Engineering reference // structural mechanics

Drawing No. EH–CS–014 // Civil & Structural Engineering

Structural Engineering Formula Sheet

A mechanics-first structural reference for preliminary member behavior. These equations describe elastic stress, deformation and idealized instability; they do not replace code-specific strength, stability, serviceability, connection, seismic, fire or load-combination requirements.

Fast reference, with engineering context

Use the equations directly for screening calculations, then open the linked EngineerHub tools for input handling and unit conversion. Formula applicability and major limitations are stated beside each relation.

Start with Axial Formula summary

Reference conventions

σ = P/AAxial normal stress

σ = My/IElastic bending stress

EIFlexural rigidity

Pcr = π²EI/(KL)²Euler elastic buckling

Simply supported beam under load

Structural formulas connect external loading to internal force resultants, stresses and elastic deformation. Boundary conditions and load paths are part of the model.

01 // Axial stress, strain and elongation

For a prismatic member under concentric axial load in the linear-elastic range, stress is uniform and elongation follows Hooke’s law.

Core relation

σ = P/A

ε = σ/E ; δ = PL/(AE)

Symbol	Meaning	SI units	US customary
P	Axial force	N	lbf
A	Cross-sectional area	m², mm²	in²
σ	Normal stress	Pa, MPa	psi, ksi
E	Young’s modulus	Pa, GPa	psi, Msi
L	Member length	m	in, ft
δ	Axial elongation	m, mm	in

Open Young’s Modulus Reference Worked example ↓

Worked example — steel tie elongation

Given: P = 120 kN, A = 2000 mm², L = 2.0 m, E = 200 GPa.

σ = 120000/0.002 = 60 MPa

δ = PL/(AE) = 0.00060 m = 0.60 mm

Answer: σ = 60 MPa; δ = 0.60 mm.

US check: ≈8.70 ksi and 0.0236 in.

Concentric loading and linear elasticity are assumed.

Eccentricity creates bending. Net-section effects, holes, yielding, fracture and connection behavior require separate checks.

02 // Common section properties

Second moment of area controls bending stiffness; section modulus connects moment directly to extreme-fiber elastic bending stress.

Core relation

I_rect = bh³/12 ; Z = I/c

I_circle = πd⁴/64 ; J_{solid circle} = πd⁴/32

Symbol	Meaning	SI units	US customary
I	Second moment of area	m⁴, mm⁴	in⁴
Z	Elastic section modulus	m³, mm³	in³
J	Polar second moment (circular shaft)	m⁴, mm⁴	in⁴
b,h,d	Section dimensions	m, mm	in
c	Neutral-axis to extreme fiber	m, mm	in

Open Beam / Column Calculator Worked example ↓

Worked example — 100×200 mm rectangle

Given: b = 0.100 m, h = 0.200 m.

I = 0.1×0.2³/12 = 6.667×10⁻⁵ m⁴

Z = I/(h/2) = 6.667×10⁻⁴ m³

Answer: I = 6.667×10⁻⁵ m⁴; Z = 6.667×10⁻⁴ m³.

Use the axis corresponding to the actual bending direction.

Principal-axis orientation, composite sections, transformed sections and local plate behavior can materially change stiffness and strength.

03 // Elastic bending stress

For Euler–Bernoulli beam bending in the linear-elastic range, normal stress varies linearly from the neutral axis.

Core relation

σ = My/I

σ_max = M/Z

Symbol	Meaning	SI units	US customary
M	Bending moment	N·m, kN·m	lbf·ft, kip·ft
y	Distance from neutral axis	m, mm	in
I	Second moment of area	m⁴	in⁴
Z	Elastic section modulus	m³	in³
σ	Bending normal stress	Pa, MPa	psi, ksi

Open Beam / Column Calculator Worked example ↓

Worked example — rectangular beam under moment

Given: M = 12 kN·m; b = 100 mm, h = 200 mm; I = 6.667×10⁻⁵ m⁴.

σmax = 12000×0.100/(6.667×10⁻⁵) = 18.0 MPa

Answer: σmax = 18.0 MPa.

US check: ≈2.61 ksi.

Formula gives elastic stress, not code design strength.

Lateral-torsional buckling, local buckling, plasticity, residual stress and compactness are not represented by My/I alone.

04 // Beam transverse shear

The general elastic beam shear relation uses first moment of area Q. For a rectangle, the maximum shear stress is 1.5 times the average.

Core relation

τ = VQ/(It)

Rectangle: τ_max = 3V/(2A)

Symbol	Meaning	SI units	US customary
V	Internal shear force	N	lbf
Q	First moment of area about neutral axis	m³	in³
I	Second moment of area	m⁴	in⁴
t	Local section thickness	m	in
τ	Shear stress	Pa, MPa	psi, ksi

Open Beam / Column Calculator Worked example ↓

Worked example — rectangular section

Given: V = 40 kN; rectangular area A = 0.020 m².

τavg = V/A = 2.0 MPa

τmax = 1.5×2.0 = 3.0 MPa

Answer: τmax = 3.0 MPa.

US check: ≈0.435 ksi.

The maximum for a rectangle occurs at the neutral axis.

Thin-walled open/closed sections and shear flow may be better handled with q = VQ/I. Steel-code web shear strength is a separate design check.

05 // Common elastic beam deflections

Closed-form deflection equations depend on support and loading. These examples assume constant E and I and small deflection.

Core relation

Simply supported, center P: δ_max = PL³/(48EI)

Simply supported UDL: δ_max = 5wL⁴/(384EI) ; Cantilever tip P: δ = PL³/(3EI)

Symbol	Meaning	SI units	US customary
P	Point load	N	lbf
w	Uniform line load	N/m	lbf/ft
L	Span	m	ft
E	Young’s modulus	Pa	psi
I	Second moment of area	m⁴	in⁴
δ	Elastic deflection	m, mm	in

Open Beam / Column Calculator Worked example ↓

Worked example — center-loaded simple beam

Given: P = 10 kN, L = 4.0 m, E = 200 GPa, I = 6.667×10⁻⁵ m⁴.

δmax = 10000×4³/(48×200×10⁹×6.667×10⁻⁵)

δmax ≈ 0.00100 m = 1.00 mm

Answer: δmax ≈ 1.00 mm.

US check: ≈0.0394 in.

Serviceability limits are project/code dependent.

Shear deformation, variable stiffness, composite action, cracking, creep and second-order effects can make these formulas inadequate.

06 // Circular-shaft torsion

For Saint-Venant torsion of a circular shaft, shear stress varies linearly with radius and twist depends on GJ.

Core relation

τ = Tr/J

θ = TL/(GJ)

Symbol	Meaning	SI units	US customary
T	Torque	N·m	lbf·in
r	Radius at evaluation point	m	in
J	Polar second moment	m⁴	in⁴
G	Shear modulus	Pa	psi
L	Shaft length	m	in
θ	Angle of twist	rad	rad or deg

Open Pipe Stress Calculator Worked example ↓

Worked example — 50 mm solid shaft

Given: T = 500 N·m, d = 50 mm, L = 1.0 m, G = 79 GPa.

J = π(0.05)⁴/32 = 6.136×10⁻⁷ m⁴

τmax = T(d/2)/J = 20.37 MPa

θ = TL/(GJ) = 0.01031 rad = 0.591°

Answer: τmax ≈ 20.4 MPa; θ ≈ 0.591°.

US check: τmax ≈2.95 ksi.

Noncircular torsion requires different torsion constants and stress distributions.

Keys, splines, shoulders and notches create stress concentrations and are not included in this nominal shaft solution.

07 // Euler elastic column buckling

For an ideal slender column, Euler’s critical load depends on flexural rigidity and effective length.

Core relation

P_cr = π²EI/(KL)²

λ = KL/r ; r = √(I/A)

Symbol	Meaning	SI units	US customary
Pcr	Euler critical load	N	lbf
K	Effective length factor	dimensionless	dimensionless
L	Unbraced/member length	m	ft
r	Radius of gyration	m, mm	in
E	Young’s modulus	Pa	psi
I	Buckling-axis second moment	m⁴	in⁴

Open Beam / Column Calculator Worked example ↓

Worked example — pin-ended ideal column

Given: E = 200 GPa, I = 8.0×10⁻⁶ m⁴, L = 3.0 m, K = 1.0.

Pcr = π²×200×10⁹×8×10⁻⁶ / 3² = 1.755×10⁶ N

Answer: Pcr ≈ 1.75 MN.

US check: ≈394 kip.

Euler buckling is not a general column design equation.

Real columns have yielding, residual stress, imperfections, eccentricity, local buckling and frame interaction. Use the governing design code, e.g. AISC 360-22 for structural steel.

08 // Von Mises equivalent stress

For ductile isotropic materials, von Mises stress is a useful scalar measure of multiaxial deviatoric stress for yield screening.

Core relation

σ_vm = √(σₓ² − σₓσᵧ + σᵧ² + 3τₓᵧ²)

Uniaxial + torsion: σ_vm = √(σ² + 3τ²)

Symbol	Meaning	SI units	US customary
σx,σy	Normal stresses	MPa	ksi
τxy	In-plane shear stress	MPa	ksi
σvm	Von Mises equivalent stress	MPa	ksi

Open Pipe Stress Calculator Worked example ↓

Worked example — plane-stress combination

Given: σx = 100 MPa, σy = 40 MPa, τxy = 30 MPa.

σvm = √(100² − 100×40 + 40² + 3×30²)

σvm = 101.49 MPa

Answer: σvm ≈ 101.5 MPa.

US check: ≈14.72 ksi.

Von Mises is most appropriate for ductile isotropic yielding; it is not a universal failure criterion.

Brittle materials, composites, soils, concrete and anisotropic materials need failure criteria appropriate to their behavior.

09 // Quick formula summary

Compact print reference. Use the detailed sections above for definitions and limitations.

Topic	Equation	Purpose	Tool
Axial	σ=P/A; δ=PL/AE	Axial stress/deformation	Material E
Section	I, Z, J	Geometry/stiffness	Beam/column
Bending	σ=M/Z	Elastic flexure	Beam/column
Shear	τ=VQ/It	Elastic transverse shear	Beam/column
Deflection	δ=f(P,w,L,E,I)	Serviceability	Beam/column
Torsion	τ=Tr/J; θ=TL/GJ	Circular shafts	Stress
Buckling	Pcr=π²EI/(KL)²	Ideal column instability	Beam/column
Von Mises	σvm	Ductile yield screening	Stress

10 // Assumptions & limitations

Fundamental equations are only useful when their assumptions match the actual problem.

11 // Technical references

AISC 360-22 is referenced for current structural-steel design context. The sheet intentionally stays with fundamental mechanics rather than reproducing code design equations without their applicability limits.

Structural Engineering Formula Sheet

Fast reference, with engineering context

Reference conventions

Simply supported beam under load

01 // Axial stress, strain and elongation

Concentric loading and linear elasticity are assumed.

02 // Common section properties

Use the axis corresponding to the actual bending direction.

03 // Elastic bending stress

Formula gives elastic stress, not code design strength.

04 // Beam transverse shear

The maximum for a rectangle occurs at the neutral axis.

05 // Common elastic beam deflections

Serviceability limits are project/code dependent.

06 // Circular-shaft torsion

Noncircular torsion requires different torsion constants and stress distributions.

07 // Euler elastic column buckling

Euler buckling is not a general column design equation.

08 // Von Mises equivalent stress

Von Mises is most appropriate for ductile isotropic yielding; it is not a universal failure criterion.

09 // Quick formula summary

10 // Assumptions & limitations

11 // Technical references

12 // Related EngineerHub tools

Related resources