July 3, 2025

Jessica A. Dennehy

11th Grade

Williamsville East High School

Introduction

The earliest known evidence of bridge-building dates back to ancient Babylon around 4000 BC, where a massive reservoir system was engineered to control the overflow from the Euphrates River. While the purpose of a bridge may seem straightforward, building one that lasts for any extended period of time is anything but. Constructing a durable structure requires interfield collaboration; from local architects and engineers to material scientists and local government, everyone has to work together to ensure that a bridge is built to be structurally sound. With this in mind, every subcomponent matters in the construction of a bridge: every beam, truss, bolt, nut, and weld plays a role, and mathematics ensures that these individual components are able to withstand the test of time.

Bridges come in many forms, each designed with a specific purpose in mind. Most modern bridges are a combination of many different simple bridges, but for introductory purposes, here’s a quick breakdown of the most common simple bridges and what makes them unique.

Beam Bridges

A traditional beam bridge consists of two abutments, which serve as a means of support, and a level, horizontal deck, which spans the two ends. These are the simplest style of bridge, being the first ever built. However, even with its apparent simplicity, their design is grounded in an understanding of both physics and mathematics. All loads of a beam bridge should be applied to its level surface. Fundamentally, when this occurs, the top of the beam will compress, being squeezed downwards, while the bottom will be under tension, being stretched outwards; depending on the materials used and the intended use of the bridge, these forces must be precisely calculated to ensure safety and stability. To ensure that the deck is able to withstand loads, the weight of the load can not remain entirely on the flat surface; instead, most of it is transferred vertically through the beam and down into the supports, where they are then compressed. With different materials, shapes, and supports, there are a multitude of variables that must be accounted for. Mathematics is used in order to calculate and rely on equations from statics and material science to calculate bending moments, shear forces, and stress distribution. Math is used to model these internal forces, predicting how the beam will bend before the bridge is ever built.

Truss Bridges 

Truss bridges utilize a network of accordingly named trusses, or load-bearing structures composed of a series of triangles, in order to efficiently distribute weight loads. Triangles are used not only because they provide an effective material consumption-to-supportive weight ratio, but also because a triangle cannot actually be distorted by stress. That is why a truss provides a stable form in stress-loading applications, allowing for the support of considerable external loads over a large area. Every subcomponent is designed to endure only so much stress in a particular direction. Engineers, accordingly, must utilize systems of equations to model how forces travel through each part of the bridge. If even one component is overloaded with force, or the force experienced is not what it handles, it will fail; even a singular failure can result in the entire bridge’s collapse. Math, usually a combination of linear algebra and trigonometry, will often be used to determine how the forces acting on the bridge will be distributed in the trusses. A statically determinate truss bridge is one where the force on each truss is known, and engineers can then conclude which materials can be used for construction to withstand that force.

Calculating stress on a truss isn't the most intuitive. Most forces are applied on a truss joint, where individual components of the bridge intersect. These intersections are also often referred to as panel joints. Every truss, forming a triangle, will consist of connected pieces; the connected pieces forming the top and bottom of the truss are referred to, respectively, as the top and bottom chords. Meanwhile, the sloping and vertical pieces connecting the chords are collectively referred to as the web of the truss. All of these components are stressed primarily in either axial tension or axial compression. As one component of the truss experiences a force, another component will experience a force in the opposite direction. They may not have the same magnitude, with there being multiple components of a truss, but one must keep in mind that a structurally sound bridge’s net force will also be equal to zero. For example, bending occurs, which compresses the top chords. The bottom chords will experience tension, and depending on their orientation, the vertical/diagonal members will experience either tension or compression.

Arch Bridges 

An arch bridge is unique in that it carries loads primarily by compression, which then exerts both vertical and horizontal forces on the foundation. These were relatively easy to construct due to them being made from small, easily carried blocks of brick or stone, while still being able to span wide openings. An arch can also carry a much greater load than a horizontal beam can, due to the pressure it creates downward on the arch. A downward stress on an arch pushes onto the curvature of the materials underneath it; this squashes them tightly together and forces the wedge-like components that form the arch, called voussoirs, closer together. The load on the arch bridge is then diverted through the "keystone," the component center of the arch, around the curve of other stones, and down into the abutments. Only then does the ground push back upward and inward, allowing for the arch to hold. However, there comes a catch: the compressive forces that hold the arch together also tend to squeeze the blocks outward radially, but when a load is applied, these outward forces, which hold the arch, are diverted downwards in a diagonal force called a thrust. This will cause the arch to collapse if it is not properly buttressed. Hence, the vertical supports, or posts, upon which an arch rests must be massive enough to buttress the thrust and conduct it into the foundation, where they must therefore prevent both vertical settling and horizontal sliding.

Bridge Failure

Bridges, regardless of material or construction, fail for the same reason: the forces within the bridge become unbalanced. The net force of a structurally stable bridge will be zero, so when a force becomes too great for even one component in the bridge, whether it's as large as an adamant or as small as a singular rivet, the component will fail. After that, it's a domino effect: the load on the bridge has to then be shared by fewer components, which might overexceed its limit. Then another component fails, and then another. Eventually, the entire bridge collapses as a result of cascadingly failing materials. As such, those behind the construction of the bridge will back up continuousness in the initial construct and also utilize mathematics to account for those critical variables (load weight, wind forces, temperature fluctuations, etc.) that can push the bridge over its limit.