Analysis of Stress and Strain Review: - Axially loaded Bar - Torsional shaft p tnt sn q txy h P P q tyx 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 an arbitrary 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 sy sy State of stress at one point: tyx tyx tyz txy txy txy Stress element: y tzy sx sx sx sz tzx txz tyx x sy z 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 subject to stresses, and all stresses are parallel to the x and y axes. Stresses on inclined planes txy q sx tyx sy Transformation equations for plane stress

Transformation Equations angle between x1 and x axes, measured counterclockwise

Plane Stress – Special Cases Uniaxial Stress: sx sx tyx Pure Shear: txy txy tyx sy Biaxial Stress: sx sx sy

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 15o 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 The angle defines the orientation of the principal planes.

Principal Stresses OR

Shear Stress Shear stresses on the principal planes: Example 2: Principal stresses in pure shear case: tyx txy txy tyx

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 Circle For Plane Stress – Equations of Mohr’s Circle Transformation equations: (1)2 + (2)2

Two Forms of Mohr’s Circle

Construction of Mohr’s Circle Approach 1: For the given state of stresses, calculate and R. The center Of the circle is ( , 0) and the radius is R.

Construction of Mohr’s Circle Approach 2: Find points corresponding to q = 0 and q = 90o and then draw a line. The intersection is the origin of the circle.

Applications of Mohr’s Circle Example 4: 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 40o, (b) the principal stresses and (c) the maximum shear stress.

Plane Strain Definition: Only x and y faces are subject 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: