# Allowable shear - indeterminate Torsion

• naggy
In summary, we have a hollow steel shaft ACB with an outside diameter of 50 mm and inside diameter of 40 mm. It is held against rotation at ends A and B, with horizontal forces P applied at the ends of a vertical arm welded to the shaft at point C. The maximum permissible shear stress is 45 MPa, and using the relation between shear stress and torque, we can determine that the maximum shear stress occurs at point C and is equal to s = 2P*200mm*(50-40)/(π * (50^4 - 40^4)/32). Solving for P gives us an allowable value of 22.5 kN for the forces P.
naggy
This is an indterminate torsional member.

A hollow steel shaft ACB of outside diameter 50 mm and inside diameter 40 mm is held against rotation at ends A and B. Horizontal forces P are applied at the ends of a vertical arm that is welded to the shaft at point C. Determine the allowable value of the forces P if the maximum permissible shear stress is 45 MPa

I can't find a picture of this problem, it's in my textbook. I'll try to make it as clear as possible with my limited knowledge.

|P
A |--------C-------|B
L1 |P L2​
The arms touch the bar, naturally.
L1 (AC) = 600 mm
L2 (CB) = 400 mm
The length from C to the force P is 200 mm on both sides. It produces a torque that is pointing to the right, toward B.

Where is the maximum shear stress in the shaft and why? I know the relation between shear stress (s) and torque (T)

s = T*radius of bar/I

I is the polar moment of inertia

And I know that 2P * 200mm = Torque about point C

because of the two forces P. Therefore, the maximum shear stress occurs at point C and is equal to s = 2P*200mm*(50-40)/(π * (50^4 - 40^4)/32) Since the maximum permissible shear stress is 45 MPa, then solving for P gives us: P = 45MPa * (π * (50^4 - 40^4)/32) / (2 * 200mm) Therefore, the allowable value of the forces P is: P = 22.5 kN

I would approach this problem by first identifying the key variables and equations involved. In this case, we have a hollow steel shaft with known dimensions and a force applied at a specific point. The maximum permissible shear stress is also given as 45 MPa.

To determine the allowable value of the forces P, we can use the equation for shear stress in a circular shaft, which is s = T*r/I, where T is the torque applied, r is the radius of the shaft, and I is the polar moment of inertia.

In this case, the torque applied is equal to 2P * 200mm, as both forces P act at a distance of 200mm from point C. The polar moment of inertia can be calculated using the known dimensions of the shaft and the equation I = (pi/2)*(D^4 - d^4), where D is the outer diameter and d is the inner diameter.

Once we have calculated the polar moment of inertia, we can rearrange the equation for shear stress to solve for the allowable value of P. It is important to note that the maximum shear stress will occur at the point of maximum torque, which in this case is at point C.

Therefore, the maximum allowable value of P can be determined by setting the calculated shear stress equal to 45 MPa and solving for P. This will give us the maximum force that can be applied at each end of the vertical arm without exceeding the permissible shear stress in the shaft.

In conclusion, by using the equation for shear stress in a circular shaft and considering the key variables and known dimensions of the problem, we can determine the allowable value of the forces P in this indeterminate torsional member.

## 1. What is allowable shear in the context of indeterminate torsion?

Allowable shear is the maximum amount of shear stress that a material can safely withstand before experiencing failure or deformation. In the context of indeterminate torsion, it refers to the maximum shear stress that can be applied to a structure without causing excessive deformation or failure.

## 2. How is allowable shear calculated in indeterminate torsion?

The calculation of allowable shear in indeterminate torsion involves considering the geometry and material properties of the structure, as well as the applied loads and boundary conditions. The specific method for calculating allowable shear will depend on the specific type of indeterminate torsion problem being analyzed.

## 3. What factors affect the allowable shear in indeterminate torsion?

The factors that affect the allowable shear in indeterminate torsion include the material properties of the structure, such as the yield strength and shear modulus, as well as the geometry and boundary conditions. Additionally, the type and magnitude of applied loads, such as torsional moments and shear forces, will also impact the allowable shear.

## 4. Why is it important to consider allowable shear in indeterminate torsion?

Considering allowable shear in indeterminate torsion is important because it helps ensure the structural integrity and safety of the system. If the allowable shear is exceeded, it can lead to excessive deformation or failure of the structure, potentially causing damage or injury.

## 5. How can the allowable shear in indeterminate torsion be increased?

The allowable shear in indeterminate torsion can be increased by using stronger materials, such as high-strength steel or reinforced concrete, or by increasing the size and dimensions of the structural elements. Additionally, appropriate structural design and detailing can also help increase the allowable shear capacity of a system.

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