Problem Statement:
A shaft with a diameter of 100 mm is subjected to a torque of 10,000 Nm, a bending moment of 5 kNm, and an axial thrust of 30 kN. The goal is to determine the maximum principal stress and shear stress in the shaft.

Solution:

Step 1: Calculate the bending stress:
The bending stress in a shaft can be calculated using the formula:
σ_bend = (M * c) / I
where,
M = Bending moment
c = Distance of the point from the neutral axis
I = Moment of inertia of the cross-section of the shaft

Given:
Bending moment (M) = 5 kNm = 5000 Nm
Diameter of the shaft = 100 mm = 0.1 m
Radius (r) = Diameter / 2 = 0.05 m
Distance from neutral axis (c) = radius = 0.05 m

The moment of inertia (I) for a solid circular section can be calculated using the formula:
I = (π * d^4) / 64
where,
d = Diameter of the shaft

Substituting the given values:
I = (π * 0.1^4) / 64 = 0.0000314 m^4

Substituting the values into the formula for bending stress:
σ_bend = (5000 * 0.05) / 0.0000314 = 796178.34 Pa

Step 2: Calculate the torsional stress:
The torsional stress in a shaft can be calculated using the formula:
τ_torsion = (T * r) / J
where,
T = Torque
r = Radius of the shaft
J = Polar moment of inertia of the cross-section of the shaft

Given:
Torque (T) = 10,000 Nm
Radius (r) = 0.05 m

The polar moment of inertia (J) for a solid circular section can be calculated using the formula:
J = (π * d^4) / 32
where,
d = Diameter of the shaft

Substituting the given values:
J = (π * 0.1^4) / 32 = 0.0000982 m^4

Substituting the values into the formula for torsional stress:
τ_torsion = (10,000 * 0.05) / 0.0000982 = 509579.38 Pa

Step 3: Calculate the axial stress:
The axial stress in a shaft can be calculated using the formula:
σ_axial = F / A
where,
F = Axial thrust
A = Cross-sectional area of the shaft

Given:
Axial thrust (F) = 30 kN = 30000 N

The cross-sectional area (A) of a solid circular section can be calculated using the formula:
A = π * r^2

Substituting the given values:
A = π * 0.05^2 = 0.0078539 m^2

Substituting the values into the formula for axial stress:
σ_axial = 30000 / 0.0078539 = 3822766.70 Pa

Step 4