Beam Deflection & Bending Stress Calculator - Cantilever & Supported: Free Beam Deflection & Bending Stress Calculator tool: calculate beam deflection, bending moment, shear force, and maximum flexural stress using.
1. Euler-Bernoulli Classical Beam Theory & Governing Differential Equations
Structural beams are prismatic engineering members subjected to transverse lateral loads that produce internal bending moments and shear stresses. Classical beam theory, pioneered by Leonhard Euler and Daniel Bernoulli in the mid-18th century, models elastic deflection by establishing the fundamental kinematic postulate: plane sections perpendicular to the longitudinal axis remain planar and perpendicular to the neutral surface after deformation.
This kinematic assumption implies that transverse shear deformations ($gamma_{xy} = 0$) are negligibly small relative to flexural bending strains. Under these conditions, the transverse deflection curve $v(x)$ is governed by the fourth-order ordinary differential equation:
V(x) = frac{d M(x)}{d x} = -E I frac{d^3 v(x)}{d x^3} quad (text{Internal Shear Force})
M(x) = E I frac{d^2 v(x)}{d x^2} quad (text{Internal Bending Moment})
theta(x) = frac{d v(x)}{d x} quad (text{Beam Rotation / Slope})
Where:
- $E$ is Young's Modulus of Elasticity ($text{N/m}^2$ or $text{Pa}$), representing material stiffness.
- $I$ is the Second Moment of Area (Moment of Inertia) of the cross-section about the neutral bending axis ($text{m}^4$).
- $E I$ represents the flexural rigidity of the beam member ($text{N}cdottext{m}^2$).
- $q(x)$ is the distributed transverse loading function per unit length ($text{N/m}$).
2. Analytical Solutions for Canonical Boundary Conditions
Integrating the Euler-Bernoulli equation and applying kinematic and static boundary conditions yields closed-form solutions for beam deflection, slope, and maximum internal forces:
| Boundary & Loading Type | Max Deflection ($delta_{max}$) | Location of $delta_{max}$ | Peak Moment ($M_{max}$) | Peak Shear ($V_{max}$) |
|---|---|---|---|---|
| Simply Supported – Point Load $P$ at Midspan | $frac{P L^3}{48 E I}$ | $x = L/2$ (Center) | $frac{P L}{4}$ | $frac{P}{2}$ |
| Simply Supported – Uniform Load $w$ (UDL) | $frac{5 w L^4}{384 E I}$ | $x = L/2$ (Center) | $frac{w L^2}{8}$ | $frac{w L}{2}$ |
| Cantilever – Point Load $P$ at Free Tip | $frac{P L^3}{3 E I}$ | $x = L$ (Tip) | $P L$ (Fixed Base) | $P$ |
| Cantilever – Uniform Load $w$ (UDL) | $frac{w L^4}{8 E I}$ | $x = L$ (Tip) | $frac{w L^2}{2}$ (Fixed Base) | $w L$ |
| Fixed-Fixed – Point Load $P$ at Midspan | $frac{P L^3}{192 E I}$ | $x = L/2$ (Center) | $frac{P L}{8}$ (Supports) | $frac{P}{2}$ |
| Fixed-Fixed – Uniform Load $w$ (UDL) | $frac{w L^4}{384 E I}$ | $x = L/2$ (Center) | $frac{w L^2}{12}$ (Supports) | $frac{w L}{2}$ |
Examining the coefficient denominators demonstrates the profound impact of boundary fixity: clamped (fixed) ends reduce maximum deflection under midspan point loading by 75% ($frac{1}{192}$ vs $frac{1}{48}$) and under UDL by 80% ($frac{1}{384}$ vs $frac{5}{384} approx frac{1}{76.8}$) compared to pinned supports.
3. Flexural Stress, Section Modulus & Serviceability Design Criteria
Once internal bending moments are determined, the normal flexural stress distribution $sigma_x(y)$ across the cross-sectional depth is given by Navier's Bending Equation:
Where $y$ is the distance from the neutral axis, $c$ is the extreme fiber distance ($c = h/2$ for symmetric shapes), and $Z = I/c$ is the Elastic Section Modulus.
Geometric Section Properties:
- Solid Rectangular Section ($b times h$): $$I = frac{b h^3}{12}, quad Z = frac{b h^2}{6}$$
- Solid Circular Section ($text{Diameter } D$): $$I = frac{pi D^4}{64}, quad Z = frac{pi D^3}{32}$$
- Hollow Rectangular Structural Section (HSS, outer $b times h$, inner $b_i times h_i$): $$I = frac{b h^3 - b_i h_i^3}{12}, quad Z = frac{2 I}{h}$$
Code Serviceability vs Ultimate Strength:
In structural building practice (AISC 360, Eurocode 3, AS 4100), beams must satisfy two distinct limit states:
- Ultimate Strength Limit State (ULS): The maximum flexural stress $sigma_{max}$ amplified by load factors must not exceed the design yield strength: $sigma_{max} le phi F_y$ (or $M_u le phi_b M_n$), ensuring an adequate Safety Factor ($text{FoS} = S_y / sigma_{max} ge 1.67$).
- Serviceability Limit State (SLS): Deflections under unfactored live and dead loads must remain within acceptable bounds to prevent aesthetic cracking of architectural finishes (plaster, drywall), vibration discomfort, or ponding water accumulation:
- $L/360$: Commercial ceilings supporting brittle plaster/gypsum drywall.
- $L/240$: General roof and floor beams with flexible membrane ceilings.
- $L/180$: Agricultural, industrial purlins, or temporary falsework.
4. Worked Engineering Calculation: Simply Supported Steel Girder
Consider a primary structural floor beam spanning $L = 6.0text{ meters}$ between two columns, supporting a concentrated central equipment load of $P = 25.0text{ kN}$ ($25,000text{ N}$).
- Cross Section: Solid rectangular steel billet, width $b = 150text{ mm}$ ($0.15text{ m}$), depth $h = 300text{ mm}$ ($0.30text{ m}$).
- Material: ASTM A36 Structural Steel, $E = 205text{ GPa} = 205 times 10^9text{ N/m}^2$, Yield Strength $F_y = 250text{ MPa}$.
Step A: Geometric Section Properties
Z = frac{b h^2}{6} = frac{0.15 times (0.30)^2}{6} = frac{0.15 times 0.090}{6} = 2.25 times 10^{-3}text{ m}^3 = 2,250text{ cm}^3
Step B: Flexural Rigidity ($EI$)
Step C: Internal Peak Shear and Bending Moment
M_{max} = frac{P L}{4} = frac{25,000text{ N} times 6.0text{ m}}{4} = 37,500text{ N}cdottext{m} = 37.50text{ kN}cdottext{m}
Step D: Maximum Extreme Fiber Bending Stress
text{Factor of Safety (FoS)} = frac{F_y}{sigma_{max}} = frac{250text{ MPa}}{16.67text{ MPa}} = 15.0
The flexural stress is well within the yield capacity of structural steel, providing ample margin against plastic hinge formation.
Step E: Midspan Elastic Deflection & Code Verification
text{Span-to-Deflection Ratio} = frac{L}{delta_{max}} = frac{6,000text{ mm}}{1.626text{ mm}} approx frac{L}{3690}
The allowable deflection threshold under strict architectural criteria ($L/360$) is:
Because $delta_{max} (1.63text{ mm}) ll 16.67text{ mm}$, the beam satisfies both strength and serviceability criteria by a wide margin.
5. Shear Force & Bending Moment Sign Conventions
In engineering mechanics, standard structural sign conventions ensure unambiguous interpretation across analytical models and finite element analysis (FEA):
- Bending Moment Sign: A positive bending moment induces compression in top fibers and tension in bottom fibers ("sagging" smile curvature). A negative bending moment induces tension in top fibers and compression in bottom fibers ("hogging" frown curvature, typical over interior continuous piers or fixed cantilever root attachments).
- Shear Force Sign: Positive shear rotates an infinitesimal beam element in a clockwise direction (upward shear on left face, downward on right face).
- Curvature Relation: At locations where internal shear passes through zero ($V(x) = 0$), the local bending moment reaches an extreme value ($frac{d M}{d x} = 0$), identifying peak critical cross-sections requiring heavy longitudinal tensile reinforcement in reinforced concrete or thick flange plates in structural steel framing.