Structural Analysis Fundamentals / reference

Notation, terms
and formulae.

Look up a concept and return to the module that explains it.

Teaching basis and review status

These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.

Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.

Read the lesson register and outstanding review notes

Anticlastic curvatureModule 9: The curvature that appears across the width of a beam when it is bent along its length, opposite in sign and equal to ν times the primary curvature. It is what makes a bent plate take up a saddle shape.
Bending momentModule 3: The internal moment at a section, equal to the algebraic sum of the moments of all forces to one side of it.Not to be confused: A bending moment diagram plots this against position along the beam; an influence line plots one fixed section's value against the position of a moving load.
Bredt's formulaModule 11: The shear flow in a closed thin-walled section under torsion, q = T/2Aₘ, constant round the perimeter.
BucklingModule 18: Failure by sudden lateral deflection under axial compression, governed by stiffness and geometry rather than by strength.Not to be confused: Squashing is a strength failure; buckling is a stability failure, and a member can buckle at a stress far below yield.
Carry-over factorModule 16: The fraction of a moment applied at one end of a member that appears at the other. It is 0.5 towards a fixed far end and zero towards a pinned one.
Castigliano's second theoremModule 15: The deflection at the point of application of a load, in its direction, equals the partial derivative of the total strain energy with respect to that load.
CompatibilityModule 16: A statement about how a structure must fit together geometrically. Compatibility conditions supply the equations that equilibrium cannot in an indeterminate structure.Not to be confused: Equilibrium is about forces balancing; compatibility is about geometry.
Composite actionModule 12: Two elements connected so that they cannot slip relative to one another, and so act as a single deeper section with one neutral axis.
Compound trussModule 4: Two or more simple trusses joined by three non-concurrent, non-parallel members or by a pin and a member. It is determinate, but the method of joints may not be able to start on it.
ContraflexureModule 3: A point where the bending moment changes sign, so the curvature reverses.
CreepModule 8: Deformation that continues to grow while the stress is held constant.Not to be confused: Relaxation is the same underlying process with the strain held constant and the stress falling.
DuctilityModule 8: The capacity to undergo large plastic deformation before fracture. Measured by percentage elongation and reduction in area.Not to be confused: Ductility is not strength — it buys warning and redistribution, not capacity.
Effective lengthModule 18: The length of an equivalent pin-ended strut with the same critical load, obtained by multiplying the actual length by a factor that depends on the end conditions.
EquilibriumModule 2: The condition that all forces and all moments on a body sum to zero. In a plane this gives three independent equations.
Euler critical loadModule 18: The axial load at which a perfectly straight, perfectly elastic pin-ended strut becomes unstable: π²EI/Lₑ².Not to be confused: It is an upper bound, not a capacity — real columns always fail below it.
FatigueModule 8: Progressive cracking under repeated loading, at stress ranges far below yield. Driven by the stress range and by the geometry of the detail.
Fixed-end momentModule 16: The moment developed at the end of a member whose ends are fully restrained against rotation, under a given loading.
Funicular shapeModule 6: The shape that carries a particular loading in pure axial force, with no bending. A parabola is funicular for a uniform load.Not to be confused: A shape is funicular for one loading only; change the load and bending appears.
Influence lineModule 17: A plot of one chosen effect at one chosen section against the position of a moving unit load.Not to be confused: A bending moment diagram fixes the load and moves the section. An influence line does the reverse.
Kinematic indeterminacyModule 16: The number of unknown joint displacements in a structure — what the stiffness method solves for.Not to be confused: Static indeterminacy counts unknown forces instead. A continuous beam is statically complex but kinematically simple; a truss is the reverse.
Macaulay bracketModule 13: The notation ⟨x − a⟩, defined as zero when x < a and as (x − a) otherwise, which lets one expression describe a beam with many loads.
Maxwell's reciprocal theoremModule 15: For a linearly elastic structure, the deflection at B due to a load at A equals the deflection at A due to the same load at B.
Modular ratioModule 12: The ratio of the two Young's moduli in a composite section, n = E₁/E₂. The transformed width is multiplied or divided by it.
Moment distributionModule 16: An iterative hand method: lock every joint, then release and balance them one at a time, distributing by stiffness and carrying over.
Müller-Breslau principleModule 17: The influence line for any effect is the deflected shape produced by releasing that effect and imposing a unit displacement.
Neutral axisModule 9: The line across a section on which the direct stress is zero. In pure bending it passes through the centroid; axial load or a product of inertia moves it.
Plastic hingeModule 9: A zone that has reached its fully plastic moment and can rotate at constant moment, allowing load to redistribute.
Poisson's ratioModule 7: The ratio of lateral contraction to axial extension, ν = −ε_lateral/ε_axial.
Principal axesModule 9: The pair of perpendicular axes about which the product of inertia vanishes, so that ordinary bending theory applies.
Principal stressesModule 14: The greatest and least direct stresses at a point, acting on the planes where the shear stress is zero.
Product of inertiaModule 9: The integral ∫xy dA. It can be negative, and it vanishes when the chosen centroidal axes are principal axes, including an axis aligned with section symmetry.
RedundantModule 16: An unknown force in excess of those equilibrium can determine. Releasing it gives a determinate structure that can be analysed.Not to be confused: Redundant means surplus to equilibrium, not useless — redundancy provides alternative load paths.
RelaxationModule 8: The fall in stress over time when the strain is held constant. The reason prestressing tendons lose force.
Second moment of areaModule 9: The integral ∫y² dA about an axis. It measures how far material is spread from that axis, and therefore the section's resistance to bending.
Section modulusModule 9: Z = I/y_max, packaging the geometry so that σ_max = M/Z.Not to be confused: The plastic modulus Z_p is the corresponding quantity for a fully yielded section, and is larger.
Shape factorModule 9: The ratio of plastic to yield moment, Z_p/Z. Exactly 1.5 for a rectangle, about 1.15 for a universal beam.
Shear centreModule 10: The point through which a transverse load must act if the section is to bend without twisting. It is where the resultant of the internal shear flows acts.Not to be confused: It coincides with the centroid only for doubly symmetric sections. For a channel it lies outside the section.
Shear flowModule 10: The shear force per unit length along a section, q = VQ/I, measured in force per unit length rather than as a stress.
Slenderness ratioModule 18: λ = Lₑ/r, the effective length divided by the radius of gyration. It decides whether a column squashes or buckles.
Slope-deflectionModule 16: A method expressing member end moments in terms of the joint rotations and sway, then solving joint equilibrium for those displacements.
Static indeterminacyModule 16: The excess of unknown forces over available equilibrium equations.
Strain energyModule 15: The recoverable energy stored in an elastic structure as it deforms, equal to the work done by the loads.
Stress trajectoryModule 14: A curve whose tangent at every point follows a principal stress direction. The two families cross everywhere at right angles.
SuperpositionModule 3: The principle that responses to separate loads may be added, valid for a linearly elastic structure undergoing small deflections.Not to be confused: It fails for slender members, where the axial force acting through the deflection adds moment.
Tension coefficientModule 4: t = T/L for a member, which turns joint equilibrium into sums of t·Δx and t·Δy with no trigonometry.
Transformed sectionModule 12: A composite section redrawn in one material, by scaling the width of the other material by the modular ratio.
Virtual workModule 15: The principle that for a structure in equilibrium, external virtual work equals internal virtual work — pairing an equilibrium force system with an unrelated compatible displacement system.
WarpingModule 11: Out-of-plane distortion of a cross-section under torsion. It occurs in every shape except a circle, and is why circular-shaft theory does not generalise.

Notation used throughout the course
Symbol	Meaning	Unit	Introduced
A	Cross-sectional area	mm²	Module 7
Aₘ	Area enclosed by the mean wall line of a closed section	mm²	Module 11
E	Young's modulusAbout 205 000 for steel, 70 000 for aluminium	N/mm²	Module 7
EI	Flexural rigidityResistance to bending; the product, not either factor alone	N·mm²	Module 13
G	Shear modulusG = E/2(1 + ν); about 79 000 for steel	N/mm²	Module 7
GJ	Torsional rigidity	N·mm²	Module 11
H	Horizontal thrust of a cable or arch	kN	Module 5
I	Second moment of area	mm⁴	Module 9
I_xy	Product of inertiaZero when the chosen centroidal axes are principal axes, including an axis aligned with section symmetry; can be negative	mm⁴	Module 9
I₁, I₂	Principal second moments of area	mm⁴	Module 9
J	Polar second moment of area, or torsion constantπd⁴/32 for a solid circle only; other shapes need their own constant	mm⁴	Module 11
K	Bulk modulusK = E/3(1 − 2ν)	N/mm²	Module 8
L	Length or span	mm or m	Module 3
Lₑ	Effective length of a columnActual length times the end-condition factor	mm	Module 18
M	Bending moment	kNm or N·mm	Module 3
M_p	Fully plastic momentσ_y Z_p	kNm	Module 9
M_y	Yield momentσ_y Z	kNm	Module 9
N	Normal (axial) forceTension positive throughout this course	kN	Module 3
P	Point load or axial force	kN	Module 2
P_cr	Euler critical load	kN	Module 18
Q	First moment of area about the neutral axisFor the part of the section beyond the level considered	mm³	Module 10
R	Reaction, or radius of curvature	kN or mm	Module 2
T	Torque	kNm or N·mm	Module 11
U	Strain energy	N·mm or J	Module 15
V	Shear force	kN	Module 3
Z	Elastic section modulusI divided by the distance to the extreme fibre	mm³	Module 9
Z_p	Plastic section modulusFirst moment of the two equal areas about the equal-area axis	mm³	Module 9
a	Distance to a section or load	m	Module 3
b	Width	mm	Module 9
d	Depth, diameter, or effective depth	mm	Module 9
e	Eccentricity	mm	Module 9
h	Rise of an arch, or sag of a cable, or height	m	Module 5
j	Number of joints in a truss	—	Module 4
k	Stiffness, or shear form factor	varies	Module 16
m	Number of members in a truss	—	Module 4
n	Modular ratio, or member force under a unit load	—	Module 12
q	Shear flowForce per unit length along the section, not a stress	N/mm	Module 10
r	Radius, or radius of gyrationr = √(I/A)	mm	Module 18
t	Thickness, or tension coefficient	mm or kN/m	Module 4
v	Deflection	mm	Module 13
w	Distributed load intensity	kN/m or N/mm	Module 3
x, y, z	Coordinates	mm or m	Module 1
α	Coefficient of thermal expansion, or an angleAbout 12 × 10⁻⁶ per °C for steel	per °C	Module 8
γ	Engineering shear strainThe change in a right angle. Use γ/2 in any transformation	radians	Module 7
δ	Deflection or extension	mm	Module 7
ε	Direct strainDimensionless; often quoted in microstrain, 10⁻⁶	—	Module 7
η	Influence line ordinateDimensionless for reaction and shear; length for moment	varies	Module 17
θ	Rotation, or angle of twist	radians	Module 11
λ	Slenderness ratioλ = Lₑ/r	—	Module 18
ν	Poisson's ratioAbout 0.30 for steel; cannot exceed 0.5	—	Module 7
σ	Direct stressTension positive	N/mm²	Module 7
σ_y	Yield stress	N/mm²	Module 8
τ	Shear stress	N/mm²	Module 7
ψ	Sway angle in slope-deflection	radians	Module 16

Direct stress

Stress and strain · Module 7

\sigma = \frac{P}{A}

Assumes: Uniform stress over the area, away from load introduction points

Direct strain

Stress and strain · Module 7

\varepsilon = \frac{\delta}{L}

Assumes: Small, uniform strain

Hooke's law

Stress and strain · Module 7

\sigma = E\varepsilon

Assumes: Elastic behaviour only, below the limit of proportionality

Axial extension

Stress and strain · Module 7

\delta = \frac{PL}{AE}

Assumes: Constant P, A and E along the member

Hooke's law in shear

Stress and strain · Module 7

\tau = G\gamma

Assumes: Elastic behaviour

Poisson's ratio

Stress and strain · Module 7

\nu = -\frac{\varepsilon_{lat}}{\varepsilon_{ax}}

Assumes: Isotropic material, uniaxial stress

Shear modulus from E and ν

Elastic constants · Module 8

G = \frac{E}{2(1+\nu)}

Assumes: Isotropic, linearly elastic material

Bulk modulus from E and ν

Elastic constants · Module 8

K = \frac{E}{3(1-2\nu)}

Assumes: Isotropic material; ν cannot exceed 0.5

Thermal strain

Elastic constants · Module 8

\varepsilon_T = \alpha\,\Delta T

Assumes: The member is free to move, so no stress arises

Fully restrained thermal stress

Elastic constants · Module 8

\sigma = E\alpha\,\Delta T

Assumes: Complete restraint — an upper bound. Independent of length and area

Impact factor

Elastic constants · Module 8

n = 1 + \sqrt{1 + \frac{2h}{\delta_{st}}}

Assumes: Elastic behaviour, no energy lost. Gives 2 for a suddenly applied load with no fall

Thin cylinder, hoop stress

Pressure vessels · Module 7

\sigma_h = \frac{pr}{t}

Assumes: Thin wall, r/t above about 20, away from ends

Thin cylinder, longitudinal stress

Pressure vessels · Module 7

\sigma_l = \frac{pr}{2t}

Assumes: As above. Always exactly half the hoop stress

Load sharing between parallel members

Pressure vessels · Module 7

P_i = P\,\frac{k_i}{\sum k}, \quad k_i = \frac{A_iE_i}{L_i}

Assumes: The members deform together

Truss determinacy, plane

Trusses · Module 4

m + r = 2j

Assumes: Necessary but not sufficient — arrangement matters too

Truss determinacy, space

Trusses · Module 4

m + r = 3j

Assumes: As above, in three dimensions

Truss chord force from beam actions

Trusses · Module 4

F = \frac{M}{d}

Assumes: Parallel chords, loads applied at joints

Bending stress

Bending · Module 9

\sigma = \frac{My}{I}

Assumes: Elastic, plane sections remain plane, symmetric about the plane of loading

Section modulus

Bending · Module 9

Z = \frac{I}{y_{max}}, \quad \sigma_{max} = \frac{M}{Z}

Assumes: As above

Combined axial and bending

Bending · Module 9

\sigma = \frac{P}{A} \pm \frac{My}{I}

Assumes: Small deflections; fails for slender members

Middle-third rule

Bending · Module 9

e \le \frac{Z}{A} = \frac{h}{6}

Assumes: Rectangular section; the condition for no tension anywhere

General bending formula

Bending · Module 9

\sigma = \frac{(M_xI_{yy} - M_yI_{xy})y + (M_yI_{xx} - M_xI_{xy})x}{I_{xx}I_{yy} - I_{xy}^2}

Assumes: Centroidal axes, no twist. Reduces to My/I when I_xy = 0

Principal second moments

Bending · Module 9

I_{1,2} = \frac{I_{xx}+I_{yy}}{2} \pm \sqrt{\left(\frac{I_{xx}-I_{yy}}{2}\right)^2 + I_{xy}^2}

Assumes: Same algebra as Mohr's circle for stress

Plastic moment

Bending · Module 9

M_p = \sigma_y Z_p

Assumes: Ductile material, section stocky enough not to buckle locally

Shear stress in a beam

Shear · Module 10

\tau = \frac{VQ}{It}

Assumes: Shear stress uniform across the width at that level

Shear flow

Shear · Module 10

q = \frac{VQ}{I}

Assumes: Force per unit length, not a stress

Peak shear stress, rectangle

Shear · Module 10

\tau_{max} = \frac{3V}{2A}

Assumes: Rectangular section; 1.5 times the average

Torsion formula

Torsion · Module 11

\frac{T}{J} = \frac{\tau}{r} = \frac{G\theta}{L}

Assumes: CIRCULAR section only — solid or hollow

Polar second moment, solid circle

Torsion · Module 11

J = \frac{\pi d^4}{32}

Assumes: Circular section only

Bredt shear flow

Torsion · Module 11

q = \frac{T}{2A_m}

Assumes: Closed thin-walled section

Torsion constant, thin open section

Torsion · Module 11

J = \tfrac{1}{3}\sum b t^3

Assumes: Narrow rectangles, free warping. Note the cube on the thickness

Torsion constant, closed thin section

Torsion · Module 11

J = \frac{4A_m^2 t}{s}

Assumes: Closed loop; hundreds of times stiffer than the same material slit

Fully plastic torque, solid circle

Torsion · Module 11

T_p = \frac{2\pi\tau_y r^3}{3}

Assumes: Ductile material; shape factor is exactly 4/3

Transformed width

Composite sections · Module 12

b_{tr} = \frac{b}{n}, \quad n = \frac{E_1}{E_2}

Assumes: Bonded interface, so strain is continuous

Cracked RC neutral axis

Composite sections · Module 12

\frac{bx^2}{2} = nA_s(d-x)

Assumes: Concrete carries no tension; elastic behaviour

Moment–curvature

Deflection · Module 13

\frac{1}{R} = \frac{M}{EI}

Assumes: Elastic, small deflections

Euler–Bernoulli beam equation

Deflection · Module 13

EI\frac{d^2v}{dx^2} = M

Assumes: Small slopes; bending deformation only

Cantilever, end load

Deflection · Module 13

\delta = \frac{PL^3}{3EI}

Assumes: Uniform EI, bending only

Simply supported, central point load

Deflection · Module 13

\delta = \frac{PL^3}{48EI}

Assumes: Uniform EI

Simply supported, full UDL

Deflection · Module 13

\delta = \frac{5wL^4}{384EI}

Assumes: Uniform EI

Shear deflection, cantilever

Deflection · Module 13

\delta_s = \frac{kVL}{GA}

Assumes: k = 6/5 rectangle, 10/9 circle. Negligible unless the member is stubby

Stress on an inclined plane

Complex stress · Module 14

\sigma_\theta = \frac{\sigma_x+\sigma_y}{2} + \frac{\sigma_x-\sigma_y}{2}\cos 2\theta + \tau_{xy}\sin 2\theta

Assumes: Plane stress. The angle is doubled

Principal stresses

Complex stress · Module 14

\sigma_{1,2} = \frac{\sigma_x+\sigma_y}{2} \pm \sqrt{\left(\frac{\sigma_x-\sigma_y}{2}\right)^2 + \tau_{xy}^2}

Assumes: Plane stress

Mohr's circle of strain

Complex stress · Module 14

R = \sqrt{\left(\frac{\varepsilon_x-\varepsilon_y}{2}\right)^2 + \left(\frac{\gamma_{xy}}{2}\right)^2}

Assumes: Uses HALF the engineering shear strain — the commonest error in the topic

Von Mises equivalent stress

Complex stress · Module 14

\sigma_e = \sqrt{\sigma_1^2 - \sigma_1\sigma_2 + \sigma_2^2}

Assumes: Ductile material, plane stress

Strain energy, axial

Energy methods · Module 15

U = \frac{P^2L}{2AE}

Assumes: Constant force along the member

Strain energy, bending

Energy methods · Module 15

U = \int \frac{M^2}{2EI}\,dx

Assumes: Bending only

Strain energy, torsion

Energy methods · Module 15

U = \frac{T^2L}{2GJ}

Assumes: Constant torque

Unit-load method, trusses

Energy methods · Module 15

\delta = \sum \frac{NnL}{AE}

Assumes: Pin-jointed, axial force only

Unit-load method, beams

Energy methods · Module 15

\delta = \int \frac{Mm}{EI}\,dx

Assumes: Bending dominates; consistent sign convention for M and m

Castigliano's second theorem

Energy methods · Module 15

\delta = \frac{\partial U}{\partial P}

Assumes: The load must act at the point wanted, or use a dummy load