Beam Deflection & Bending Moment Cantilever Load Solver
Calculate exact cantilever beam deflection, slope, shear force, bending moments, and stress profiles with multi-load physics analysis and real-time structural diagrams.
Beam & Load Configuration
Span: L / 444
Shear V₀: 10.0 kN
Fiber c = 100 mm
Real-Time Structural Visualization
Euler-Bernoulli Beam Theory & Cantilever Deflection Formulas
The structural mechanics of cantilever beams are derived directly from classical Euler-Bernoulli beam theory, which presumes that planar cross-sections perpendicular to the neutral axis remain flat and perpendicular after deformation. When rotational couples or cantilever end moments are transferred into the member, you can evaluate the concentrated rotational input using our torque and lever arm force calculator before computing the resulting flexural profile. For an elastic beam member with a constant cross-section and flexural rigidity EI, the fourth-order differential equation relating transverse deflection v(x) to distributed lateral load q(x) is:
| Loading Scenario | Max Deflection (δmax at tip x = L) | Max Slope (θmax at tip) | Max Moment (Mmax at root x = 0) | Max Shear (Vmax at root) |
|---|---|---|---|---|
| Concentrated Tip Load (P) | (P · L³) / (3 · E · I) | (P · L²) / (2 · E · I) | -P · L | P |
| Intermediate Point Load (P at x = a) | [P · a² · (3L - a)] / (6 · E · I) | (P · a²) / (2 · E · I) | -P · a | P |
| Uniformly Distributed Load (w) | (w · L⁴) / (8 · E · I) | (w · L³) / (6 · E · I) | -(w · L²) / 2 | w · L |
| Applied End Moment (M₀) | (M₀ · L²) / (2 · E · I) | (M₀ · L) / (E · I) | -M₀ | 0 |
| Triangular Load (q₀ at root to 0) | (q₀ · L⁴) / (30 · E · I) | (q₀ · L³) / (24 · E · I) | -(q₀ · L²) / 6 | (q₀ · L) / 2 |
Cross-Section Geometry & Second Moment of Area (I)
A beam's resistance to bending depends fundamentally on how its material is distributed relative to the neutral bending axis. The second moment of area, denoted as I, penalizes distance quadratically: ∫ y² dA. Depth h plays an overwhelmingly dominant role compared to width b:
Neutral axis lies at mid-height (h/2). Doubling the depth increases flexural stiffness by a factor of 8 (2³ = 8).
Symmetrical in all transverse axes. Outer fiber distance c = d/2. Section modulus Z = (π · d³) / 32.
Highly efficient structural shape that maximizes second moment of area while minimizing dead weight and material costs.
Concentrates cross-sectional area in heavy flanges far from the neutral axis, creating maximum flexural efficiency.
Engineering Serviceability Limits (L/δ) & Structural Codes
In structural engineering practice, beams rarely fail by plastic collapse before exceeding maximum allowable deflection limits. Excessive flexure results in cosmetic damage to finishes, cracked drywall, ponding water on cantilevered roofs, and unacceptable human perceptible vibration. Building codes (AISC 360, Eurocode 3, IBC) define serviceability criteria based on span-to-deflection ratios:
| Application Type | Standard Criterion | Typical Code Basis | Engineering Rationale |
|---|---|---|---|
| Cantilever Roof Overhangs | δ ≤ L / 180 | IBC Chapter 16 / ASCE 7 | Prevents roof water ponding and gutter drainage failures under snow/wind loads. |
| Commercial Balconies & Decks | δ ≤ L / 240 | Eurocode EN 1990 Annex A1 | Ensures psychological comfort for occupants and avoids visual perception of sagging. |
| Beams Supporting Plaster / Masonry | δ ≤ L / 360 | AISC Design Guide 3 | Prevents unsightly tension cracking and brittle spalling in rigid interior gypsum finishes. |
| Precision Machinery & Glass Facades | δ ≤ L / 500 to L / 1000 | ISO 10137 Serviceability | Protects architectural curtain-wall glass from edge-loading stresses and optical distortion. |
Step-by-Step Worked Engineering Example
Follow this worked step-by-step problem illustrating how to calculate the deflection, internal moments, bending stress, and factor of safety for a steel cantilever beam supporting a concentrated tip load:
- Step 1: Compute Second Moment of Area (I)
I = (b · h³) / 12 = [0.100 m · (0.200 m)³] / 12 = 0.0008 / 12 = 6.6667 × 10⁻⁵ m⁴ = 66,666,667 mm⁴ - Step 2: Determine Flexural Rigidity (EI)
EI = (200 × 10⁹ N/m²) × (6.6667 × 10⁻⁵ m⁴) = 13,333,400 N·m² - Step 3: Calculate Maximum Tip Deflection (δmax)
δmax = (P · L³) / (3 · E · I) = [15,000 N × (2.5 m)³] / (3 × 13,333,400 N·m²) = 234,375 / 40,000,200 = 0.005859 m = 5.86 mm - Step 4: Compute Maximum Bending Moment at Root (Mmax)
Mmax = -P · L = -15 kN × 2.5 m = -37.5 kN·m = -37,500 N·m - Step 5: Calculate Peak Bending Stress (σmax)
σmax = (|Mmax| · c) / I = (37,500 N·m × 0.100 m) / (6.6667 × 10⁻⁵ m⁴) = 56,250,000 Pa = 56.25 MPa - Step 6: Evaluate Factor of Safety (FoS) & Span Ratio
FoS = σyield / σmax = 250 MPa / 56.25 MPa = 4.44 (Highly Safe, FoS ≥ 2.0)
Deflection Ratio = L / δmax = 2,500 mm / 5.86 mm ≈ L / 427 (Satisfies Strict Plaster Code L / 360)
Frequently Asked Questions (FAQ)
What is the classic Euler-Bernoulli equation for cantilever beam deflection under a point load?
For a cantilever beam of length L rigidly fixed at one end with an elastic modulus E and second moment of area I, the maximum deflection under a concentrated tip load P occurs at the free end and equals δmax = (P · L³) / (3 · E · I). The maximum bending moment occurs at the fixed root and equals Mmax = -P · L.
How does a uniformly distributed load (UDL) differ from a concentrated tip load on a cantilever?
Under a uniformly distributed load w (force per unit length), the maximum tip deflection is δmax = (w · L⁴) / (8 · E · I), and the maximum root moment is Mmax = -(w · L²) / 2. Because distributed loads place their resultant center of mass closer to the fixed support, a UDL produces roughly 37.5% as much tip deflection as an equivalent total concentrated force placed at the tip.
Where do the maximum shear force and bending moment occur in a cantilever beam?
In every standard cantilever configuration subjected to transverse downward loads, the maximum internal shear force and maximum bending moment occur simultaneously at the fixed root clamp (x = 0). At the free tip (x = L), internal bending moment drops to zero unless an external concentrated moment couple is applied.
What is the second moment of area (moment of inertia) and why is depth more critical than width?
The second moment of area (I) measures a cross-section's geometric resistance to bending about its neutral axis. For a rectangular beam of width b and depth h, I = (b · h³) / 12. Because depth h is raised to the third power, doubling the beam height increases its bending stiffness eightfold (2³ = 8), whereas doubling its width only doubles stiffness.
What are standard allowable structural deflection limits in building codes?
Civil engineering building codes (such as AISC, Eurocode 3, and IBC) enforce serviceability deflection limits based on the span-to-depth ratio L/δ. Typical criteria for cantilever projections range between L / 180 for roof overhangs to L / 360 or L / 480 for floors supporting brittle plaster or architectural glass to prevent cracking.
How is structural factor of safety (FoS) computed from bending stress?
The elastic bending stress at extreme fiber is calculated using the flexure formula σ = (M · c) / I, where c is the distance from the neutral axis to the outermost fiber. The Factor of Safety is defined as FoS = Yield Strength / σmax. An FoS greater than 1.5 to 2.0 is generally specified for reliable mechanical and civil designs.
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