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Calculate maximum deflection, bending stress, and reaction forces for structural beams under point loads and uniformly distributed loads. Supports simply supported and cantilever configurations.
Beam design must satisfy two criteria: strength (stress within allowable limits) and serviceability (deflection within code limits, typically L/360 for floors). Both must pass independently — a beam can be strong enough but deflect excessively, or have acceptable deflection but overstressed cross-section.
| Beam/Property | Formula/Value | Note |
|---|---|---|
| Simply Supported + Center Load | PL³/48EI | Max deflection at center |
| Simply Supported + UDL | 5wL⁴/384EI | Uniform distributed load |
| Cantilever + End Load | PL³/3EI | 16x more than SS beam |
| Cantilever + UDL | wL⁴/8EI | Max at free end |
| Steel E | 200 GPa | All structural steel grades |
| Allowable Deflection | L/360 (live) | Floor beam standard |
Every structural beam must be checked for both strength and deflection. Strength check: maximum bending stress sigma = Mc/I must be less than allowable stress (yield strength / safety factor). For A36 steel: allowable = 250/1.67 = 150 MPa. Serviceability check: maximum deflection must be less than L/360 for live load on floor beams (L = span length).
| Section | Moment of Inertia I | c | Use |
|---|---|---|---|
| Rectangle (b×h) | bh³/12 | h/2 | Timber, concrete |
| Wide Flange (W-shape) | Tabulated AISC | d/2 | Structural steel |
| Circular solid (d) | πd⁴/64 | d/2 | Shafts, round bars |
| Hollow rect (B×H, b×h) | (BH³-bh³)/12 | H/2 | HSS tubing |
| Material | E (Elastic Modulus) | Strength | Application |
|---|---|---|---|
| Structural Steel (A36) | 200 GPa | 250 MPa yield | Most common structural |
| Aluminum 6061-T6 | 69 GPa | 276 MPa yield | Lightweight structures |
| Douglas Fir Timber | 13.1 GPa | 30-40 MPa | Residential framing |
| Reinforced Concrete | 25-30 GPa | Compression only | Needs rebar for tension |
ASD (Allowable Stress Design): allowable stress = yield / SF. AISC safety factor for bending: SF = 1.67. So for A36 steel: allowable bending stress = 250/1.67 = 150 MPa. LRFD (Load and Resistance Factor Design): factored loads (1.2D + 1.6L) must be less than factored resistance (phi × nominal). LRFD is the modern preferred method in US structural engineering practice.
Engineering design always includes safety factors to account for material variability, load uncertainty, construction tolerances, and progressive deterioration. Typical safety factors: structural steel bending 1.67 (ASD), concrete compression 1.5-2.0, bolts in tension 2.0-3.0, pressure vessels 3.5-4.0. Always verify that your design complies with applicable local codes, national standards, and industry regulations before construction or installation.
Engineering is both a science and a craft. Formulas provide exact answers when inputs are exact — but real engineering involves estimating loads, assuming material properties, and accounting for construction tolerances. The best engineers know which simplifications are conservative (on the safe side) and which might be unconservative, and they apply appropriate margins accordingly.
Always perform a sanity check on engineering calculations. Does the answer make physical sense? Is it in the right order of magnitude? A beam deflection of 50mm on a 3m span should immediately raise concern — that is L/60, far exceeding the L/360 limit. A pump power of 500 kW for a small building system similarly warrants rechecking. Dimensional analysis (checking units cancel correctly) catches the majority of formula errors.