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Chapter 4: Problem 29

Determine how the bending stress at the surface of a shaft subjected to a bending moment \(M\) changes with values for the shaft diameter \(d\).

Short Answer

Expert verified
The bending stress (\(蟽\)) at the surface of a shaft subjected to a bending moment (\(M\)) is inversely proportional to the cube of the diameter (\(d\)). Therefore, as the shaft diameter increases, the bending stress decreases and vice versa.

Step by step solution

01

Identify the Variables

In this case, the bending moment \(M\) is constant. The variable that's changing is the shaft diameter \(d\). Therefore, we need to understand how changes in \(d\) affect the bending stress \(蟽\).
02

Understand the Relationship Between 蟽 and d

From the formula \(\sigma = \frac{32M}{蟺d^3}\), we can see that \(蟽\) is inversely proportional to \(d^3\). This implies that as \(d\) increases, \(蟽\) decreases and vice versa.
03

Explain the Relationship

Since \(蟽\) and \(d\) are inversely proportional to each other, it can be said that as the diameter of the shaft increases, there is more area to distribute the bending stress; hence, the overall bending stress decreases. Conversely, if the diameter of the shaft decreases, the area to distribute the stress becomes less, resulting in an increase in the overall bending stress.

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Most popular questions from this chapter

Figure P4.70 shows a triaxial stress element having a critical three- dimensional stress state where \(\sigma_{x}=60,000, \sigma_{y}=-30,000, \sigma_{z}=-15,000, \tau_{\mathrm{xy}}=9000\). \(\tau_{\mathrm{yz}}=\) \(-2000\), and \(\tau_{\mathrm{zx}}=3500 \mathrm{psi}\). Calculate the first, second, and third stress invariants. Then solve the characteristic equation for the principal normal stresses. Also, calculate the maximum shear stress and draw the resulting Mohr circle representation of the state of stress.

Is using force flow to study stress in a component an art or a science? Can the concept of force flow be used to study problems where numerical values for loads are indeterminate?

Figure P4.65 shows a cylinder internally pressurized to a pressure of \(7000 \mathrm{psi}\). The pressure causes tangential and axial stresses in the outer surface of 30,000 and \(20,000 \mathrm{psi}\), respectively. Determine the maximum shear stress at the outer surface.

A uniformly tapered vertical cone with a height, \(h\), and a base diameter, \(d\), is cast from a urethane material having a density, \(\rho\). Determine the compressive stress at the cross section of the base, \(B\), and at the cross section, \(A\), half way up the cone, and compare the compressive stress at \(B\) with the compressive stress at \(A\). The volume of the cone is \(V_{\text {cone }}=(1 / 12) \pi d^{2} h\).

A stainless steel member has a three-dimensional state of stress at a critical location where \(\sigma_{x}=50,000, \sigma_{y}=-10,000, \sigma_{z}=15,000 . \tau_{\mathrm{xy}}=-3500, \tau_{\mathrm{yz}}=-1000\), and \(\tau_{\mathrm{zx}}=\) 2000 psi. Calculate the first, second, and third stress invariants and solve the characteristic equation for the principal normal stresses. Also, calculate the maximum shear stress and draw the Mohr circle representation of the state of stress at the critical point.
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