Statics

Lesson 17

Shear and Moment Diagrams

🟦 17.1 Introduction

🟦 17.2 A quick overview

🟦 17.3 Relationships between load, shear, and moment

🟦 17.4 Principles of graphical integration

🟦 17.5 Continuous functions and piecewise functions

🟦 17.6 Constructing shear and moment diagrams

🟦 17.7 Graphical integration: PRELIMINARY EXAMPLES

🟦 17.8 Step-by-step examples

🟦 17.9 Computing the area under higher-order curves

➜ Practice Problems

🟦 17.1 Introduction

In the last lesson, you learned how to write equations for:

V(x) internal shear force [as a function of position variable x]

M(x) internal (bending) moment [as a function of position variable x]

These equations express how internal shear force (V) and internal bending moment (M) vary along the length of a beam. V(x) and M(x) can be plotted using pure mathematics. In other words, if you were to go to Desmos or your favorite graphing calculator, you could input them manually and make a plot (or graph). That takes too much time.

Instead, we will learn a fast way to graph (or plot) those equations. It is sometimes called the graphical method for constructing shear and moment diagrams. This infographic is an orientation to the concept. Do take a quick look, but don't get overwhelmed -- the step-by-step procedure follows below.

🟦 17.2 A quick overview

Fall 2024 students: please watch “Understanding Shear Force and Bending Moment Diagrams” by YouTube user The Efficient Engineer. The video is about 16 minutes long. You don’t need to take notes; it's OK to listen.

You may have noticed that the video says that dV/dx = –w, which seemingly conflicts with the derivation below (dV/dx = w). The difference in the sign has to do with the way positive load is defined. In the video, a downward load is defined as positive (in the direction of gravity). In the derivation, an upwards load is defined as positive (in the positive y direction). This detail is trivial; it doesn't change the graphical method for constructing shear and moment diagrams.

🟦 17.3 Relationships between load, shear, and moment

A beam is a member that is subjected to bending. In beams, loads are applied transverse (perpendicular) to the member's longitudinal axis.

We will continue to use the symbol w(x) to indicate a line load on a beam. We also can think of this as the load intensity (force per distance). A loading function can be zero, constant, linear, or any function you can dream up.

The image below provides a brief derivation for the key relationships between load, shear, and moment:

w = dV/dx V = dM/dx

How to interpret the subscripts:

The subscript L means "left" (as in the plane on the left, which could alternatively be called the negative x-face).

The subscript R means "rigatoni" (as in a type of pasta that is delicious). That was a joke. R is for right.

While the formulas derived above are commonly written as derivatives, our process will be one of integration -- not differentiation. Like this:

V = ∫ w(x) dx and M = ∫ V(x) dx

We will graphically integrate the free-body diagram (which contains the loading functions) to yield the shear diagram. After that, we will graphically integrate the shear diagram to yield the moment diagram.

🟦 17.4 Principles of graphical integration

Let's review a few concepts from calculus that we will need to apply to shear and moment diagrams.

Slides 1 through 10 are a review of the the fundamental theorem of calculus.

The graphical integration process begins on Slide 11. This is how we will construct shear and moment diagrams.

Study slides 11 through 23 very carefully.

Pay extra attention to slides 21 and 22 (this concept is often challenging for Statics students).

❏ Flipbook: important review of calculus concepts

🟦 17.5 Continuous functions and piecewise functions

Calculus is the study of continuous functions, like y = f(x) in the image.

❏ A continuous function

Most shear and moment diagrams will be piecewise functions. A continuous (integrable) function will be bounded by discontinuities (or jumps).

The discontinuities will be caused by forces (in the shear diagram) and moments (in the moment diagram).

We will graphically integrate between the discontinuities.

We must simply be cognizant of the domain of each piece of the piecewise function.

❏ A piecewise function

🟦 17.6 Constructing shear and moment diagrams

In the graphical integration method (or simply the graphical method) for constructing shear and moment diagrams, we apply fundamental principles of calculus to piecewise functions (functions with discontinuities).

We use x to measure position (or length) from our origin, and graphically integrate with respect to length x.

THE BASICS:

Integrate load (w) to get shear. Integrate shear (V) to get moment (M). Start at x=0 and work from left to right.

the slope of V equals the value of w (because dV/dx = w)
the slope of M equals the value of V (because dM/dx = V)

For each integration, the exponent of x increases by 1.

When you integrate a constant function, you get a linear function.
When you integrate a linear function, you get a quadratic function (x^2).
When you integrate any function x^n, you get a higher-order function (x^(n+1)).

The area of the curve is equal to the change in the value of the function's integral (or its anti-derivative).

The area under w(x) between x1 and x2 is equal to the change in the value of shear between those same coordinates.
The area under V(x) between x1 and x2 is equal to the change in the value of moment between those same coordinates.
Positive areas mean the function's integral increases between x1 and x2.
Negative areas mean the function's integral decreases between x1 and x2.

Determine the concavity of the higher order functions (smiles vs. frowns).

If values of a function increase from left to right, then the integrated function is concave up (smiling).
If values of a function decrease from left to right, then the integrated function is concave down (frowning).

DISCONTINUITIES:

Concentrated forces (point loads) cause jumps in the shear diagram. The sign convention is intuitive. Upward force means jump UP; downward force means jump DOWN.
Concentrated moments (couple moments) cause jumps in the moment diagram. The sign convention is counterintuitive. Clockwise moment means jump UP. Counterclockwise moment means jump DOWN.