Thermal Stress Problem + Tensions Problem

In summary, the author attempted to solve a problem in which she was stuck, and found that she needed to solve for the displacements of B, C, and D as function of the displacement A and the angle the bar makes with the horizontal.
  • #1
bossmombo
3
0
[SOLVED] Thermal Stress Problem + Tensions Problem

I actually have two problems in mind, so let me lay them out clearly:

1. http://www.straydreamers.com/images/hw1.jpg [Broken]

Homework Equations


For Steel:
d=1 in
Ls= 6 + 2 + 2 = 10 in
thermal expansion coefficient (alphaS): 6.3*10^(-6)/F
Elastic Modulus: 30 Mpsi

For Aluminum:
area= 3 in^2
La= 6 in
thermal expansion coefficient (alphaA): 12.9*10^(-6)/F
Elastic Modulus: 10.4 Mpsi

ΔT=60F

Answers given at the back of the book
Stress(steel) = 1.784 ksi
Stress(alum.) = -467 psi

3. Attempt at a solution
I tried finding the first equation needed to start the problem but each time as I go through with it I don't get the answer at the back of the book.
I found the thermal expansion for:
Steel= alphaS*ΔT*Ls = 6.3*10^(-6)/F * 60 F * 10 in = 0.00378 in
Aluminum= alphaA*ΔT*La = 12.9*10^(-6)/F * 60 * 6 = 0.004644 in

I assumed there was going to be a reaction force for the aluminum equal to:
[PaLa/EaA]
and I know that the stress of steel or aluminum in the end is P/A

my problem in the end is: finding the first equation, and how to get to P. Can anyone help out?
--------------------------------------------------------------------------------------

1.http://www.straydreamers.com/images/hw1bis.jpg [Broken]

2. Attempt at a solution
I started with the sum of the forces in the y direction so:
T1+T2+T3+T4-P=0

Then did the sum of the moments about A after the deformation was done:
(L)T2+(2L)T3-(2L)P+(3L)T4=0

Then considered the deformation with similar triangles:
def.2/L=def.3/(2L)=def.4/(3L)

After which I'm stuck because there are too many unknowns..so, did I go about it all wrong or am I close but don't see it?
 
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  • #2
I managed to solve the first problem, still working on the second one though. If anyone is actually curious about the first one I can post my work.
 
  • #3
For the second problem, i recommend expressing the displacements of B, C and D as function of the displacement A and the angle the bar makes with the horizontal times the space between the wires, then use Hooke's Law to relate the loads with the displacements, and finish working it out with static equilibrium, it'll end as 2 unknowns (displacement of A and the angle times the space between the wires) 2 equations.
 
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  • #4
In problem 2 you have to assume the wires are flexible, otherwise you won't get a solution (as you already discovered).

One way of looking at it is to see that because the horizontal bars are rigid, the wire lengths will be in an arithmetic progression, so by Hooke's Law the tensions will be

TA = x
TB = x + y
TC = x + 2y
TD = x + 3y

Then find x and y by resolving and taking moments.
 
  • #5
Thanks you two, I got the solutions for that second one.
 

1. What is thermal stress and how does it impact materials?

Thermal stress is a type of stress that occurs when there is a significant difference in temperature between two parts of a material. This can cause the material to expand or contract, leading to cracks or deformations. Thermal stress can weaken materials and make them more prone to failure.

2. What factors contribute to thermal stress?

There are several factors that can contribute to thermal stress, including temperature gradients, thermal conductivity, and thermal expansion coefficients of the material. Other factors such as changes in environmental conditions, rapid temperature changes, and material properties can also play a role in thermal stress.

3. How can thermal stress be prevented or minimized?

Thermal stress can be prevented or minimized by using materials with low thermal expansion coefficients, designing structures with temperature differentials in mind, and using insulation or other techniques to control temperature gradients. Proper maintenance and regular inspections can also help identify and address potential thermal stress issues before they become a problem.

4. What are the effects of thermal stress on different materials?

The effects of thermal stress can vary depending on the type of material. For example, metals may experience changes in strength and ductility, while plastics may experience warping or cracking. In ceramics, thermal stress can cause cracks and decrease in strength. Understanding the specific properties and behaviors of different materials is important in addressing thermal stress problems.

5. How is thermal stress different from tension stress?

Thermal stress and tension stress are two different types of stress that can affect materials. Thermal stress is caused by temperature changes, while tension stress is caused by forces acting on a material. However, both types of stress can lead to material failure if not properly managed. It is important to consider both types of stress in the design and maintenance of structures and materials.

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