How are Shear Force, Bending Moment, and Deflection calculated in Beams?
In mechanics of materials and structural analysis (Euler-Bernoulli beam theory), transverse loads acting on a beam induce two primary internal actions at any cross-section $x$: Shear Force $V(x)$ (tending to shear the section vertically) and Bending Moment $M(x)$ (curving the beam by stretching bottom fibers and compressing top fibers, or vice versa). They are linked by $\frac{dM}{dx} = V(x)$, which is why bending moment reaches its peak wherever shear force crosses zero.
The vertical elastic deflection $\delta(x)$ is obtained by integrating the moment-curvature differential equation twice: $EI \frac{d^2 y}{dx^2} = M(x)$, where $E$ is the material's Young's Modulus (in $\text{GPa}$) and $I$ is the cross-section's second moment of area (in $\text{cm}^4$).
The four classical analytical formulas used in this calculator are:
• Simply Supported + Uniform Load $q$: $V_{\max} = \frac{qL}{2}$, $M_{\max} = \frac{qL^2}{8}$, $\delta_{\max} = \frac{5qL^4}{384EI}$.
• Simply Supported + Center Point Load $P$: $V_{\max} = \frac{P}{2}$, $M_{\max} = \frac{PL}{4}$, $\delta_{\max} = \frac{PL^3}{48EI}$.
• Cantilever + Uniform Load $q$: $V_{\max} = qL$, $M_{\max} = \frac{qL^2}{2}$, $\delta_{\max} = \frac{qL^4}{8EI}$.
• Cantilever + Free-End Point Load $P$: $V_{\max} = P$, $M_{\max} = PL$, $\delta_{\max} = \frac{PL^3}{3EI}$.
Practical sizing example
6-meter Steel Beam: Consider a simply supported structural steel beam ($E = 200\text{ GPa}$) of span $L = 6.0\text{ m}$ carrying a uniform load $q = 15\text{ kN/m}$ with a section inertia $I = 8,000\text{ cm}^4$. The maximum bending moment is $M_{\max} = \frac{15 \times 6^2}{8} = 67.5\text{ kN·m}$ and maximum midspan deflection is $\delta_{\max} = 15.82\text{ mm}$ ($L/379$), well within the $L/300 = 20.0\text{ mm}$ serviceability limit.
Frequently Asked Questions (FAQ)
How is the maximum bending moment calculated for a simply supported beam under uniform load?
For a simply supported beam of span $L$ carrying a uniformly distributed load $q$ (in $\text{kN/m}$), the maximum bending moment occurs at midspan ($x = L/2$) and equals $M_{\max} = \frac{q L^2}{8}$, while support reactions and maximum shear equal $V_{\max} = \frac{q L}{2}$.
What is flexural rigidity EI and how does it affect beam deflection?
Flexural rigidity is the product of the material's Young's Modulus ($E$) and the cross-section's second moment of area ($I$). Maximum deflection ($\delta_{\max}$) is inversely proportional to $EI$: doubling the section's moment of inertia cuts deflection in half.
What is a standard allowable deflection limit in building design?
Structural building codes typically limit live-load service deflection to between $L/300$ and $L/360$ for floor beams (and $L/240$ to $L/250$ for total load), preventing partition cracking and noticeable floor vibration.
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