🟦 18.1 Introduction
In the last lesson, you learned how to write equations for:
V(x) internal shear force [as a function of the position variable, x]
M(x) internal (bending) moment [as a function of the 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 provides a general orientation to the concept. Take a quick look, but don't get overwhelmed -- the step-by-step procedure follows below.
🟦 A quick overview
Mines in Amsterdam 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 a ton of notes, just try to listen. One of the example problems should look familiar (we used it in class).
🟦 18.2 Relationships between load, shear, and moment
A beam is a general term for a member that is subjected to bending. Beams are loaded such that the applied forces are transverse (perpendicular) to the member's longitudinal axis.
The loading function is a line load, so we will use the symbol w(x). We can think of this as the load intensity (force per distance).
The image below provides a brief derivation for the key relationships between load, shear, and moment:
w = dV/dx
V = dM/dx
The subscript L means "left" (as in the plane on the left, which technically is 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.
Our general process will be one of integration -- not differentiation. So, this way of thinking is most useful for us:
graphically integrate the free-body diagram (which contains the loading functions) to yield the shear diagram
graphically integrate the shear diagram to yield the moment diagram
🟦 18.3 Principles of graphical integration
Let's review a few concepts from calculus that we will need to apply to 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.
Here's the roadmap:
Load is the derivative of shear, and shear is the derivative of moment. Therefore, the slope of V is equal to the value of w. Similarly, the slope of M is equal to the value of V.
The area of the curve is equal to the change in the value of the function's integral (or its anti-derivative). Therefore the area under w(x) between x1 and x2 is equal to the change in the value of shear between those same coordinates. Similarly, 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 increase values of the integrated function. Negative areas decrease values of the integrated function.
We start at zero, work our way from left to right, and are mindful of discontinuities (or jumps) in the plots. Concentrated forces (point loads) cause jumps in the shear diagram. Upward force means jump UP; downward force means jump DOWN. Concentrated moments (couple moments) cause jumps in the moment diagram. Clockwise moment means jump UP. Counterclockwise moment means jump DOWN.
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). In turn, the quadratic integrates to cubic (x^3) ... to quartic (x^4) ... to quintic (x^5) ... etc.
The hardest part is determining the concavity of the higher order functions (smiles vs. frowns). If values increase from left to right, then the integrated function is concave up (smiling). If values decrease from left to right, then the integrated function is concave down (frowning).
🟦 18.4 Piecewise functions
Calculus is the study of continuous functions, like y = f(x) in the photo.
Most of our shear and moment diagrams will be piecewise functions. We can still graphically integrate between discontinuities (caused by forces in the shear diagram and caused by moments in the moment diagrams). But we have to be cognizant of the domain of each piece of the piecewise function.
🟦 18.5 Graphical integration: PRELIMINARY EXAMPLES
These three preliminary example problems are fully solved for you.
Please inspect these three examples carefully.
Your objective is just to correlate these solutions to the theory (in section 18.3) and/or the infographic (in section 18.1).
🟦 18.6 A simply-supported beam (STEP-BY-STEP EXAMPLE)
This lengthy flipbook shows you EVERY SINGLE STEP of the shear and moment diagram construction process. Enjoy!
🟦 18.7 A cantilever beam (STEP-BY-STEP EXAMPLE)
Here is another LOOOOONG step-by-step example.
🟦 18.8 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:
➜ Practice Problems
Mines in Amsterdam: there are 12 problems to work. Unless your name is Trenton (who has definitely earned a pass for slacking if he wants to use it), please make a good faith effort to solve ALL of these before class. When we get to class, we will decide whether to focus on some of these problems or move on to some new challenges. Shear and moment diagrams require quite a bit of time on task for mastery. You just have to practice; it's the only way to learn how to do this.
Construct shear and moment diagrams of this beam. It is in static equilibrium.
2. Construct shear and moment diagrams of this beam. It is in static equilibrium.
3. Construct shear and moment diagrams of this beam. It is in static equilibrium.
4. Construct shear and moment diagrams of this beam. It is in static equilibrium.
5. Construct shear and moment diagrams of this beam.
Be sure to compute the reactions first.
6. Construct shear and moment diagrams of this beam.
Of course, you don't need me to remind you to compute the reactions first.
7. Construct shear and moment diagrams of this beam.
8. Construct shear and moment diagrams of this beam.
9. Construct shear and moment diagrams of this beam.
10. Construct shear and moment diagrams of this beam. As shown, please model the effects of the soil particles on the bottom of the foundation as a constant line load.
11. Construct shear and moment diagrams of this beam. It's a canoe!
Please note that this is a very rough and rudimentary model of a canoe. We know that the water pressure function is more complex (Pascal's Law plus the curved hull). On top of that, a more realistic model would not lump the weight of the boat right in the middle of the canoe.
Sometimes it's OK for engineers to use rough, approximate, and preliminary models in order to get rough, approximate, and preliminary answers.
12. Construct shear and moment diagrams of this beam.