Beam Deflections

The perpendicular distance from the original neutral axis to the deformed neutral axis is the deflection. Calculating this vertical displacement is essential for Serviceability Limit State (SLS) design to ensure structures remain functional and aesthetically acceptable under everyday loads.

The deflection of a beam is related to the bending moment by the differential equation of the elastic curve:

Boundary conditions are known physical constraints at the supports that allow us to solve for the integration constants.

• Pinned or Roller Support: Deflection is zero ($y = 0$). Slope ($\theta$) is generally non-zero.
• Fixed Support: Deflection is zero ($y = 0$) AND Slope is zero ($\theta = 0$ or $y' = 0$).
• Free End: Deflection and slope are unconstrained. Moment and shear are zero only when no end couple or end force is applied; otherwise the boundary values follow those applied actions.

Locating Maximum Deflection

An interior deflection extremum occurs where the slope of the elastic curve is zero ($y' = \theta = 0$). Also check interval endpoints: a cantilever's maximum deflection can occur at its free end with nonzero slope.

• For a symmetrically loaded simply supported beam, this is always at the midspan ($x = L/2$).
• For an asymmetrically loaded beam (e.g., a point load off-center), the maximum deflection is not at the load, nor is it at the midspan. You must set the slope equation to zero ($y' = 0$) and solve for $x$ to find the location, then plug that $x$ into the deflection equation.

For linear elastic beams, the deflection due to multiple loads is the sum of the deflections caused by each load individually. This allows us to use standard formulas for simple cases (e.g., point load, uniform load) and add them up. This method is incredibly powerful because it turns complex loading problems into the addition of simple, standard cases, circumventing the need for complex calculus.

Common Formulas (Max Deflection)

• Simply Supported (Uniform Load $w$): $\delta_{max} = \frac{5wL^4}{384EI}$ (at center).
• Simply Supported (Point Load $P$ at center): $\delta_{max} = \frac{PL^3}{48EI}$.
• Cantilever (Point Load $P$ at end): $\delta_{max} = \frac{PL^3}{3EI}$.
• Cantilever (Uniform Load $w$): $\delta_{max} = \frac{wL^4}{8EI}$.

The Area-Moment method is a semi-graphical technique that relates the slope and deflection of a beam directly to the properties of the $M/EI$ diagram.

• First Area-Moment Theorem: The change in slope between any two points on the elastic curve equals the area of the $M/EI$ diagram between those two points. $\theta_{B/A} = \text{Area}_{M/EI}$.
• Second Area-Moment Theorem: The vertical deviation (vertical distance) of point B from the tangent line drawn at point A is equal to the "moment of area" of the $M/EI$ diagram between A and B, taken about point B. $t_{B/A} = (\text{Area}_{M/EI}) \cdot \bar{x}_B$.

The conjugate beam method is an ingenious technique that maps the problem of finding slopes and deflections (a geometric problem) entirely into a problem of finding shear and moments (a statics problem) on a fictitious "conjugate" beam.

The Fundamental Analogy:

1. The "Load" ($w$) on the conjugate beam is defined exactly as the $M/EI$ diagram of the real beam.
2. The internal "Shear" ($V$) in the conjugate beam at any point equals the Slope ($\theta$) of the real beam at that point.
3. The internal "Bending Moment" ($M$) in the conjugate beam at any point equals the Deflection ($y$) of the real beam at that point.

Because deflection and slope correspond directly to moment and shear, the boundary conditions of the real beam must be mathematically transformed into new boundary conditions for the conjugate beam.

The Virtual Work method, or Unit Load method, is an energy-based technique to calculate the deflection or slope at a specific point on a structure. It is highly versatile and can be applied to beams, trusses, and frames. To find the deflection $\delta$ at a point, a fictitious "virtual" unit load ($1$) is placed at that point in the direction of the desired deflection.

The external virtual work done by the unit load is equated to the internal virtual strain energy stored in the body due to the real loads.

When a beam has multiple different loads (point loads, partial uniform loads), writing a single moment equation $M(x)$ across the entire length is impossible with standard algebra. Macaulay's Method uses singularity functions (indicated by angled brackets $\langle x-a \rangle^n$) to write a single, continuous equation for the entire beam, making integration much simpler.

• $\langle x-a \rangle^n = (x-a)^n$ if $x \ge a$
• $\langle x-a \rangle^n = 0$ if $x < a$
• Integration rule: $\int \langle x-a \rangle^n dx = \frac{\langle x-a \rangle^{n+1}}{n+1}$ (for $n \ge 0$)

Implementation

When using Macaulay's method, you do not expand the bracket $\langle x-a \rangle$. You treat it as a single variable during integration. If a uniform load starts at $a$ and ends at $b$, it is mathematically modeled as a continuous load starting at $a$, with a counteracting negative continuous load starting at $b$.