How far is a bridge too far? Matthew Gilbert PhD 5th July 2021

This was every bit an inform­at­ive talk laced with some quite chal­len­ging math­em­at­ical con­cepts which inform on best bridge design.

Computational mod­el­ling and optim­isa­tion are able to pre­dict how a bridge’s com­pon­ents and struc­tures will behave under spe­cific con­di­tions, and thereby improv­ing design, effi­ciency and man­u­fac­tur­ing tech­nique e.g. with 3D print­ing

What are the span limits for any bridge form?


For example, those made from stone have a very lim­ited span. Steel fares a bit better. So what altern­at­ive forms are more effi­cient and allow for a greater span?


Robert Hooke 1635–1703 invest­ig­ated the shape of an arch by hanging weights from a straight beam. He then inver­ted this shape to deduce the optimal arch shape for effi­cient bridge con­struc­tion.

Forces gen­er­ated by the weight of a stone arch bridge limit span to just 120m.

Other mater­i­als e.g. rein­forced con­crete or metal can extend that limit to around 400m.The steel rail bridge at Newcastle upon Tyne would be a good example.

Fig 1 Types of arches 

Cable sup­por­ted

The most usual form is the sus­pen­sion bridge, for example the Humber Bridge

Fig 2 Humber Bridge

The main towers were built first and linked together with a pair of cables spun each side of the bridge and ver­tical cables hanging down to sup­port the deck. Deck sec­tions were then made to com­plete it. It was opened 40 years ago and was the inspir­a­tion for Dr Gilbert’s future career in civil engin­eer­ing on a school trip.

Other examples are the planned Halsafjorden in Norway which will span 2050m. Wind engin­eer­ing aspects are import­ant with a bridge of this size, with wind tunnel aero­dy­namic test­ing, wind buf­feting, flut­ter and vortex induced vibra­tion lead­ing to ongo­ing modi­fic­a­tion of the bridge deck geo­metry. There will be a float­ing tower in the middle anchored to the fjord bed below.

The span of a simple sus­pen­sion bridge is the dis­tance of sus­pen­ded road­way between towers and the 2km span of the Akashi Kaikyõ Bridge in Japan is the limit.

Cable-stayed form

Long span arches need cable-stays. Cables dir­ectly con­nect the towers to the road­way by a series of diag­onal stays between tower and deck. This bridge form is effi­cient as the main con­struc­tion steps are sim­ul­tan­eous. The main tower with pro­gress­ive lat­eral deck exten­sion on each side increases the span. Deck exten­sions then meet and are joined together.

Fig 3 Queensferry Crossing

The Queensferry Forth Replacement Crossing is a 4-span cable-stayed bridge with main spans of 650m

Hybrid sus­pen­sion and cable-stayed bridges are even more effi­cient. Diagonal cables are dropped to sup­port the deck

while being exten­ded. The middle part is sup­por­ted by main cables. The longest bridge in the world, 164.8km long, is the hybrid Danyang Kunshan Grand Bridge in China

Danyang Kunshan Grand Bridge in China









The best form of bridge

The most import­ant ques­tion is the optimal min­imal volume of mater­ial required to build the most resi­li­ent struc­ture. As bridge span increases, pro­gress­ively more of the struc­ture is neces­sary to simply carry its own weight, thereby lim­it­ing the span.

Steel cur­rently is the mater­ial of choice, so the only altern­at­ive at present is to adapt the effi­ciency of bridge design. A simple bracket with an attached weight is a good ana­logy. A large dif­fer­ence in volume is achieved using an ortho­gonal bracket shape (Fig 5).

Fig 5 — orthogaonal is on the left

Computer mod­el­ling has shown that com­plex ortho­gonal lay­outs are best for achiev­ing equi­lib­rium of bridge struc­ture (Fig 6).

Fig 6

But not all ten­sion and com­pres­sion ele­ments are ortho­gonal because the weight of the struc­ture itself needs to be factored in.

Development of the ideal is ongo­ing. There are eco­nomic issues, and factors such as wind load­ing and mater­i­als; when sub­sti­tu­tion of steel by carbon graphene becomes tech­nic­ally feas­ible, the app (“try it your­self”) sug­gests the pos­sib­il­ity of a 10 km span.

Most of the forces on a bridge are dir­ec­ted down the tower. A new bridge form needs less mater­ial largely because the forces from the deck are trans­mit­ted from the super­struc­ture to the found­a­tions. This is achieved by keep­ing load paths short and main­tain­ing wide angles between the tensile and com­press­ive ele­ments. Sharp angles give large com­pres­sion forces. Changing the compression/tension ratio optim­ises change in shape and type of bridge

A bridge too far

A bridge has been pro­posed to link the UK with Ireland. Bridge length is a bigger prob­lem fur­ther south between Wales and Eire.  Scotland on the other hand adds to jour­ney time from England and the con­tin­ent. There are con­sid­er­able fin­an­cial, struc­tural and polit­ical chal­lenges but it is feas­ible. A com­bined tunnel bridge would be pos­sible.