Structural Engineering & Statics
where the load goes
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The load must go somewhere

Every force applied to a structure reaches the ground by some route. Choosing that route is most of what design is, and tracing it is most of what analysis is.
A beam, its loads and its reactions. A free body: the beam cut away from everything it touches, with the forces that were doing the touching drawn on it. The reactions are computed from the loads, so their arrows are to scale relative to each other. Equilibrium

Everything adds to nothing, and that is the whole of statics

A structure that stays put obeys two statements — the forces on it sum to zero, and so do the moments. Every number in the subject comes out of those two sentences.

 The same beam, cut at x = 5. A beam separated at one station. On the exposed face a shear force and a bending moment appear, equal and opposite on the two pieces, with values obtained by summing the forces on whichever piece is easier. Equilibrium

The free body is a choice, and choosing it well is the whole skill

Cutting a structure open is not a step in the method. It is the method — and where the cut is made decides whether the answer takes one line or twenty.

 A truss cut through panel 3. The truss severed through one panel, with everything to the right removed and the three cut member forces drawn on the exposed faces. Taking moments about the marked joint removes two of the three unknowns, so one equation gives the third: -52.94. Equilibrium

Answering one question without solving the rest

A truss of fifty members can be interrogated about one of them. Cut through three, take moments about the point where two of them meet, and the third falls out in a single line.

 A Pratt truss of 6 panels. A Pratt truss under equal panel-point loads. The joint equilibrium equations were assembled and solved; 10 members came out in tension, 9 in compression and 2 carrying nothing. Structural form

The triangle that cannot fold, and everything built out of it

A square of pinned bars is a mechanism. A triangle is not, and that single fact is the reason trusses exist and the reason they look the way they do.

 The cable and the arch are the same curve. The shape a cable takes under a uniform load is a parabola, and it carries that load in pure tension. Reflected, the identical curve carries the same load in pure compression, which is what an arch is. Structural form

The shape that carries itself, and the arch that is its reflection

Hang a chain and it takes the one shape that carries its load in pure tension. Turn the shape upside down and it carries the same load in pure compression. That is what an arch is.

 A line of thrust, and the masonry it has to stay inside. An arch ring of 9% of the span in thickness, rising 28% of the span, under its own weight as a uniform load. Any horizontal thrust between 3.85 and 5.23 puts a line of compression entirely inside the masonry, so the arch stands — and which of them it actually takes is not decided by statics. The two extremes are drawn: the minimum-thrust line, which rides high at the crown and low at the haunches, and the maximum-thrust line, which does the opposite. Structural form

The line that must stay inside

A masonry arch does not stand because its shape is right. It stands because some line of compression can be drawn inside the stonework — any one will do, and there are infinitely many to choose from.

 The same beam, cut at x = 5. A beam separated at one station. On the exposed face a shear force and a bending moment appear, equal and opposite on the two pieces, with values obtained by summing the forces on whichever piece is easier. Internal forces

What a cut reveals, and why it was there all along

Cut a beam anywhere and two quantities appear on the face — a shear force and a bending moment. Nothing was applied there. They are what the material was already doing.

 Load, shear and moment — a simple span. The applied load, the shear force it produces and the bending moment that follows, drawn one above another to the same horizontal scale. Shear is the integral of the load and moment is the integral of shear. Internal forces

The diagram is an integral, and that is why it can be drawn by eye

Load, shear and moment are one function and its two integrals. Once that is seen, the diagrams stop being things to calculate and become things to sketch.

 Where to put the supports. Peak sagging and hogging moment for a uniformly loaded beam, against how far the supports are moved in from the ends. The best arrangement is where the two curves cross, and it is nowhere near the ends. Internal forces

Where to put the supports, which is not at the ends

Moving the supports of a uniformly loaded beam inward by about a fifth of its length halves the worst bending moment. The load has not changed and nor has the beam.

 3 continuous spans against 3 simple ones. The bending moment in a continuous beam, solved by the stiffness method, drawn over the moment in the same spans made simply supported. The peak sagging moment falls from 30.6 to 19.6, and a hogging moment of 24.5 appears over the supports where there was none. Internal forces

The moment over the support, and what it buys

Run a beam over its supports instead of stopping at each one, and the mid-span moment falls by a third while a new moment appears where there was none. Nothing was added but continuity.

 Influence line for the bending moment at x = 3. The bending moment at one fixed station, plotted against the position of a unit load walking across the span. The beam was re-solved at 301 load positions. The worst position is x = 3.00, giving 2.100. Internal forces

The worst place to stand

A bridge is not designed for a load. It is designed for a load that moves, and for every station along it there is a different position of that load that does the most damage.

 One slit, and the torsional stiffness falls by a factor of hundreds. A 200 by 200 box of 8 mm wall, drawn closed and then slit along its length. Closed, the torque runs round the wall as a shear flow and the torsion constant is 5.66×10⁷ mm⁴; slit, the loop is broken and only each wall's own thickness resists, giving 1.31×10⁵ mm⁴. The ratio is 432 to one, so the same torque twists the slit section 432 times as far and raises a peak shear stress 36 times as high. Nothing about the material changed. Internal forces

The moment that will not lie flat

A plane cut exposes three actions. A real cut exposes six, and the fourth of them behaves unlike the others — torsion is resisted by a loop of shear, and one slit down the length of a tube destroys it.

 The worst position is not the obvious one. Three axles totalling 320 units, marched across a span of 20 in steps of 0.02. The envelope is the largest moment each station ever sees; its peak is 1160.2 at 9.88 along the span, which is 0.12 off midspan and occurs under the axle nearest the resultant rather than under the heaviest one. Barré's construction, which places midspan halfway between that axle and the resultant, independently gives 1160.3 at 9.88. The dashed curve is the envelope the same total weight would produce as one load rather than three: its peak is 1600.0, which is 38% more — spreading a load out is worth something. Internal forces

The train that is worse than its heaviest axle

An influence line says where to stand one load. A vehicle is several loads at fixed spacings, and the worst arrangement never puts the heaviest one at the peak.

 Shear stress across a section. The distribution of shear stress over an I-section, computed as VQ/It by accumulating the first moment of the area above every height. The peak is 0.41 against a mean of 0.19 — a ratio of 2.15 — and it falls at the neutral axis, where the bending stress is zero. Sections and stress

The shear nobody draws

A stack of loose planks slides at its ends when it is loaded. Glue them and the sliding stops — and whatever the glue is now carrying is a stress that no bending calculation contains.

 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. Sections and stress

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.

 Shear stress across a section. The distribution of shear stress over an I-section, computed as VQ/It by accumulating the first moment of the area above every height. The peak is 82.77 against a mean of 38.46 — a ratio of 2.15 — and it falls at the neutral axis, where the bending stress is zero. At the junction between flange and web the flow is 820.0 per unit length, so connectors of 20000 each have to be spaced no further apart than 24 — which is what turns two pieces into one section. Sections and stress

The section made of pieces

Two plates bolted together are twice as strong as one plate. Two plates made to act as one section are eight times as stiff. The difference is entirely in the joint, and the joint has a calculable spacing.

 Where the peak moment goes when a restraint is added. The same uniformly loaded beam with three sets of restraints, and the bending moment in each. Adding restraint moves moment from mid-span to the supports and lowers the peak — but only the first case can be solved by statics. Deflection

One support too many, and what it costs to know

Add a redundant restraint and the load has two routes to the ground. Equilibrium cannot say how it splits, and the answer turns out to depend on stiffness — which is a different kind of question.

 The water that will not run off. Water depth against sag at the middle of the bay, for a roof bay starting with a reversed construction camber of 0.01. There is no critical value to reach: the deflection grows from the first increment, slowly at first and then without bound as the ratio approaches 1.00 — which is the depth at which the bay fills faster than it stiffens, and which the roof bay therefore never attains. The perfect roof bay, drawn for comparison, sits on the vertical axis until it arrives there and then has no answer at all. Deflection

The water that will not run off

A flat roof deflects, the deflection makes room for water, the water deepens the deflection. It is the same equation as a buckling column, with rain in place of the axial load.

 Two materials pulled until they stop. Two stress-strain curves — mild steel, cast iron — plotted to a strain of 2.0%. One of them has a plateau, so the stress at which yielding starts is something the specimen does rather than something anyone chooses. No offset construction is drawn. Materials

The property that appears in none of the equations

Ductility is in no design formula on this site. Every method on this site depends on it — and a brittle structure does not merely fail early, it makes the analysis wrong.

 A I-section at 80% of its plastic moment. The same I-section drawn three ways: the shape, the strain across its depth, and the stress that strain produces in mild steel. The strain diagram is a straight line, because plane sections stay plane whatever the material is doing. The stress diagram is not: 0% of the area has yielded, working inward from both faces, and the neutral axis sits at 70.9 mm against a centroid at 100.0 mm. The compression resultant is 322.5 kN and the tension resultant 322.5 kN, on a lever arm of 180.1 mm, which multiplies back to the 58.1 kNm the section is carrying. A rolled residual stress pattern of ±30% of yield is locked in before any load arrives. Materials

The stress that was there before the load

A rolled steel section leaves the mill carrying eighty N/mm² of stress with nothing applied to it, in a pattern that sums to no force and no moment. It is invisible to every calculation and it is the knee in every column curve.

 Two ways to fail, and the curve between them. The exact plastic interaction between axial force and moment for two sections of identical area, both normalised by their own squash load and their own plastic moment. The rectangle stands 25.0% of its plastic moment outside the straight line at an axial ratio of 0.50; The I-section stands 6.0% of its plastic moment outside the straight line at an axial ratio of 0.12. Every section here is symmetric about its centroid, so the equal-area axis and the centroid coincide and it makes no difference which the moments are taken about. The straight line is the rule that says the two capacities share out in proportion, and everything between it and a curve is capacity that rule gives away. Materials

Two ways to fail, and the curve between them

A column carrying both compression and bending has two capacities and a rule for sharing them out. The rule is a straight line, the truth is a curve, and for a rectangle the straight line gives away a quarter of the plastic moment at half the squash load.

 What is left when the load comes off. Mild steel taken to a strain of 0.60% and then unloaded to zero stress, at which point the strain has not returned to zero: 0.469% of it is permanent. Materials

What is left when the load comes off

Unload a section that has yielded and it does not return to nothing. It returns to a self-equilibrating stress field it did not have before, a permanent set, and an elastic range wider than the one it started with.

 Three details, and no material anywhere on the plot. Stress range against cycles to failure for three detail categorys — 160, 90, 36 N/mm² at two million cycles. The lines are parallel because they share a slope of three, and the spread between them is a factor of 4.4 in stress and therefore 88 in life. Nothing on this plot depends on the strength of the steel: the same detail in a grade twice as strong lies on the same line. At a stress range of 70 N/mm² the lives are 160: 6.8e+7, 90: 4.3e+6, 36: 2.7e+5 cycles. The knee in each line is the constant-amplitude limit, past which the slope becomes five. Materials

The load that never came near failing anything

A detail survives sixty-eight million cycles at a stress range that another detail in the same steel survives two hundred and seventy thousand of. The two lie a factor of two hundred and fifty apart, and the material is not on the plot anywhere.

 The hour that is really a temperature. The retention factors for carbon steel against temperature: the yield stress and the elastic modulus. The modulus falls away first — at 500°C the steel has kept 78% of its strength and 60% of its stiffness — so a member's failure mode can change during a fire. A member working at 60% of its cold capacity runs out of strength at 558°C, and out of the stiffness for the same ratio at 500°C, 58 degrees earlier. There is nothing about time in any of it: a fire rating is a temperature the member must not reach, converted into the minutes a particular fire takes to get it there. Materials

The hour that is really a temperature

A fire rating is quoted in minutes and there is no time in the physics anywhere. What decides is a temperature, and the stiffness reaches its limit sixty degrees before the strength does — so the way a member fails can change while it is burning.

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