# Mechanics of Materials & Principal of superposition

• djeitnstine
In summary: The problem requires a much more different approach than that.In summary, the conversation discusses a problem in mechanics of materials where a steel tube is placed in a vice and forces are applied to it. The question arises about the direction of the reaction forces and the convention for drawing forces in mechanics. It is clarified that in mechanics, the convention is different and a tensile force is positive and a compressive force is negative regardless of the direction it points in the global coordinate system. The conversation also discusses the use of free-body diagrams and element local coordinate systems in determining the direction of forces.

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## Homework Statement

A steel tube with a 32mm outer diameter and a 4mm thickness is placed in a vice and is adjusted so that the jaws tough the ends of the tube without exerting any pressure on them. The two forces shown are then applied to the tube. After these forces are applied, the vice is adjusted to decrease the distance between the jaws by 0.2mm. Determine the forces exerted by the vice on the tube at A and D.

http://img193.imageshack.us/img193/2700/13357119.th.jpg [Broken]

## The Attempt at a Solution

http://img51.imageshack.us/img51/2055/76147557.th.jpg [Broken]

Ok I did the question and got the answer after the professor solved it in class. However what I do not understand is why does he take the reaction forces as radially outward (in tension) and when he did the solution, he treated left as positive and right as negative.

This seems to go against convention. Even the solution manual has the same technique as he does. His explanation was that $$R_a$$ was in tension. Clearly from convention its in compression and so is $$R_d$$. Their answer also 'miraculously' show that it is in compression.

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In statics, you can draw any arrow in any direction. (If it's the "wrong" direction to produce a positive answer, you'll get a negative answer, of course.) The key is to sum the forces correctly. Here we'd want $-R_A-42+30+R_D=0$ or $R_A+42-30-R_D=0$.

He may have been saying that $R_A$ corresponds to a hypothetical tensile force, which it does. In the end, $R_A$ will turn out to be a negative number, indicating that the force was a positive compressive force all along. Does this answer your question?

Hmm, well that's only half of the answer. What I failed to mention was that when I used the right as positive and left as negative convention I got a rubbish answer. The work here isn't exactly statics, its mechanics of materials. This problem as you should see is statically indeterminate.

The way the problem is solved is by considering reduction in length $$\delta$$ created by the forces. Where $$\delta = \frac{FL}{AE}$$ (Where F is the force(s) in question, L is the length of the tube, A is its cross sectional area and $$E=\sigma \epsilon$$). We find the $$\delta$$ for each section (AB BC CD) and realize that total $$\delta_T= \delta_{AB} \delta_{BC} \delta_{CD} = 0$$

I.e. $$\Sigma_A^D \frac{FL}{AE}$$

When I used the regular convention For F (right as positive and left as negative) my answer came out totally wrong.

In mechanics the convention is different; a tensile force is positive and a compressive force is negative, regardless of the direction the force points in the global coordinate system. (These signs correspond to arrows pointing out of the beam.) Nevertheless, there should be no problem in assuming an unknown force to be tensile or compressive; the answer will reveal whether the guess is right. It takes some getting used to.

djeitnstine: As Mapes mentioned, you can establish any free-body diagram (global) coordinate system you wish. Then you can draw any particular unknown vector in any direction you wish (the positive or negative direction). If you draw an unknown vector in the negative direction, and its magnitude turns out positive, then the force is in the global negative direction. After you obtain a force result from this free-body diagram, you then transform the force to the element local coordinate system for each element. A universal convention for the element coordinate system is, tension positive, compression negative. You therefore finally see, from the element local, not global, coordinate system perspective, what force or stress is actually on the material.

Ok thanks, I also discussed it with my professor and found out that made a mistake when I worked the problem out using convention.

## 1. What is the concept of mechanics of materials?

Mechanics of materials is a branch of mechanics that deals with the behavior of solid objects subjected to external forces. It involves the study of stress, strain, and deformation of materials under various loading conditions.

## 2. What is the principle of superposition?

The principle of superposition states that the response of a material to multiple applied loads is equal to the sum of the individual responses to each load acting alone. This principle is applicable to linear elastic materials and allows for the analysis of complex loading scenarios.

## 3. How is the principle of superposition used in mechanics of materials?

In mechanics of materials, the principle of superposition is used to simplify the analysis of structures subjected to multiple loads. By breaking down the total load into individual components and analyzing each one separately, the overall response of the structure can be determined more easily.

## 4. What is the difference between stress and strain in mechanics of materials?

Stress is a measure of the internal forces acting on a material, while strain is a measure of the resulting deformation or change in shape of the material. Stress is typically measured in units of force per unit area, while strain is a dimensionless quantity.

## 5. How does the mechanics of materials apply to real-world applications?

Mechanics of materials is used in a wide range of engineering and scientific fields, including civil and mechanical engineering, material science, and physics. It is used to design and analyze structures such as buildings, bridges, and airplanes, as well as to understand the behavior of materials under various conditions.