Where did torsional shear stress in the x-y plane came from?

In summary, the conversation discusses the derivation of the maximum torsional shear stress equation (4-14) for a circular rod under torsion. It involves calculating the shear strain and stress between adjacent cross sections, integrating the differential torque on each cross section, and utilizing the polar moment of inertia to determine the maximum shear stress at a given location.
  • #1

Homework Statement


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Homework Equations


maximum torsional shear stress = (Torque*radius)/polar moment of inertia

The Attempt at a Solution


I am lost on equation(4-14), I looked through the textbook but didn't find a derivation.
 

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  • #2
The circular rod is being twisted about its axis. There is a rotation of each cross section by an angle ##\theta## that varies linearly with distance along the rod axis. If you work out the shear strain between adjacent cross sections, you obtain ##r\frac{d\theta}{dx}##. For this shear strain, you can then work out the shear stress. Knowing the shear stress, you can then integrate the differential torque on each area of cross section about the axis. This will involve the polar moment of inertia. Knowing the actual torque, you can then determine ##d\theta/dx##. This will be sufficient to determine the maximum shear stress at r = R, the location of maximum shear stress.
 

1. What is torsional shear stress in the x-y plane?

Torsional shear stress in the x-y plane refers to the force that is applied to an object in a twisting or rotational motion, causing it to experience shear stress in the x-y direction. This type of stress is commonly seen in structural engineering, as it can affect the stability and strength of a structure.

2. How is torsional shear stress calculated?

Torsional shear stress is calculated by taking the torque applied to an object and dividing it by the polar moment of inertia, which is a measure of an object's resistance to torsion. The resulting value is the shear stress in the x-y plane.

3. What factors can contribute to torsional shear stress in the x-y plane?

There are several factors that can contribute to torsional shear stress in the x-y plane, including the shape and material properties of an object, the magnitude and direction of the applied torque, and the boundary conditions of the object.

4. How is torsional shear stress different from other types of stress?

Torsional shear stress is different from other types of stress, such as tensile or compressive stress, because it is caused by a twisting or rotational force rather than a pulling or pushing force. It also acts in a different direction, causing shear forces along a plane rather than normal forces along an axis.

5. How can torsional shear stress in the x-y plane be reduced?

Torsional shear stress in the x-y plane can be reduced by using materials with higher shear strength, increasing the size or shape of an object to increase its resistance to torsion, or by adding reinforcement such as bracing or cross-sections to distribute the stress more evenly.

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