Why a parameterized library is more useful than a tile for every picture.
A central point load, an off-centre point load, and three unequally spaced point loads do not need three separate statics engines. The same reaction and sectional-equilibrium equations apply after the load magnitudes and positions are entered. CivilKits therefore keeps recognisable named cases while also providing flexible calculators for load zones and multiple point loads.
current coverage
19 beam-response tools across simply supported, cantilever, fixed-fixed, propped cantilever, overhang, and two-span continuous support families. These sit within a 23-tool structural-analysis branch.
1. Simply supported beams
Vertical reactions follow ΣV = 0 and ΣM = 0. The internal bending moment at a section is obtained from the actions on either side of the cut; extrema occur at boundaries, concentrated-load positions, or where the shear is zero.
- Full-span UDL + central point loadClosed-form maximum shear, moment, and midspan deflection.
- Full-span UDL + point load at any positionAsymmetric reactions and the critical positive-moment section.
- One-way full-span triangular loadLinearly varying load with a declared zero and peak end.
- Symmetric central triangular loadA tent-shaped load rising to a midspan peak.
- One or two partial UDL zonesCovers a central patch, either end patch, patches at both ends, and separated or overlapping zones.
- Up to three point loadsCovers central, arbitrary, equal, unequal, equally spaced, and unequally spaced point loads.
- UDL + point load + prescribed end momentsUses signed member-end actions; it does not infer joint moments from frame stiffness.
2. Cantilever beams
The fixed support supplies vertical reaction and moment. Constant-EI elastic movements are calculated from standard Euler–Bernoulli solutions and linear superposition; the actual connection must be capable of providing the assumed rotational restraint.
- Free-end point loadThe classic PL, PL²/(2EI), and PL³/(3EI) case.
- Point load at any positionIncludes load-point deflection and rigid movement of the unloaded tail.
- Full-span UDLFixed shear and moment, tip rotation, and tip deflection.
- Triangular load rising to the free endThe orientation is explicit because reversing the triangle changes the coefficients.
- UDL + point load + free-end momentSame-sense component superposition with each deflection contribution reported.
3. Fixed, propped, overhang, and continuous beams
These cases add compatibility or continuity conditions to equilibrium. Perfect fixity, an unyielding prop, and constant EI are strong assumptions; settlement, joint flexibility, cracking, or stiffness changes can materially alter the actions.
- Fixed-fixed beam with full-span UDLSymmetric reactions, end hogging, midspan sagging, and deflection.
- Fixed-fixed beam with central point loadSymmetric fixed-end and midspan response.
- Fixed-fixed beam with point load at any positionOff-centre reactions, end moments, load-point sagging, and load-point deflection.
- Propped cantilever with full-span UDLCompatibility reaction, fixed-end hogging, positive moment, and elastic deflection.
- Propped cantilever with point loadProp reaction and moment response as the load moves along the span.
- Single or double overhang beamMove both supports inside the beam and combine a full-length UDL with one point load.
- Two-span continuous beam with independent UDLsThree-moment solution for centre-support hogging, reactions, and each span’s positive maximum.
The equations behind the library
Statics for determinate beams
For a simply supported span L carrying point loads Pi at xi and distributed-load resultants Wj at centroids x̄j, RB = [Σ(Pixi) + Σ(Wjx̄j)]/L and RA = ΣPi + ΣWj − RB. Sectional shear and moment are then evaluated piecewise.
Elastic deflection
The closed-form movement tools assume small deflection, linear elasticity, and constant flexural rigidity EI. They use EI d²v/dx² = M(x), with integration constants set by the support conditions. Linear superposition is valid only while those assumptions remain reasonable.
Compatibility and continuity
Fixed and propped cases enforce zero rotation or zero displacement at the restrained points. The two-span UDL tool uses the three-moment relationship with simple outer supports and constant EI across both spans.
Choose the diagram carefully.
- Match the support condition before matching the load shape.
- Check whether a triangular load rises left-to-right, right-to-left, or to a central peak.
- Keep the sign convention visible when prescribed end moments are entered.
- Do not treat a perfect fixed support, prop, or continuous joint as automatically representative of the real connection.
- Add self-weight and load combinations explicitly; these calculators do not invent omitted actions.
- Use the results as analysis inputs, not as a substitute for strength, stability, serviceability, connection, or code checks.