Formula sheet

Structural Loads and Beam Analysis Formula Sheet

Structural load formulas for load factors, reactions, shear and moment, bending stress, section properties, deflection, Euler buckling, and serviceability checks.

Branch: Civil and Construction Engineering
Content: Formula sheet
Updated: May 29, 2026
Revision: v1.0.5 · reviewed

This formula sheet collects common first-pass relationships for structural loads and beam analysis. It is intended for conceptual checks, hand verification, and interpretation of structural models. It does not replace governing building codes, bridge standards, material-specific design rules, or project specifications.

Use consistent units and state the design basis. Distinguish service loads, factored loads, ultimate limit states, serviceability limit states, temporary works, and construction-stage cases.

Factored design load

Single load action:

F_d=\gamma_F F_k

where $F_d$ is design load, $F_k$ is characteristic or nominal load, and $\gamma_F$ is load factor.

For a simplified combination:

E_d=\sum_i \gamma_i F_{k,i}

where $E_d$ is design effect such as shear, moment, axial force, or deflection demand. Actual load combinations must follow the relevant standard.

Do not mix load factors from one design format with allowable stresses or resistance factors from another without a valid basis.

Distributed load resultants

Uniformly distributed load over length $L$ :

W=wL

Resultant location:

\displaystyle x=\frac{L}{2}

Linearly varying triangular load from zero to $w_{max}$ :

\displaystyle W=\frac{1}{2}w_{max}L

Resultant location from the zero-load end:

\displaystyle x=\frac{2L}{3}

Distributed loads must be converted carefully when moving between area load, line load, and point-load idealizations.

Simple beam reactions

Simply supported beam with central point load $P$ :

\displaystyle R_A=R_B=\frac{P}{2}

Simply supported beam with full-span uniform load $w$ :

\displaystyle R_A=R_B=\frac{wL}{2}

Cantilever with end point load $P$ :

R=P

Fixed-end moment at support:

M=PL

Cantilever with full-span uniform load $w$ :

R=wL

Fixed-end moment at support:

\displaystyle M=\frac{wL^2}{2}

Always verify that vertical forces and moments balance before using stress or deflection results.

Shear and bending moment

Simply supported beam with central point load $P$ :

\displaystyle V_{max}=\frac{P}{2}

\displaystyle M_{max}=\frac{PL}{4}

Simply supported beam with full-span uniform load $w$ :

\displaystyle V_{max}=\frac{wL}{2}

\displaystyle M_{max}=\frac{wL^2}{8}

Cantilever with end point load $P$ :

V_{max}=P

M_{max}=PL

Cantilever with full-span uniform load $w$ :

V_{max}=wL

\displaystyle M_{max}=\frac{wL^2}{2}

These values assume ideal support conditions and simple loading. Overhangs, partial loads, moving loads, continuity, and frame action require separate analysis.

Bending stress

Elastic bending stress:

\displaystyle \sigma=\frac{My}{I}

Maximum bending stress:

\displaystyle \sigma_{max}=\frac{Mc}{I}

Section modulus:

\displaystyle S=\frac{I}{c}

so:

\displaystyle \sigma_{max}=\frac{M}{S}

Allowable-stress check:

\sigma_{max}\leq\sigma_{allow}

Limit-state checks may instead compare factored demand with factored resistance according to the governing code.

Common section properties

Rectangular section about centroidal strong axis:

\displaystyle I=\frac{bh^3}{12}

Section modulus:

\displaystyle S=\frac{bh^2}{6}

Solid circular section:

\displaystyle I=\frac{\pi d^4}{64}

Hollow circular section:

\displaystyle I=\frac{\pi(D^4-d^4)}{64}

For built-up sections, use the parallel-axis theorem:

I=\sum_i\left(I_i+A_id_i^2\right)

where $d_i$ is distance from each component centroid to the composite neutral axis.

Shear stress

Average shear stress:

\displaystyle \tau_{avg}=\frac{V}{A}

Beam shear formula:

\displaystyle \tau=\frac{VQ}{Ib}

where $V$ is shear force, $Q$ is first moment of area above or below the point, $I$ is second moment of area, and $b$ is local width.

For many steel I-sections, web shear is often approximated by:

\displaystyle \tau_{web,avg}\approx\frac{V}{A_w}

where $A_w$ is web area. Detailed design must follow the relevant material standard, including web buckling, bearing, stiffeners, and local load introduction.

Beam deflection

Simply supported beam with central point load:

\displaystyle \delta_{max}=\frac{PL^3}{48EI}

Simply supported beam with full-span uniform load:

\displaystyle \delta_{max}=\frac{5wL^4}{384EI}

Cantilever with end point load:

\displaystyle \delta_{max}=\frac{PL^3}{3EI}

Cantilever with full-span uniform load:

\displaystyle \delta_{max}=\frac{wL^4}{8EI}

Deflection formulas assume linear elastic behaviour, small deflection, ideal support conditions, and Euler-Bernoulli beam assumptions. Deep beams, shear deformation, cracking, creep, connection flexibility, and composite action may require adjusted models.

Slope

Simply supported beam with central point load, slope at support:

\displaystyle \theta_A=\frac{PL^2}{16EI}

Simply supported beam with full-span uniform load, slope at support:

\displaystyle \theta_A=\frac{wL^3}{24EI}

Cantilever with end point load, end rotation:

\displaystyle \theta=\frac{PL^2}{2EI}

Cantilever with full-span uniform load, end rotation:

\displaystyle \theta=\frac{wL^3}{6EI}

Slope can govern facade alignment, bearing rotation, machinery support, and serviceability even when vertical deflection is acceptable.

Serviceability ratios

Deflection limits are often expressed as span ratios:

\displaystyle \delta_{allow}=\frac{L}{R}

where $R$ may be a project or code-defined ratio such as 240, 360, 500, or another value depending on member type and consequence.

Utilization:

\displaystyle U_\delta=\frac{\delta_{calc}}{\delta_{allow}}

The load combination and time basis must match the limit. Total deflection, live-load deflection, long-term deflection, and differential deflection are not interchangeable.

Euler buckling

Ideal elastic buckling load:

\displaystyle P_{cr}=\frac{\pi^2EI}{(KL)^2}

Slenderness ratio:

\displaystyle \lambda=\frac{KL}{r}

Radius of gyration:

\displaystyle r=\sqrt{\frac{I}{A}}

where $K$ is effective length factor. Euler buckling is an ideal model; real design must address imperfections, end restraint, residual stress, material nonlinearity, bracing, local buckling, and code column curves.

Thermal movement and restraint

Free thermal expansion:

\Delta L=\alpha L\Delta T

Free thermal strain:

\epsilon_{th}=\alpha\Delta T

Fully restrained thermal stress:

\sigma=E\alpha\Delta T

Structures often have partial restraint, sliding bearings, expansion joints, creep, cracking, or staged construction. Thermal load effects should be modelled with realistic restraint conditions.

Moving point load influence

For a simply supported beam, the maximum bending moment from a single moving point load $P$ occurs when the load is at midspan:

\displaystyle M_{max}=\frac{PL}{4}

The maximum reaction occurs when the point load is close to the support:

R_{max}\approx P

Influence lines are used for more complex moving loads such as vehicles, cranes, trains, and construction equipment.

Mini example: simply supported floor beam

A simply supported beam has span:

L=6\ \text{m}

and service uniform load:

w=12\ \text{kN/m}

Maximum moment:

\displaystyle M_{max}=\frac{wL^2}{8}=\frac{12(6)^2}{8}=54\ \text{kN m}

Maximum shear:

\displaystyle V_{max}=\frac{wL}{2}=\frac{12(6)}{2}=36\ \text{kN}

If:

E=200\ \text{GPa}

and:

I=80\times10^6\ \text{mm}^4=8.0\times10^{-5}\ \text{m}^4

Maximum elastic deflection:

\displaystyle \delta_{max}=\frac{5wL^4}{384EI}

Using $w=12\,000\ \text{N/m}$ :

\displaystyle \delta_{max}=\frac{5(12\,000)(6^4)}{384(200\times10^9)(8.0\times10^{-5})}

\delta_{max}=12.7\ \text{mm}

If the serviceability limit is:

\displaystyle \frac{L}{360}=\frac{6000}{360}=16.7\ \text{mm}

then the calculated deflection is below this simple limit. A real design would still check strength, shear, lateral stability, connections, vibration, long-term effects, construction loads, and the governing code.

Common cautions

Do not use service-load deflection formulas with factored ultimate loads unless the check specifically requires it. Do not assume supports are perfectly pinned or fixed without connection evidence. Do not ignore self-weight, construction loads, or temporary bracing. Do not use beam formulas for deep beams, slabs, frames, shells, or soil-supported members without checking assumptions. Do not accept a software result unless equilibrium, deflected shape, and boundary conditions make physical sense.

REF

Disciplines