Question

The steel shaft has a diameter of 2 in. It is supported on smooth journal bearings A and B, which exert only vertical reactions on the shaft. Determine the absolute maximum bending stress in the shaft if it is subjected to the pulley loadings shown.

Answer 1

To determine the absolute maximum bending stress in the steel shaft, we need to analyze the pulley loadings exerted on the shaft and consider the vertical reactions from the smooth journal bearings. By using the bending stress formula and the section modulus formula, we can calculate the bending stress.

Step-by-step explanation:

The absolute maximum bending stress in the steel shaft can be determined by analyzing the pulley loadings exerted on the shaft. To calculate the bending stress, we need to know the vertical reactions exerted by the smooth journal bearings A and B on the shaft. Additionally, we need to consider the diameter of the shaft.

Once we have this information, we can use the bending stress formula: bending stress = moment / section modulus.

The moment can be calculated by multiplying the load applied by the distance from the centerline of the shaft to the point of application of the load. The section modulus is a geometric property of the cross-sectional shape of the shaft and can be calculated using the formula: section modulus = (pi * (diameter/2)³) / 4.

Answer 2

The absolute maximum bending stress in the shaft is 1250 psi.

Step 1: Draw the shear force and bending moment diagrams

Shear force diagram:

Shear force at the left end (just left of A) is +500 lb (due to the upward reaction at A).

Shear force remains constant at +500 lb between A and the first pulley.

Shear force drops by 500 lb at the first pulley (due to the downward load).

Shear force remains constant at 0 lb between the pulleys.

Shear force increases by 500 lb at the second pulley (due to the upward load).

Shear force remains constant at +500 lb until the right end (just before B).

Bending moment diagram:

Bending moment is zero at both ends (simply supported bearings).

Bending moment increases linearly from 0 to -1250 lb-ft between the left end and the first pulley (due to the 500 lb shear force acting at 25 in from the left end).

Bending moment remains constant at -1250 lb-ft between the first and second pulleys.

Bending moment decreases linearly from -1250 lb-ft to 0 at the right end (due to the 500 lb shear force acting at 25 in from the right end).

Step 2: Determine the critical section

The critical section for maximum bending stress is at the first pulley, where the bending moment is -1250 lb-ft.

Step 3: Calculate the section properties

The diameter of the shaft is 2 in, which gives a radius of 1 in.

The area of the circular cross-section is A = πr², so A = π(1 in)² = 3.14 in².

Step 4: Calculate the bending stress

The bending stress (σ) at the critical section is calculated using the formula: σ = M/c, where:

M is the bending moment (-1250 lb-ft)

c is the distance from the neutral axis to the outermost fiber (in this case, the radius, which is 1 in).

Therefore, σ = (-1250 lb-ft) / (1 in) = -1250 psi.

Step 5: Determine the absolute maximum bending stress

Since the bending stress is negative throughout the shaft, the absolute maximum bending stress is simply the magnitude of the calculated value: |σ| = |-1250 psi| = 1250 psi.

Complete the Question:

The steel shaft has a diameter of 2 in. It is supported on smooth journal bearings A and B, which exert only vertical reactions on the shaft. Determine the absolute maximum bending stress in the shaft if it is subjected to the pulley loadings shown.