Strength of materials vs Theory of Elasticity

[ramzerimar]

In mechanical engineering we have various courses in strenght of materials, and I've noticed that graduate students learn the Theory of Elasticity. I've researched a little bit about it, and I know that the Theory of Elasticity is more general than strenght of materials. But I have some doubts on the range of applications of elasticity theory. When would I choose to use the more complicated methods of elasticity instead of the simple formulas that we learn in strength of materials?

 

[Chestermiller]

In mechanical engineering we have various courses in strenght of materials, and I've noticed that graduate students learn the Theory of Elasticity. I've researched a little bit about it, and I know that the Theory of Elasticity is more general than strenght of materials. But I have some doubts on the range of applications of elasticity theory. When would I choose to use the more complicated methods of elasticity instead of the simple formulas that we learn in strength of materials?

Good question. In Strength of Materials, we make educated guesses about the kinematics of the deformation, and then are able to do the remainder of the analysis with no difficulty. However, Strength of Materials is applicable only to relatively simple problems, and, even then, in many cases (such as beam bending), it is not exact (although it is typically very accurate). Still, Strength of Materials can be applied to a large number of different kinds of problems that are encountered in practice. However, it can't be applied to very complex loading and deformation problems, and for that, we need to use the Theory of Elasticity. Theory of elasticity involves solving the stress equilibrium equations (in differential form) combined with the 3d version of Hookes law (relating stress and strain) and the strain displacement kinematic equations. It does not require any simplifying assumptions regarding the kinematics of the deformation, but it is often computationally intense. Such solutions are often obtained using finite element mechanics (FEM) software.

 

[ramzerimar]

Good question. In Strength of Materials, we make educated guesses about the kinematics of the deformation, and then are able to do the remainder of the analysis with no difficulty. However, Strength of Materials is applicable only to relatively simple problems, and, even then, in many cases (such as beam bending), it is not exact (although it is typically very accurate). Still, Strength of Materials can be applied to a large number of different kinds of problems that are encountered in practice. However, it can't be applied to very complex loading and deformation problems, and for that, we need to use the Theory of Elasticity. Theory of elasticity involves solving the stress equilibrium equations (in differential form) combined with the 3d version of Hookes law (relating stress and strain) and the strain displacement kinematic equations. It does not require any simplifying assumptions regarding the kinematics of the deformation, but it is often computationally intense. Such solutions are often obtained using finite element mechanics (FEM) software.

Thank your for the answer. In strength of materials, we learn how to deal with some structural components like beams, rods, etc, and how to calculate stress and strain in them. We know that we simplify some real life structures so that they can fit in the models that we have and apply the equations of strength of materials. But in industry, do engineers really use those equations? I mean, I have the impression that people tend to use FEM software for most things today. Do engineers still use those simple equations to design real life structures?

 

[Chestermiller]

In industry, we used strength of materials very extensively to analyze and improve the performance of product parts and processing equipment. It is accurate and saves time if used within its constraints. However, when a part or piece of equipment is geometrically complicated and/or has complicated loading, the tool of choice is FEM. So it all depends on the specific situation, but it greatly helps to have experience in both areas.

 

[ramzerimar]

In strength of materials, we learn how to calculate stress and strain in some basic structures: shafts for torsion, beams for bending... Also, we have the theory of thin shells to calculate stress on structures like pressure vessels and airplane fuselages, or maybe a ship hull. The way I see, strength of materials teaches you about some useful "building blocks" that you can use to build something more complicated. Looks like strength of materials is more applicable to strucutures, but parts are generally more complicated. A gear, for example. How would I proceed to calculate stress on a gear without FEM, just by using strength of materials?

Is this correct? Pure strength of materials is more easily applicable to structures?

 

[Nidum]

How do you think we managed to design complex components and assemblies before FEA became available ?

 

[ramzerimar]

How do you think we managed to design complex components and assemblies before FEA became available ?

Yes, I know that. My question is: how is strength of materials applied to those components today. We learn how to calculate stress on simples elements: rods, beams, shafts... A gear is a complicated thing, and I have no idea how to model something like that using strength of materials.

And my other question still stands: if engineers tend to use the formulas of strength of materials to design basic structures, like beams, thin plates, that form more complicated assemblies, while FEA is reserved only for more complicated situations.

 

[Chestermiller]

Yes, I know that. My question is: how is strength of materials applied to those components today. We learn how to calculate stress on simples elements: rods, beams, shafts... A gear is a complicated thing, and I have no idea how to model something like that using strength of materials.

And my other question still stands: if engineers tend to use the formulas of strength of materials to design basic structures, like beams, thin plates, that form more complicated assemblies, while FEA is reserved only for more complicated situations.

The answer to your question is yes. Strength of materials is still the first tool to consider using.

 

[Dr.D]

How would I proceed to calculate stress on a gear without FEM, just by using strength of materials?

Specifically to this question, the approach is to think of the gear tooth as a cantilever beam first, with the bending stress calculated as Mc/I. The critical stress is usually located in the fillet, and this is then modified by an empirical factor known as the Lewis form factor. The Lewis factor is available in tabular form in various machine design textbooks.

Gear design is a highly refined art, and those who specialize in it have developed a host of very specialized tools. It is not a good area for amateurs to dabble.

 

[afreiden]

Chestermiller's first response is great. I wanted to make one additional comment regarding "design."
Strength of materials equations for beams, columns, etc., are still used extensively in the field of structural engineering. Structural engineers design the skeletons of skyscrapers using strength of materials equations such as Mc/I. There are at least two reasons why they still do this despite the availability of high fidelity FEA: 1) Due to the high computational cost of high fidelity FEA, it is not yet feasible to use it to predict the response of an entire skyscraper to relatively slow acting loads like earthquakes.
2) In the structural engineering industry, "design" largely consists of choosing beam and column sizes so that you are as economical as possible while making sure the building doesn't fall down. There might be 1,000 different cross-sections that you can choose from in the building code, each with a particular moment of inertia, I. As an example, you can rearrange the equation Mc/I to find the "I" required and then pick the one out of 1,000 cross-sections that is appropriate. While high fidelity FEA provides precise analyses, it doesn't lend itself so nicely to "design."