材料力学07_应力变换

2
Two principal stresses and two principal planes: which plane is subject to max or min?
Procedure • Calculate p1,p2 first. • 1
x y x y
• Simplifying the above equations to give the following stress transformation relations for plane stress
x y
x y x y
2 x y 2 2 x y 2
Two angles p1 and p2 are 1 defined. p1 and p2 are 90 apart, i.e., |p1 p2| = 90
x y
2 x y 2 xy 2
2
2
• max, min
y A sin sin xy A sin cos 0
Fy 0 :
xyA x A cos sin xy A cos cos
y A sin cos xy A sin sin 0
• From Topic One, we know that the most general state of stress at a point may be represented by 6 components,
Normal Stress x , y , z Shear Stress xy , yz , zx ( xy yx , yz zy , zx xz )
School of Engineering Mechanical Engineering
TOPIC
ENGR 323
07
Mechanics of Deformable Bodies
Stress Transformation
© Tulong Zhu, All rights reserved.
Introduction
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x , y , xy 0 and z zx zy 0.
06 - 3
Transformation of Plane Stress: Introduction
• In this topic, we will investigate the stress transformation of plane stress in different coordinate systems, and to determine the maximum stresses.
06 - 9
Example 1
Problem For the state of plane stress shown, determine (a) the principal planes, (b) the principal stresses, (c) the maximum shearing stress and the corresponding normal stress, and (d) plot all the principal stresses in an appropriately orientated element. • Find the element orientation for the principal stresses: 2 xy p1 and p2 tan 2 p x y • Determine the principal stresses from x y x y 1 cos 2 p1 xy sin 2 p1 2 2 x y x y 2 cos 2 p 2 xy sin 2 p 2 2 2 • Calculate the maximum shearing stress with x y max sin 2 s1 xy cos 2 s1 s1 p1 45 2
Purpose Stress Transform. Principal Shear Stresses:
• • To obtain the maximum shearing stress.
• xy
x y
2
sin 2 xy cos 2
x y • tan 2 s 2 xy
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06 - 6
We don’t know max or min is for p1 or p2.
Principal Stresses: Procedure
Principal Stresses • max, min
x y
2
2 xy x y 2 xy , tan 2 p1, p 2 2 x y
x y 2 • max xy 2
2
x y tan 2 s 2 xy
Two principal shear stresses and s: how to pair? Procedure
• Calculate s1 first:
06 - 7
• p
( x y ) 2
sin 2 p xy cos 2 p 0
from tan 2 p
2 xy
x y
Shear stress equals zero on principal planes.
Principal Shear Stress
s1 p1 45
• max
x y
2
sin 2 s1 xy cos 2 s1
No need to calculate the principal shear stress for s2 (why?). Normal Stress on Planes of Principal Shear
06 - 10
Solution Strategy
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• Plot the elements with principal normal and shearing stresses.
Example 1: Principal Stresses
Define two angles s1 and s2, separated by 90. In calculation, one angle is enough, and s p 45
d /d = 0
x y 2 • max xy 2
Plane Stress
• Plane Stress - state of stress in which two faces of the cubic element are free of stress. For the illustrated example, the state of stress is defined by
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• Normal Stress does not vanish on these two planes
x y max min ave 2 2
Normal stress does not vanish on planes of principal shear stresses.
cos 2 xy sin 2 cos 2 xy sin 2
xy
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x y
2
sin 2 xy cos 2
06 - 5
Positive shear stress: Counterclockwise
Principal Stresses
Purpoபைடு நூலகம்e Stress Transformation
2 x y
2

• 2

2 x y
2
cos 2 p1 xy sin 2 p1
cos 2 p 2 xy sin 2 p 2
One is max and the other is min.
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Shear Stress on Principal Plane
• To obtain the maximum normal stress. • x
x y x y
2 2
cos 2 xy sin 2
Principal Stresses:
dx’ /d =0
• tan 2 p
2 xy
x y
2
2
1
1
06 - 4
(a) Stress Status
(b) Geometry
Transformation of Plane Stress
• From the equilibrium of the triangular-shaped element, we have
Fx 0 :
xA x A cos cos xy A cos sin
• If we rotate the coordinate system, a different set of stress components the same stress state will be obtained for the SAME STRESS STATE.
Which set of stress components will be used in design and analysis? Is there a coordinate system in which some stress components achieve maximum?
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• To investigate the stress transformation, we can consider the equilibrium of a portion (a corner) of the square element, as shown. The faces of this prismatic element are perpendicular to the x, y, and x’ axes.
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2
Normal Stress
06 - 8
• Normal Stress does not vanish on these two planes x y 1 2 ave 2 2
Principal Shear Stresses: Procedure
Principal Shear Stresses
06 - 2
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Plane Stress
General Stresses • The stress transformation for general stresses is pretty complicated. We will be focusing on a special case – plane stress.
• We need to determine the relations between the stress components in the xyz system and those in the x′y′z′ system, and determine the maximum stress components.