A solid shaft of diameter d = 10 cm is subjected to a tensile force P = 10,000 N and a torque T = 5000 N cm. At point A on the surface,determine the principal stresses, the octahedral shearing stress and the maximum shearing stress.

ANSWER:

Given:

  • Diameter of the shaft (d) = 10 cm
  • Tensile force (P) = 10,000 N
  • Torque (T) = 5,000 N cm

Step 1: Calculate the normal stresses

The normal stress due to the tensile force is:

σx = P / A

where A is the cross-sectional area of the shaft. The cross-sectional area of a circle is:

A = πr²

where r is the radius of the shaft. The radius of the shaft is half the diameter, so r = 5 cm. Therefore, the cross-sectional area is:

A = π(5 cm)² = 25π cm²

Plugging in the values for P and A, we get:

σx = 10,000 N / (25π cm²) ≈ 127.4 N/cm²

The normal stress due to the torque is:

σy = -T / (πr³)

Plugging in the values for T and r, we get:

σy = -5,000 N cm / (π(5 cm)³) ≈ -63.7 N/cm²

Step 2: Calculate the shear stress

The shear stress is:

τxy = T / (2πr³)

Plugging in the values for T and r, we get:

τxy = 5,000 N cm / (2π(5 cm)³) ≈ 31.85 N/cm²

Step 3: Calculate the principal stresses

The principal stresses are the two normal stresses that act perpendicular to each other at a point on the surface of the shaft. The principal stresses can be calculated using the following equation:

σ1,2 = (σx + σy) / 2 ± √((σx - σy)² / 4 + τxy²)

Plugging in the values for σx, σy, and τxy, we get:

σ1 = 31.85 N/cm²
σ2 = -127.4 N/cm²

Step 4: Calculate the octahedral shearing stress

The octahedral shearing stress is a measure of the intensity of the shearing stresses acting at a point on the surface of the shaft. The octahedral shearing stress can be calculated using the following equation:

τoct = √((σ1 - σ2)² / 3 + 3τxy²)

Plugging in the values for σ1, σ2, and τxy, we get:

τoct = 103.25 N/cm²

Step 5: Calculate the maximum shearing stress

The maximum shearing stress is the largest of the three shear stresses acting at a point on the surface of the shaft. The maximum shearing stress can be calculated using the following equation:

τmax = (σ1 - σ2) / 2

Plugging in the values for σ1 and σ2, we get:

τmax = 98.55 N/cm²

Therefore, the principal stresses at point A on the surface of the shaft are 31.85 N/cm² and -127.4 N/cm², the octahedral shearing stress is 103.25 N/cm², and the maximum shearing stress is 98.55 N/cm².

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