THE MODULUS OF ELASTICITY

The Modulus of Elasticity



The Modulus of Elasticity

1. Introduction

modulus of elasticity of concrete is a parameter necessary in structural analysis for determination of strain distributions and displacements, especially when design of structure is based on elasticity considerations. This property is conventionally measured using standardized tests based on small specimens subjected to uniaxial compressive loading. specimen dimensions are at least three times maximum aggregate size of concrete. Furthermore, empirical expressions developed from experimental studies are available to estimate modulus of elasticity from compressive strength, which is a standard measure for characterizing concrete.

A procedure similar to that for calculating compressive strength has at times been adopted for experimental determination of modulus of elasticity, where tests are performed on conventional-size specimens made from wet-screened concrete [4]. More often, empirical expressions for conventional concrete are used to estimate modulus of elasticity from compressive strength. Such approaches neglect effect of aggregate and specimen size on modulus, which can be significant and vary with age of dam concrete [5].

The modulus of elasticity refers to a material's stiffness. This can also be thought of as the amount of deformation a material undergoes when subjected to a load. Experimentally, the modulus of elasticity, or Young's modulus, is found by determining the slop of the stress versus strain curve.

With excessive loading, the stress-strain curve initially begins linearly, followed by a dramatic change of slope. The phenomena occurring during this sudden change in slope is known as plastic deformation and is beyond the scope of this lab. For the purposes of testing the Young's modulus, the applied load should be kept below the yield strength, the pressure as which a material begins to experience plastic deformation.

A simple way of determining the Young's modulus is to create a uniaxial stress state. This is achieved by supporting a beam in a cantilever setup while applying pressure to a point on the beam. A strain gage should be located perpendicular, as well as a known distance, from the applied force.

With a known force, beam, and strain, and resulting stress can be calculated. To do so, the flexure formula can be used.

Equation 1

Where M is the bending moment at the point of interest (measured in inch-pounds or Newton-meters), c is the distance from the neutral axis to the surface (measured in inches or meters), and I is the centroidal moment of inertia measured around the horizontal axis (inches4 or meters4).

Since all three terms are calculated, it is easier to replace each term with terms representing terms physically measured. I is dependant on the beam geometry, and in this case is equal to:

Equation 2

where b is the width and t is the thickness.

c is replaced by half of the beam's thickness.

M refers to the bending moment and in an elementary uniaxial setup is equal to the applied force P multiplied by the effective length, Le.

Putting all three terms together, equation 1 becomes:

Equation 3

Equation 3 is only valid for the surface of an end-loaded cantilever beam with a rectangular ...