Figure 2.30(a) shows an element under axial stress 20 MPa and 50 MPa along with shear stress 40 MPa in x-y plane. Determine the state of stress in x' - y' plane at 30° counter-clockwise to x-y plane.

Question

Figure 2.30(a) shows an element under axial stress 20 MPa and 50 MPa along with shear stress 40 MPa in x-y plane. Determine the state of stress in [latex]x^{\prime} - y^{\prime}[/latex] plane at 30° counter-clockwise to x-y plane.

Accepted Answer

Equation (2.11b) can be used to cal-culate stresses in [latex]\acute{x}-y^{\prime}[/latex] plane and Eq. (2.13a) will give principal stresses. The directive cosines in transfer matrix are drived from Fig. 2.30(b) as follows.

[latex] au_{x y} imes c_{n x}+\left( au_{y y}- au_{n n} ight) imes c_{n y}+ au_{z y} imes c_{n z}=0 [/latex] (2.11b)

[latex]\begin{vmatrix} {\left( au _{x x} - au _{n n} ight)} & { au _{y x}} & { au _{z x}} \ { au _{x y}} & {\left( au _{y y} - au _{n n} ight)} & { au _{z y}} \ { au _{x z }} & { au _{z y}} & {\left( au _{z z} - au _{n n} ight)} \end{vmatrix}=0[/latex] (2.13a)

[latex]c_{x^{\prime} x}=\cos heta [/latex]

[latex]c_{x^{\prime} y}=\cos \left(90^{\circ}+ heta ight)=-\sin heta [/latex]

[latex]c_{y^{\prime} x}=\cos \left(90^{\circ}- heta ight)=\sin heta [/latex]

[latex]c_{y^{\prime} y}=\cos heta[/latex]

Substitute these values in Eq. (2.10b) for stresses in [latex]x^{\prime} - y^{\prime}[/latex] reference axes as follows.

[latex] au_{n n}= au_{x x} imes\left(c_{n x} ight)^2+ au_{y y} imes\left(c_{n y} ight)^2+ au_{z z} imes\left(c_{n z} ight)^2 [/latex]

[latex]+2 au_{x y}\left(c_{n x} c_{n y} ight)+2 au_{x z}\left(c_{n x} c_{n z} ight) [/latex]

[latex]+2 au_{y z}\left(c_{n y} c_{n z} ight)[/latex] (2.10b)

[latex]\begin{vmatrix} {\sigma_{x^{\prime}}}& { au _{x^{\prime}y^{\prime}}} \ { au _{y^{\prime} x^{\prime}}} & {\sigma_{y^{\prime}y^{\prime}}} \end{vmatrix}[/latex]

[latex]=\begin{vmatrix} {c_{x^{\prime}x}} & {c _{x^{\prime}y}} \ {c_{y^{\prime} x}} & {c_{y^{\prime} y}} \end{vmatrix}[/latex]

[latex]\begin{vmatrix} {\sigma _{x x} } & { au _{x y}} \ { au _{y x}} & {\sigma _{y y}} \end{vmatrix}[/latex]

[latex]\begin{vmatrix} {c_{x^{\prime}x}} & {c_{y^{\prime} x}} \ {c_{x^{\prime}x}} & {c_{y^{\prime} y}} \end{vmatrix}[/latex]

[latex]=\begin{vmatrix} {0.866 } & {-0.5} \ {0.5} & {0.866} \end{vmatrix}[/latex]

[latex]\begin{vmatrix} {20 } & {40} \ {40} & {-50} \end{vmatrix}[/latex]

[latex]\begin{vmatrix} {0.866 } & {0.5} \ {-0.5} & {0.866} \end{vmatrix}[/latex]

[latex]=\begin{vmatrix} {-32.14 } & {50.3} \ {50.3} & {2.14} \end{vmatrix}[/latex]

Principal stresses are given by Eq.(2.13a),

[latex]=\begin{vmatrix} {-32.14-\sigma } & {50.3} \ {50.3} & {2.14 - \sigma } \end{vmatrix}=0[/latex]

which gives principal stresses

[latex]\sigma= 38.14 MPa and -68.14 MPa [/latex]

Question 2.22: Figure 2.30(a) shows an element under axial stress 20 MPa an...

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