The magnitudes of normal stresses on two mutually perpendicular planes at a point in an elastic body are 60 MPa compressive and 80 MPa tensile. Find the magnitudes of shearing stresses on these planes, if the magnitude of one of the principal stresses is 100 MPa tensile. Find also the magnitude of

Question

The magnitudes of normal stresses on two mutually perpendicular planes at a point in an elastic body are 60 MPa compressive and 80 MPa tensile. Find the magnitudes of shearing stresses on these planes, if the magnitude of one of the principal stresses is 100 MPa tensile. Find also the magnitude of the other principal stress at this point.

Accepted Answer

Let us show the stresses on a differential element with unknown shear stress assumed arbitrarily as shown in Figure 4.18(a). Also, we show the Mohr’s circle in the Figure 4.18(b). The circle is drawn by locating its centre,

[latex] C\left\lgroup \frac{\sigma_{x x}+\sigma_{y y}}{2}, 0 ight group [/latex]

that is, (10, 0) MPa and the point (100, 0) MPa.

Clearly, the radius of the circle, R is

R = (100 -10) MPa = 90 MPa

Therefore,

[latex] (80-10)^2+ au_{x y}^2=90^2 \Rightarrow au_{x y}^2=3200 \Rightarrow au_{x y}=\pm 56.57 MPa [/latex]

Therefore, the magnitude of shear stress is

[latex] au_{x y}=\pm 56.57 MPa [/latex]

If [latex] \sigma_2 [/latex] be the other principal stress, then

[latex] \sigma_1+\sigma_2=\sigma_{x x}+\sigma_{y y}=-60+80=20 [/latex]

or [latex] \sigma_2=20-\sigma_1=20-100=-80 [/latex]

The other principal stress, [latex] \sigma_2=80 [/latex] 80 MPa (compressive).

Question 4.7: The magnitudes of normal stresses on two mutually perpendicu...

Related Answered Questions

Verified Answer:

Derive an expression for the maximum shear stress τmax in the plane of the wall of the conical tank filled with water which is supported along its upper edge as shown in Figure 4.22. ...

Verified Answer:

At a point in a two-dimensional stress system, the normal stresses on two mutually perpendicular planes are σxx and σyy and the shear stress is τxy . At what value of the shear stress, one of the principal stresses will become zero? ...

Verified Answer:

Verified Answer:

At a point in a loaded structure, a pure shear stress state of τ = ± 400 MPa prevails on two given planes at right angles. (a) What would be the state of stress across the planes of an element taken at ± 45° to the given planes? (b) What are the magnitudes of these stresses? ...

Verified Answer:

Two mutually perpendicular planes of an element are subjected to normal stresses of 10.5 MPa (tensile) and 3.5 MPa (compressive) and shear stress of 7 MPa (Figure 4.15). Find out the magnitudes and direction of principal stresses. ...

Verified Answer:

Draw the Mohr’s circle for the biaxial stress element shown in Figure 4.12. From it, compute the principal stresses and locate the principal planes. ...

Verified Answer:

Consider Figure 4.11(c) and take σx = 350 kg/cm² and σy = −700 kg/cm² (compressive). Find out the value of Φ defining the plane on which no normal stress acts. What is the magnitude of shear stress on that plane? ...

Verified Answer:

Construct Mohr’s circle for the case of (a) simple tension where σy = 0 [Figure 4.10(a)]; (b) biaxial stress where σx is tension, σy compression and |σy| = 2|σx [Figure 4.10(b)]; (c) biaxial stress where σy = −σx [Figure 4.10(c)]. ...

Verified Answer:

At a certain point in a strained material, there exists a stress condition as shown in Figure 4.8. Find (a) the normal and shear stresses on the inclined plane AB; (b) principal stresses and principal planes; (c) maximum shear stresses and their planes. ...

Verified Answer: