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Analysis of Stress and Strain

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Analysis of Stress and Strain. Review:. - Torsional shaft. - Axially loaded Bar. t nt. p. s n. q. t xy. h. P. P. q. t yx. Questions: (1) Is there any general method to determine stresses on any arbitrary plane

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Analysis of Stress and Strain

Review:

- Torsional shaft

- Axially loaded Bar

tnt

p

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.

Shear Stress

Shear stresses on the principal planes:

Example 2: Principal stresses in pure shear case:

tyx

txy

txy

tyx

Plane Stress

Example 3: Find the principal stresses and maximum shear stresses and

show them on a sketch of a properly oriented element.

Transformation equations:

(1)2 + (2)2

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: