Sheer stress in circular membrane is an essential concept in the field of solid mechanics. It arises when a force is applied parallel to the surface of a circular membrane, causing it to deform. The magnitude of the shear stress depends on the force applied, the radius of the membrane, and the thickness of the membrane. The membrane’s material properties also play a role in determining the shear stress.
Sheer Stress in Circular Membrane
Consider a circular membrane of radius ‘a’ subjected to a uniform pressure ‘p’. The membrane is fixed along its circumference.
Assumptions:
- The membrane is thin and behaves as a two-dimensional sheet.
- The material of the membrane is linearly elastic and obeys Hooke’s law.
- The membrane is in a state of plane stress, i.e., no bending or twisting.
Radial Stress (σr):
The radial stress is the stress acting in the radial direction. It is given by:
σr = -p * (a^2 / r^2)
where:
* σr is the radial stress
* p is the pressure
* a is the radius of the membrane
* r is the distance from the center of the membrane
Tangential Stress (σt):
The tangential stress is the stress acting in the tangential direction. It is given by:
σt = -p * (a^2 / 2r^2)
where:
* σt is the tangential stress
Sheer Stress (τrθ):
The sheer stress is the stress acting in the plane of the membrane and parallel to the circumference. It is given by:
τrθ = 0
Table Summarizing Stresses:
| Stress Type | Equation |
|---|---|
| Radial Stress | σr = -p * (a^2 / r^2) |
| Tangential Stress | σt = -p * (a^2 / 2r^2) |
| Sheer Stress | τrθ = 0 |
Explanation:
The radial stress is compressive and decreases with increasing distance from the center of the membrane. The tangential stress is also compressive but is half the magnitude of the radial stress. The sheer stress is zero, indicating no slipping or sliding within the membrane.
Question 1:
What causes sheer stress in a circular membrane?
Answer:
Sheer stress in a circular membrane is caused by a force that is applied tangent to the membrane’s surface. This force causes the membrane to deform, resulting in a strain that is parallel to the direction of the applied force. The amount of sheer stress is directly proportional to the magnitude of the applied force and inversely proportional to the thickness of the membrane.
Question 2:
How is the sheer stress in a circular membrane related to the membrane’s radius?
Answer:
The sheer stress in a circular membrane is inversely proportional to the square of its radius. This means that as the radius of the membrane increases, the sheer stress decreases. This is because the larger the membrane, the more it is able to deform in response to an applied force, resulting in a lower level of stress.
Question 3:
What factors affect the sheer stress in a circular membrane?
Answer:
The sheer stress in a circular membrane is affected by several factors, including the magnitude of the applied force, the thickness of the membrane, the radius of the membrane, and the material properties of the membrane. The sheer stress is directly proportional to the magnitude of the applied force and inversely proportional to the thickness and radius of the membrane. Additionally, the material properties of the membrane, such as its stiffness and flexibility, can also influence the level of sheer stress.
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