Beam Deflection Calculator

The maximum deflection (δ_max) for a simply supported beam with a uniformly distributed load (w) is δ_max = (5 × w × L⁴) / (384 × E × I), where L is beam length, E is modulus of elasticity, and I is moment of inertia. This closed-form solution assumes small deformations, linear elastic behavior, and Euler–Bernoulli beam theory. It is codified in Eurocode 2 (EN 1992-1-1 §7.4.1) for serviceability limit state (SLS) checks and referenced in AISC Design Guide 1 (2nd ed., Eq. 2.1) and ASCE/SEI 7-22 Annex C for preliminary deflection estimation. Note: This formula applies *only* to prismatic beams with pinned–pinned end conditions — fixed or cantilever supports require different coefficients. Always verify boundary assumptions before application.

The calculator provides high theoretical accuracy (>99% for idealized conditions) since it implements the exact Euler–Bernoulli analytical solution. However, real-world accuracy depends on input fidelity: material property variability (e.g., ASTM A6 tolerances allow ±15% yield strength variation), geometric imperfections (out-of-straightness per AISC 360 Table B4.1), and unmodeled effects like shear deformation (negligible for L/h > 10 but contributes ~3–5% error in short beams). For structural steel design per AISC 360-22, this tool is suitable for preliminary SLS screening — but final designs must account for camber, composite action, creep (in concrete), and load combinations per LRFD or ASD methods. Always cross-check with FEA for complex geometries or non-prismatic sections.

For aluminum 6061-T6, use E ≈ 68.9 GPa (68,900,000,000 Pa) — per ASTM B209 and MMPDS-18 §3.2.1. This value is valid at 20°C; E decreases by ~0.025% per °C rise above ambient (MMPDS-18 Fig. 3.2.1.1). At 60°C, E drops to ~67.8 GPa — introducing ~1.6% deflection overestimation if ignored. Humidity has negligible effect on aluminum’s E, unlike timber. Importantly, aluminum’s lower E (vs. steel’s ~200 GPa) means 3× greater deflection for identical geometry and loading — making serviceability often govern over strength. Always confirm temper condition and reference MMPDS or manufacturer datasheets, as E varies by ±2 GPa across heat lots.

Yes — but with critical adjustments. Timber requires using adjusted modulus of elasticity (E′) per ANSI/AWC NDS-2018 §3.3.3: E′ = E × C_M × C_t × C_i × C_r, where C_M < 1.0 for wet service, C_t accounts for temperature, and C_i adjusts for incising. Also, NDS mandates using 5th-percentile E (E_min) for deflection control (§3.3.2), not nominal E. For example, Southern Pine SS has nominal E = 1.6 million psi, but E_min = 1.2 million psi — a 25% reduction increasing predicted deflection. Additionally, time-dependent effects (creep) require deflection amplification: total deflection = instantaneous × (1 + λ), where λ = 1.0–2.0 per NDS Table 3.3.4. Never substitute steel E values for wood.

Because deflection and strength are governed by separate limit states: ACI 318-19 Chapter 24 controls serviceability (deflection), while Chapter 22 governs strength. Allowable deflections depend on member type — e.g., flat roofs not supporting nonstructural elements: L/180; floors supporting partitions: L/480 (Table 24.2.2). Your calculation may be technically correct, but exceed these thresholds due to insufficient stiffness (low I or E), excessive span (L), or unaccounted long-term effects (creep/shrinkage adds ~2–3× instantaneous deflection in concrete). ACI permits deflection calculations using effective moment of inertia (I_e) per Eq. 24.2.3.5 — not gross I. If your inputs used gross I, results are nonconservative. Always apply I_e and verify against Table 24.2.2 limits.

No — this tool computes only static, quasi-static deflection under uniform dead/live loads per Euler–Bernoulli theory. It does not include dynamic amplification, resonance, or human comfort criteria. For crane girders, ISO 20816-5 specifies velocity limits (<4 mm/s RMS for office environments); for footbridges, AASHTO LRFD §3.6.2.5 requires natural frequency > 3 Hz to avoid pedestrian-induced resonance. Impact factors (e.g., 1.25–2.0 per ASCE/SEI 7-22 Table 4-1 for moving equipment) must be applied *externally* to the uniform load input. Vibration analysis requires modal analysis (not static deflection), so always supplement with FEA or specialized tools like SAP2000 or STAAD.Pro for dynamic serviceability assessment.

Validate by replicating the analytical solution: for w = 1000 N/m, L = 5 m, E = 210 GPa, I = 1e−5 m⁴ → δ_max = (5 × 1000 × 5⁴) / (384 × 210e9 × 1e−5) = 0.00386 m. Compare to hand calcs (ensure unit consistency: Pa = N/m², I in m⁴). Then run a simple FEA model in software like SkyCiv or FreeFEM: use 10+ beam elements, pinned supports, and distributed load — expect <0.5% deviation from analytical result if mesh is refined. Discrepancies >2% indicate modeling errors (e.g., incorrect support restraints, inconsistent units, or missing axial/bending coupling). Per ASME V&V 10, such verification satisfies Level 1 validation for static linear analysis.