mechanics of materials chp10
TRANSCRIPT
Stress transformation is about:
- Changing the orientation of the axes used to describe stress so that you
can get the stress components in other directions, in particular, the
orientation in which maximum stresses occur. - We will only be considering Plane Stress which is the common assumption used in engineering practice. Plane stress assumes all stresses
can be analyzed in a single plane. Therefore, we need only consider three
stress components and not all six. . .
Chapter 10 - Stress Transformation
2
Stress Transformation Procedure is pretty simple actually. You just have to cut off the sample element along
the inclined direction you want to know the stresses and then do a free body diagram
of the remaining wedge – use equilibrium and solve for the forces along the inclined
plane
Equals...
3
General Equations of Plane Stress Transformations
Positive sign convention
4
Principle Stresses and Maximum In-Plane Shear Stress
The MAXIMUM and MINIMUM normal stress and MAXIMUM shear stress by
differentiating the previous equations with respect to theta and setting the result to
zero . .
IN-PLANE PRINCIPLE STRESSES
Which solves to give. . .
5
IN-PLANE PRINCIPLE STRESSES CONTINUED
Shear stress is ZERO on the principal planes, ie
6
MAXIMUM IN-PLANE SHEAR STRESSES
Similarly, by differentiating the equation for shear stress with respect to theta and
setting the result to zero we get . .
This equation has two roots (solutions) for theta that are 45 degrees apart –
which means that
The planes for maximum shear stress can be determined by orienting an element 45 degrees from the position of an element than defines the planes of principal stress. The solution is:
There is a normal stress on the planes of maximum shear stress and it is found via
substitution to be equal to:
7
Recall from the past the following:
The standard equation for a circle is
(x - h)2 + (y - k)2 = r2
where h and k are the x- and y coordinates of the center of the circle and r is the
radius.
8
Otto Mohr rewrote the equations for normal and shear stress as
Notice the form of these
stress equations looks like
that of a circle?
(x - h)2 + (y - k)2 = r2
9
The coordinate system requires sigma positive to the right and tau positive
downwards
Procedure for analysis
The following steps are required to draw and use Mohr’s circle
10
11
12
13
14
Problem 9-1: The state of stress at a point in a member is
shown on the element. Determine the stress components acting
on the inclined plane AB. Solve the problem using the stress-
transformation equations.
15
Solution X’
y’
16
Problem 9-2: The state of stress at a point in a member is
shown on the element. Determine the stress components acting
on the inclined plane AB. Solve the problem using Mohr’s circle.
17
Solution
ksiR xy
yx0623.88
2
35
2
2
2
2
2
ksiyx
average 42
35
2
875.82
2
35
8tan
2
tan 11
yx
xy
18
P