Beam deflection describes how a structural beam bends under applied loads, and it is one of the most fundamental calculations in civil and mechanical engineering. This simulator uses Euler-Bernoulli beam theory, which assumes that plane cross-sections remain plane during bending, giving closed-form formulas for deflection, bending moment, and shear force under common loading conditions. The key relationship is EI d²y/dx² = M(x), where E is Young's modulus, I is the second moment of area, and M(x) is the bending moment distribution.
You can select from three boundary conditions — simply supported (pinned at both ends), cantilever (fixed at one end, free at the other), and fixed-fixed (both ends fully restrained) — and apply either a concentrated point load or a uniformly distributed load. The canvas plots the deflected shape, the bending moment diagram, and the shear force diagram in real time as you adjust span length, load magnitude, and material properties.
What is the formula for maximum deflection of a simply supported beam with a central point load?
For a simply supported beam of length L, Young's modulus E, second moment of area I, and central load P, the maximum deflection at midspan is δ = PL³/(48EI). If the load is off-centre at position a from the left support (b = L − a), the maximum deflection is approximately δ = Pa²b²/(3EIL), which reduces to the central formula when a = b = L/2.
How does fixing both ends of a beam reduce deflection?
A fixed-fixed beam is four times stiffer than a simply supported beam under a central point load: δ = PL³/(192EI) versus PL³/(48EI). The end fixity introduces restraining moments at the supports that partially counteract the applied load, halving the peak bending moment and dramatically reducing midspan deflection. This is why bridge abutments and floor beams are often designed as continuous rather than simply supported.
What is the second moment of area and how do I choose a good cross-section?
The second moment of area I (also called the moment of inertia) measures how far material is distributed from the neutral axis. For a solid rectangle b × h, I = bh³/12. An I-beam achieves a high I value for its weight by concentrating material in the flanges far from the neutral axis, which is why steel I-beams with I ≈ 8000–30000 cm⁴ are standard in construction. Increasing h by a factor of 2 increases I by a factor of 8.
Shear force V at a cross-section is the net transverse force on one side of the cut, while bending moment M is the net torque. They are related by V = dM/dx: the shear force equals the slope of the bending moment diagram. For a simply supported beam with a central point load, the shear force is constant at P/2 on each side and jumps discontinuously at the load point, while the bending moment peaks at PL/4 at midspan.
The simulator includes presets for wood (E ≈ 12 GPa, I = 3000 cm⁴, typical timber joist), steel I-beam (E = 200 GPa, I = 8000 cm⁴, a standard 254 × 146 UB section), reinforced concrete (E ≈ 30 GPa, I = 25000 cm⁴), and aluminium alloy (E = 70 GPa, I = 4000 cm⁴). You can also enter any custom E and I combination to model composite or exotic materials.
A cantilever of the same span and cross-section deflects 16 times more than a simply supported beam under the same central load: δcant = PL³/(3EI) versus PL³/(48EI). This is because only one end is fixed, so the beam cannot share the reaction between two supports. Cantilevers are used where columns cannot be placed at one end, such as balconies, diving boards, and aircraft wings.
A UDL (w kN/m) represents a load spread evenly along the beam, such as the self-weight of a floor slab, snow load on a roof, or wind pressure on a wall. For a simply supported beam under UDL, maximum deflection is δ = 5wL⁴/(384EI) — occurring at midspan — and maximum bending moment is M = wL²/8 at midspan. The fourth-power dependence on span means doubling the span increases deflection 16-fold.
Building codes typically limit live-load deflection to span/360 (for floors) or span/180 (for roofs), preventing cracking in plaster ceilings and discomfort to occupants. Engineers achieve this by increasing the second moment of area (deeper beams), using higher modulus materials, shortening the effective span with intermediate columns, or designing beams as continuous rather than simply supported, which reduces deflection by roughly 80%.
EI is called the flexural rigidity of the beam. It combines the material stiffness (E) with the geometric efficiency of the cross-section (I). A steel I-beam with E = 200 GPa and I = 8000 cm⁴ has EI = 1600 MN·m². Increasing EI is the primary design lever for reducing deflection without changing the span or load. The stiffness EI/L³ displayed in the panel gives an intuitive sense of how resistant the beam is to a central point load.
The neutral axis is the longitudinal line within the cross-section at which bending stress is zero. Above it (on the compression side) the material shortens; below it (tension side) it elongates. Stress varies linearly: σ = My/I, where y is distance from the neutral axis and M is the local bending moment. The extreme fibres carry the highest stress, which is why cracks in concrete beams first appear at the tensile face where concrete is weakest.
Euler-Bernoulli theory is highly accurate for slender beams where span-to-depth ratio exceeds about 10. For deep beams (span/depth < 5), shear deformation becomes significant and Timoshenko beam theory should be used, which adds a shear correction factor κ. Thin-walled and curved beams require further corrections. In all cases, Euler-Bernoulli over-predicts stiffness — i.e., under-predicts deflection — for stocky members.