RHCES Beam Deflection Calculator

Inputs

Load Case

Span L (m)

Load magnitudekN/m or kN (point)

Distance a (m)used by ssptN only

E (MPa)

I (mm⁴)

Closed-form deflection

Simple UDL: δ = 5wL⁴/(384EI).
Simple PT (mid): δ = PL³/(48EI).
Cantilever PT: δ = PL³/(3EI).
Cantilever UDL: δ = wL⁴/(8EI).
Fixed UDL: δ = wL⁴/(384EI).

Results

Mid-span Mmax—

Vmax—

δ_max—

L/δ ratio—

About this calculator

This tool returns maximum moment, shear, and deflection for six classical, statically-determinate beam load cases — simply-supported and cantilever beams under a uniformly distributed load (UDL) or point load, plus a fixed-fixed beam under UDL — using closed-form elastic beam theory. It is meant for quick preliminary strength and serviceability checks.

Method & formulas

All cases follow standard Euler-Bernoulli beam theory (see Hibbeler, or Gere & Timoshenko, Mechanics of Materials):

Simple beam, UDL w: M_max=wL²/8 (midspan) V_max=wL/2 δ_max=5wL⁴/(384EI) (midspan)

Simple beam, point load P at midspan: M_max=PL/4 V_max=P/2 δ_max=PL³/(48EI)

Simple beam, point load P at distance a from a support: solved by the standard double-integration / Macaulay's-method solution for an off-center load — the maximum deflection is not generally under the load itself unless a=L/2.

Cantilever, point load P at the free tip: M_max=PL (fixed end) V_max=P δ_max=PL³/(3EI) (tip)

Cantilever, UDL w: M_max=wL²/2 (fixed end) V_max=wL δ_max=wL⁴/(8EI) (tip)

Fixed-fixed beam, UDL w: M_support=wL²/12 V_max=wL/2 δ_max=wL⁴/(384EI) (midspan)

E is the elastic modulus and I the second moment of area of the cross-section, taken as constant along the span in every case.

Worked example

Using the tool's own defaults for the simple-beam UDL case — L=6 m, w=20 kN/m, E=200,000 MPa, I=180×10⁶ mm⁴: M_max = 20×6²/8 = 90 kN·m and V_max = 20×6/2 = 60 kN. For deflection, converting to consistent N-mm units (w=20 kN/m = 20 N/mm, L=6000 mm): δ_max = 5×20×6000⁴/(384×200,000×180×10⁶) ≈ 9.4 mm, giving a span/deflection ratio of about L/640 — comfortably within typical L/360 floor serviceability limits.

Assumptions & limitations

Classical linear-elastic, small-deflection theory (plane sections remain plane, shear deformation neglected) — not valid for deep beams (span/depth roughly < 4–5).

I is treated as constant (uncracked, prismatic) along the full span; for reinforced-concrete serviceability, an effective cracked-section I_e should be used instead of the gross I_g, or deflection will be under-predicted.

Only single, isolated determinate spans are covered — continuous multi-span beams require superposition or a full continuous-beam analysis.

Long-term effects (creep and shrinkage under sustained load) are not included; apply an additional long-term multiplier separately.

Verify results against the governing code and a licensed engineer's judgment.

FAQ

Which E and I should I use for a reinforced-concrete beam?

Use the concrete elastic modulus E_c (roughly 4700√f'c MPa) and, for cracked-section serviceability, an effective moment of inertia I_e between the gross I_g and the cracked-transformed I_cr — using I_g alone over-estimates stiffness and under-predicts real deflection.

How do I know if a computed deflection is acceptable?

Compare δ_max to the applicable code deflection limit for the member's use — commonly L/360 for floors supporting non-structural elements, L/240 for roofs, or L/480 where attached partitions could be damaged by deflection.

Why is deflection so much more sensitive to span than moment is?

Deflection scales with L⁴ (or L³ for a point load) while moment scales with only L¹ or L², so doubling the span multiplies deflection roughly 16-fold even though peak moment only doubles or quadruples — this is why serviceability, not strength, often governs long-span members.

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