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Calculate reaction forces, moments, and deflection for cantilever beams under various loading conditions. Analyze point loads, distributed loads, and triangular loads with support for multiple materials.
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A cantilever beam is a horizontal structural member that is fixed at one end (the support) and free at the other end. Unlike simply supported beams, the cantilever carries all its load back to the fixed support, creating significant bending moments and deflection at the free end.
Cantilever beams are commonly used in:
Common formulas for cantilever beams with different loading conditions:
| Load Type | Max Moment | Tip Deflection | Slope at Tip |
|---|---|---|---|
| Point Load at End (P) | M = P×L | δ = PL³/(3EI) | θ = PL²/(2EI) |
| Point Load at Distance (P at a) | M = P(L-a) | δ = Pa²(3L-a)/(6EI) | θ = Pa²/(2EI) |
| UDL (w) | M = wL²/2 | δ = wL⁴/(8EI) | θ = wL³/(6EI) |
| Triangular Load (max w) | M = wL²/6 | δ = wL⁴/(30EI) | θ = wL³/(24EI) |
Where: P = Point load (lbs), w = Distributed load (lbs/ft), L = Cantilever length (inches), E = Modulus of elasticity (psi), I = Moment of inertia (in⁴), δ = Deflection (inches), θ = Slope (radians)
Deflection refers to the vertical displacement of the beam under load. While a cantilever can physically deflect more than other beam types, excessive deflection causes problems:
L/240 means a 10-foot cantilever can deflect max 0.5 inches (120÷240)
Building balconies are typical cantilever applications where the structure extends from the building facade. The fixed support is the building structure itself, while the free end can support people and furniture.
Deck overhangs create pleasant outdoor spaces and can overhang pools or water features. These must be carefully designed to manage deflection under snow loads and occupant loads.
Shelves mounted to walls are simple cantilevers. The shelf bracket is the fixed support, and the shelf itself is the cantilever. Load capacity depends heavily on the cantilever length.
Some bridge designs use cantilever arms extending from towers or supports. Famous examples include certain pedestrian bridges and harbor-spanning structures.
A cantilever is fixed at one end and free at the other, with reactions (force and moment) at the fixed end. A simply supported beam rests on two supports and has only vertical reactions. Cantilevers typically deflect more and have maximum moment at the fixed support.
The acceptable deflection depends on the application. For a residential balcony, L/240 is typical, meaning a 10-foot cantilever could deflect about 0.5 inches. For aesthetic or comfort reasons, many designers target tighter limits like L/180 or L/360.
Deflection increases dramatically with length. Because deflection is proportional to L³ or L⁴ depending on load type, doubling the length increases deflection by 8 to 16 times! This is why cantilever length is limited.
For residential decks and balconies, maximum cantilevers are typically 12-16 feet for wood and 20-30 feet for steel, depending on loads. Industrial applications might go longer with engineered designs. Local building codes establish limits.
Counterweights can reduce net deflection by placing load on the fixed end side, but this is rarely practical for permanent structures. The better approach is to increase beam strength (larger section or better material) or reduce the free length.
An over-designed (over-sized) cantilever will be stiff and have minimal deflection, which is generally good. The downside is increased cost and material use. Engineers aim for efficiency by designing to meet deflection and strength requirements without excessive material.
For most residential projects like decks and balconies, local building codes may allow some simple cantilever designs without an engineer. However, for anything non-standard, long spans, heavy loads, or commercial applications, professional design is essential and often required by law.
Deflection is inversely proportional to moment of inertia (I). Increasing I reduces deflection significantly. Using a deeper beam (larger in the bending direction) is more effective than a wider beam—depth affects I as the cube, while width affects it linearly.
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