Beam Calculator: Deflection, Shear and Moment

The solver is derived here from equilibrium and the Euler–Bernoulli relation $EIy''=M$, with $y$ positive upward. It follows the Macaulay bracket and superposition approach illustrated by D. J. Dunn, Deflection of Beams, pp.11–15, and the statics relationships on FIRGELLI's beam page.

For $\langle x-a\rangle^n$, negative arguments contribute zero; order zero is a right-continuous unit step. Let $J_n(x)$ be the sum of the point-load terms $P\langle x-a\rangle^n/n!$, the patch-load term $q[\langle x-s\rangle^{n+1}-\langle x-e\rangle^{n+1}]/(n+1)!$, and the self-weight term $q_{sw}x^{n+1}/(n+1)!$.

For pin + roller, impose $M(0)=M(L)=y(0)=y(L)=0$. For a cantilever impose $y(0)=y'(0)=0$ and free-end equilibrium including any tip load. For fixed ends impose $y=y'=0$ at both ends. For the propped cantilever impose $y(0)=y'(0)=y(L)=M(L)=0$. These conditions give the readable reaction and integration-constant steps.

The span is split at load discontinuities. Exact polynomial roots of $V=0$ and $y'=0$, together with every interval end, locate the absolute moment and deflection peaks. Charts use 41 points per nonzero interval; duplicated positions retain the two shear limits at point loads. The peak results do not depend on this plotting grid.

Section formulas are $I=bh^3/12$ for a rectangle and $I=\pi D^4/64$ for a solid circle, subtracting the inner shape for tubes. IPE properties come from the platform catalog including root fillets. Stress is $\sigma_b=|M|_{max}c_{max}/I$. Self-weight is $q_{sw}=\rho A g$ when selected.

• Straight, prismatic, homogeneous beam with constant $EI$ and loads in one bending plane. No applied concentrated couples or axial force.
• Ideal supports prevent the selected translations and rotations, with no settlement or support flexibility. A bearing requiring a negative reaction needs a hold-down.
• Small rotations, linear elasticity and plane sections remaining plane. Shear deformation and geometric stiffness are omitted.
• The rectangle's entered depth is vertical. IPE strong/weak axes and the dressed 2x4 orientation can be switched. Tubes have ideal sharp corners.
• Custom section stress uses the entered farthest fibre distance; the outline depth is represented as $2c_{max}$ only for reporting.
• Metal properties are platform reference-table values. Timber moduli and air-dry shipping densities come from Oregon State's table; these are illustrative clear-wood averages, not allowable stresses or graded-lumber stiffness guarantees.
• The equations and support-condition solution are derived here; the named publications supply independent checks. The diagram separates load arrows into lanes above the beam and shows the exaggerated deflection on a separate baseline below it, so upward deflection stays clear of load labels.

Use for elastic bending estimates and teaching. The warning at length/depth < 10 is a prompt to assess shear deformation, not a universal accuracy limit. A displacement over 2% of beam length triggers a small-deflection warning; the actual acceptable limit depends on the application.

No Eurocode, AWC, AISC or other code compliance, strength capacity or safe-load claim is made. No yield, shear, local or lateral-torsional buckling, fatigue, torsion, bearing, connection, stability, creep, vibration, dynamic amplification, load factors or combinations are checked. Timber grading, moisture, duration of load and long-term deformation need a separate assessment. IPE dimensions follow the platform's EN 10365:2017-based catalog; this does not make the calculation a code check. Check against the standard in force when using catalog data for a project.

The inspection shear is the right-hand value at an internal point load and the left-hand value at the right end. Equal absolute peaks report the first position found. Signed deflection can be upward or downward; the maximum card is always a magnitude. Equilibrium residuals retain floating-point roundoff; the normalized right-end displacement residual is rounded to 12 decimal places.