Direct tensile stresses of 120 MN/m² and 70 MN/m² act on a body on mutually perpendicular planes. What is the magnitude of shearing stress that can be applied so that major principal stress at the point does not exceed 135 MN/m²? Determine the value of major principal stress and the maximum shear

Question

Direct tensile stresses of 120 MN/m² and 70 MN/m² act on a body on mutually perpendicular planes. What is the magnitude of shearing stress that can be applied so that major principal stress at the point does not exceed 135 MN/m²? Determine the value of major principal stress and the maximum shear stress.

Accepted Answer

Let us refer to Figure 4.18(a) of the previous problem in which we have

[latex] \sigma_{x x}=+120 MN / m ^2 \quad ext { and } \quad \sigma_{y y}=+70 MN / m ^2 [/latex]

Let [latex] au_{x y} [/latex] be the shear stress acting on the body. Therefore,

[latex] \begin{aligned} \sigma_1 & =\frac{1}{2}\left(\sigma_{x x}+\sigma_{y y} ight)+\sqrt{\left\lgroup \frac{\sigma_{x x}-\sigma_{y y}}{2} ight group^2+ au_{x y}^2} \ 135 & =0.5(120+70)+\sqrt{\{0.5(120-70)\}^2+ au_{x y}^2} \end{aligned} [/latex]

This implies

[latex] au_{x y}=\pm 31.22 MPa \Rightarrow\left| au_{x y} ight|=31.22 MN / m ^2 [/latex]

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

[latex] \sigma_1+\sigma_2=\sigma_{x x}+\sigma_{y y}=120+70=190 [/latex]

or [latex] \sigma_2=190-135=55 MPa [/latex]

If [latex] au_{\max } [/latex] be the maximum shear stress, then

[latex] au_{\max }=\sqrt{\left\lgroup \frac{\sigma_{x x}-\sigma_{y y}}{2} ight group^2+ au_{x y}^2} [/latex]

or [latex] au_{\max }=\sigma_1-\frac{1}{2}\left(\sigma_{x x}+\sigma_{y y} ight)=[135-0.5(120+70)]=40 MPa [/latex]

Question 4.8: Direct tensile stresses of 120 MN/m² and 70 MN/m² act on a b...

Related Answered Questions

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:

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: