Stress (continuum mechanics)

[sara_87]

Homework Statement

A catilever beam with rectangular cross-section occupies the region -a $$\leq$$x1 $$\leq$$a , -h $$\leq$$ x2 $$\leq$$ h , 0 $$\leq$$ x3 $$\leq$$ L
The end x3=L is built-in and the beam is beant by a force P applied at the free end x3=0 and acting in the x2 direction. The stress tensor has components:

(T)=(0 0 0, 0 0 A+B(x2)$$^{2}$$, 0 A+B(x2)$$^{2}$$ C(x2)(x3))

where A, B, C are constants.
Question: express the resultant force on the free end x3=0 in terms of A, B, C.

Homework Equations

The Attempt at a Solution

I don't know how to do this question. can someone please help me and tell me how to start please. thank you

 

[mighty2000]

Dear forum post,

Thank you for reaching out for help with this problem. I am happy to assist you in finding a solution.

First, let's review the given information. We have a cantilever beam with a rectangular cross-section, occupying the region -a ≤ x1 ≤ a, -h ≤ x2 ≤ h, and 0 ≤ x3 ≤ L. This means that the beam is fixed at one end (x3=L) and free at the other end (x3=0). The beam is being bent by a force P applied at the free end in the x2 direction. We are given a stress tensor with components (0 0 0, 0 0 A+B(x2)^2, 0 A+B(x2)^2 C(x2)(x3)), where A, B, and C are constants.

Now, to find the resultant force on the free end x3=0, we need to use the equilibrium equations for a 3D body. These equations state that the sum of all forces in each direction (x1, x2, and x3) must equal zero, and the sum of all moments about any point must also equal zero.

In this case, we only have a force acting in the x2 direction, so we can ignore the x1 and x3 equations. This leaves us with the following equation for the x2 direction:

ΣF2 = 0

To find the resultant force in the x2 direction, we need to integrate the stress tensor over the cross-sectional area of the beam. This can be done using the following equation:

F2 = ∫σ22 dA

Where σ22 is the stress component in the x2 direction and dA is the differential area over which we are integrating. In this case, the differential area can be represented as dA = dx1dx3.

Now, we can substitute the given stress tensor components into the above equation and integrate over the cross-sectional area of the beam. This will give us the resultant force on the free end in the x2 direction, which we can then express in terms of A, B, and C. I will leave the actual integration to you, but please let me know if you need further assistance.

I hope this helps you get started on solving this problem. If you have any further questions, please don't hesitate to ask.