Applied Mechanics

Introduction

Solid mechanics or mechanics of materials is the study of the relations between forces acting on a solid body and causing strains and stresses. In this chapter we will introduce mechanical properties in order to be able to dimension parts to withstand given loads without failure.

Compared to forces, stress is a better measure to gauge the strength of a material from which the body is made of. Using force (or the completely misplaced term weight in some forums) as a measure of how much force a solid body can carry before it fails is troublesome since that depends on the size of the object, more precisely it depends on the load carrying area. By dividing the force by the area we get stress which is a better measure for comparing mechanical properties of a material. More on this topic can be found in the next section.

A thorough understanding of the fundamentals of solid mechanics is crucial for the design of any machine or structure.

Conducting analysis by hand as the primary tool for structural analysis is a thing of the past, modern engineers need to focus on the study of the problem statements and primarily creating and verifying the models. The human factor, the engineer, sets up the problem by posing the governing equations, load cases thorough boundary conditions and loads, sets up assumptions and simplifications, in short, creates a computational model of the physical model. Then we must let the computer do the actual computations, the solution of the resulting system of equations and any parameter studies and visualizations. This is what machines excel at. The computations are less error prone, more accurate and much much faster. The engineer needs to focus on the computational thinking to produce results. In this course we shall focus on the modeling and we will be freeing up time to explore more fun problems, real-life problems that traditional courses lack since it is hard or impossible to solve them by hand and instead we simplify problems and create these academic problems which are never found in real life engineering applications.

After this course in the basics of solid mechanics a natural step is to take a course in the Finite Element Method (FEM), which is the dominating computer method for solving structural mechanics problems as well as other similar problems. Ask the authors about such a course at JTH.

Solid mechanics always starts with a mechanical analysis and a free body diagram. Let us use a simple example using a body which has two or three supports and is subjected to a force. A body using two supports can be solved using mechanics alone, the reaction forces can easily be determinate from force-, and moment-equilibrium.

We get two equations and two unknowns, $R_A$ and $R_B$. This is called statically determinate.

syms R_A R_B F a b
eqs = [R_A+R_B - F == 0
       a*-F + (a+b)*R_B == 0]

[R_A, R_B] = solve(eqs, [R_A, R_B])

Adding another support to the body, we get

Now we have three unknowns $R_A$, $R_B$ and $R_C$, but still only two equations. In linear algebra this is known as an underdetermined system, but in mechanics and solid mechanics we use the term statically undeterminate structure. We need more equations to be able to solve this! We need to include relations about the deformation of the material under load. Going from pure mechanics, where we looked at how bodies act under external forces alone, we shall move towards solid mechanics and focus more on inner loads and deformations caused by external loads. Thus we no longer can treat bodies as un-deformable (rigid). In addition to equilibrium of forces, we need to add material relations and compatibility relations. These concepts boil down to relations summarized in the golden quad which we will get to in later sections. To summarize: The goal in solid mechanics is to determine if a design can withstand the given loads and how it deforms.