Change in length LSolve Engineering Problems: P, Brass, Steel

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In summary, the first conversation discusses a problem involving a rigid beam supported by a pin and two cables, with a load applied causing tension in the cables and rotation of the beam. The relevant parameters and calculations are provided for finding the force and stress at points A and C, the change in length of the cables, and the support force at point B. The second conversation involves a brass tube and steel rod attached to common end plates, with a change in temperature causing a change in distance between the plates. The relevant equations and parameters are given for determining the resulting stresses in both materials.
  • #1
I'm trying to study for an exam, and there are a few problems that i cannot figure out:

1) A rigid beam is supported at point B by a pin/hole connection and by two cables at points A and C, respectively.
Initially (before load P is applied), there is no slack nor tension in either of the two cables. Under influence of load "P" both cables will be under tension and the beam will have rotated slightly in counter-clockwise direction. Neglect the weight of the beam in all of your calculations.
Force P=500 [kN]
Modulus of Elasticity E=200 [GPa] (same for both cables)
Lengths on beam : a=1.6 [m] ; b=0.8 [m]
Lengths of cables : L1 = 0.75 [m] ; L2 = 0.6 [m]
Cross-section cables : A1 = 1100 [mm2] ; A2 = 1300 [mm2]

Following Questions need to be answered:
Force @ Point A
Stress @ Point A
change in length of L1
Force @ Point C
Stress @ Point C
change in length of L2
angle formed by beam and horizontal
Support force B?


2)Both, a brass tube and the steel rod located inside the brass tube are solidly attached to common end plates A and B at either end.
The respective diameters are :

Do = 5.2 [cm]
Di = 3.0 [cm]
d= 2.9[cm]
Brass:
alpha=19x10^-6/degree celcius
E=210GPa
Steel:
alpha=12x10^-6/degree celcius
E=105GPa
Initially the assembly is stress free.
Now the temperature increases simultaneously for the brass tube and steel rod by 60oC. As a result the distance L between the two end plates will change.
Determine the resulting stresses in the brass tube and the steel rod, counting tension as positive and compression as negative.

Following Questions to be answered:
Stress in Brass in MPa
Stress in Steel in MPa
 
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  • #2
Welcome to the PF. An important rule here is that you must show your work in order for us to help you. Also, the Homework Posting Template (that you apparently deleted instead of using) asks you to list the relevant equations and concepts that apply to your problems.

Please do both so that we can provide some assistance.
 
  • #3


I would first like to commend you for seeking help in understanding these problems and preparing for your exam. I can see that you are working on problems related to mechanics and material properties, which are important concepts in engineering.

In the first problem, we are dealing with a rigid beam supported by cables and a load applied at point P. The first step in solving this problem would be to draw a free body diagram and apply the equations of equilibrium to find the unknown forces at points A, B, and C. From the given information, we know that the load P is 500 kN and the lengths and cross-sectional areas of the cables.

To find the force at point A, we can use the equation ΣFy=0, where ΣFy is the sum of the forces in the y-direction. We know that the only force acting in the y-direction is the cable at point A, which is under tension. Therefore, the force at point A would be equal to 500 kN.

To find the stress at point A, we can use the equation σ=F/A, where F is the force at point A and A is the cross-sectional area of the cable at point A. From the given information, we can calculate the stress at point A to be 500 kN/1100 mm2 = 0.45 MPa.

To find the change in length of L1, we can use the equation ΔL=PL/AE, where P is the load, L is the length of the cable, A is the cross-sectional area, and E is the modulus of elasticity. Substituting the given values, we can calculate the change in length of L1 to be 0.75 mm.

Similarly, we can find the force, stress, and change in length at point C using the same equations and substituting the respective values. The angle formed by the beam and the horizontal can be found using trigonometry, by considering the geometry of the beam and the lengths of the sides.

For the second problem, we are dealing with the thermal expansion of a brass tube and a steel rod. To solve this problem, we need to consider the change in length of both materials due to the increase in temperature. We can use the equation ΔL=αLΔT, where α is the coefficient of thermal expansion, L is the initial length, and ΔT is the change in temperature.

For the brass tube
 

1. What is the difference between P, Brass, and Steel in terms of change in length?

P, Brass, and Steel are all materials commonly used in engineering. Each material has a different coefficient of thermal expansion, which determines how much its length will change when the temperature changes. P has the lowest coefficient of thermal expansion, followed by Brass, and then Steel. This means that P will experience the least amount of change in length with temperature fluctuations, while Steel will experience the most.

2. How do you calculate the change in length for a specific material?

The change in length for a material can be calculated using the formula ΔL = αLΔT, where ΔL is the change in length, α is the coefficient of thermal expansion, L is the original length of the material, and ΔT is the change in temperature. This formula can be used for any material, including P, Brass, and Steel.

3. Can the change in length of a material be controlled or prevented?

The change in length of a material with temperature is a natural phenomenon and cannot be completely prevented. However, it can be controlled to some extent by using materials with lower coefficients of thermal expansion, such as P or Brass, or by designing structures that can accommodate the expected changes without causing damage or failure.

4. How does the change in length of a material affect its performance in engineering applications?

The change in length of a material can have a significant impact on its performance in engineering applications. If a material experiences a large change in length with temperature, it can lead to structural stress, distortion, or failure. This is why it is important to consider the coefficient of thermal expansion when selecting materials for specific engineering applications.

5. Are there any other factors besides temperature that can cause a change in length of a material?

Yes, besides temperature, there are other factors that can cause a change in length of a material. These include mechanical stress, moisture, and chemical reactions. Mechanical stress can cause a material to expand or contract, while moisture can cause swelling or shrinking. Chemical reactions can also cause changes in the molecular structure of a material, resulting in a change in length. These factors should also be taken into consideration when selecting materials for engineering applications.

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