Coulomb theory or Tresca theory. Assumes that yielding occurs when the maximum shear stress in a material element reaches the value of the maximum shear stress that would be observed at the instant when yielding occurred if the material were subject to uniaxial tension. Assume that a material is subjected to a uniaxial tension until yielding. Stress at yielding  max =  yp /2 this theory Then says that for any object subjected to a combo of normal and shear stresses for which the max shear stress is calculated, yielding will begin when the max shear =  max

Maximum distortion energy theory states that the conditions for yielding occurs when the RMS of the differences between the principal stresses is equal to the yield strength of the material that is found in a simple tension test. Also known as Von Mises yield theory or the Mises-Hencky theory. If:  y = SQRT (  max 2 -  max  min +  min 2 ) Fails!

Maximum normal stress theory is based on the assumption that failure by yielding occurs whenever the largest principal stress is equal to the yield strength (  y ) or by rupture whenever the largest principal stress is equal to the ultimate strength (  u ). Easiest theory to use. For comparison of these theories, failure occurs on or outside the regions shown below. Most Accurate Most conservative wrt MDET

Working stress is the maximum possible stress in an object or structure that must be designed to withstand. To account for uncertainties in loading dynamics the allowable stress must be set considerably lower than the ultimate strength of the material thus providing for a margin of safety while allowing for an efficient use of what could be an expensive material. Safety factor n is usually defined as the ratio of the ultimate 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.Temperature 2.Friction 3.Wear and tear 4.corrosion

Fatigue and Endurance relative to repeated loadings and unloadings. Failure may occur after a few cycles or after a million cycles depending on the amplitude of the applied load, the physical propertes of the material, size of the structure etc. Fracture due to repeated loading is called fatigue leading to complete structural failure. uu Periodic stress applied until failure after N cycles. Higher the amplitude the lower the value of N.

Higher amplitude Fatigue strength for given N Fatigue life Endurance limit

Glass, ceramics

Stress concentration or stress raising  = F / ( A – A h ) Note that stress around a hole increases or is raised 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, and the idea of stress-relieving drilling of holes as shown on the right. The above equation provides the tensile stress in the wall at the ends of the major axis of the hole.  Note that is b = a then SCF = 3 as before for a circular hole, but if b > 3, by adding holes at the end of the crack you can at least knock the stress down to a SCF = 3 and stop crack from growing.