Structural Engineering & Statics
where the load goes
Concept

Modular ratio — where it appears

The ratio of two materials' elastic moduli, which is what decides their shares of a moment and has nothing to do with their strengths. It changes with time as one material creeps and the other does not, so a composite section has one set of properties on the day it is built and another a decade later.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

The neutral axis is wherever the first moment vanishes. A 300 by 500 section with 1200 mm² of steel at a depth of 450, carrying 150 kNm after the tension side has cracked. The neutral axis is no longer at mid-depth: it has risen to 137.0 mm from the top, which is where the first moment of the compression zone plus the transformed steel vanishes. The compression is 18.0 N/mm² at the top fibre and the steel carries 309 N/mm²; the resulting couple is 371 kN on a lever arm of 404 mm, which multiplies back to the 150 kNm applied. The uncracked section would have had 3422×10⁶ mm⁴ against the cracked 1139×10⁶ — a loss of 67% of the stiffness.

When half the section has given up

Bending theory puts the neutral axis through the centroid. That is a consequence, not a rule — and when the tension side cracks, the same reasoning moves the axis somewhere else entirely.

sections · Bending stress The deflection that arrives years late. The multiplier on a concrete member's deflection under a sustained load, against time. The elastic deflection arrives on the day the load does and is the 1.0 at the left. After a year it has been multiplied by 3.00, after five years by 3.29, and it approaches 3.38. Nothing has been added to the load and nothing about the strength has changed: this is a serviceability failure arriving on a structure that passed every strength check on the day it was built.

The deflection that arrives three years late

A concrete beam that passes every check on the day it is built goes on deflecting for a decade, and ends up three times where it started. Nothing about the load changed, and nothing about the strength was ever in question.

materials · Creep The same strain, two moduli, and a width multiplied to say so. A timber section with a steel plate in it, carrying 20.0 kNm. Plane sections stay plane, so the strain at a height is the same in both materials; Hooke's law then puts the stresses in the ratio of the moduli, which here is 19.09. Multiplying the stiffer material's WIDTH by that ratio gives a fictitious section of one material with the same neutral axis and the same forces — 595.2×10⁶ mm⁴ of it, against 351.0 for the same shape with the moduli ignored. The steel plate is 3.8% of the area and carries 43% of the moment, at 96 N/mm² against the timber's 5.0. The transform is not an approximation: it is compatibility and Hooke's law written down.

A section made of two materials, one of them pretended away

Multiplying a material's width by the ratio of the moduli produces a fictitious section of one material with the right neutral axis and the right forces. It is not a trick — it is compatibility and Hooke's law written down — and it says a stiff material takes what its modulus asks for.

sections · Transformed section The props decide where the stress ends up. Bottom-fibre stress in the steel of a 12 m composite beam carrying 6 kN/m of wet concrete and 9 kN/m afterwards. Unpropped, the bare steel takes the first stage alone and reaches 146 MPa; propped, the finished composite section takes everything and reaches 93 MPa — a ratio of 1.57. 62% of the unpropped beam's final stress was locked in before the slab was structural at all. The deflections differ by 1.73 times for the same reason, and no drawing of the finished beam distinguishes the two.

The section that changed while it was being loaded

A stress is computed from a moment and a section modulus. When part of the moment arrived while the section was a different shape, there is no single section modulus to divide by — the stresses add and the properties do not, and two identical finished beams can differ by half again in stress with nothing on the drawing to say which is which.

sections · Staged section The neutral axis is wherever the first moment vanishes. A 300 by 600 section with 1800 mm² of steel at a depth of 540, carrying 250 kNm after the tension side has cracked. The neutral axis is no longer at mid-depth: it has risen to 234.5 mm from the top, which is where the first moment of the compression zone plus the transformed steel vanishes. The compression is 15.4 N/mm² at the top fibre and the steel carries 301 N/mm²; the resulting couple is 541 kN on a lever arm of 462 mm, which multiplies back to the 250 kNm applied. The uncracked section would have had 6673×10⁶ mm⁴ against the cracked 3809×10⁶ — a loss of 43% of the stiffness.

Where the steel is, not how much of it

A section in bending resists a moment with a couple, and a couple is a force times a distance. The force is bought — it is an area of steel at a stress. The distance is free, decided by where the bars were put, and it is the variable almost nobody optimises because it does not appear on an order.

sections · Lever arm The least reliable number in the material decides the answer, briefly. What a twenty per cent error in the concrete's tensile strength does to a computed deflection, against how far past cracking the beam is. Well past the cracking moment it does almost nothing — at 1.9 times M_cr the spread is 56 per cent — because the section is nearly fully cracked and the interpolation has run out. Just above cracking it does everything: at 1.19 times M_cr the same twenty per cent moves the deflection by 3658 per cent. Tensile strength is the property with the widest scatter and the least direct test, and a beam designed to sit near its cracking moment has put the answer on it.

The curvature nobody applied

Concrete shrinks as it dries, by about half a millimetre in every metre. In a symmetrically reinforced member that is a shortening and nothing else. In a member with more steel in one face than the other — which is every beam and every slab — the steel holds one side back and the section bends, with no load on it at all.

deflection · Shrinkage curvature

Named alongside it

The objects these essays reach for when they reach for this one.

Cracked sectionCreepNeutral axisPlane sectionsSecond moment of areaTransformed sectionCamberComposite actionDeflectionDifferential shorteningElastic modulusLever arm

All concepts