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{(72 MPa )+(36 MPa )}{2}=54 MPa \&R=\sqrt{(18 MPa )^{2}+(153 MPa )^{2}}=154.0552 MPa\end{aligned}[/latex]

[latex]\begin{aligned}&\sigma_{p 1}=C+R=54 MPa +154.0552 MPa =208 MPa \&\sigma_{p 2}=C-R=54 MPa -154.0552 MPa =-100.1 MPa \& au_{\max }=R=154.1 MPa \&\sigma_{ ext {avg }}=C=54 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{153 MPa }{|(72 MPa )-(54 MPa )|}=\frac{153 MPa }{18 MPa }=8.5000 \quad herefore 2 heta_{p}=83.2902^{\circ} \quad ext { thus, } heta_{p}=41.6^{\circ}[/latex]

By inspection, the angle [latex] heta_{p}[/latex] from point x to point 1 is turned counterclockwise. 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 30.96° 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(30.96°) = 61.93° clockwise from point x. The angle between point n and point 2 is

[latex]\beta=180^{\circ}-83.2902^{\circ}-61.9275^{\circ}=34.7823^{\circ}[/latex]

The σ coordinate of point n is found from:

[latex]\begin{aligned}\sigma_{n} &=C-R \cos \beta \&=54 MPa -(154.0552 MPa ) \cos \left(34.7823^{\circ} ight)=72.5 MPa ( C )\end{aligned}[/latex]

The τ coordinate of point n is found from:

[latex]\begin{aligned} au_{m t} &=R \sin \beta \&=(154.0552 MPa ) \sin \left(34.7823^{\circ} ight)=87.9 MPa\end{aligned}[/latex]

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

(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_{ ext {abs max }}= au_{\max }=154.1 MPa[/latex]

Question 12.87: Consider a point in a structural member that is subjected to...

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