Statics

Lesson 17

Shear and Moment Diagrams

🟦 17.1 Introduction

🟦 17.2 Principles of graphical integration

🟦 17.3 Continuous vs. piecewise functions

🟦 17.4 Relationships between load, shear, and moment

🟦 17.5 How to construct shear and moment diagrams

🟦 17.6 Graphical integration: PRELIMINARY EXAMPLES

🟦 17.7 Step-by-step examples

🟦 17.8 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. You could derive the equations and use your favorite graphing calculator to plot them.

That approach is much too time-consuming.

This lesson will teach you a fast way to plot the V(x) and M(x) equations. It is sometimes called the graphical method for constructing shear and moment diagrams. We will be doing graphical integration.

🟦 17.2 Principles of graphical integration

Let's review a few concepts from calculus that are the basis for graphical integration.

Slides 1 through 10 review the fundamental theorem of calculus.

Slides 11 through 23 show how we will construct shear and moment diagrams. Study them carefully, and pay special attention to slides 21 and 22 (the concept that often proves challenging for Statics students).

❏ Flipbook: important review of calculus concepts

🟦 17.3 Continuous vs. 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.

❏ A piecewise function

🟦 17.4 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 also be called the negative x-face). The subscript R means "right" (which could be called the positive x-face).

While the formulas derived above are commonly written as derivatives, our process will be one of integration.

V = ∫ w(x) dx

M = ∫ V(x) dx

Our process has three steps:

(1) Draw a FBD of the beam and solve reactions.

(2) Construct the shear diagram by graphically integrating the load function.

(3) Construct the moment diagram by graphically integrating the shear diagram.

🟦 17.5 How to construct 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 (distance from the origin) and we integrate graphically with respect to length x. Begin each function at (0,0), work from left to right, and end at (L,0).

DISCONTINUITIES (JUMPS):

Forces (point loads) cause jumps in the shear diagram.

An upward force means jump UP
A downward force means jump DOWN.

Moments (couple moments) cause jumps in the moment diagram.

The sign convention will feel awkward:

A clockwise moment means jump UP.
A counterclockwise moment means jump DOWN.

FUNCTIONS (BETWEEN DISCONTINUITIES):

The area under the curve of the source function is equal to the change in value of the function's integral.

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

These are not jumps; the change is gradual.

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

- - If a function's values increase from left to right, the function's integral is concave up (smiling).
  - If a function's values decrease from left to right, the function's integral is concave down (frowning).

Statics students often struggle with this concept.

Label each function with the power of x (zero or constant, linear, quadratic, cubic, etc.).

- - The integral is always one power of x higher than the source function (xⁿ → xⁿ⁺¹)
  - For example, when you integrate a linear function (a polynomial with x¹), you get a quadratic function (a polynomial with x²)

When a function crosses from positive to negative values, you must solve for the x-intercept in order to compute area under the curve. To do this, apply these relationships:

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).

The slope of V(x) is equal to w(x) and also equal to rise over run (V1 / x1).

🟦 17.6 Graphical integration: PRELIMINARY EXAMPLES

Now that you know the basics, inspect these example problems. They are fully solved for you.

Your objective is just to correlate these solutions to the theory in 17.5.

The image below is the same information, consolidated into a single image.

❏ Flipbook: Preliminary Examples 1, 2, and 3

❏ Solutions: Preliminary Examples 1, 2, and 3

🟦 17.7 Step-by-step examples

The only way to master shear and moment diagrams is to practice them.

Watch me solve shear and moment diagrams for this beam, one step at a time.

❏ Step-by-step example 1

Here's another step-by-step example of a shear and moment diagram problem.

❏ Step-by-step example 2

🟦 17.8 Computing the area under higher-order curves

On slide 12 of the last flipbook, you may have noticed a formula for calculating the area under a higher order curve.

These geometric relationships are useful to memorize and apply:

❏ Area formulas to memorize and apply

➜ Practice Problems

INSTRUCTIONS: Solve shear and moment diagrams for each of the beams below. You must solve for all values, and label each function (constant, linear, quadratic, etc.). Be sure to include units (either with every value, or in the legend).

Here is a solution video for all 10 of the practice problems. You may use it to check your work, or as a learning tool. It's not here for you to copy, though. You will only master shear and moment diagrams by putting in the work yourself.

Problem 1.

Problem 2.

Problem 3.

Problem 4.

Problem 5.

Problem 6.

Problem 7.

Problem 8.

Problem 9.

Problem 10.

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