Two-way Slab Deflection
Computes max deflection of a simply supported rectangular slab under uniform load.
—
Deflection of a two-way simply supported slab
Take a rectangular slab that rests on all four edges and carries a uniformly distributed load. Its largest deflection sits at the centre and follows δ = α·w·L⁴ / (E·h³). Here w is the load per unit area, L the shorter span, E the modulus of elasticity and h the slab thickness. The coefficient α comes from tables (Marcus, Czerny or Bares) and changes with the aspect ratio Ly/Lx and the bending direction. A square slab gives α ≈ 0.00406 when uncracked and ν = 0.2. Run an example with w = 5 kN/m², L = 4 m, E = 25 GPa, h = 0.12 m and you land at δ ≈ 7 mm.
The Brazilian code NBR 6118 draws the serviceability line at L/300 for visual acceptability, and at L/250 for the total deflection once creep is counted. Tighten that to L/350 or L/500 when the slab carries fragile partitions. Just keep your units lined up: w in N/m², E in Pa, L and h in metres. That way δ comes out in metres.
Applications
Think solid reinforced-concrete slabs in homes and offices, prefabricated panels, ribbed slabs modelled as an equivalent solid plate for early sizing, the slabs of stairs and balconies, and SLS checks you run before committing to a detailed FEM model. When Ly/Lx ≤ 2, the two-way formula spares you the overly cautious assumption that the slab bends in just one direction.
FAQ
Why use h³ instead of I? A strip of unit width has a second moment of area of I = h³/12. The constant 12 simply gets folded into the coefficient α, which leaves the formula working with h³ on its own.
Where do I find α? The Czerny, Marcus and Bares tables list α across a range of Ly/Lx ratios and boundary conditions. For concrete (ν = 0.2) and a square slab on four simple supports, α ≈ 0.00406.
Does it account for cracking? No. Once the section cracks, multiply δ by I_g/I_eff using Branson's approach, or fall back on the bi-linear stiffness in NBR 6118 §17.3.2.
What about long-term effects? Take the instantaneous δ and multiply it by (1 + α_f). For slabs with no compression reinforcement, α_f ≈ 1.32 per NBR 6118 §17.3.2.1.2.
Related Tools
Beam Deflection (Center Load)
Compute max deflection δ = PL³/(48EI) of a simply-supported beam with center point load.
Crane Load Moment
Calculate a crane's load moment, M = W·R, from the load weight W (N) and the operating radius R (m, the horizontal distance from the crane's rotation center to the load). The load moment is the product of the load weight and its distance to the crane's rotation axis, and it GOVERNS the TIPPING stability — the most feared and catastrophic crane failure mode. A crane tips when the load moment (tending to overturn it forward, toward the load) exceeds the STABILIZING moment (the crane's own weight and counterweight, acting backward). The genius — and danger — is in the RADIUS: the SAME load generates a much larger moment when far (boom extended) than near (boom retracted). So a crane's capacity is NOT a single number, but a LOAD CHART that drops drastically as the radius grows — a crane lifting 50 tonnes at 5 m radius may lift only 5 tonnes at 30 m. Exceeding the maximum load moment (the 'load curve') is the main cause of crane tipping, so cranes have load moment indicators (LMI) locking operation near the limit. Computing the load moment and comparing it with the allowable moment for that radius is the fundamental safety check in every crane operation. Enter the load weight and the operating radius.
Capacitor RC Charge Time Calculator
Computes the time to charge a capacitor from the time constant tau equals R times C and the target percentage.
Joint Stiffness Constant
Calculate a bolted joint's stiffness constant, C = k_b ÷ (k_b + k_m), from the bolt stiffness k_b (N/mm) and the members' (clamped parts) stiffness k_m (N/mm). The constant C (also called bolt load fraction) is the heart of bolted-joint analysis: it tells what FRACTION of an external tensile load is carried by the BOLT, the rest (1−C) being carried by decompression of the MEMBERS. The value of C reveals the elegant, protective behavior of a preloaded joint: since the (massive) members are usually much stiffer than the (thin) bolt, k_m >> k_b, so C is SMALL (typically 0.2-0.4). This means that when an external load P is applied, only a small portion C·P adds to the bolt tension — most of the load (1−C)·P is absorbed by RELIEF of the compression between the parts. That is why the bolt stress variation is small (good fatigue resistance) and why preload is so beneficial. The smaller C (stiff parts, flexible bolt), the better the bolt protection. The constant C appears in all subsequent formulas: bolt load, residual member force, separation load and fatigue safety factor. Enter the bolt stiffness and the members' stiffness.
Volumetric Organic Loading Rate
Calculate the volumetric organic loading rate (OLR) of a biological reactor, OLR = BOD load ÷ volume, dividing the influent organic load (kg BOD/day) by the reactor's working volume (m³). The result, in kg BOD/(m³·day), shows how much organic matter is applied per unit volume and is central to sizing lagoons, trickling filters, UASB and activated sludge: high loads demand more biomass and oxygen, while low loads indicate an oversized reactor. Enter the daily BOD load and the reactor volume.
Wire Rope Working Load Limit (WLL)
Calculate a wire rope's allowable working load, WLL = MBL ÷ SF, from the minimum breaking load (MBL, N) and the required safety factor. The working load (WLL — Working Load Limit, or SWL — Safe Working Load) is the MAXIMUM load that can be safely applied to a rope, fitting or lifting equipment — the information STAMPED on slings, shackles, hooks and equipment plates, and what the operator uses to decide whether a given load can be lifted. It is obtained by dividing the breaking load (the real strength that would break the component) by the code safety factor (5 for general lifting, more for special situations). Respecting the WLL is an absolute safety rule in lifting and material-handling: exceeding the working load dangerously approaches the component to rupture, eliminating the safety margin covering dynamic effects, wear and uncertainties. The WLL is not the rope's strength — it is the SAFE fraction of it. Every rigging operation starts by checking that the load to lift is below the WLL of each component in the load line (rope, slings, shackles, hook, eye), since the chain is only as strong as its weakest link. Enter the breaking load and the safety factor.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.