Consider a point in a structural member that is subjected to plane stress. Normal and shear stresses acting on horizontal and vertical planes at the point are shown. (a) Draw Mohr’s circle for this state of stress. (b) Determine the principal stresses and the maximum in-plane shear stress acting at

Question

Consider a point in a structural member that is subjected to plane stress. Normal and shear stresses acting on horizontal and vertical planes at the point are shown.
(a) Draw Mohr’s circle for this state of stress.
(b) Determine the principal stresses and the maximum in-plane shear stress acting at the point. Show these stresses on an appropriate sketch (e.g., see Figure 12.15 or Figure 12.16).
(c) Determine the normal and shear stresses on the indicated plane and show these stresses on a sketch.
(d) Determine the absolute maximum shear stress at the point.
Instructors: Problems 12.85-12.88 should be assigned as a set.

Accepted Answer

(b) The basic Mohr’s circle is shown.

[latex]\begin{aligned}&C=\frac{(30 MPa )+(90 MPa )}{2}=60 MPa \&R=\sqrt{(30 MPa )^{2}+(105 MPa )^{2}}=109.2016 MPa\end{aligned}[/latex]

[latex]\begin{aligned}&\sigma_{p 1}=C+R=60 MPa +109.2016 MPa =169.2 MPa \&\sigma_{p 2}=C-R=60 MPa -109.2016 MPa =-49.2 MPa \& au_{\max }=R=109.2 MPa \&\sigma_{ ext {avg }}=C=60 MPa ( T )\end{aligned}[/latex]

The magnitude of the angle [latex]2 heta_{p}[/latex] between point x (i.e., the x face of the stress element) and point 2 (i.e., the principal plane subjected to [latex]\sigma_{p 2}[/latex]) is found from:

[latex] an 2 heta_{p}=\frac{105 MPa }{|(30 MPa )-(60 MPa )|}=\frac{105 MPa }{30 MPa }=3.5000 \quad herefore 2 heta_{p}=74.0546^{\circ} \quad ext { thus, } heta_{p}=37.0^{\circ}[/latex]

By inspection, the angle [latex] heta_{p}[/latex] from point x to point 2 is turned clockwise. The orientation of the principal stresses and the maximum in-plane shear stress is shown in the sketch below.

(c) To determine the normal and shear stresses on the indicated plane, we must first determine the orientation of the plane relative to the x face of the stress element. Looking at the stress element, we observe that the normal to the indicated plane is oriented 50° clockwise from the x axis. In Mohr’s circle, all angle measures are doubled; therefore, point n (which represents the state of stress on the n plane) on Mohr’s circle is rotated 2(50°) = 100° clockwise from point x. The angle between point n and point 2 is

[latex]\beta=100^{\circ}-74.0546^{\circ}=25.9454^{\circ}[/latex]

The σ coordinate of point n is found from:

[latex]\begin{aligned}\sigma_{n} &=C-R \cos \beta \&=60 MPa -(109.2016 MPa ) \cos \left(25.9454^{\circ} ight) \&=-38.1954 MPa =38.2 MPa ( C )\end{aligned}[/latex]

The τ coordinate of point n is found from:

[latex]\begin{aligned} au_{n t} &=R \sin \beta \&=(109.2016 MPa ) \sin \left(25.9454^{\circ} ight)=47.7773 MPa =47.8 MPa\end{aligned}[/latex]

Since point n is above the σ axis, the shear stress acting on the plane surface tends to rotate the stress element clockwise.

(d) Since the point in a structural member is subjected to plane stress

[latex]\sigma_{z}=\sigma_{p 3}=0[/latex]

Three Mohr’s circles can be constructed to show stress combinations in the [latex]\sigma_{p 1}-\sigma_{p 2}[/latex] plane, the [latex]\sigma_{p 1}-\sigma_{p 3}[/latex] plane, and the [latex]\sigma_{p 2}-\sigma_{p 3}[/latex] plane. These three circles are shown below.

In this case, the absolute maximum shear stress occurs in the [latex]\sigma_{p 1}-\sigma_{p 2} [/latex] plane (which is also the x-y plane).
Therefore

[latex] au_{ abs \max }= au_{\max }=109.2 MPa[/latex]

Question 12.88: Consider a point in a structural member that is subjected to...

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