Straight Beam — Shear, Moment, Slope & Deflection

Single-span elastic beam — 6 load types x 6 end restraints, including thermal and imposed-displacement loading

Straight Beam — Shear, Moment, Slope & Deflection

This calculator solves a single-span, uniform elastic straight beam. It uses classical Euler-Bernoulli beam theory — the closed-form solution of the bending differential equation EI\,y'''' = w(x) — so the results are exact for the assumed model, not an approximation. You can combine any of six load types with any of six end-restraint combinations, which gives 36 cases in total.

Load types (choose one; the beam is linear, so you can superpose several by running it once per load):

• Concentrated transverse load W at a distance a from the left end
• Distributed load varying linearly from w_a at x = a to w_l at the right end
• Concentrated moment M_o at x = a
• Imposed angular displacement \theta_o at x = a — a forced kink in the beam
• Imposed lateral displacement \Delta_o at x = a — a forced step, as with a settled support
• Uniform top-to-bottom temperature difference between T_1 and T_2, from x = a to the right end

End restraints (named left-to-right; a guided end cannot rotate but is free to move up and down, like a slider in a track):

• Free - fixed (the cantilever)
• Guided - fixed
• Pinned - fixed (the propped cantilever)
• Fixed - fixed
• Pinned - pinned (the simply supported beam)
• Guided - pinned

You enter the load type, the end restraints, the span l, the load position a, the quantities for the chosen load, and the section stiffness through the elastic modulus E and the second moment of area I. The calculator picks the load terms for your load type, solves the end values (R_A, M_A, \theta_A, y_A) from the two end-restraint conditions, then evaluates one singularity-function master equation along the beam to give the shear V(x), bending moment M(x), slope \theta(x) and deflection y(x). It also reports the end reactions, end moments, end slopes, and the location and value of the maximum moment and maximum deflection.

Why some loads produce no internal force: the first three load types are mechanical — they create internal forces directly. The last three (angular displacement, lateral displacement, temperature difference) are geometric or thermal: they impose a deformation rather than a force, so their moment load term is exactly zero. On a statically determinate beam (free-fixed or pinned-pinned) these three give M = 0 everywhere and zero reactions — the beam simply deforms, carrying no force. Only a statically indeterminate beam develops internal forces, because the restraints must work against the imposed deformation. This is exactly where thermal stress comes from: switch the same thermal input from pinned-pinned to fixed-fixed and the moments appear.

Sign convention: the origin is at the left end (x = 0), with x running right to l. Loads are positive downward; deflection y is positive upward (so a downward load gives a negative deflection); the bending moment is positive when the lower fibre is in tension (sagging).

Model limits: first-order linear-elastic slender-beam theory — single span, constant EI, small deflections, one load at a time. It does not include shear deformation, P-Δ second-order effects, large deflection, or axial force. Because the system is linear, you can superpose several loads by running the calculator once per load and adding the results. The outputs are structural response quantities (reactions, internal forces, deformations); there is no pass/fail safety check.

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