Archimedes and freeboard (надводний борт)
A floating module of waterplane area A = L·B displaces ΔV = A·T of water, where T is the draft (осадка). Floating means ρ g A T = W, so a module of weight W sits at T = W/(ρ A) and has a freeboard f = H − T. Every extra newton of load needs 1/(ρ g A) metres of extra draft, so an isolated module loaded at its centre carries at most P = ρ g A (H − f_min) − W_module. The page checks its solver against this closed form to the last digit. The module is idealised as a box with straight sides; real pontoons have raked ends and chines.
Hinged chain
Neighbouring modules are connected by pins at deck level that carry shear but no bending. Each module has two degrees of freedom, heave z and pitch θ, with restoring stiffness k_h = ρ g A and k_θ = ρ g B L³/12. Force and moment balance of each module and equal deck height at every hinge give a tridiagonal system for the hinge shears (2 S_{j-1} + 8 S_j + 2 S_{j+1} = ...); the page solves it for every frame. The ends of the chain are the shore ramps (a ramp is a simple beam from the bank seat to the end hinge, so the end module is held only by its own buoyancy) and, where the modules touch the river bed, a one-sided bed support found by an active-set iteration. Dynamics is a second-order lag of the quasi-static solution with a heave period T = 2π √((m + m_a)/(ρ g A)), added mass m_a = ½ ρ π (B/2)² L (an order-of-magnitude estimate); no waves or wakes are modelled.
Heel and stability
A vehicle off the centre line applies a heeling moment P·e. For a wall-sided box the righting arm is GZ = sin φ (GM + ½ BM tan²φ) with BM = B²/(12 T), KB = T/2 and GM = KB + BM − KG; the heel angle solves Δ·GZ = P e cos φ. The edge freeboard is f − (B/2) tan φ. KG is the mass-weighted centre of module and vehicle (vehicle centre of gravity about 1.0 m above the deck).
Current, drag and anchors
Drag on a module is F = ½ ρ C_d A_p v² with A_p = L·T the wetted area facing the flow (the long side of each module faces upstream) and a debris factor that adds area. Drag grows with the draft. The river velocity follows Manning's formula for a wide channel, v = (1/n) h^{2/3} √S, with n = 0.035 and a slope S you choose. The water piles up on the upstream face by about v²/2g; the page subtracts that from the freeboard as a conservative rule. An anchor line of scope s (length over height) is inclined at α = asin(1/s); the tension is T = F/cos α and the line also pulls the module down with F tan α, which the solver feeds back into the draft. Holding power of a fluke anchor is taken as m g × k with k a representative bed factor (mud 8, sand 14, gravel 6); real holding depends strongly on the soil and on setting the anchor.
Panel truss (ферма): method of joints
Each girder is a Pratt truss of 10 panels, 3.0 m long and 1.5 m high: pins at the joints, so every member carries only axial force. The page writes the two equilibrium equations at each joint and solves all member forces and the support reactions as one linear system; an independent direct-stiffness solve of the same truss gives the same forces and the joint displacements. Rough check: a chord carries about M/h. Allowable tension is 0.6 f_y A (f_y = 355 MPa); allowable compression is the smaller of that and the Euler load π² E I / L² divided by 1.8. Member sections are representative values chosen to give a sensible capacity, not manufacturer data.
Launching with a nose
The assembled bridge is pushed over rollers. A light launching nose (orange) cuts the cantilever moment: with the tip a distance x beyond the roller R1 the structure behaves as a beam with piecewise weight and stiffness, solved by a finite-element beam with unilateral roller supports. The overturning check is a moment balance about R1: the structure tips when the moment of everything ahead of R1 exceeds that of everything behind it (plus the counterweight). The smallest counterweight for the tip just reaching the far roller is W = (M_front − M_rear)/b with b the distance from R1 to the tail. Truss shear flexibility makes the equivalent beam softer; the factor eta is measured on the truss itself.
Not a design tool
This is an educational model. Real emergency bridges are designed, checked and approved by qualified engineers with the manufacturer's data, soil and river surveys, load testing and local rules. Modular panel bridges of this general kind (the best known is the Bailey bridge) and floating bridges have been used worldwide for civil crossings after floods. The models here are generic and not affiliated with any manufacturer. Follow local authority guidance.