Structural Analysis Fundamentals

Reference

Every symbol, term and formula the course uses, in one searchable place.

This is a companion to the lessons, not a substitute for them. Every entry names the module that teaches it, so you can go back to where the idea is explained rather than take a formula on trust.

Each formula carries its assumptions. That is deliberate: most errors in structural analysis come not from getting the algebra wrong but from using a correct formula outside the conditions it was derived under. The torsion formula applied to a rectangle, or superposition applied to a slender column, will both give you a confident and wrong answer.

Search filters all three sections at once — the tab counts update as you type, so you can see where else a term appears.

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 whenever the section has an axis of 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 whenever the section has an axis of 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

σ = \frac{P}{A}

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

Direct strain

Stress and strain · Module 7

ε = \frac{δ}{L}

Assumes: Small, uniform strain

Hooke's law

Stress and strain · Module 7

σ = E ε

Assumes: Elastic behaviour only, below the limit of proportionality

Axial extension

Stress and strain · Module 7

δ = \frac{P L}{A E}

Assumes: Constant P, A and E along the member

Hooke's law in shear

Stress and strain · Module 7

τ = G γ

Assumes: Elastic behaviour

Poisson's ratio

Stress and strain · Module 7

ν = - \frac{ε _{l a t}}{ε _{a x}}

Assumes: Isotropic material, uniaxial stress

Shear modulus from E and ν

Elastic constants · Module 8

G = \frac{E}{2 ( 1 + ν )}

Assumes: Isotropic, linearly elastic material

Bulk modulus from E and ν

Elastic constants · Module 8

K = \frac{E}{3 ( 1 - 2 ν )}

Assumes: Isotropic material; ν cannot exceed 0.5

Thermal strain

Elastic constants · Module 8

ε_{T} = α Δ T

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

Fully restrained thermal stress

Elastic constants · Module 8

σ = E α Δ T

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

Impact factor

Elastic constants · Module 8

n = 1 + 1 + \frac{2 h}{δ _{s t}}

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

Thin cylinder, hoop stress

Pressure vessels · Module 7

σ_{h} = \frac{p r}{t}

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

Thin cylinder, longitudinal stress

Pressure vessels · Module 7

σ_{l} = \frac{p r}{2 t}

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}, k_{i} = \frac{A _{i} E _{i}}{L _{i}}

Assumes: The members deform together

Truss determinacy, plane

Trusses · Module 4

m + r = 2 j

Assumes: Necessary but not sufficient — arrangement matters too

Truss determinacy, space

Trusses · Module 4

m + r = 3 j

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

σ = \frac{M y}{I}

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

Section modulus

Bending · Module 9

Z = \frac{I}{y _{ma x}}, σ_{ma x} = \frac{M}{Z}

Assumes: As above

Combined axial and bending

Bending · Module 9

σ = \frac{P}{A} \pm \frac{M y}{I}

Assumes: Small deflections; fails for slender members

Middle-third rule

Bending · Module 9

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

Assumes: Rectangular section; the condition for no tension anywhere

General bending formula

Bending · Module 9

σ = \frac{( M _{x} I _{y y} - M _{y} I _{x y} ) y + ( M _{y} I _{xx} - M _{x} I _{x y} ) x}{I _{xx} I _{y y} - I _{x y}^{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 _{y y}}{2} \pm (\frac{I _{xx} - I _{y y}}{2})^{2} + I_{x y}^{2}

Assumes: Same algebra as Mohr's circle for stress

Plastic moment

Bending · Module 9

M_{p} = σ_{y} Z_{p}

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

Shear stress in a beam

Shear · Module 10

τ = \frac{V Q}{I t}

Assumes: Shear stress uniform across the width at that level

Shear flow

Shear · Module 10

q = \frac{V Q}{I}

Assumes: Force per unit length, not a stress

Peak shear stress, rectangle

Shear · Module 10

τ_{ma x} = \frac{3 V}{2 A}

Assumes: Rectangular section; 1.5 times the average

Torsion formula

Torsion · Module 11

\frac{T}{J} = \frac{τ}{r} = \frac{Gθ}{L}

Assumes: CIRCULAR section only — solid or hollow

Polar second moment, solid circle

Torsion · Module 11

J = \frac{π d ^{4}}{32}

Assumes: Circular section only

Bredt shear flow

Torsion · Module 11

q = \frac{T}{2 A _{m}}

Assumes: Closed thin-walled section

Torsion constant, thin open section

Torsion · Module 11

J = \frac{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{4 A _{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 π τ _{y} r ^{3}}{3}

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

Transformed width

Composite sections · Module 12

b_{t r} = \frac{b}{n}, n = \frac{E _{1}}{E _{2}}

Assumes: Bonded interface, so strain is continuous

Cracked RC neutral axis

Composite sections · Module 12

\frac{b x ^{2}}{2} = n A_{s} (d - x)

Assumes: Concrete carries no tension; elastic behaviour

Moment–curvature

Deflection · Module 13

\frac{1}{R} = \frac{M}{E I}

Assumes: Elastic, small deflections

Euler–Bernoulli beam equation

Deflection · Module 13

E I \frac{d ^{2} v}{d x ^{2}} = M

Assumes: Small slopes; bending deformation only

Cantilever, end load

Deflection · Module 13

δ = \frac{P L ^{3}}{3 E I}

Assumes: Uniform EI, bending only

Simply supported, central point load

Deflection · Module 13

δ = \frac{P L ^{3}}{48 E I}

Assumes: Uniform EI

Simply supported, full UDL

Deflection · Module 13

δ = \frac{5 w L ^{4}}{384 E I}

Assumes: Uniform EI

Shear deflection, cantilever

Deflection · Module 13

δ_{s} = \frac{k V L}{G A}

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

Stress on an inclined plane

Complex stress · Module 14

σ_{θ} = \frac{σ _{x} + σ _{y}}{2} + \frac{σ _{x} - σ _{y}}{2} cos 2 θ + τ_{x y} sin 2 θ

Assumes: Plane stress. The angle is doubled

Principal stresses

Complex stress · Module 14

σ_{1, 2} = \frac{σ _{x} + σ _{y}}{2} \pm (\frac{σ _{x} - σ _{y}}{2})^{2} + τ_{x y}^{2}

Assumes: Plane stress

Mohr's circle of strain

Complex stress · Module 14

R = (\frac{ε _{x} - ε _{y}}{2})^{2} + (\frac{γ _{x y}}{2})^{2}

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

Von Mises equivalent stress

Complex stress · Module 14

σ_{e} = σ_{1}^{2} - σ_{1} σ_{2} + σ_{2}^{2}

Assumes: Ductile material, plane stress

Strain energy, axial

Energy methods · Module 15

U = \frac{P ^{2} L}{2 A E}

Assumes: Constant force along the member

Strain energy, bending

Energy methods · Module 15

U = \int \frac{M ^{2}}{2 E I} d x

Assumes: Bending only

Strain energy, torsion

Energy methods · Module 15

U = \frac{T ^{2} L}{2 G J}

Assumes: Constant torque

Unit-load method, trusses

Energy methods · Module 15

δ = \sum \frac{N n L}{A E}

Assumes: Pin-jointed, axial force only

Unit-load method, beams

Energy methods · Module 15

δ = \int \frac{M m}{E I} d x

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

Castigliano's second theorem

Energy methods · Module 15

δ = \frac{\partial U}{\partial P}

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