Statics

Lesson 8

Beams: Members Subjected to Bending

🟦  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:

❏ 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.Β 


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

Β 

🟦  8.3 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 of the FBD. On the FBD, you may replace the actual loading with another system that is statically equivalent. 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).

We do not draw the supports themselves on the FBD; instead, we replace them with their effect on the beam (forces and moments).

Use qualitative equilibrium analysis to try to deduce the directionality of each reaction. 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. Please note that if the loading isn't complicated, you can often predict the directionality of the reaction forces (and moments). However, if the loading is complicated, a qualitative equilibrium analysis is not reasonable. In that case, make a prediction, and draw the vectors on the FBD accordingly.

It is perfectly OK to assume the incorrect directions for any (or all) of the unknown forces (and moments). In fact, many instructors ask students to always draw reactions in the positive direction by default. 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 can use planar analysis procedures to solve beam reactions, because they lie in a plane. We can simply elevate (or project) the front face of the beam; we do not need to draw the cross-section.

The planar E.o.E. are:

Select an equation that allows you to immediately solve for one of the unknowns. In general, this will usually be 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. For instance, say that you predicted that Ay was upward. Then, you calculated a positive value for Ay, thereby confirming that 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 ↑

Here is the best way to handle a negative answer. Let's say we assumed Ay was upwards, but in our final answer we got a negative sign. That means Ay is downwards. You would simply write the following:

Ay = -5 kN ↑ ∴ Β 5 kN ↓

Important. Never go back, erase, and correct the incorrect assumption you drew on the FBD. That makes it impossible to understand what you did. 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.

🟦  8.4 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

🟦  8.5 Principle of Transmissibility

The principle of transmissibility states that we can slide a vector anywhere along its line of action to form a statically equivalent system.Β 

This idea can be advantageous in solving certain beam problems.

🟦  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.Β 

The solution also explores various options provided by the Principle of Transmissibility related to the 80 kip angled load.

❏ Flipbook: a cantilever beam

➜ Practice Problems