# Modulus of Elasticity: Steel, Concrete & Aluminum

In this lesson, we are going learn about the modulus of elasticity, a property of materials, and we are going to look at some examples using the modulus of elasticity of steel, concrete, and aluminum.
What is the Modulus of Elasticity?
Have you ever taken a spring out of a broken toy? If so, you probably stretched it with a gentle force and released it to see if it would go back to its original shape, and it did. However, if you apply too large of a force, the spring will not go back to its original shape. It is now ruined. Elastic materials, such as steel, work just like this spring.
When a force is applied to an elastic substance, the substance will be strained. It will elongate in the direction of the force. However, if the force is removed, it will return to its original length. This is called elasticity. However, depending on the material, there is a limit to the magnitude of force that can be used. If it is exceeded, it will cause the substance to not be able to return to its original length upon removal of the force. This limit is called yielding.
According to Hook’s law, stress is proportional to strain. Strain is the ratio of elongation of the substance to its original length. The ratio of stress to strain of an elastic substance is the modulus of elasticity. The relationship is as follows:
equation
In words: Sigma equals the modulus of elasticity times Epsilon.
Sigma = stress caused by an external force, which is measured in N/m2 or Pa for the SI system, and psi for the English system.
E = the modulus of elasticity of the material, which is also measured in N/m2 or Pa for the SI system, and psi for the English system.
Epsilon = strain caused by the stress and isn’t measured in units.
This means that materials with a high modulus of elasticity will require more stress to elongate the same amount of strain as compared to a material with a lower modulus of elasticity.
The modulus of elasticity is material dependent. For example, the modulus of elasticity of steel is about 200 GPa (29,000,000 psi), and the modulus of elasticity of concrete is around 30 GPa (4,350,000 psi). The modulus of elasticity of aluminum is 69 GPa (10,000,000 psi).
Modulus of Elasticity Example
Two loads of the same magnitude and directions are applied to two beams of the same cross-section. One beam is steel and the other is aluminum. The two beams are both 10 ft long. Since both beams are under the same magnitude of stress, we get:
FORMULAe
Substituting in for sigma gives us:
ESubS
And since the beams have a different modulus of elasticity, the values of strains must be different.
Since stress is equal to the force divided by the sectional area:
Sigma = F / A
Strain is equal to the difference between the difference in length and the original length:
Epsilon = delta / L
Using these formulas we can rewrite the formula Sigma = E * Epsilon as follows:
Formula1
When we put the values for sigma and epsilon into the equation, we get:
Eequals
Reinforced Concrete Example
As an example, in steel-reinforced concrete when a load is applied to the section, steel bars strain the same amount as the concrete. But, since both materials have a different modulus of elasticity, the steel bars carry more stress than the concrete. This is one of the reasons why concrete is reinforced with steel bars in construction. Since concrete is very weak in tension, the reinforcing steel bars carry most of the tensile stress.

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