Stress Strain, Principal Axes, and Mohr's Circle

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1 Stress Strain, Principal Axes, and Mohr's Circle General Stress State in Principal Axes System: Mohr's Circle This is a general state, we will rotate about the -axis and compare the result to a general two-dimensional stress state. i sprinc xx 0 0 stensordiag = 0 sprinc yy sprinc k stensordiag êê MatrixForm y ; { i sprinc xx 0 0 y 0 sprinc yy 0 k 0 0 sprinc { Rotation about -axis by angle q 2 i Cos@qD -Sin@qD 0 rotmat@q_d := Sin@qD Cos@qD 0 k 0 0 rotmat@qd êê MatrixForm i Cos@qD -Sin@qD 0 y Sin@qD Cos@qD 0 k 0 0 { y ; {

2 2 Lecture-0.nb Transformation to general two-dimensional stress state coordinate system by rotating the principal system by q around -axis 3 srot = srot êê MatrixForm i Cos@qD 2 sprinc xx + Sin@qD 2 sprinc yy Cos@qD Sin@qD I-sprinc xx + sprinc yy M 0 Cos@qD Sin@qD I-sprinc xx + sprinc yy M Sin@qD 2 sprinc xx + Cos@qD 2 sprinc yy 0 k 0 0 sprinc y { Writing the same equation in a slightly different way... 4 srotalt = Collect@srot êê TrigReduce, qd, Sin@2 qd<d; srotalt êê MatrixForm i ÄÄÄÄ Cos@2 qd Isprinc 2 xx - sprinc yy M + ÄÄÄÄ Isprinc 2 xx + sprinc yy M ÄÄÄÄ Sin@2 qd I-sprinc 2 xx + sprinc yy M 0 ÄÄÄÄ Sin@2 qd I-sprinc 2 xx + sprinc yy M ÄÄÄÄ Cos@2 qd I-sprinc 2 xx + sprinc yy M + ÄÄÄÄ Isprinc 2 xx + sprinc yy M 0 k 0 0 sprinc y { Naming the coefficients of the two-dimensional state in the rotated principal system i slab xx slab xy slab x y 5 slabmat = slab xy slab yy slab y = srotalt; k slab x slab y slab { slabmat êê MatrixForm i ÄÄÄÄ Cos@2 qd Isprinc 2 xx - sprinc yy M + ÄÄÄÄ Isprinc 2 xx + sprinc yy M ÄÄÄÄ Sin@2 qd I-sprinc 2 xx + sprinc yy M 0 ÄÄÄÄ Sin@2 qd I-sprinc 2 xx + sprinc yy M ÄÄÄÄ Cos@2 qd I-sprinc 2 xx + sprinc yy M + ÄÄÄÄ Isprinc 2 xx + sprinc yy M 0 k 0 0 sprinc y {

3 Lecture-0.nb 3. slab xx in laboratory system rotated by q from principal axis system 6 slab xx ÄÄÄ Ä 2 Cos@2 qd Isprinc xx - sprinc yy M + ÄÄÄÄ 2 Isprinc xx + sprinc yy M 2. slab yy in laboratory system rotated by q from principal axis system 7 slab yy ÄÄÄ Ä 2 Cos@2 qd I-sprinc xx + sprinc yy M + ÄÄÄ Ä 2 Isprinc xx + sprinc yy M 3. slab xy in laboratory system rotated by q from principal axis system 8 slab xy ÄÄÄ Ä 2 Sin@2 qd I-sprinc xx + sprinc yy M All - components remain ero except the original diagonal term slab Equations, 2, and 3 are the equations that express a circle known 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. Notice that there are two invariants of the general stress :

4 4 Lecture-0.nb The trace (or twice the offset of Mohr's Circle): 9 Simplify@slab xx + slab yy D sprinc xx + sprinc yy And the determinant 0 Simplify@slab xx slab yy - Hslab xy L^2D sprinc xx sprinc yy These last two results are precisely the trace and determinant of the x and y terms in the original diagonal form of the stress state, thus illustrating the invariance of these quantities under rotation of coordinate rotations. Example of Mohr's circle for two-dimensional body in uniaxial tension with sprinc xx = 0 MPa and all other stress components equal to ero uniaxial0 = 8sprinc xx -> 0, sprinc yy -> 0< 9sprinc xx Æ 0, sprinc yy Æ 0=

5 Lecture-0.nb 5 2 ParametricPlot@8slab xx, slab xy< ê. uniaxial0, 8q, 0, p<, AxesLabel Ø 8"normal stress", "shear stress"<, AspectRatio Ø, PlotLabel Ø " \t \t Mohr Circle for 0 MPa Uniaxial Tension", PlotStyle Ø 8Thickness@0.0D, Hue@D<D

6 6 Lecture-0.nb shear stress Mohr Circle for 0 MPa Uniaxial Tension normal stre -2

7 Lecture-0.nb 7-4

8 8 Lecture-0.nb Ö Graphics Ö 3 uniaxialother = 8sprinc xx -> 30, sprinc yy -> 0< 9sprinc xx Æ 30, sprinc yy Æ 0=

9 Lecture-0.nb 9 4 ParametricPlot@8slab xx, slab xy< ê. uniaxialother, 8q, 0, p<, AxesLabel Ø 8"normal stress", "shear stress"<, AspectRatio Ø, PlotRange -> 880, 40<, 8-20, 20<<, PlotLabel Ø " \t \t Mohr Circle for sprinc xx = 30 sprinc yy =0", PlotStyle Ø 8Thickness@0.0D, Hue@D< D

10 0 Lecture-0.nb shear stress Mohr Circle for sprinc xx = 30 sprinc yy = normal stress

11 Lecture-0.nb -20

12 2 Lecture-0.nb Ö Graphics Ö Comparing this plot with Figure 0-3 in the lecture notes, we see that the maximum and minimum tensile stresses are 0 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).

Lesson 18: There Is Only One Line Passing Through a Given Point with a Given Slope

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