• Simply Supported: Pinned at one end, Roller at the other. Statically Determinate.
• Cantilever: Fixed at one end, Free at the other. Statically Determinate.
• Overhanging: Supported at two points with one or both ends extending beyond the supports. Statically Determinate.
• Continuous: Supported at more than two points. Statically Indeterminate.
• Propped Cantilever: Fixed at one end, Roller at the other. Statically Indeterminate.

Standard Sign Convention:

• Positive Shear ($V$): Tends to rotate the cut segment clockwise. (e.g., pointing DOWN on the right face of a left-hand segment).
• Positive Moment ($M$): Tends to bend the segment concave upward, creating a "smiling" shape. This puts the top fibers in compression and bottom fibers in tension.

The relationship between Load ($w$), Shear ($V$), and Moment ($M$) is defined by differential calculus:

1. Slope of Shear Diagram = Load Intensity

2. Slope of Moment Diagram = Shear Force

This means the shear value at any point is the slope of the moment curve at that point.

The Area Method provides a graphical and rapid way to construct shear and moment diagrams by calculating the areas under the respective curves:

• Change in Shear: $V_2 - V_1 = \int_{x_1}^{x_2} w \, dx$ (The change in shear between two points equals the area under the load diagram between those points).
• Change in Moment: $M_2 - M_1 = \int_{x_1}^{x_2} V \, dx$ (The change in moment between two points equals the area under the shear diagram between those points).

Curve Degrees The relationship dictates that each successive diagram increases in polynomial degree:

• If load $w(x)$ is degree $n$.
• Then Shear $V(x)$ is degree $n+1$.
• Then Moment $M(x)$ is degree $n+2$.

For example: A Uniform Load (degree 0, flat line) $\rightarrow$ Linear Shear (degree 1, sloped line) $\rightarrow$ Parabolic Moment (degree 2, curve).

Because the shear force is the mathematical derivative of the bending moment ($\frac{dM}{dx} = V(x)$), the bending moment curve will reach a local maximum or minimum value exactly at the points where the shear diagram crosses the zero axis ($V = 0$).

In structural design, finding these "points of zero shear" is the most critical step, as the beam must be sized to resist the maximum moment occurring at these locations.

At this point, the beam's curvature changes from concave upward (positive moment, sagging) to concave downward (negative moment, hogging).

Significance in Design:

• At this point, flexural stress is zero.
• In reinforced concrete, this point dictates where top longitudinal reinforcement (for negative moment) can be terminated or shifted to the bottom (for positive moment).
• It is a common location to place splices or hinges in continuous structures, as they will only need to transmit shear forces, not bending moments.

While static loads are fixed in position, moving loads (like trucks on a bridge or crane hoists on a runway beam) change position over time. To design structures carrying moving loads, engineers must find the absolute maximum internal forces (shear and moment) that occur as the load system traverses the entire span.

For a system of moving point loads, the maximum shear usually occurs at the supports when the heaviest load is positioned directly over or adjacent to the support.

For a single point load, the maximum moment clearly occurs when the load is at the midspan of a simply supported beam. However, for a train of wheel loads (like an AASHTO HL-93 design truck passing over a bridge girder), the point of Absolute Maximum Bending Moment must be calculated using a specific positioning rule.

Rule: The absolute maximum bending moment occurs under a specific wheel load when the centerline of the beam span perfectly bisects the distance between the Resultant of all moving loads and that specific Wheel Load.

Typically, the absolute maximum moment will occur under the heaviest wheel load that is located nearest to the resultant of the load group.

A singularity function is defined by pointed brackets. It effectively "turns on" only when $x > a$.

Definition:

• If $x < a$: $\langle x - a \rangle^n = 0$
• If $x \ge a$: $\langle x - a \rangle^n = (x - a)^n$