Engineering Mechanics

Introduction

Engineering mechanics is the application of mechanics to solve problems involving common engineering elements. As with mechanics, it is dealt with in two stages--the simpler statics and then the more involved dynamics.

This book assumes familiarity with high school physics and calculus, although the mathematics used is fairly elementary.

Statics

We describe the motion of bodies using Newton's second law of motion

$\text{Force = Mass * Acceleration}\text{or}𝐟=m𝐚$

Statics deals with the situation where the acceleration ($𝐚$) is zero - which happens when a body is at rest. That means that the total force on a body at rest must be zero. In other words, the sum of the forces on the body must equal zero.

$\text{Sum of Forces = 0}\text{or}\sum 𝐟=\mathrm{𝟎}$

Engineers typically draw what is called a "free body diagram to show all forces on a body at rest. These forces are then broken down into vectors consistent with a useful coordinate system and summed in sets (components parallel to each basis vector) which are then set to zero to meet the static constraint of no acceleration being present.

This typically results in sets of equations which can be solved using simple linear algebra techniques or even simple algebra and substitution.

Insert a png example freebody diagram and calculations here.

Truss

Forces act along the members, and there are no shear forces or moments. The ends of a truss are pinned, so that they don't carry moments. The only reactions at the ends of a truss member are forces. External forces on trusses act only on the end points. Truss problems are solved by the method of sections, where an imaginary section is taken normal to truss members, or by the method of joints, in which a single joint is isolated and analyzed and the resulting forces are transferred to adjacent joints, where the process is repeated.

Chains and Cables

Chains and cables are attached at end points and have a continuous load on them due to self weight or external loads. Let a cable of length $L$ have a load of $w$ acting per unit distance between the supports. If the tension in the cable at any point is $T(x)$, then we have, for an infinitesimal length of the cable making an angle $\theta (x)$ with the horizontal,

$\begin{array}{c}\frac{d}{dx}[T\cos \theta ]=0\\ \frac{d}{dx}[T\sin \theta ]=w\end{array}$

Thus, we have,

$\begin{array}{c}T\cos \theta =\text{constant}=K\\ T\sin \theta =\int wdx+C_1\end{array}$

or

$\tan \theta =\frac{1}{K}[\int wdx+C_1]$

Now

$\tan \theta =\frac{dy}{dx}.$

So we have a differential equation for $y$ in terms of $x$:

$\frac{dy}{dx}=\frac{1}{K}[\int wdx+C_1]$

Solving for $y$,

$y=\frac{1}{K}[\int (\int wdx)dx+C_{1}x+C_2]$

If $w$ is a constant, then we have,

$y=\frac{1}{K}[\frac{w{x}^2}{2}+C_{1}x+C_2]$

which is the equation of a parabola.

For a rope where the loading is given in terms of the length of the rope (much more common), i.e., $T\equiv T(s)$ and $\theta \equiv \theta (s)$, we have,

$\begin{array}{cc}\frac{d}{ds}[T\cos \theta ]& =0\\ \frac{d}{ds}[T\sin \theta ]& =w(s)\end{array}$

where

$ds=\sqrt{d{x}^2+d{y}^2}$

Dynamics

While statics deal with the part of mechanics where all objects are stationary, dynamics deals with moving objects. It should be mentioned that objects moving with zero acceleration can be looked at as stationary objects. Dynamics may be broken down into kinematics and kinetics. Kinematics deal with displacement, velocities and accelerations without concern with the forces involved. Kinetics deal with the forces and moments involved in making the body move.