4.3. Shear Force and Bending Moment Diagrams

The following step-by-step procedure can be used for determining the member end forces as well as the shears, bending moments, and axial forces in members of plane statically determinate frames:

This section outlines a systematic approach to analyze statically determinate frames, which are structures where the internal forces and reactions can be determined using only the equations of equilibrium. The analysis consists of several steps that must be followed in order to achieve accurate calculations of forces and moments within the structure.

Think of analyzing a statically determinate frame like assembling a puzzle. Each piece (frame member) must fit together without conflict (ensuring equilibrium) to complete a picture (the overall force distribution). If one piece is wrong, the entire picture becomes distorted.

1. Check for static determinacy. Using the procedure described in the preceding section, determine whether or not the given frame is statically determinate. If the frame is found to be statically determinate and stable, proceed to step 2. Otherwise, end the analysis at this stage.

The first step is to confirm whether the frame can be analyzed using static equilibrium. A statically determinate frame allows the calculation of internal forces using only external forces and reactions. If the frame is not statically determinate, it needs additional analysis methods—not covered in this procedure.

Imagine trying to balance a seesaw; if it's too heavy on one side (indicating instability), it won't be statically determinate. You can't analyze it properly until it's balanced.

2. Determine the support reactions. Draw a free-body diagram of the entire frame, and determine reactions by applying the equations of equilibrium and any equations of condition that can be written in terms of external reactions only.

The next step involves creating a free-body diagram (FBD) which visually represents the structure and its external forces. By applying the equations of equilibrium (sum of forces in any direction equals zero), you can find the support reactions at fixed supports and roller joints. If the frame is internally unstable, not all reactions may be calculable.

Consider a bridge supported at both ends; understanding how much weight it can hold at its center (how much reaction force each support needs to provide) is like creating an FBD. You sketch it to visualize where forces act and solve for unknowns.

3. Determine member end forces. It is usually convenient to specify the directions of the unknown forces at the ends of the members of the frame by using a common structural (or global) XY coordinate system...

In this step, you establish a coordinate system to analyze forces at each member's ends. By drawing free-body diagrams for each member, you can identify internal forces and moments. Assumptions about force directions can be made for calculation purposes, but adjustments will be necessary to respect Newton's third law when assessing opposing forces.

Think of a tug-of-war: each end of the rope experiences forces that must be understood in terms of direction. If one team pulls left (one end’s force), the opposing team’s pull creates tension (the other end’s reaction), which is akin to calculating member forces in a frame.

4. For each member of the frame, construct the shear, bending moment, and axial force diagrams as follows...

Once the member end forces are known, you can draw shear force, bending moment, and axial force diagrams. These diagrams visually represent how forces change along the length of each member. The shear diagram indicates how forces vary between the member's ends, while the bending moment diagram shows how moments apply along the length.

Think of a swimming pool's water surface in response to waves: at different points in the pool, the height varies (shear) and the force that causes the water to push up or down varies (bending). Placing forces and moments accurately illustrates how materials will react under load.

5. Draw a qualitative deflected shape of the frame. Using the bending moment diagrams constructed in step 4, draw a qualitative deflected shape for each member of the frame...

This final step involves visually approximating how the structure will deform under the loads applied. You'll connect deflected shapes of individual members at the joints, ensuring the structure’s angles and supports are maintained. Although bending and shear are shown, axial deformations are usually negligible and often disregarded.

Visualize how a trampoline deforms when you jump—while it bends in the middle (bending moments), its stretch in length (axial forces) is minimal. Similarly, the deflected shape of a structure illustrates the impact of applied forces.

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