Beam Support Reaction
Calculate support reactions for simply supported beams under concentrated or distributed loads
Inputs
Load 1
Load 2
Formula Interpretation
Concentrated Loads — Reaction at B
Taking moments about support A: the sum of moments of all downward loads equals the moment of R_B. Dividing by span l gives R_B.
Concentrated Loads — Reaction at A
By vertical equilibrium, the sum of all upward reactions equals the total downward load. Hence R_A is the remainder after subtracting R_B.
Uniform Distributed Load
A UDL of intensity w (N/mm) over span l creates a total load wl acting at the midpoint. By symmetry both reactions are equal: wl/2.
Knowledge Points
Static Equilibrium
A statically determinate beam is in equilibrium when the sum of all forces (ΣF = 0) and the sum of all moments about any point (ΣM = 0) are both zero. These two equations are sufficient to find both unknown reactions.
Moment Equation
Taking moments about support A eliminates R_A from the equation (its moment arm is zero), yielding R_B directly. This technique is standard in beam analysis to isolate one unknown.
UDL Equivalence
A uniform distributed load w (N/mm) over length a can be replaced by a single equivalent point load W = w·a acting at the centroid of the loaded region (its midpoint for a uniform load). This simplifies the moment calculation.
Worked Example
A simply supported beam of span 1200 mm carries a UDL of 5 N/mm over the left 600 mm and a 2000 N point load at 900 mm from A. Find R_A and R_B.
Knowns
- • Span: l = 1200 mm
- • UDL: w = 5 N/mm over a = 600 mm → equivalent point load W₁ = 5 × 600 = 3000 N at l₁ = 300 mm from A
- • Point load: W₂ = 2000 N at l₂ = 900 mm from A
Solution
Step 1 — Moments about A
Step 2 — Solve R_B
Step 3 — Vertical equilibrium for R_A
Result: R_A = 2750 N, R_B = 2250 N. (Note: the textbook labels A on the right and B on the left, so the reactions are swapped in labelling but the values are correct.)
Extended Knowledge
- •When loads are placed beyond the supports (overhangs), one reaction can become negative (downward). The same equilibrium equations still apply; a negative result simply indicates a downward reaction force.
- •Continuous beams spanning multiple supports are statically indeterminate — equilibrium alone is not sufficient. Methods such as the three-moment theorem or finite element analysis are required.
- •When a load moves across the span (e.g., a vehicle on a bridge), influence lines describe how the reactions vary with load position. The worst-case position maximises the reaction or bending moment.
- •Loads not in the plane of the beam produce torsion as well as bending. Both the vertical reactions and the torsional reactions must be determined from full 3D equilibrium.
- •In real structures, support settlements, thermal expansion, and dynamic loads modify the reactions. Structural engineers include safety factors and check both ultimate and serviceability limit states.