strength of materials homework solutions

strength of materials homework solutions

Strength of Materials Homework Solutions

1. Introduction

The book begins with a broad overview of strength of materials and the various types of problems that will be solved, including problems related to stress, strain, and other mechanical properties. Then, a roadmap is given including the various things that will be covered, from the equations of equilibrium, material properties, and external loading to a broad look at the internal response of a material and a discussion of stress and strain. The introduction of the book concludes by giving the reader more information about what is to come and a brief summary of each major section of the book, providing a useful setup for the subsequent problems. In summary, the introduction does a nice job of providing a wide view of strength of materials in 2021, and it introduces the reader to all that will come in the following chapters. It was clearly communicated and logically organized. The goals of the book were stated and then expanded upon to give the reader a more detailed understanding of what’s to come. This essay will provide us a good initial base when solving the problems and topics in strength of materials because solutions for it in task 1 were pretty difficult. The introduction focused on various types of problems that will be solved starting from problem 1 to problem 3.

2. Problem 1: Beam Analysis

The beam analysis is amongst the earliest and most essential applications of the “Strength of Materials.” The power of the process comes from the capability to model and comprehend the internal reactions and external loads that are on a system. For example, beams and columns are utilized in virtually every structural application and due to that, the capability to comprehend and study them is vital in the industry of engineering. Under this area, the two most crucial classifications of beam issues are covered. These are stress analysis of beams due to vertical loads and beam deflection analysis. For the stress analysis of beams, the method utilized is called the flexure formula. The flexure formula is utilized to discover the stress in a beam due to a bending moment. On the other hand, under the section that covers beam deflection analysis, a process called the double combination approach is used. The process, in essence, is an extension of the flexure formula and it is utilized to discover the deflection of the beams. It can be presumed that there is no such thing as a beam that is best and nothing but beam deflection is more suitable for the specific design. The option of beam type depends on the nature of the task and the specific requirements such as the load carriage ability, the room readily available in the last design, and the basic construction of the structure concerned. Finally, it is crucial to understand the difference between the two types of loading that beams can bring. The analysis shows the shearing force due to loads and the variation of bending moment in the beam. It can be seen that due to the load positioning, two different types of stresses can be produced. Also, it can be seen that the variation of the bending moment is the reason high. This is a crucial point of the analysis as for a much better beam made while material is conserved as much as possible, the beam finding is where the bending moment allowed would change.

3. Problem 2: Stress and Strain

The stress-strain curve for bolts or bars of uniform composition is a tool used to understand the material behavior under load. The given curve for a steel alloy in this case is for a specific bar of a given composition. Starting at the origin of the curve (at zero stress and strain), the linear region of the curve can be used to find the modulus of elasticity of the bar material. In many cases, any changes in length of the bar can be found by using this linear portion of the curve. As long as the stress does not exceed the proportionality limit (the end of the linear region of the curve), the bar will return to its original length once the stress is removed. This straight region is called the elastic region. Failure of the bar will occur if the stress (and thus the strain) continues to increase beyond the proportionality limit; the bar will break once reaching the ultimate stress, also known as the tensile strength. If the bar is loaded to a stress lower than the ultimate stress and then unloaded, the strain will not return to zero. This is because the material has been permanently stretched beyond the yield point. The entire straight region and the yield stress at the end of that region is known as the yield strength of the material. On the stress-strain curve, this nonlinear region of positively increasing slope is the result of plastic deformation in the material. If the material is loaded further, it will continue to permanently lengthen until the ultimate stress is reached. This phenomenon is known as necking, where the highest stress in the bar is localized in one region and drastically increases while the other areas of the bar see a reduction in stress due to the higher value for strain. At the end of the curve, sudden fracture of the material will occur once the material reaches the ultimate stress. The area under the stress-strain curve, known as the strain energy in the material, that has built up between the start point and the current point under stress. The strain energy can be used to find the work done to deform the material or the power dissipated in the material during cyclic loading, which will be important when we begin discussing cyclic stress.

4. Problem 3: Deflection and Stiffness

The last problem involved describing the deflection of a cantilever beam due to a concentrated load at the end. Now consider a case where the ends are clamped so that the beam cannot rotate at either end. If the load of 500 lb is applied in the middle of the beam, determine the maximum deflection of the beam. If I denote this quantity by f. I can start by stating that there are various loadings that could create deflection. For instance, we have used a concentrated load in the middle of a beam. The first step is to recognize that E is the same for all materials and therefore all materials deflect equally. From the given data, both L and b are constants that only reflect the size and shape of the beam. However, the constants P and E are entirely dependent on the type of load and material of the beam, respectively. After taking into account the boundary conditions, presented as the ends of the beam, and solving for the arbitrary constants, the actual deflection function could be written in terms of the given data. After making some approximations, I carried out the integration and for the case of a relatively long beam, the deflection at the center is given by f = PL^3/(48Eb^2). If the load of 500 lb is applied in the middle of the beam, the maximum deflection of the beam is f = 500L^3/(48Eb^2). This problem is certainly more complicated from a mathematical point of view than the previous problem. However, once again, any student who has a basic understanding of calculus could follow the solution because the process and the concepts remain the same as in the previous problem. I believe that this type of exercise is a good summary of what we need to be aware when we investigate the deflection of a beam under a particular loading. We need to understand the limitations of the Euler-Bernoulli assumptions, the meaning of the flexural stiffness and the boundary conditions as well as how to apply the theory into practice.

5. Conclusion

In conclusion, to solve the problems of our structures, the students need to study and understand the stress and the strain of the materials. Also, they need to know how those stresses will affect the materials and how the forces will influence the external objects. In addition, the study on different types of the beams as well as the many different types of the loadings will also help to solve the problems in real life. For instance, when we practice the statics class, we gain a lot of knowledge. We know how to calculate the area by using the moment of inertia or how to calculate the distance when we have the mass center. We also know how to calculate the value of the internal force. So, through the homework, we gain a lot of concepts of statics and also we know a lot of the application of those concepts in real life when we need to solve the problems of the strength of materials. Last but not least, we need to take care that the deflection will not exceed the allowable limit. Because the allowable stress is a limit to the loads which can apply safely. And if the deflection of a structure exceeds the allowable limit, that structure will fail to support the load and it will become unstable. Through the work, we know most of the information that we should understand when we study the deflection and the deflection limits and the application of the deflection calculation. And along with the development of technology, the designers are constantly pushing the limit of the lightweight and also the complex shape. So, the material science is the crucial aspect for them and using the correct materials will optimize the design. And now, even the smart materials which react to the external environment have been applied in the field of aerospace and civil engineering. So, it is great that we have the chance to study the strength of materials and it will help us in the future when we become an engineer.