To ensure the beam meets both the maximum shear stress and angle of twist criteria, calculate the minimum thickness using the formulas for shear stress and angle of twist. Then, select the larger thickness value from both calculations to satisfy both criteria.
To calculate the minimum thickness of the beam walls for a thin-wall circular section beam that is experiencing a torque, two conditions need to be considered: the maximum allowable shear stress and the maximum allowable angle of twist.
First, we use the maximum shear stress criterion to find the minimum required thickness. The torque (T) experienced by the beam causes a shear stress (τ) which should not exceed the maximum allowable shear stress (τ_max). For a thin-walled circular tube, the shear stress is given by:
τ = Τ * T / (2 * Π * r^3 * t)
Where:
- Τ is the torque applied to the beam
- Π is the constant pi
- r is the radius of the beam
- t is the thickness of the beam wall
Rearranging for t, we have:
t = T / (2 * π * r^3 * τ_max)
Substituting the given values T = 15 kN-m, r = 100 mm (half of the diameter), and τ_max = 200 MPa, we can calculate t.
Maximum Allowable Angle of Twist
Next, we check whether the thickness found satisfies the maximum allowable angle of twist. The angle of twist (θ) in radians for a length (L) of the beam can be expressed as:
θ = TL / (G * J)
Where:
- L is the length of the beam
- G is the shear modulus of the material
- J is the polar second moment of area, which for a thin-walled tube can be approximated as J ≈ 2 * π * r^3 * t
We can substitute θ_max = 2° and solve for t to ensure it meets this condition as well.
Finally, we take the larger thickness value from both calculations to ensure the beam meets both the shear stress and angle of twist criteria.