Beam deflection

This calculator uses static, linear Euler–Bernoulli beam theory for a straight, prismatic beam with constant flexural rigidity EI. Deflections are positive downward and x is measured from the left support. The slope dy/dx approximates the section rotation in radians only for small slopes. The displayed deflection curve comes from the selected beam solution; its vertical scale is changed relative to the length scale and the factor is shown explicitly.

A cantilever is fixed on the left and free on the right. A simply supported beam has a pin on the left and a roller on the right. Point loads act at the free end, at midspan or at the entered offset 0 ≤ a ≤ L. The uniform line load acts over the whole span. Only fields used by the selected load and section are checked. Loads are nonnegative, including zero. A point load applied directly at a support transfers there without internal bending in this ideal model. A flat beam has no unique maximum-deflection location.

All inputs use SI units: lengths in m, E in Pa, P in N, w in N/m and I in m⁴. Scientific notation and a decimal point or comma are accepted. For example, 200 GPa = 200e9 Pa. Rectangular I = bh³/12 uses depth h in the bending direction and width b perpendicular to it. For a solid circle I = πd⁴/64; for a circular tube I = π(dₒ⁴ − dᵢ⁴)/64, with 0 ≤ dᵢ < dₒ. A directly entered I must be the second moment of area about the relevant bending axis, not a mass moment of inertia.

Beam curvature is related to bending moment by EI. The solution enforces zero deflection and slope at a clamped end, or zero deflection at the two simple supports. For an off-centre point load, the largest deflection is generally not at the load. Its location is found from the zero-slope condition. The result also gives the largest absolute slope, bending moment and internal shear. Displayed values are rounded; the calculations use unrounded values.

The model omits shear deformation, large deflections, yielding, buckling, dynamics, support compliance and load redistribution. Self-weight is included only if you add it to w. There is no strength, stability, serviceability or building-code approval, and no universal deflection limit is applied. Calculate is explicit; editing removes the previous result. Reset clears the result while keeping the inputs. The last valid configuration is retained across navigation.

Reference: MIT OpenCourseWare, 1.050 Solid Mechanics, Chapter 10: Deflections due to Bending.