2.8. Theory of Structures

The procedure for determining internal forces at a specified location on a beam can be summarized as follows:

This introduction sets the stage for understanding how to analyze beams in structural engineering. It suggests that there is a systematic approach to determining internal forces, which is essential for ensuring structures are stable and safe.

Think of analyzing a beam like checking the stability of a bridge. Just as engineers need to consider various forces acting on the bridge, we must analyze internal forces within the beams that hold it up.

1- Compute the support reactions by applying the equations of equilibrium and condition (if any) to the free body of the entire beam. In cantilever beams, this step can be avoided by selecting the free, or externally unsupported, portion of the beam for analysis.

In this step, we calculate the reactions at the supports of the beam. These reactions are the forces exerted by the supports to keep the beam in equilibrium. If we are analyzing a cantilever beam (which is supported at only one end), we can simplify things by focusing on the unsupported portion.

Imagine a seesaw. The point where it rests (the support) reacts to all the weight on either side. If one side is unsupported, a lot of the analysis can focus on just that side, making calculations easier.

2- Pass a section perpendicular to the centroidal axis of the beam at the point where the internal forces are desired, thereby cutting the beam into two portions.

Here, we are instructed to make an imaginary cut in the beam where we want to analyze internal forces. This cut divides the beam into two parts, enabling us to examine the forces and moments acting on either side of the cut.

Think of slicing a cake. If you want to check how much icing is on a specific piece, you cut a slice and look at the frosting on that piece rather than the whole cake.

3- Although either of the two portions of the beam can be used for computing internal forces, we should select the portion that will require the least amount of computational effort, such as the portion that does not have any reactions acting on it or that has the least number of external loads and reactions applied to it.

Choosing which side of the beam to analyze is critical. Selecting the portion that simplifies calculations reduces potential errors and saves time. This is particularly important when the chosen side has no reactions or fewer loads.

Imagine you have a heavy backpack. It's easier to analyze and see if you can lift it if you only examine one side instead of trying to assess both sides at once.

4- Determine the axial force at the section by algebraically summing the components in the direction parallel to the axis of the beam of all the external loads and support reactions acting on the selected portion.

This step involves calculating the axial forces that act along the length of the beam. We do this by adding up all the loads and reactions acting parallel to the beam's axis to determine the resultant axial force at the cut section.

Think of a tug-of-war game. The total force exerted by both teams pulls in different directions. When assessing the strength of the rope, we need to sum how strongly each team is pulling.

5- Determine the shear at the section by algebraically summing the components in the direction perpendicular to the axis of the beam of all the external loads and reactions acting on the selected portion.

In this step, we focus on calculating the shear forces, which act perpendicular to the beam's length. Just like axial forces, we sum all the forces acting in the vertical direction at the section to find the total shear force.

Imagine a stack of books on a table. If someone pushes the top book to one side, the pushing force represents shear. We calculate this force to understand how much resistance the stack can handle.

6- Determine the bending moment at the section by algebraically summing the moments about the section of all the external forces plus the moments of any external couples acting on the selected portion.

The bending moment indicates how much a beam bends under load. We calculate it by summing the moments, produced by external forces and couples, about the cut section. Moments are calculated as the force multiplied by the distance from the point of interest.

Picture a seesaw again. If one side is pushed down, the moment around the pivot changes. Understanding how these moments work helps us ensure that the seesaw remains balanced.

7- To check the calculations, values of some or all of the internal forces may be computed by using the portion of the beam not utilized in steps 4 through 6. If the analysis has been performed correctly, then the results based on both left and right portions must be identical.

Verification is a crucial step in the analysis. By calculating the internal forces on the other side of the cut section, we ensure consistency in our results. If both calculations yield the same results, our analysis is likely correct.

Think of double-checking your homework answers. If the same question yields the same answer regardless of which route you take to get there, you can be more confident in your solution.

For the following examples, determine the axial forces, shears, and bending moments at points A and B of the structure shown.

After discussing the procedure for analyzing internal forces in beams, the text provides examples to apply this knowledge. Working through examples reinforces the learned concepts and demonstrates their practical application.

Consider learning to ride a bicycle. Initially, you learn the theory of pedaling, balancing, and steering. By practicing on a real bike, you can apply what you've learned – this transition from theory to practice is similar to what happens when students work with examples.

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