stress_transform_math-1 - Stress, strain, principal axes,...

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Stress, strain, principal axes, stress transfomation, and mohr circle Consider a the state of stress of a body in a generic lab frame x-y-z s l = s l xx s l xy s l xz s l xy s l yy s l yz s l xz s l yz s l zz . If we want to find the corresponding stress components in a different frame of reference, the the stress components in that frame s g is given by: s g = R s l R T . Here, in the simplest case of two dimensional stresses I s l xz =s l yz = 0 M the rotation matrix is given by: R = cos H q L sin H q L 0 - sin H q L cos H q L 0 00 1 . Here q is the rotation of the new frame of reference about the z axis. Upon transformation the stress components become. Rot @ θ _ D : = Cos @ θ D Sin @ θ D 0 Sin @ θ D Cos @ θ D 0 1 ; σ lab = σ l xx σ l xy 0 σ l xy σ l yy 0 σ l zz ; σ g @ θ _, σ lab_ D : = Collect @ Rot @ θ D . σ lab.Transpose @ Rot @ θ DD êê FullSimplify êê TrigReduce, 8 Cos @ 2 θ D , Sin @ 2 θ D<D ; MatrixForm @ σ g @ θ , σ lab DD Sin @ 2 θ D σ l xy + 1 2 Cos @ 2 θ DI σ l xx −σ l yy M + 1 2 I σ l xx l yy M Cos @ 2 θ D σ l xy + 1 2 Sin @ 2 θ l xx l yy M Cos @ 2 θ D σ l xy + 1 2 Sin @ 2 θ l xx l yy M Sin @ 2 θ D σ l xy + 1 2 Cos @ 2 θ l xx l yy M + 1 2 I σ l We will now use these relations to obtain the mohr circle. The center of the circle is given by: C = s l xx + s l yy 2 , and the radius of the mohr circle is given by, R = J s l xx -s l yy 2 N 2 +s l xy 2 . MohrCenter @ σ _ D : = H σ @@ 1, 1 DD @@ 2, 2 DDL 2 MohrRadius @ σ _ D : = . σ @@ 1, 1 DD @@ 2, 2 DD 2 2 @@ 1, 2 DD 2 Note that the center and the radius is completely independent upon the frame of reference we use to obtain the co-ordinates of the stresses.
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MohrCenter @ σ g @ θ , σ lab DD êê FullSimplify MohrRadius @ σ g @ θ , σ lab DD êê FullSimplify 1 2 I σ l xx l yy M 1 2 4 σ l xy 2 + I σ l xx −σ l yy M 2 Sample Mohr circle is drawn here for s xx = 1, s xy = 3, and s yy = 2. ParametricPlot A Evaluate A8 σ g @ θ , σ lab D@@ 1, 1 DD , σ g @ θ , σ lab D@@ 1, 2 DD< ê . 9 σ l xx 1, σ l yy 2, σ l xy 3 =E , 8 θ ,0 , π < , AxesLabel 8 "normal stress", "shear stress" < , PlotLabel "\t \t Mohr Circle for different values of σ ", AspectRatio 1, PlotStyle 8 Thickness @ 0.01 D , Hue @ 1 D<E - 1 1 2 3 4 normal stress - 3 - 2 - 1 1 2 3 shear stress Mohr Circle for different values of s Now we will draw a well illustrated proper Mohr circle which also tells us (in graphical form) the way to draw the Mohr circle and the way to interpret the principal directions. This procedure draws the Mohr circle by taking the stress components in a given (lab) frame x-y-z as its input. The following discussion is heavily borrowed from Prof. Craig Carter's course available at: http- ://pruffle.mit.edu. 2 Lecture-1-Sep.nb
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mohr @ off_, rad_ D : = 8 Red, Thick, Circle @8 off, 0 < , rad D< s12graph @ s11_, s12_ D : = 8 Darker @ Orange D , Arrow @88 s11, s12 < , 8 0, s12 <<D , Text @ s12, 8 0, s12 < , 8 2.5, 1.5 < , Background White D< s22graph @ s22_, s12_ D : = 8 Blue, Arrow @88 s22, s12 < , 8 s22, 0 <<D , Text @ s22, 8 s22, 0 < , 8 0, 1 < , Background White D< s11graph @ s11_, s12_ D : = 8
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stress_transform_math-1 - Stress, strain, principal axes,...

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