Beam Reaction Calculator

Support reactions are the forces a beam pushes back with at its supports, and for a statically determinate member they follow from equilibrium alone. A simply supported beam has a pin at one end and a roller at the other: three unknown reaction components matched by three equilibrium equations, so no material property, elastic modulus, or section size enters the answer. Summing vertical forces gives Ra + Rb equal to the total applied load; summing moments about the left support isolates Rb directly. A uniform load w over span L splits evenly at wL/2 per end. A point load P at distance a from the left divides in inverse proportion to distance, Ra = P(L-a)/L and Rb = Pa/L, so a load one quarter along the span sends 75 percent of itself to the near support. Once reactions are known, shear at any section is the running sum of forces to the left of the cut, and bending moment is the accumulated area under the shear diagram. Maximum moment occurs where shear crosses zero, at midspan for a uniform load and at the load point for a single concentrated force. Reactions size the columns, bearing plates and footings beneath the beam, while internal shear and moment size the beam itself. The method assumes small deflections, rigid non-settling supports, and genuinely free rotation at each end. Continuous beams over three or more supports, fixed ends, and overhanging spans are statically indeterminate and require compatibility or stiffness methods rather than equilibrium alone.

The principle behind reaction calculation is the law of the lever, stated by Archimedes in the third century BCE, and the equilibrium of moments generalized by Pierre Varignon and by Isaac Newton in the Principia around 1687. What was missing for centuries was the systematic idea of isolating a body and accounting for every force acting on it. That free-body approach matured through the eighteenth and nineteenth centuries and became a formal teaching method in engineering schools, notably at the Ecole Polytechnique after its founding in 1794. Applying statics rigorously to beams and trusses was largely driven by the nineteenth-century railway boom. Bridge engineers needed reactions and internal forces for structures far larger than anything earlier builders had faced, and they needed them for loads that moved. Karl Culmann published Die graphische Statik in Zurich in 1866, giving graphical constructions for reactions and moment diagrams that dominated design offices for decades. Otto Mohr and others developed influence lines in the 1860s and 1870s so an engineer could find the worst position of a passing locomotive. The shear and moment diagram conventions taught today, and the tabulated beam cases reproduced in the AISC Steel Construction Manual and American Wood Council literature, descend directly from that work, later condensed into the closed-form reaction expressions this calculator evaluates.