Structural Engineering & Statics
where the load goes
The collection

Every essay — page 8

Essays 169 to 181 of 181, in the same order.
1200 kN214 kN of tiestruts at 20°0.5 huniform from here onGuyon: 0.25 P (1 − a/h) = 214 kN — the same number, from the same lever armbearing 20.0 N/mm² · once spread 5.7 N/mm² Internal forces

The force that splits what it pushes on

A prestressing tendon delivers its whole force through a plate a fraction of the section deep. One depth further along the stress is uniform, and the spreading in between requires a transverse tension nobody applied — the force that splits end blocks, and the only number in the design that no equilibrium equation on the member can see.

9 figures · anchorage zone facescore — 25 N/mm² in sheard = 61 mm between the face centroids41 N/mm²41 N/mm²the core carries none of itbending stressD = 91.35 × 10⁹ N·mm² · 98.8% of it is the separation termwrinkling at 236 N/mm², which contains no length at all Sections and stress

Two skins and the space between them

A sandwich panel is a section made of a material that carries the bending and a material that carries none of it. The parallel-axis term is not a correction here — it is 98.8 per cent of the second moment — and the shear deflection is not a correction either.

9 figures · sandwich section 0501001502002503003504000.00.10.20.30.40.50.6eccentricity below the centroid (mm)10⁶ ÷ P (so lower is more prestress)transfer, toptransfer, bottomservice, topservice, bottom1029 kNthe kernthe wedge runs from e = 0 to 400 mm Sections and stress

Four inequalities and a wedge

A prestressed section has to satisfy two stress limits when the force is largest and the load smallest, and two more when the force has relaxed and the load has arrived. Each is linear in one over the force — which turns a search for a prestress into a region on a page, and turns an impossible section into an empty one.

9 figures · prestress limits storey shearthe link — 800 mm0.69 kN in each diagonal per kN of shearlink shear 0.47 kN · storey drift 0.008 mm · deflection magnified 5435× Structural form

The part that is meant to be weak

A braced frame is stiff and has nowhere to yield. A moment frame yields everywhere and is soft. Move the two diagonals a metre apart along the beam and the whole storey shear has to pass through the segment between them — which keeps most of the stiffness and puts every yielding in one member the designer chose.

9 figures · eccentric brace the split, at 5.1 mvolume 0.100 × 10⁹ mm³ against 0.204 for a fan from the base3 of 3 members sized by buckling Structural form

The tree that strength does not ask for

A branching column carries a roof on many points and reaches the ground on one. Size every member by its stress and the optimum tree turns out to have no trunk at all — the best answer is a fan of straight struts from the base. Put buckling in and the trunk appears, at 57 per cent of the height.

9 figures · branching structure 1968 kNthe restraint: 0.35 N/mm per mmtwo half-waves, each 6000 mmunrestrained 173 kN in one half-wave, drawn faintly · effective length 3555 mm Stability

Held everywhere, and it forgets its length

A brace at a point divides a member's buckling length. A restraint spread along the whole member does something else — the member chooses its own number of half-waves, and past a few of them the critical load stops depending on the length at all.

8 figures · continuous restraint 0.100.323.1610.000%5%10%15%20%25%30%local critical load ÷ global critical load (logarithmic)capacity lost below the weaker single modethe two coincide hereworst: 23% at 0.47at coincidence the loss is 2% Stability

Two ways of buckling at once

A thin-walled column can bow as a whole or ripple in its plates, and each has its own critical load. The received advice is that the worst arrangement is the one where the two are equal. The arithmetic says the opposite — at coincidence the interaction costs two per cent, and the expensive region is where the plates go first.

8 figures · mode interaction 2700 kNthe leaf, 900 kN3600 kN on the trunnionno moment, at any angle9 mW·a = 8100 kNm and C·r = 8100 kNm — a product, not a weightinertia 9908 t·m² against 7431 for the leaf alone Equilibrium

Balanced, and four times as heavy

A counterweight cancels a moment about a pivot, and that is the only thing it cancels. The bearing beneath carries both weights, the inertia rises as the square of the radius, and a load that moves cannot be balanced at more than one position at all.

7 figures · counterweight 02468101214161820-200-100100distance along the beam (m)bending moment (kNm, sagging up)the sagging envelopethe hogging envelopeeach case exact to 3e-16 · the envelope out by 23% Equilibrium

The envelope is not a structure

A continuous beam whose imposed load may sit on any span has eight load cases, and every one of them is a genuine state of equilibrium. The curve the design is made against is not one of them — it is assembled from different cases at different stations, and it fails the identity all eight satisfy exactly.

8 figures · load arrangement 012345678-0.03-0.02-0.0100.010.020.03along the span (m)deflection, each scaled to the same mid-span drop3.20 · a uniform load3.00 · a load at mid-span2.99 · a triangular load3.60 · a load on half the spanθ = C · δ/L, and the same δ gives 1.21× the rotation across these Deflection

The angle nobody limits

Every serviceability rule in this collection limits a displacement. What a bearing, a joint and a cladding gap actually have to accommodate is an angle — and the angle is locked to the displacement by a coefficient that contains no material, no section and no span.

9 figures · end rotation 050100150200250300350400020406080100120sideways movement (mm)height (m)bendingracking366 mmroof drift 1 in 328 · worst storey 1 in 296 Deflection

Two motions with one name

A tall building's sway is two movements added. A frame racks like a stack of parallelograms, worst at the bottom; a cantilever bends about its base, worst at the top. The total at roof level says nothing about which storey is worst, and on this building it is neither.

8 figures · drift components 0.0000.0050.0100.0150.0200.0250.030010203040compressive strainstress (N/mm²)44.4 N/mm² at 0.0068unconfined: 30 at 0.002still carryingstrength × 1.48 · ultimate strain × 8.0 · toughness × 11 Materials

Squeezed sideways into a different material

Concrete in a cylinder test fails by splitting apart sideways under a load pushing it down. Put a hoop round it and the splitting has to stretch steel — and a lateral pressure of a twelfth of the strength raises the strength by half and the ultimate strain by eight.

8 figures · confinement 10321003161000316201234size (mm, logarithmic)nominal strength (N/mm²)the specimen: 3.10the structure: 1.14the plastic limitfracture mechanicsthe test overestimates by 2.71× · D₀ = 120 mm Materials

The bigger one is the weaker one

Two geometrically similar beams of the same concrete should fail at the same nominal stress, because a strength is supposed to be a material property. They do not. The large one fails at less, and the reason is that a crack releases energy in proportion to a volume and consumes it in proportion to an area.

8 figures · size effect