# 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?

Beams

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?

Arches

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

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.

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

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).

Computer mod­el­ling has shown that com­plex ortho­gonal lay­outs are best for achiev­ing equi­lib­rium of bridge struc­ture (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 Layopt.com (“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.