Analysis of Stress and Strain Review: - Axially loaded Bar - Torsional shaft Questions: (1) Is there any general method to determine stresses on any arbitrary plane at one point if the stresses at this point along some planes are known? (2) For a general loaded member, how many planes on which stresses are known are required to determine the stresses at any plane at one point?

Analysis of Stress and Strain State of stress at one point: Stress element: - Use a cube to represent stress element. It is infinitesimal in size. - (x,y,z) axes are parallel to the edges of the element - faces of the element are designated by the directions of their outward normals. Sign Convention : - Normal stresses: “+” tension; “-” compression. - Shear stresses: “+” the directions associated with its subscripts are plus-plus or minus-minus “-” the directions associated with its subscripts are plus-minus or minus-plus

Plane Stress Definition: Only x and y faces are subjected to stresses, and all stresses are parallel to the x and y axes. Stresses on inclined planes

Transformation Equations angle between x 1 and x axes, measured counterclockwise

Plane Stress – Special Cases Uniaxial Stress : Pure Shear : Biaxial Stress :

Plane Stress Example 1: A plane-stress condition exists at a point on the surface of a loaded structure, where the stresses have the magnitudes and directions shown on the stress element of the following figure. Determine the stresses acting on an element that is oriented at a clockwise angle of 15 o with respect to the original element.

Principal Stresses Principal stresses: maximum and minimum normal stresses. Principal planes: the planes on which the principal stresses act

Principal Stresses Shear stress on the principal planes:

Principal Stresses Example 2: Find principal stresses of an element which is in pure shear.

Maximum Shear Stresses

Plane Stress Example 3: Find the principal stresses and maximum shear stresses and show them on a sketch of a properly oriented element.

Mohr’s Circile For Plane Stress – Equations of Mohr’s Circle Transformation equations:

Two Forms of Mohr’s Circle

Construction of Mohr’s Circle

Applications of Mohr’s Circle - Stresses on inclined element - Principal stresses and maximum shear stresses

Applications of Mohr’s Circle Example: An element in plane stress at the surface of a large machine is subjected to stresses Using Mohr’s circle, determine the following quantities: (a) the stresses acting on an element inclined at an angle of 40 o, (b) the principal stresses and (c) the maximum shear stress.

Plane Strain Definition: Only x and y faces are subjected to strains, and all strains are parallel to the x and y axes. Note: Plane stress and plane strain do not occur simultaneously.

Plane Strain Transformation Equations: Principal Strains:

Measurement of Strain – Electrical Resistance Strain Gages Principle: length changes electrical resistance change Measurement of Principal Strains: need three gages! Why? How?

Hooke’s Law For Plane Stress Assumptions: -

Hooke’s Law For Plane Stress Example: A 45 0 strain rosette consists of three strain gages arranged to measure strain in two perpendicular directions and also at a 45 0 angle between them, as shown in the Following figure. The rosette is bounded to the surface before it is loaded. It is measured that the strains in three gages (A,B,C) are , and respectively. Find principal stresses at this point.

Strain Energy Density Strain Energy: