Fixed-fixed beam center load calculator

A fixed-fixed beam reacts very differently from a simply supported beam because both ends restrain rotation. This calculator applies the classical Euler–Bernoulli closed-form solution for a prismatic span with one point load exactly at its center. Enter the load, span, Young’s modulus, and second moment of area to obtain the two support moments, the center bending moment, and the downward midspan deflection. The result uses an explicit bending-moment sign convention and consistent units, making it suitable for hand-calculation checks, preliminary sizing, teaching, and deterministic engineering software tests.

Understand the fixed-fixed center-load result

A fixed-fixed beam has translational and rotational restraint at both ends. Under a point load placed exactly at midspan, symmetry makes the vertical reactions equal and gives identical bending behavior in the two halves of the span. The restraints create hogging moments at the supports, while the middle of the beam carries a sagging moment.

With sagging defined as positive, the left and right end moments are each minus P times L divided by eight, and the center moment is plus P times L divided by eight. The equal magnitudes are a special feature of this symmetric load case; they should not be assumed for an off-center point load or an asymmetric beam. The maximum vertical displacement occurs at the center and has magnitude P times L cubed divided by 192 E I.

Compared with the familiar simply supported central-load expression, whose denominator is 48 E I, rotational restraint reduces the idealized midspan deflection to one quarter. These equations assume a straight, prismatic member, constant flexural rigidity E I, linear-elastic material behavior, small displacement, and supports that truly prevent end rotation.

Enter a coherent set of beam properties

Load is the concentrated force P applied at the precise center of the clear fixed-to-fixed span. Length is that full span L. The calculator does not attach or convert units, so every input must belong to one coherent system.

For example, newtons, metres, pascals, and metres to the fourth produce moments in newton-metres and deflection in metres. Using newtons and millimetres instead requires E in newtons per square millimetre and I in millimetres to the fourth, producing newton-millimetres and millimetres. The response reports support moments as negative, the center moment as positive, and deflection as a positive magnitude paired with the direction downward, so consumers do not need to guess the convention.

Use the calculation within its engineering limits

This capability is best used as a transparent closed-form check, not as a substitute for a complete structural model. It does not calculate reactions, shear diagrams, stresses, buckling, natural frequency, plastic redistribution, support flexibility, shear deformation, cracking, composite action, or second-order effects. It also does not cover distributed loads, multiple loads, eccentric point loads, varying section properties, partial fixity, or spans with releases.

Real connections often provide less than perfect rotational restraint, so a measured structure can deflect more and develop different end moments than the ideal fixed-fixed solution. For preliminary work, compare the returned moments with section resistance and the returned deflection with the governing serviceability criterion using the applicable code, load combinations, safety factors, and material rules.