🟦 8.1 Types of beams
Simply put, the word beam is used to describe a member that is primarily subjected to bending.
This flipbook illustrates types of beams, and provides new vocabulary and terminology:
simply-supported beam
simply-supported with an overhang
cantilever beam
You'll need to remember the characteristics of the "big three" types of engineering connections:
the pin connection (constrains translation, and therefore can develop force reactions in the x- and y-direction)
the pin-roller connection (it acts like a pin, except that it releases translation parallel to a surface; it there can only develop a force reaction that is perpendicular to that surface)
the fixed connection (acts like a pin but also constrains rotation, and therefore can develop a couple moment as a reaction)
❏ Flipbook: Types of Beams
🟦 8.2 Beams in the wild
❏ Laboratory testing of a simply-supported beam
In the animation, a simply-supported reinforced concrete beam is loaded in a laboratory.
The two (hydraulic) cylinders on top are applying vertical downward forces to the beam at the third-points.
The load is incrementally increased until the beam fails.
Prior to failure, we see the beam develop cracks, as the material becomes overstressed.
Source: https://makeagif.com/i/ezDabV and https://www.youtube.com/watch?v=SdpjUunqel4
❏ Beams in a steel-framed building
Beams are often (but not always) used in buildings to support floors, which in turn support people, furniture, and more!
This picture shows a series of cantilevered steel beams in a building (under construction).
Source: https://www.structuremag.org/wp-content/uploads/2019/10/1119-nsc-6.jpg
🟦 8.3 Actions and reactions
Beams carry applied loads. You can thiink of these as the actions or the input.
The loads could be any of these:
Forces (kN, kips, pounds, N, etc.)
Line loads (kN/m, kips/foot, etc.)
Could be constant
Could be linear (shaped like a triangle)
Could be higher-order functions (x^2, x^3, etc.)
Couples
couple moments (kN-m, kip-feet, kip-inches, etc.)
force couples (two equal forces, parallel, opposite in direction)
To go from the loading diagram to the free-body diagram, we free the body from the supports.
In that step, we replace the engineering connections (pins, pin-rollers, and fixed connections) with their effects on the beam (reaction forces or reacting moments).
These are simply called reactions or beam reactions.
I sometimes use a hatch (∤) through the reaction vectors to distinguish them from applied loads.
🟦 8.4 General procedure for calculating beam reactions
❏ Loading diagram
First, sketch the beam's loading diagram. Include the supports (pins, pin-rollers, and fixed connections) and the loading (point loads, line loads, etc.). On this diagram, we show the actual loading (that is, we do not replace it with a statically equivalent system).
❏ Free-body diagram
Your second picture is the FBD. You may replace the actual loading with a statically equivalent system if you like. For instance, if you have have a distributed load (force per distance, or force intensity), you can replace it with a statically equivalent resultant force (or forces).
Be sure not to draw the support symbols on the FBD. Instead, replace them with their effect on the beam (forces and moments).
On the FBD, please draw the reactions in the direction that you think is correct, based on qualitative equilibrium analysis and/or your own intuition. If the loading isn't complicated, you can often predict the directionality of the reaction forces (and moments). If the loading is complicated, a qualitative equilibrium analysis is not reasonable. In that case, make an assumption as to the direction of the reaction vectors.
It is perfectly OK to assume the incorrect directions for any (or all) of the unknown forces (and moments). The act of making a prediction will help you build intuition and become a better engineer. That said, the math will always correct you if your prediction turns out to be incorrect.
❏ Equations of equilibrium
We use the planar E.o.E. to solve beam reactions. We simply elevate (or project) the side view of the beam; we do not need to draw the cross-section.
The planar E.o.E. are:
the summation of forces in the x-direction must equal zero
the summation of forces in the y-direction must equal zero
the summation of moments about any z-axis must equal zero
Select an equation that allows you to immediately solve for one of the unknowns. This is usually the moment equilibrium equation.
Be strategic: sum moments about a point (really, an axis) that is coincident to one or more unknown forces.
Use the remainder of the E.o.E. to solve for the remainder of the unknowns.
❏ Reporting final answers
When you use the E.o.E., a final answer that is positive confirms that the direction you assumed for the vector is correct. A final answer that is negative contradicts your assumption.
For instance, say that you assumed that Ay was upward, and then calculated a positive value for Ay. The positive assumption is validated by the positive sign; your assumption was correct. In your final answer, use an arrow to emphatically communicate that you understand the true direction of the reaction on the beam, like this:
Ay = +5 kN ∴ 5 kN ↑
What do you do if you get a negative answer? Read this carefully. Say we assumed Ay was upwards, but got a negative sign in our final answer we got a negative sign. That means Ay is downwards. You would simply write the following:
Ay = -5 kN ↑ ∴ 5 kN ↓
Important. Do NOT erase, and correct the incorrect assumption you drew on the original FBD. Instead, if you would like to redraw the FBD with the arrows pointing in the correct directions, you may do that as the very last step.
🟦 8.5 Superposition for beams
Superposition is a fancy word for a simple idea. It can help you solve beam reactions quickly.
Consider this simply-supported beam that is subjected to two applied loads.
Sometimes it is advantageous to partition the beam into two subsystems, solve them separately, and then superimpose the results back together.
❏ Flipbook: How to use superposition
🟦 8.6 Example: a simply-supported beam
In this flipbook, you'll solve reactions for a simply-supported beam.
In the solution, you'll see how you can manipulate the vectors by using principles of equivalent systems.
Additionally, this solution highlights the benefits of using qualitative equilibrium analysis.
❏ Flipbook: A simply-supported beam
🟦 8.7 Example: a simply-supported beam with an overhang
Beam AB is simply-supported with an overhang. It supports an applied moment at A, an applied force at an angle, and a line load.
This solution highlights the use of superposition as a problem-solving shortcut.
❏ Flipbook: a simply-supported beam with an overhang
🟦 8.8 Example: a cantilevered beam
Cantilever beam ABC supports a trapezoidal (or linear) line load and an angled force.
Even though the beam is "bent," it is still a beam. Our approach doesn't change just because there is a bend in the beam.
This solution includes a procedure for breaking the trapezoidal load into two point loads.
The solution also explores various options provided by the Principle of Transmissibility related to the angled load.