At a point in a stressed body, the known stresses are $\sigma_{x}$ = 18 ksi (C), $\sigma_{y}$ = 15 ksi (C), $\sigma_{z}$ = 12 ksi (C), $\tau_{x y}$ = −15 ksi, $\tau_{y z}$ = +12 ksi, and $\tau_{z x}$ = −9 ksi. Determine:
(a) the principal stresses and the absolute maximum shear stress at the point.
(b) the orientation of the plane on which the maximum tensile stress acts.

Question

At a point in a stressed body, the known stresses are [latex]\sigma_{x}[/latex] = 18 ksi (C), [latex]\sigma_{y}[/latex] = 15 ksi (C), [latex]\sigma_{z}[/latex] = 12 ksi (C), [latex] au_{x y}[/latex] = −15 ksi, [latex] au_{y z}[/latex] = +12 ksi, and [latex] au_{z x}[/latex] = −9 ksi. Determine:
(a) the principal stresses and the absolute maximum shear stress at the point.
(b) the orientation of the plane on which the maximum tensile stress acts.

Accepted Answer

The known stresses are

[latex]\begin{array}{lll}\sigma_{x}=-18 ksi & \sigma_{y}=-15 ksi & \sigma_{ z }=-12 ksi \ au_{x y}=-15 ksi & au_{y z}=12 ksi & au_{x x}=-9 ksi\end{array}[/latex]

(a) The principal stresses can be obtained from the roots of the cubic equation [Eq. (12.27)]

[latex]\sigma_{p}^{3}-I_{1} \sigma_{p}^{2}+I_{2} \sigma_{p}-I_{3}=0[/latex]

The three invariants have values of

[latex]\begin{aligned}I_{1} &=\sigma_{x}+\sigma_{y}+\sigma_{z} \&=(-18)+(-15)+(-12)=-45 \I_{2} &=\sigma_{x} \sigma_{y}+\sigma_{y} \sigma_{z}+\sigma_{z} \sigma_{x}- au_{x y}^{2}- au_{y z}^{2}- au_{z x}^{2} \&=(-18)(-15)+(-15)(-12)+(-12)(-18)-(15)^{2}-(12)^{2}-(-9)^{2}=216 \I_{3} &=\sigma_{x} \sigma_{y} \sigma_{z}+2 au_{x y} au_{y z} au_{z x}-\left(\sigma_{x} au_{y z}^{2}+\sigma_{y} au_{z x}^{2}+\sigma_{z} au_{x y}^{2} ight) \&=(-18)(-15)(-12)+2(-15)(12)(-9)-\left[(-18)(12)^{2}+(-15)(-9)^{2}+(-12)(-15)^{2} ight]=6,507\end{aligned}[/latex]

therefore, Eq. (12.27) is

[latex]\sigma_{p}^{3}-(-45) \sigma_{p}^{2}+(216) \sigma_{p}-(6,507)=\sigma_{p}^{3}+45 \sigma_{p}^{2}+216 \sigma_{p}-6,507=0[/latex]

The three roots of this cubic equation are the principal stresses:

[latex]\begin{aligned}\sigma_{p 1} &=9.15 ksi ( T ) \\sigma_{p 2} &=22.4 ksi ( C ) \\sigma_{p 3} &=31.7 ksi ( C )\end{aligned}[/latex]

The absolute maximum shear stress at the point is found from

[latex] au_{ abs \max }=\frac{\sigma_{\max }-\sigma_{\min }}{2}=\frac{9.1477-(-31.7286)}{2}=20.4 ksi[/latex]

(b) From Eqs. (b) in Section 12.10,

[latex]\begin{aligned}&\left(\sigma_{x}-\sigma_{p} ight) l+ au_{x y} m+ au_{z x} n=0 \&\left(\sigma_{y}-\sigma_{p} ight) m+ au_{y z} n+ au_{x y} l=0 \&\left(\sigma_{z}-\sigma_{p} ight) n+ au_{z x} l+ au_{y z} m=0\end{aligned}[/latex]

Since we are interested in the orientation of the maximum tensile stress, set [latex]\sigma_{p}=\sigma_{p 1}=9.1477 ksi[/latex] and rewrite these equations as:

[latex]\begin{aligned}&{[(-18)-(9.1477)] l+(-15) m+(-9) n=0} \&{[(-15)-(9.1477)] m+(12) n+(-15) l=0} \&{[(-12)-(9.1477)] n+(-9) l+(12) m=0}\end{aligned}[/latex]

which can be simplified and rearranged to these three equations.

[latex]\begin{aligned}&-27.1477 l-15 m-9 n=0 (a) \&-15 l-24.1477 m+12 n=0 (b)\&-9 l+12 m-21.1477 n=0 (c)\end{aligned}[/latex]

Consider Eqs. (a) and (b). Eliminate n from these two equations to obtain the relationship

[latex]m=-1.15967 l[/latex] (d)

Repeat the process with Eqs. (b) and (c), eliminating m to obtain the relationship

[latex]n=-1.08362 l[/latex] (e)

The direction cosines are related by the identity:

[latex]l^{2}+m^{2}+n^{2}=1[/latex]

Substitute Eqs. (d) and (e) into this identity and solve for the direction cosine l:

[latex]\begin{gathered}l^{2}+(-1.15967 l)^{2}+(-1.08362 l)^{2}=3.51908 l^{2}=1 \ herefore l=0.5331\end{gathered}[/latex]

Backsubstitute this result into Eqs. (d) and (e) to determine direction cosines m and n:

[latex]\begin{aligned}&m=-1.15967(0.5331)=-0.6182 \&n=-1.08362(0.5331)=-0.5776\end{aligned}[/latex]

From l, m, and n, determine the angles α, β, and γ:

[latex]\begin{array}{ll}\cos \alpha=0.5331 & \alpha=57.8^{\circ} \\cos \beta=-0.6182 & \beta=128.2^{\circ} \\cos \gamma=-0.5776 & \gamma=125.3^{\circ}\end{array}[/latex]

Question 12.99: At a point in a stressed body, the known stresses are σx = 1...

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