For most of the twentieth century bridges were designed by allowable stress: compute the stress under the expected loads and keep it below some fraction of the material strength. Simple, but blind to the fact that a dead load is known within a few percent while a truck load is not, and that a brittle failure deserves more margin than a ductile one.

Load and Resistance Factor Design (LRFD) separates the uncertainties. Each load is multiplied by a factor γ that grows with how uncertain it is; each resistance is multiplied by a factor φ below one that shrinks with how variable the material or failure mode is. The structure passes when Σγ_iQ_i ≤ φR_n. The factors were calibrated so that a component has a target reliability index β of about 3.5 at the strength limit state, roughly a probability of failure of 2 in 10,000 over the design life, for a component, before redundancy and system effects.

A limit state is a condition beyond which the bridge no longer does its job. AASHTO LRFD checks four kinds: strength, service, fatigue and fracture, and extreme events. The rest of this chapter walks through the loads, then how they combine at each limit state.

The factors were not chosen by instinct. For AASHTO LRFD, Nowak's calibration (published as NCHRP Report 368, 1999) took about 200 representative bridges designed to the old Standard Specifications and computed the reliability index of their girders from measured statistics of loads and resistances; it scattered widely with span and girder type. The target β = 3.5 was chosen to represent past practice, and γ and φ were then chosen to bring every girder type as close to it as possible. The statistics explain the numbers: the weight of factory-made components has a coefficient of variation of about 8% and cast-in-place components about 10%, an asphalt overlay about 25%, and the live load was built from heavy-truck survey data extrapolated to the largest load expected in 75 years.

Two consequences follow. The factors belong to the load model: the 1.75 is calibrated with HL-93 and cannot be applied to a different design vehicle without recalibration. And β = 3.5 is a target for members, not for the bridge, which in a redundant system is safer. Eurocode targets β = 3.8 over a 50-year reference period for the normal consequence class (EN 1990, Annex C), a comparable level expressed over a different period.

Permanent loads are what is there all the time. AASHTO names them by initials because they get different factors:

barrierwearing surfacedeckgirdersutility

DC

Dead load of structural components and non-structural attachments, girders, deck, barriers, diaphragms. Well known; γ = 1.25 (or 0.90 when a smaller value governs, as for uplift or overturning).

DW

Wearing surface and utilities. Less certain, because overlays get added over the years; γ = 1.50.

EV, EH, ES

Vertical earth fill, horizontal earth pressure, surcharge, the loads on abutments, retaining walls and culverts.

Locked-in effects

Prestressing (PS), creep and shrinkage (CR, SH), settlement (SE), and forces built in during staged construction (EL). These are not "loads" in the everyday sense but they move the structure and must be tracked.

For a long concrete bridge, permanent load is 70–85% of the total, which is why self-weight is the first number every designer estimates and why lighter materials pay for themselves at long spans.

Nobody designs for actual trucks; codes replace traffic with a notional model chosen so that its effects envelope what real traffic produces. AASHTO's model is HL-93:

35 kN (8 kip)145 kN (32 kip)145 kN (32 kip)4.3 m4.3–9.0 m, variable

• The design truck (three axles, 325 kN total) or the design tandem (two 110 kN axles), whichever is worse, plus the lane load of 9.3 kN/m. Truck and lane act together.
• A dynamic allowance IM of 33% on the truck or tandem (not the lane) for strength and service; 15% for fatigue; 75% for deck joints.
• Multiple presence factors of 1.20, 1.00, 0.85 and 0.65 for one, two, three and four or more loaded lanes, because it is unlikely that every lane carries a maximum truck at once.
• For negative moment over interior supports, 90% of two trucks 15 m apart plus 90% of the lane load.
• The fatigue truck: the design truck with a fixed 9.0 m rear spacing, one lane, no multiple presence.

How much of a lane's load goes to one girder is the distribution factor, tabulated in AASHTO 4.6.2.2 for common bridge types. Other codes work the same way with different numbers: Eurocode LM1 uses a 300/200/100 kN tandem per lane plus a 9 kN/m² UDL in the first lane; India's IRC uses Class A and Class 70R vehicles; New Zealand's HN-HO-72; Abu Dhabi's TR-516 adopts AASHTO with local vehicle checks.

Wind, a static pressure on the superstructure and substructure, derived in the 10th edition from a 3-second-gust design wind speed with exposure and elevation adjustments, plus wind on live load. For slender or long spans the static pressure is only the beginning: flutter, vortex shedding, galloping and buffeting are checked by wind-tunnel testing and dynamic analysis.

Temperature, a uniform rise and fall (a design range of the order of 40–70 °C depending on climate and material) that moves the deck and sets joint and bearing sizes, and a vertical gradient through the section (hot top, cool bottom) that bends it, important for concrete boxes.

windheatearthquakeflood

Earthquake, AASHTO designs for ground motion with a 7% probability of exceedance in 75 years (about a 1,000-year return period), site-adjusted by soil class. The philosophy is that a bridge may be damaged but should not collapse: columns are detailed to hinge in a controlled way and decks must not fall off their seats.

Water, stream pressure, ice, debris and buoyancy, but above all scour: foundations are checked for the scour of a 100-year flood (design) and for the 500-year or overtopping flood (check) using HEC-18. Vessel collision (AASHTO Guide Specifications) sizes pier protection on navigable waters.

A limit state is a load combination with its own factors. AASHTO LRFD Table 3.4.1-1 lists them; the ones that matter most:

servicestrengthfatigueextreme event

Strength I

Normal vehicular use, no wind. 1.25 DC + 1.50 DW + 1.75 (LL+IM). Governs most girders.

Strength III

Design wind with no live load on the bridge (the wind is too strong for traffic).

Strength V

Normal traffic with a moderate wind: LL factor 1.35, wind reduced.

Extreme Event I

Earthquake with a reduced live load, γ_EQ typically 0.5.

Extreme Event II

Ice, vessel or vehicle collision, check flood, with 0.5 live load.

Service I

Everything at factor 1.0, deflection, crack width in RC, settlement, slope stability.

Service III

Tension in prestressed concrete: 1.0 DC + 1.0 DW + 0.8 (LL+IM) in the 10th edition for most cases. This is the check that sizes prestressed girders.

Fatigue I / II

Infinite-life (1.75 × fatigue truck) and finite-life (0.80 ×) stress ranges on steel details and reinforcement.

The habit to build: for every member, ask which combination governs and why. In a prestressed girder it is Service III at midspan and Strength I for shear; in a steel girder it is often the deck-pour construction stage or fatigue at a cross-frame connection, not Strength I at all.

Construction stages deserve their own combinations. AASHTO 3.4.2 requires that during construction DC and DW be factored by at least 1.25, construction loads, equipment and their dynamic effects by at least 1.5, and wind during construction by at least 1.25, with other factors at 1.0 unless the owner specifies otherwise. The steel girder carrying wet concrete before the deck hardens is the classic case: its compression flange is braced only at the cross-frames, the overhang brackets twist the exterior girders, and the pour sequence on a continuous unit can momentarily put more moment into a span than the finished bridge ever will.

Segmental bridges built in balanced cantilever add loads that exist only during erection, listed in the segmental provisions of AASHTO Section 5: an out-of-balance segment or form traveller, a differential dead load of 2% on one cantilever, unequal construction live load on the two arms, wind uplift on one arm, and the dynamic effect of a precast segment released accidentally, taken as 100% of its weight. The pier and its temporary stability system are often sized by these combinations rather than by the finished bridge. Only a construction-stage analysis that follows every segment, tendon and traveller move, with creep and shrinkage, shows whether the as-built geometry and stresses will match the drawings.

The same physics, different calibrations. What a designer moving between jurisdictions needs to know first:

• Eurocode (EN 1990, EN 1991-2, EN 1992/1993/1994-2): partial factors γ_G = 1.35 and γ_Q = 1.35 on traffic; Load Models 1–4; ψ combination factors; national annexes change the numbers, so the country matters as much as the code.
• IRC (India, IRC:6, IRC:112): Class A, Class 70R and Class AA loading; impact factors that depend on span; a distinct RC/PSC code (IRC:112) modelled on Eurocode 2.
• NZTA Bridge Manual: HN-HO-72 loading, a strong seismic and durability focus, and explicit importance levels.
• Abu Dhabi TR-516 / TR-511 (and Dubai practice): AASHTO LRFD as the base with Gulf-specific temperature ranges, chloride durability requirements and local vehicle checks. Most Gulf highway bridges are designed to AASHTO.
• KDS 24 (Korea): a limit-state code that grew from AASHTO LRFD with Korean live-load models (KL-510) and seismic maps; the cable-stayed and suspension bridges of Korea's coast were designed to its predecessors.
• JRA (Japan): distinct in its seismic philosophy (Level 1 / Level 2 ground motions after Kobe 1995) and in its heavy use of steel.

The References page on this site links the official homes of each of these.

Numbers fix ideas better than rules. Take a single 30 m simple span with a 200 mm concrete deck on five precast prestressed I-girders at 2.4 m centres. The deck is 11.6 m wide with a 0.4 m barrier on each edge, which leaves a 10.8 m roadway; AASHTO 3.6.1.1.1 divides it into INT(10.8 / 3.6) = 3 design lanes. We want the live-load moment at midspan and the shear at the support for an interior girder, and then the Strength I demand.

Step 1, the moments for one lane. On a simple span the rear axle spacing of the design truck stays at its minimum of 4.3 m. The largest moment occurs under the middle 145 kN axle when midspan lies halfway between that axle and the resultant of all three. The 325 kN resultant sits 1.46 m from the middle axle, so the axle goes 0.73 m off midspan and M = 2,056 kNm. With the middle axle exactly at midspan the moment is 2,050 kNm, only 0.3% less, which is why midspan values are good enough at this span. The tandem gives 1,585 kNm; it governs simple-span moment only below about 12 m. The lane load gives 9.3 × 30² / 8 = 1,046 kNm. With the dynamic allowance on the truck alone, one lane carries M_LL+IM = 1.33 × 2,056 + 1,046 = 3,781 kNm.

Step 2, the distribution factor. For an interior beam of a concrete deck on concrete girders with two or more lanes loaded, AASHTO Table 4.6.2.2.2b-1 gives g = 0.075 + (S/2900)^0.6(S/L)^0.2(K_g/Lt_s³)^0.1, with S, L and t_s in mm. The longitudinal stiffness is K_g = n(I + Ae_g²). For a girder with A = 0.50 m², I = 0.14 m⁴, e_g = 0.95 m between the girder and deck centroids, and n = 1.25 (a 50 MPa girder under a 32 MPa deck, since E grows with √f'_c), K_g = 7.4 × 10¹¹ mm⁴ and the stiffness term is 1.12. Then g = 0.075 + 0.893 × 0.603 × 1.12 = 0.68 lanes per girder. The one-lane equation gives 0.48, so the multi-lane case governs. The multiple presence factors are already built into these equations, and applying 0.85 for three lanes on top of them is a common error.

Step 3, the girder demand. Live load plus impact per girder is 0.68 × 3,781 = 2,563 kNm. Permanent load per girder: deck 0.2 × 2.4 × 25 = 12.0 kN/m, girder 0.50 × 25 = 12.5 kN/m, haunch and diaphragms about 1.0 kN/m, and the two 7 kN/m barriers shared equally by the five girders, 2.8 kN/m. So DC = 28.3 kN/m and M_DC = 3,184 kNm. A wearing-surface allowance of 1.5 kN/m² gives DW = 3.6 kN/m and M_DW = 405 kNm.

Three things stand out. Live load is half the factored moment, so the distribution factor deserves a real calculation. Service III, not Strength I, will set the number of strands. And the exterior girder needs its own factor, from the lever rule (with the one-lane multiple presence factor of 1.2 applied by hand) or from the correction factor e of Table 4.6.2.2.2d-1; with a wide overhang it often carries more than an interior girder.

Traffic does not only press down. It pushes along the deck when it brakes and outward on a curve, and it includes people on foot, whose loads are small but whose footsteps can set a light structure moving.

Braking force (BR, AASHTO 3.6.4) is the greatest of 25% of the axle weights of the design truck or tandem, or 5% of the truck or tandem plus the lane load. It acts in every lane carrying traffic in the same direction, with the multiple presence factors, horizontally at 1.8 m above the deck. Per lane, 25% of the truck is 81 kN, and the 5% options overtake it only when the loaded length exceeds about 140 m. A three-lane one-way bridge therefore sees 3 × 81 × 0.85 = 207 kN, which the articulation must deliver to the fixed supports or, with elastomeric bearings, share among all supports in proportion to their stiffness. Eurocode EN 1991-2 treats braking as a single force for the whole deck, Q_lk = 0.6α_Q1(2Q_1k) + 0.10α_q1q_1kw₁L, kept between 180 and 900 kN. For a 30 m deck with the α factors at 1.0 that is 360 + 81 = 441 kN, about twice the AASHTO figure, so the fixed bearings and their piers must be rechecked whenever a design moves from one code to the other.

Centrifugal force (CE, AASHTO 3.6.3) on a curved bridge is a fraction C = f v² / (gR) of the truck or tandem axle weights, with f = 4/3 for all combinations except fatigue (where f = 1.0), v the highway design speed and R the radius. The lane load is left out because vehicles at speed are assumed to be widely spaced. At 80 km/h on a 300 m radius, C = 4/3 × 22.2² / (9.81 × 300) = 0.22, a radial force of 73 kN per truck applied 1.8 m above the deck. Besides loading the bearings and piers radially, its overturning effect shifts wheel load toward the outside of the curve, onto girders that the curvature has already made the heaviest loaded; superelevation may be counted against it.

Pedestrian load (PL) on the sidewalks of a road bridge is 3.6 kPa on any sidewalk wider than 0.6 m, acting together with the vehicle lanes (AASHTO 3.6.1.6). Footbridges follow the AASHTO LRFD Guide Specifications for the Design of Pedestrian Bridges (2009): 4.3 kPa (90 psf) with no reduction for loaded area and no dynamic allowance, plus a maintenance vehicle where the deck is wide enough to admit one. Eurocode recommends 5 kPa on the footways of road bridges and, for footbridges, a value that falls with loaded length from 5.0 to 2.5 kPa.

For a footbridge the static load rarely governs; vibration does. The Guide Specifications ask for a vertical fundamental frequency above 3.0 Hz and a lateral one above 1.3 Hz, or else a dynamic check. Walking excites about 1.6–2.4 Hz vertically and half that sideways, which is the range that locked the London Millennium Bridge in 2000. European practice (the Sétra guide of 2006, and EN 1990 Annex A2) sets limits on acceleration instead, with recommended maxima of 0.7 m/s² vertically and 0.2 m/s² horizontally in normal use.

Several entries in Table 3.4.1-1 are not forces but strains the structure wants to undergo: uniform temperature TU, temperature gradient TG, creep CR, shrinkage SH and settlement SE. A statically determinate structure accommodates them freely. A restrained one, continuous over its supports, fixed to stiff piers or integral with its abutments, turns them into forces. That is the price of continuity, paid in movement at the joints and bearings and in restraint forces in the members.

Uniform temperature (AASHTO 3.12.2) is taken from a climate table (Procedure A) or from contour maps of maximum and minimum design bridge temperature (Procedure B). With a coefficient of 10.8 × 10⁻⁶ /°C for concrete and 11.7 × 10⁻⁶ /°C for steel, a 100 m concrete deck fixed at mid-length over a 45 °C design range moves 10.8 × 10⁻⁶ × 50,000 × 45 = 24 mm at each end, and creep and shrinkage add a comparable one-way shortening in a prestressed deck. The load factor on TU is 0.50 when computing forces, because restrained concrete cracks and relaxes, and 1.20 when computing movements for joints and bearings, because running out of movement capacity is expensive.

The temperature gradient (3.12.3) describes a deck heated by the sun from above. The positive gradient falls from T₁ at the top surface to T₂ at 100 mm depth and to zero at 400 mm; in the hottest of the four solar radiation zones T₁ = 30 °C and T₂ = 7.8 °C. The negative gradient is the positive one multiplied by −0.30 for a bare concrete deck and −0.20 under an asphalt overlay. Because the profile is nonlinear, even a simple span develops self-equilibrating stresses through the depth. A continuous one also develops restraint moments: a hot top makes the beam want to hog upward off its interior supports, the supports hold it down, and the result is tension in the bottom fibre over the piers, a classic cause of cracking in continuous concrete boxes. The load factor γ_TG is project-specific; AASHTO's default is 0 at the strength limit states, 1.0 at service without live load and 0.50 with it. Eurocode EN 1991-1-5 combines the uniform and gradient components with the factors ω_N = 0.35 and ω_M = 0.75.

Creep and shrinkage matter most where the structural system changes during construction. A balanced cantilever bridge carries its self-weight as pairs of cantilevers; once the closures are cast it is continuous, and creep slowly pushes the dead-load moments toward the values it would have had if cast in one piece on falsework. With the age-adjusted effective modulus method the final moment is approximately M_∞ = M_I + (M_II − M_I) φ/(1 + χφ), where M_I is the as-built moment, M_II the monolithic one, φ the creep coefficient and χ ≈ 0.8 the ageing coefficient. With φ = 2.0 the moment travels about three-quarters of the way, creating positive moment near midspan that the cantilever stage never had; continuity tendons are there to carry it. Precast girders made continuous for live load show the same effect in miniature: creep of the prestress keeps cambering them upward after the connection is made and can crack the bottom of the pier diaphragm (NCHRP Report 519, 2004).

Settlement is a load only for indeterminate structures. If the middle support of a two-span continuous girder settles by δ, the support moment changes by 3EIδ/L². For 40 m spans, a composite girder with EI = 33 GPa × 0.40 m⁴ and δ = 25 mm, that is 620 kNm per girder, roughly a fifth of the live-load moment at that pier, before creep relaxes part of it. The commentary to AASHTO 10.5.2.2 suggests tolerable angular distortion between adjacent foundations of 0.004 for continuous spans and 0.008 for simple spans, and γ_SE is taken as 1.0 unless the project decides otherwise. On soft ground, in mining areas or over karst, the simple span is often the right answer precisely because it does not care.

Fatigue is damage by repetition: a stress range far below yield, applied millions of times, grows a crack from a weld toe or a hole until the member fractures. The load model therefore aims at the stress range a detail sees from ordinary heavy trucks, and how often it sees it, rather than at the rare worst truck.

AASHTO 3.6.1.4 uses one fatigue truck: the design truck with its rear spacing fixed at 9.0 m, in one lane, with IM = 15% and no multiple presence. Because the approximate one-lane distribution factors contain the 1.2 multiple presence factor, they are divided by 1.2 for fatigue. Two combinations use the truck:

• Fatigue I, load factor 1.75, represents the largest stress range the detail will see in its life. If that stays below the constant-amplitude fatigue threshold of the detail, cracks do not grow: infinite life.
• Fatigue II, load factor 0.80, represents the effective stress range of the whole truck spectrum, compared with the finite-life resistance (A/N)^1/3 for N cycles.

The cycle count is N = 365 × 75 × n × ADTT_SL. Here n is the number of stress cycles per truck passage (1.0 for simple spans over 12 m, 2.0 for shorter ones, 1.5 near the interior supports of continuous spans over 12 m, 5.0 for cantilevers) and ADTT_SL = p × ADTT is the truck traffic in a single lane, with p = 1.00, 0.85 or 0.80 when one, two, or three or more lanes are available to trucks. Where the truck count is unknown, AASHTO suggests a fraction of total traffic: 20% on rural interstates, 15% on urban interstates and other rural roads, 10% on other urban roads.

Which combination applies depends on the traffic. AASHTO tabulates, for each detail category, the single-lane ADTT above which the infinite-life check is used (Table 6.6.1.2.3-2); below it, the finite-life check is the economical one. For the 30 m span of section 4.7 the fatigue truck produces 1,744 kNm per lane. With IM and the one-lane factor 0.48/1.2 an interior girder sees 795 kNm, and Fatigue I asks for 1.75 × 795 = 1,390 kNm. If that moment acted on a steel girder whose critical detail was a C' stiffener weld toe with a threshold of 82.7 MPa, the section modulus at that detail would need to be at least 1,390 × 10⁶ / 82.7 = 16.8 × 10⁶ mm³. Section 5.9 gives the categories.

Two notes on the factors. They were recalibrated in the 2010s from weigh-in-motion records of real truck traffic, which showed heavier tails than the earlier values of 1.50 and 0.75 had assumed. And fatigue is an absolute check, not a relative one: a weld toe does not care how strong the girder is, only how much its stress changes, so lighter girders of higher-strength steel are more fatigue-sensitive, not less.

Concrete and its reinforcement are less often critical. AASHTO 5.5.3 limits the stress range in prestressing strand to 125 MPa where the tendon radius exceeds 9 m and 70 MPa where it is 3.6 m or less, and exempts concrete deck slabs in multi-girder bridges from fatigue checks. Eurocode EN 1991-2 offers five fatigue load models, from FLM1 (a fraction of LM1) to FLM5 (recorded traffic). FLM3, a single four-axle vehicle with 120 kN axles, is the usual model for steel, used with the damage-equivalence factors λ of EN 1993-2 to reach an equivalent stress range at 2 million cycles.

Earthquake design differs from every other load in this chapter in one respect: the bridge is not expected to stay elastic. Keeping an ordinary bridge elastic in a 1,000-year earthquake would make its piers several times stronger and its foundations far larger, a cost no owner pays for an event with a 7% chance in 75 years. The codes accept damage and prohibit collapse. For an ordinary bridge the performance target at the design event is life safety: the bridge may need to be closed and even replaced, but it must stand. Critical and essential bridges, those needed for emergency access afterwards, are held to tighter targets, typically remaining open to emergency traffic after the design event.

The hazard comes from mapped spectral accelerations adjusted for the site soil class (A to F): PGA, S_S at 0.2 s and S₁ at 1.0 s become A_s = F_pgaPGA, S_DS = F_aS_S and S_D1 = F_vS₁, which define the design spectrum. S_D1 then sorts the bridge into Seismic Zones 1–4 in AASHTO LRFD, or Seismic Design Categories A–D in the AASHTO Guide Specifications for LRFD Seismic Bridge Design, with breakpoints at 0.15, 0.30 and 0.50 g. The two horizontal directions are combined as 100% of one plus 30% of the other (3.10.8).

Two routes follow. The force-based route of AASHTO LRFD 3.10 divides elastic forces by a response modification factor R that reflects the ductility the substructure can deliver: from 1.5, for every substructure of a critical bridge, up to 5.0 for multiple-column bents of an ordinary one. Connections, which must not be the weak link, get R = 0.8 or 1.0, so they are designed for more than the elastic force or all of it. The displacement-based route of the Guide Specifications estimates the displacement demand Δ_D from the elastic spectrum and checks it against the displacement capacity Δ_C of each bent, from simplified expressions in Categories B and C or, in Category D, a pushover analysis based on the moment–curvature behaviour of the hinges. It rests on the observation, developed by Priestley and his co-workers, that damage follows displacement more closely than force.

Both routes depend on capacity design. The designer chooses where yielding will happen, normally at the top and bottom of the columns, details those plastic hinges for ductility, and designs everything else, column shear, cap beams, joints, footings and piles, for the overstrength moment the hinges can actually develop, about 1.2–1.4 times their nominal or plastic strength. The overstrength factor matters because real reinforcement is stronger than its specified yield; a column that is accidentally too strong pushes the damage somewhere brittle. Section 7.6 covers the detailing.

The last line of defence is seat width. Unseated spans were among the defining failures of San Fernando (1971), Loma Prieta (1989) and Kobe (1995). AASHTO 4.7.4.4 sets a minimum support length N = (200 + 0.0017L + 0.0067H)(1 + 0.000125S²) mm, with L the length of deck to the next expansion joint and H the pier height, both in mm, and S the skew in degrees, increased to 150% in Seismic Zones 2–4. For L = 100 m, H = 10 m and no skew, N = 437 mm, or 656 mm with the increase; a 30° skew adds a further 11%. Eurocode 8 Part 2 (EN 1998-2) follows the same philosophy with behaviour factors q of up to 3.5 for ductile concrete piers and q ≤ 1.5 for limited-ductile design.

On 9 May 1980 the freighter Summit Venture, driven off course in a squall, struck a pier of the Sunshine Skyway Bridge across Tampa Bay. A large part of the southbound span fell and 35 people died. Few bridges had been designed for ship impact before it. The accident led to the AASHTO Guide Specification and Commentary for Vessel Collision Design of Highway Bridges (1991, second edition 2009), whose method is now Article 3.14 of AASHTO LRFD. The collapse of the Francis Scott Key Bridge in Baltimore on 26 March 2024, struck by a container ship of more than 100,000 tonnes deadweight, showed what happens when a bridge built before these rules meets a modern ship.

The method is a risk calculation. For each pier exposed to navigation, and for each vessel type, size and loading condition, the annual frequency of collapse is AF = N × PA × PG × PC × PF:

• N, the number of vessels a year;
• PA, the probability that a vessel goes aberrant through human error, mechanical failure or weather, starting from a base rate of 0.6 × 10⁻⁴ for ships and 1.2 × 10⁻⁴ for barges and adjusted for the bridge location, currents and traffic density;
• PG, the geometric probability that an aberrant vessel reaches the pier;
• PC, the probability that the impact collapses the bridge, which falls as the pier's capacity rises relative to the impact force;
• PF, the reduction from protection.

Summed over all piers and vessels, AF must not exceed 0.0001 a year (1 in 10,000) for a critical or essential bridge, or 0.001 for a typical one. The designer distributes that budget among the piers and so finds, for each, the impact force it must resist or the protection it needs.

The head-on impact force of a ship is P_s = 1.2 × 10⁵ V √DWT newtons, with V the impact speed in m/s and DWT in tonnes, an empirical fit to German collision tests. A 40,000 DWT bulk carrier at 2.5 m/s (about 5 knots) gives 60 MN; a 100,000 DWT ship at 3 m/s gives 114 MN. These are tens of times the wind or braking forces on the same pier. The design speed falls from the transit speed in the channel to the drift speed at a distance of three ship lengths from the channel centreline, so moving piers away from the channel is often the most effective protection. Barges have their own formula based on bow crush depth, and the superstructure must be checked wherever a mast or deckhouse can reach it.

Protection takes four forms: a pier strong enough on its own, which at 50–100 MN means a massive footing; fenders, which absorb small impacts but not large ships; independent dolphins or protection cells; and artificial islands, as around the main piers of the rebuilt Sunshine Skyway (1987). Eurocode EN 1991-1-7 treats ship impact as an accidental action with indicative forces by vessel class; major crossings get their own probabilistic study.