he state of stress at a point in xyz reference is as follows.
$\tau_{i j}=\begin{vmatrix} {80} & {-60} & {0} \\ {-60} & {-40} & {0} \\ {0} & {0} & {40} \end{vmatrix} MPA$
Determine the stress tensor in the direction $x^{\prime}y^{\prime}z^{\prime}$ by rotating xyz through 30° anticlockwise about z-direction as shown in Fig. 2.26.

Question

he state of stress at a point in xyz reference is as follows.

[latex] au_{i j}=\begin{vmatrix} {80} & {-60} & {0} \ {-60} & {-40} & {0} \ {0} & {0} & {40} \end{vmatrix} MPA [/latex]

Determine the stress tensor in the direction [latex]x^{\prime}y^{\prime}z^{\prime}[/latex] by rotating xyz through 30° anticlockwise about z-direction as shown in Fig. 2.26.

Accepted Answer

Direction cosines,

[latex]c_{x^{\prime} x}=\cos 30^{\circ}=0.866,[/latex]

[latex]c_{x^{\prime} y}=\cos 60^{\circ}=0.5, [/latex]

[latex]c_{x^{\prime} z}=\cos 90^{\circ}=0, [/latex]

[latex]c_{zx^{\prime} } =\cos 90^{\circ}=0, [/latex]

[latex]c_{z z^{\prime}}=\cos 0^{\circ}=1, [/latex]

[latex]c_{y x^{\prime}}=\cos 120^{\circ}=-0.5, [/latex]

[latex]c_{ y^{\prime}y}=\cos 30^{\circ}=0.866, [/latex]

[latex]c_{ y^{\prime}z}=\cos 90^{\circ}=0, [/latex]

[latex]c_{z y^{\prime}}=\cos 90^{\circ}=0 [/latex]

Using Eq. (2.10),

[latex] au_{n s}=\sum_{i j} au_{i j} imes c_{n i} imes c_{s j} [/latex]

where all combinations of n, s = [latex]x^{\prime},y^{\prime}, z^{\prime}[/latex] and i, j = x, y, z are exhausted.

(1) n = [latex] x^{\prime}, s=x^{\prime} [/latex]

[latex] au _{x^{\prime} x^{\prime}}=c_{x^{\prime} x} c_{x^{\prime} x} au_{x x}+c_{x^{\prime} x} c_{x^{\prime} y} au_{x y}+c_{x^{\prime} x} c_{x^{\prime} z} au_{x z}[/latex]

[latex]+c _{x^{\prime} y} c_{x^{\prime} x} au_{y x}+c_{x^{\prime} y} c_{x^{\prime} y} au_{y y}+c_{x^{\prime} y} c_{x^{\prime} z} au_{y z}[/latex]

[latex]+c_{x^{\prime} z} c_{x x^{\prime} x} au_{z x}+c_{x^{\prime} z} c_{x^{\prime} y} au_{y z}+c_{x^{\prime} z} c_{x^{\prime} z} au_{z z}[/latex]

[latex]=0.866^2 imes 80-0.866 imes 0.5 imes 60+0-[/latex]

[latex] 0.5 imes 0.866 imes 60-0.5^2 imes 40+0 =-1.96 MPa[/latex]

(2) n = [latex] x^{\prime}, s=y^{\prime} [/latex]

[latex] au_{x^{\prime} y^{\prime}}=c_{x^{\prime} x} c_{y^{\prime} x} au_{x x}+c_{x^{\prime} x} c_{y^{\prime} y^{\prime}} au_{x y}+c_{x^{\prime} x} c_{y^{\prime} z^{\prime}} au_{x z}[/latex]

[latex] +c_{x^{\prime} y} c_{y^{\prime} x} au_{y x}+c_{x^{\prime} y} c_{y^{\prime} y} au_{y y}+c_{x^{\prime} y} c_{y^{\prime} z} au_{y z}[/latex]

[latex]+c_{x^{\prime} z} c_{y^{\prime} x} au_{z x}+c_{x^{\prime} z} c_{y^{\prime} y} au_{y z}+c_{x^{\prime} z} c_{y^{\prime} z} au_{z z}[/latex]

[latex]=-0.866 imes 0.5 imes 80-0.866^2 imes 60+0.5 imes 0.5 imes 60-0.5 imes 0.866 imes 40[/latex] = -81.96 MPa

(3) n = [latex]x^{\prime}, s=z^{\prime}[/latex]

[latex] au_{x^{\prime} z^{\prime}}=c_{x^{\prime} x} c_{z^{\prime} x} au_{x x}+c_{x x^{\prime}} c_{z^{\prime} y} au_{x y}+c_{x^{\prime} x} c_{z^{\prime} z} au_{x z}[/latex]

[latex]+c_{x^{\prime} y} c_{z^{\prime} x} au_{y x}+c_{x y^{\prime}} c_{z y^{\prime}} au_{y y}+c_{x^{\prime} y} c_{zz^{\prime}} au_{y z}[/latex]

[latex]+c_{x^{\prime} z} +c_{z^{\prime} x} au_{z x}+c_{x z^{\prime}} c_{z^{\prime} y} au_{y z}+c_{x^{\prime} z} c_{z^{\prime} z^{\prime}} au_{z z}=0[/latex]

(4) [latex]n=y^{\prime}, s=y^{\prime}[/latex]

[latex] au_{y^{\prime} y^{\prime}}=c_{y^{\prime} x} c_{y^{\prime} x} au_{x x}+c_{y^{\prime} x} c_{y^{\prime} y^{\prime}} au_{x y}+c_{y^{\prime} x} c_{y^{\prime} z} au_{x z}[/latex]

[latex]+c_{y y^{\prime}} c_{y^{\prime} x} au_{y x}+c_{y^{\prime} y} c_{y^{\prime} y} au_{y y}+c_{yy^{\prime}} c_{y^{\prime} z} au_{y z}[/latex]

[latex]+c_{y^{\prime} z} c_{y^{\prime} x} au_{z x}+c_{y^{\prime} z} c_{y^{\prime} y^{\prime}} au_{y z}+c_{y^{\prime} z^{\prime}} c_{y^{\prime} z} au_{z z}[/latex]

[latex]=0.5^2 imes 80+0.866 imes 0.5 imes 60+0+0.5 imes 0.866 imes 60-0.866^2 imes 40+0=41.96 MPa [/latex]

(5) [latex]n=y^{\prime}, s=z^{\prime}[/latex]

[latex] au_{y^{\prime} z^{\prime}}= c_{y^{\prime} x} c_{z^{\prime} x} au_{x x}+c_{y^{\prime} x} c_{z^{\prime} y} au_{x y}+c_{y^{\prime} x} c_{z^{\prime} z} au_{x z} [/latex]

[latex]+c_{y^{\prime} y} c_{z^{\prime} x} au_{y x}+c_{y^{\prime} y} c_{z^{\prime} y} au_{y y}+c_{y^{\prime} y} c_{z z^{\prime}} au_{y z} [/latex]

[latex]+c_{y^{\prime}z^{\prime}} c_{z^{\prime} x} au_{z x}+c_{y^{\prime} z^{\prime}} c_{z^{\prime} y} au_{y z}+c_{zz^{\prime} } c_{z z^{\prime}} au_{z z}=0 [/latex]

(6) [latex] n= z^{\prime}, s=z^{\prime}[/latex]

[latex] au_{z^{\prime} z^{\prime}}= c_{z^{\prime} x} c_{z^{\prime} x} au_{x x}+c_{z^{\prime} x} c_{z^{\prime} y} au_{x y}+c_{z^{\prime} x} c_{z z^{\prime}} au_{x z} [/latex]

[latex]+c_{z^{\prime} y} c_{z^{\prime} x} au_{y x}+c_{z^{\prime} y} c_{z^{\prime} y} au_{y y}+c_{z^{\prime} y} c_{z z^{\prime} } au_{y z} [/latex]

[latex]+c_{z^{\prime} z} c_{z^{\prime} x} au_{z x}+c_{z^{\prime} z} c_{z^{\prime} y} au_{y z}+c_{z^{\prime} z} c_{z^{\prime} z} au_{z z} . [/latex]

[latex]=1 imes 1 imes 40=40 MPa[/latex]

Since the stress tensor is symmetric, all the terms can now be written as follows.

[latex] au_{n s}=\begin{vmatrix} {-1.96} & {-81.96} & {0} \ {- 81.96} & {41.96} & {0} \ {0} & {0} & {40} \end{vmatrix}[/latex]

Alternately these calculations can be done by matrix multiplication as follws.

Stress tensor, [latex]\left[ au_{i j} ight]=\begin{vmatrix} {80 } & {-60} & {0} \ {-60} & {-40} & {0} \ {0} & {0} & {40} \end{vmatrix} MPa[/latex]

Direction cosine matrix for transformation,

[latex]\left[C ight]=\begin{vmatrix} {0.866} & {0.5} & {0} \ {- 0.5} & {0.866} & {0} \ {0} & {0} & {1} \end{vmatrix}[/latex]

Using Eq. (2.10d) in matrix notation,

[latex]\left[ au_{n s} ight]=\left[c ight] imes \left[ au_{i j} ight] imes\left[c ight]^{T}[/latex]

or, [latex]\left[ au_{n s} ight]=\begin{vmatrix} {0.866} & {0.5} & {0} \ {- 0.5} & {0.866} & {0} \ {0} & {0} & {1} \end{vmatrix}[/latex][latex]\begin{vmatrix} {80 } & {-60} & {0} \ {-60} & {-40} & {0} \ {0} & {0} & {40} \end{vmatrix}[/latex][latex]\begin{vmatrix} {0.866} & {-0.5} & {0} \ {- 0.5} & {0.866} & {0} \ {0} & {0} & {1} \end{vmatrix}[/latex]

or [latex]\left[ au_{n s} ight]=\begin{vmatrix} {-1.96} & {-81.96} & {0} \ {-81.96} & {41.96} & {0} \ {0} & {0} & {40} \end{vmatrix} MPa[/latex]

Question 2.16: he state of stress at a point in xyz reference is as follows...

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