## We’ll Cross That Bridge….

Previous articles introduced the concept of stress as a fundamental engineering calculation. Another useful analysis determines the deformation, or strain, experienced in structural members or assemblies. While the term “strain” can loosely refer to processes involving pasta and muscles, structural strain is defined much more rigorously. Strain in construction materials can easily lead to destruction if applied stresses become larger than anticipated, or dead loads unexpectedly become live loads. A very visual example of this is the classic film of the Tacoma Narrows Bridge collapse in 1940. Although designed to withstand winds of up to 120mph, a relatively light wind of around 40mph apparently set up a live “flutter” oscillation not anticipated by the design parameters. It becomes crucial then to know what strain or deformation can be expected in a structure when worst case design loads are applied.

## Engineering Strain

Engineering strain can be defined as the deformation of a material as the result of an applied force or load. This may be the result of static, constant load application and/or from dynamic, variable loading. Several theories or models are used to analyze these deformations. In the small strain model the material displacements are arbitrarily small and may be considered elastic, where the material recovers from the deformed state to its original dimensions. The large strain model considers the difference between the original material state and deformed state to be large and macroscopic, which may result in viscous flow, meaning the deformation is permanent. A combination of permanent and recoverable deformation, called visco-elastic strain, may also be considered. While many materials experience some form of visco-elastic deformation under load, for obvious reasons materials used in structural engineering are designed to experience only elastic, small strains. Permanently deforming a structure such as a bridge or building is usually considered to be an undesirable situation, as previously discussed.

## Strain Analysis

The general mathematical equation for strain is:

Strain = dimensional change / original dimension, or ε = Δ R/R_{o}

Where R is a dimension such as length, width, angular displacement, etc., R_{o }is the original dimension, and Δ R is the difference between the deformed dimension and the original dimension.

Just as with stresses, strains are considered axial, or normal, if they occur in line with the applied load. These are termed tensile or compressive strains. If the deformation occurs parallel to the plane or area of applied stress, it is termed shear strain. A simplified example of this deformation is the motion of the top face of a square block of rubber with respect to its parallel, anchored bottom face. Applying a shear stress to the top face results in sideways displacement of the top of the block. Dividing that displacement by the original height of the block gives the shear strain. However, structural strain rarely so simple, as it usually involves complex deformations with both normal and shear components. But utilizing these straightforward models of stress and strain in materials has led to the computational design, modeling, and analysis tools so useful in the structural design process.

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