Beam Deflection Formulas

Why This Matters

Beam deflection analysis sits at the heart of structural engineering—it's where statics, material properties, and geometry converge into a single predictive framework. You're being tested on your ability to recognize how support conditions constrain movement, how load distribution affects curvature, and why the same beam under the same total load can deflect dramatically different amounts depending on where and how that load is applied. These formulas aren't just equations to memorize; they reveal fundamental relationships between stiffness, boundary conditions, and structural behavior.

When you encounter deflection problems on exams, the real challenge isn't plugging numbers into formulas—it's selecting the correct formula based on support type and loading pattern. Every formula in this guide encodes specific assumptions about how a beam is restrained and loaded. Don't just memorize the expressions; understand what each variable controls and why cantilevers deflect more than fixed beams under identical loads. That conceptual understanding is what separates students who struggle from those who excel.

---

Simply Supported Beams

Simply supported beams represent the baseline case in deflection analysis—free rotation at both ends with vertical restraint only. Because the supports cannot resist moments, these beams experience maximum curvature and deflection compared to fixed-end configurations under the same loading.

Simply Supported Beam with Point Load at Center

• Maximum deflection $\delta = \frac{PL^3}{48EI}$ occurs at midspan—the symmetry of the loading guarantees this location
• Equal reactions of $\frac{P}{2}$ develop at each support, making this the simplest case for equilibrium analysis
• Cubic relationship with span length means doubling $L$ increases deflection by a factor of 8—span is the dominant variable

Simply Supported Beam with Uniformly Distributed Load

• Maximum deflection $\delta = \frac{5wL^4}{384EI}$ occurs at midspan, with the coefficient $\frac{5}{384}$ reflecting the distributed nature of the load
• Fourth-power span dependence makes UDL cases extremely sensitive to beam length—even small span increases dramatically affect deflection
• Equal support reactions of $\frac{wL}{2}$ each, despite the "distributed" nature of loading, because symmetry is preserved

Simply Supported Beam with Point Load at Any Location

• Asymmetric loading produces maximum deflection at a location that shifts toward the load—not necessarily at midspan
• Superposition principle allows you to break complex loading into simpler cases and sum the deflections algebraically
• Unequal reactions develop at supports, calculated using moment equilibrium about each support in turn

---

Cantilever Beams

Cantilever beams feature full fixity at one end and complete freedom at the other. This boundary condition produces the largest deflections of any standard beam type because there's no support to restrain the free end's movement.

Cantilever Beam with Point Load at Free End

• Maximum deflection $\delta = \frac{PL^3}{3EI}$ at the free end—note the coefficient $\frac{1}{3}$ is much larger than the simply supported case
• Fixed support reactions include both a vertical force $P$ and a moment $PL$ that resist the applied load
• Linear moment distribution from zero at the free end to maximum at the fixed support creates a characteristic deflected shape

Cantilever Beam with Uniformly Distributed Load

• Maximum deflection $\delta = \frac{wL^4}{8EI}$ occurs at the free end, with the $\frac{1}{8}$ coefficient reflecting load distribution effects
• Fixed support moment equals $\frac{wL^2}{2}$—this reaction moment is critical for connection design
• Parabolic moment diagram produces a deflected shape that curves more gradually than the point load case

---

Fixed-End Beams

Fixed-end (or built-in) beams have moment resistance at both supports, which dramatically reduces deflection compared to simply supported cases. The additional restraint increases stiffness but also introduces fixed-end moments that must be considered in design.

Fixed-End Beam with Point Load at Center

• Maximum deflection $\delta = \frac{PL^3}{192EI}$ at midspan—exactly one-quarter the simply supported deflection due to end fixity
• Fixed-end moments of $\frac{PL}{8}$ develop at each support, redistributing the bending stress along the beam
• Inflection points appear between the supports and midspan where the moment changes sign—a key characteristic of fixed beams

Fixed-End Beam with Uniformly Distributed Load

• Maximum deflection $\delta = \frac{wL^4}{384EI}$ at midspan—five times smaller than the simply supported case with identical loading
• Fixed-end moments of $\frac{wL^2}{12}$ at each support provide the additional stiffness that reduces deflection
• Same span dependence ($L^4$) as simply supported UDL, but the coefficient drops from $\frac{5}{384}$ to $\frac{1}{384}$

---

Hybrid Support Conditions

Propped cantilevers and other hybrid configurations combine elements of different support types, creating intermediate stiffness behavior. These cases often require compatibility equations to solve for redundant reactions.

Propped Cantilever with Point Load at Free End

• Reduced deflection compared to a pure cantilever because the prop provides vertical restraint at the otherwise-free end
• Statically indeterminate structure requires compatibility conditions—the prop reaction is found by setting deflection at that point to zero
• Combined formula reflects superposition: cantilever deflection minus the effect of the prop reaction, yielding a net deflection less than $\frac{PL^3}{3EI}$

---

Analytical Methods

Beyond direct formulas, two powerful methods handle complex cases where standard formulas don't apply—varying cross-sections, multiple loads, or unusual support conditions.

Moment-Area Method

• Graphical approach uses the area under the $\frac{M}{EI}$ diagram to find slopes and deflections at any point
• First theorem states that the change in slope between two points equals the area under the $\frac{M}{EI}$ diagram between those points
• Second theorem gives the deflection of one point relative to the tangent at another—essential for non-prismatic beams

Conjugate Beam Method

• Fictitious beam loaded with the $\frac{M}{EI}$ diagram as a distributed load, where shear represents slope and moment represents deflection
• Support transformation rules: fixed becomes free, free becomes fixed, internal hinge becomes internal support—memorize these conversions
• Powerful for complex cases including beams with varying $EI$, internal hinges, or overhangs where direct formulas fail

---

Quick Reference Table

---

Self-Check Questions

1. A simply supported beam and a fixed-end beam have identical spans, loads, and cross-sections. By what factor does fixing both ends reduce the maximum deflection for a center point load?

2. Which two beam configurations in this guide share the same $L^4$ span dependence but differ in their coefficients due to support conditions? Explain why the coefficients differ.

3. Compare and contrast the cantilever with point load at the free end versus the simply supported beam with center point load. Which deflects more under the same load $P$, and why does the support condition cause this difference?

4. An FRQ presents a beam with a varying moment of inertia along its length. Which analytical method would you choose, and what would you load your conjugate beam with?

5. A propped cantilever deflects less than a pure cantilever under identical loading. Explain the structural principle that accounts for this reduction and identify what type of analysis (determinate or indeterminate) the propped case requires.