Stress Strain, Principal Axes, and Mohr's Circle
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General Stress State Rotated from the Principal Axes
System
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Two-Dimensional Stress under Rotation: Mohr's
Circle
1. slabxx in laboratory system rotated by q from principal axis system
1
slabxx
1
2
Cos@2 qD Isprincxx - sprincyy M +
1
2
Isprincxx + sprincyy M
2. slabyy in laboratory system rotated by q from principal axis system
2
slabyy
1
2
Cos@2 qD I-sprincxx + sprincyy M +
1
2
Isprincxx + sprincyy M
3. slabxy in laboratory system rotated by q from principal axis system
slabxy
1
2
3
Sin@2 qD I-sprincxx + sprincyy M
All z - components remain zero except the original diagonal term slabzz
The above three equations can be graphically represented as Mohr's circle of stress (see accompanying class notes).
The equations show the way in which the stress tensor components in a two-dimensional state of stress (a "biaxial" stress
state) vary with orientation of the coordinate system in which the stresses are described.
Example of Mohr's circle for two-dimensional body in uniaxial tension with sprincxx = 10 MPa and all other stress
components equal to zero
uniaxial10 = 9sprincxx -> 10, sprincyy -> 0=
9sprincxx Ø 10, sprincyy Ø 0=
4
2
ParametricPlot@8slabxx , slabxy < ê. uniaxial10
, 8q, 0, p<, AxesLabel Ø 8"normal stress", "shear stress"<, AspectRatio Ø 1,
PlotLabel Ø " \t \t Mohr Circle for 10 MPa Uniaxial Tension",
PlotStyle Ø 8Thickness@0.01D, Hue@1D<D
5
Mohr Circle for 10 MPa Uniaxial Tension
shear stress
4
2
2
4
6
8
10
normal stress
-2
-4
uniaxialother = 9sprincxx -> 30, sprincyy -> 10=
9sprincxx Ø 30, sprincyy Ø 10=
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Lecture-10-evaluated.nb
ParametricPlotA8slabxx , slabxy < ê. uniaxialother
, 8q, 0, p<, AxesLabel Ø 8"normal stress", "shear stress"<,
AspectRatio Ø 1, PlotRange -> 880, 40<, 8-20, 20<<,
PlotLabel Ø " \t \t Mohr Circle for sprincxx = 30
sprincyy =10",
7
PlotStyle Ø 8Thickness@0.01D, Hue@1D<E
Mohr Circle for sprincxx = 30 sprincyy =10
shear stress
20
10
0
10
20
30
40
normal stress
-10
-20
Comparing this plot with Figure 10-3 in the lecture notes, we see that the maximum and minimum tensile stresses are 10
and 30 MPa (from intercepts with x axis), as expected, and the maximum shear stress is ±5 MPa and it is experienced on
a plane oriented at 2q = 90° or q = 45° to the tensile axis (remember that angles on Mohr's circle plots are twice the angle
in the body).
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Visualization Example: Graphics for Mohr's Circle
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Visualization Example (cont'd) : Interacting with
Mohr's Circle