Simply Supported Beam UDL

Assumptions & Limitations

• The beam is prismatic — constant cross-section and uniform EI along the full span.
• Euler–Bernoulli beam theory is assumed — shear deformation is neglected (valid when L/d > 10).
• The UDL is applied over the full span; partial loading or patch loads require a different set of formulae.
• The analysis is elastic and linear — no plasticity, no second-order (P-Δ) effects.
• Self-weight of the beam is not included separately — add it to w if required.
• No lateral–torsional buckling (LTB) check is included; this must be assessed separately for unrestrained beams (EN 1993-1-1 §6.3.2).
• Support conditions are idealised as perfect pins — no partial fixity or settlement is modelled.

Simply Supported Beam — Uniformly Distributed Load (UDL)

This worksheet calculates the mid-span deflection, maximum bending moment, and maximum shear force for a simply supported beam carrying a uniformly distributed load (UDL) over its full span.

The analysis is based on Euler–Bernoulli beam theory, which assumes that plane sections remain plane, the material is linearly elastic, and deflections are small relative to the span. The governing equations come from classical structural mechanics and are widely used in BS EN 1993 (Eurocode 3) and similar codes.

The three key results are:

• Maximum bending moment at mid-span: M = wL²/8
• Maximum shear force at the supports: V = wL/2
• Maximum mid-span deflection: δ = 5wL⁴ / (384EI)

These are the exact closed-form solutions for a prismatic, simply supported beam under a full-span UDL.

Inputs

The inputs below define the beam geometry, loading, and material/cross-section stiffness. For structural steel beams, E is typically 200 GPa. The second moment of area I depends on the chosen section profile — consult section tables (e.g. SCI Blue Book) for standard UB/UC sections. The UDL w should include all characteristic or design loads as appropriate to the design stage.

Calculations

Support Reactions

For a simply supported beam with a full-span UDL, the loading is symmetric. The total applied load is the UDL multiplied by the span, and by symmetry each support carries exactly half of this total. These reactions form the starting point for the shear force and bending moment diagrams.

Total load: W_total = w × L. Each reaction: R = W_total / 2 = wL/2.

Maximum Shear Force

The shear force diagram for a simply supported beam under a full-span UDL is linear. It is maximum at the supports (equal to the reaction R = wL/2) and reduces to zero at mid-span. The shear force at any distance x from the left support is V(x) = R − wx.

The maximum shear force governs the shear capacity check of the web in a steel section (EN 1993-1-1 §6.2.6) or the shear design of a concrete beam.

Maximum Bending Moment

The bending moment diagram is parabolic, with zero moment at both simply supported ends and a maximum at mid-span. The governing equation is derived by integrating the shear force diagram, or directly from statics:

M_max = wL² / 8

where w is the UDL (force per unit length) and L is the span. This is the most critical cross-section for bending — the beam must be checked for its moment capacity M_c,Rd ≥ M_max (Eurocode 3) or M_u ≥ M_max (ACI/BS 8110 equivalent).

Flexural Stiffness

The product EI is the flexural rigidity or bending stiffness of the beam — it governs how resistant the beam is to bending deformation. A higher EI means a stiffer beam and smaller deflections. E is the elastic modulus of the material and I is the second moment of area of the cross-section about its bending axis.

Maximum Mid-Span Deflection

The deflection of the beam is found by integrating the moment–curvature relationship M(x) = EI·d²y/dx² twice, applying the boundary conditions y(0) = y(L) = 0 (zero deflection at both supports). For a full-span UDL the result is a quartic curve, with maximum deflection at the centre given by the exact formula:

δ_max = 5wL⁴ / (384EI)

where:

• w = UDL (force per unit length)
• L = beam span
• E = Young's modulus of the material
• I = second moment of area of the cross-section
• 384 = exact constant from the integration (= 8 × 48)

This deflection is compared against the serviceability limit state (SLS) deflection limit, commonly L/360 for live load or L/250 for total load under EN 1990 / BS EN 1993.

Deflection Limit Check (Serviceability)

Eurocode (EN 1993-1-1) and most national codes limit the total mid-span deflection to a fraction of the span — commonly L/250 for the total load deflection of a beam supporting a floor or roof. This is the serviceability limit state (SLS) check. The allowable deflection δ_allow = L/250 is compared to the calculated δ_max.

Results

The four key results are summarised by the calculations above:

• Support reactions: R_A is the vertical reaction at each end — used to design the connections and bearing plates.
• Maximum shear force V_max occurs at the supports. This governs the shear design of the web or stirrups.
• Maximum bending moment M_max occurs at mid-span. This is the critical section for bending capacity — select a section with M_c,Rd ≥ M_max.
• Mid-span deflection δ_max is the SLS check. If the utilisation ratio δ_ratio exceeds 1.0, a deeper or stiffer section is needed, or the span must be reduced.

For the values entered, with a 10 kN/m UDL on a 6 m span with EI = 200 GPa × 1.56×10⁻⁴ m⁴ = 31.2 MN·m², the deflection is well within the L/250 limit for a typical UB305 or UB356 section.