Principal Planes, Principal Stresses, Max Shear Stresses

• Ref: Survey University Lecture Notes

• The simplest failure criteria such as the normal stress criteria require that a material will fail when the normal stress in a component exceeds the yield strength (ductile materials) or the ultimate strength (brittle materials) of the material.

• As we have seen that this normal stress can have various values depending on the choice of coordinate axis, thus for accurate determination of failure initiation we need to compare the maximum value of normal stress for a given loading with the material strength.

• The stress transformation equations can be used to determine this maximum value, which is know as the principal stress value.

• And the angle at which this value occurs in relation to the global coordinate system is called the angle of principal plane or principal angle.

Principal Planes, Principal Stresses, Max Shear Stresses

• In order to find the principal angle we use the transformation equation as follows:

Principal Planes, Principal Stresses, Max Shear Stresses

• Now from the expression for , i.e.

• Using this in transformation equation (1) gives

Principal Planes, Principal Stresses, Max Shear Stresses

Principal Planes, Principal Stresses, Max Shear Stresses

Principal Planes, Principal Stresses, Max Shear Stresses

Principal Planes, Principal Stresses, Max Shear Stresses

Stress Invariants – The first invariant

• Home work : reading section 7.3

Example Problem 7-2 Shames

• Solution on board

Mohr Circle (Mec Movies)

• 12-15 Driving Equation’s for Mohr’s Circle (Click here to open slides)

• 12-16 Drawing Mohr’s Circle (Click here to open slides)

• 12-17 Principal Stress using Mohr’s Circle (Click here to open slides)

• 12-18Max shear stress using Mohr’s Circle (Click here to open slides)

Design Problems – Thin walled Pressure Vessels

• Cylindrical vessel weld stresses (Click here to open slides)

• Compound Cylinder (Steel and Brass – MIT notes)

• Bolt clamped pressure vessel (MIT notes)• Critical pressure in cylindrical tank

(Click here to open slides)• Strain gage to measure tank pressure

(Click here to open slides)

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