Stress vs Strain, Elasticity Modulus

After discussing stress and strength, the next important item to know is the relation of stress versus deformation. Imagine a material which is somehow perfectly rigid. In other words, it never deforms, no matter how large, even infinite, load you apply on it. In nature no such material exists. Every material, no matter how strong, deforms when stresses act on it. For every material, that relationship between stress vs. deformation, is unique characteristic of that material.

In engineering, instead of deformation, (such as in inches) we like to express the deformation in terms of unit deformation, in other words, deformation per its original length, which is called strain (such as inch / inch), which is unitless. That is more informative for engineers.  

So,

Strain = ε = Deformation Amount / Original Dimension

So for instance, if we had a steel rod of 100 cm long, and when we pull it, it deforms by 1 cm, then the strain will be 1 cm / 100 cm = 0.01 , a unitless value.

And the relation between stress and strain is:

σ = E. ε

Here, σ is the stress, as we saw before and ε is the strain, as we just defined it.

And “E” is called “Elasticity”, which is also called Modulus of Elasticity. The the higher the elasticity, the stronger, stiffer the material and closer to being rigid. So for a perfectly rigid material, E would be infinite, which does not exist in nature.

So looking at the above formula, if we have a very stiff material, it means, much more stress is needed to deform it for a given amount of strain.

Different materials have different stress strain relationships.

For concrete, a typical curve looks like this:

Typical Concrete Stress Strain Curve

The failure is is sudden, which is because concrete is a brittle material and fails suddenly, in a crushing manner. Note that this is for compression only as concrete is weak in tension and we do not consider concrete tension in design.

fc’ is the compressive strength of concrete. 0.5fc’ is the half of this value. As you see here, unlike the steel curve we will show below, the curve almost never has a stable slope, except in low stresses, in other words, modulus of elasticity of concrete varies with compressive strength, unlike steel. So to calculate it, we make approximation and can use different methods, such as the initial slope of the curve or the tangent or secant of the line at 0.5fc’.

The stress strain curve of concrete is also dependent to a certain degree in various factors including the age of concrete, type and size of specimens, rate of loading, aggregate and cement properties. Typical ultimate strain value of concrete (where concrete breaks) is taken as 0.003 but can range between 0.002 and 0.004 depending on type of concrete and other factors including above. 

For steel, a typical curve may look like this:

Typical Stress – Strain curve for steel

Both stresses and strains are much higher than the values in concrete graph here, which means steel can withstand much more stress and also deformation, before failing. The failure here is not sudden but comes with plenty of warning, because steel is ductile material. It can absorb lot of energy and deform a lot, before breaking (the area under stress strain curve represent absorbed energy and here the area is much higher and wider than was in concrete graph).  

It means, as it approaches its final failing point, it deforms greatly, which means it gives us considerable warning. This means saving lives. This is why, in reinforced concrete structures for example, we want the steel to fail first, and not the concrete, because concrete is brittle, and suddenly breaks without warning, which can mean loss of lives. So, in a reinforced concrete structure, we do not put in too much reinforcing steel, because if we do, steel will not fail but concrete will. We do not want that. So we limit the quantity of steel we put into concrete members by a maximum amount. This way, steel will fail first, and not the concrete, which will give us considerable warning before ultimate collapse.

Elastic means, the material can totally recover back to its original position. So in both of the graphs above, for small strains, the curves can practically be assumed as straight lines, which means strain increases directly proportional to stress, with a ratio of E, and for these portions, the material is elastic. In other words, upon release of load, it will return to its original geometry, with no permanent deformation remaining.

For steel, upon reaching elastic limit, if we keep increasing the stress, the material now enters plastic zone. In plastic region, the material deforms more, with same load increase and if the stress increase continues, the material simply fails. In plastic zone, even if the load is totally released, some permanent deformation will remain, unlike elastic region. After the plastic region, the material reaches its ultimate strength. And then it finally completely fails.

For example, a steel beam may exceed its elastic limit but can continue carry the load, as can be seen in the graph, but the plastic failure (and therefore permanent deformation) already occurred. This is a very important concept to know in civil engineering and other disciplines dealing with materials. In our structural design, we do not want the stresses to get very close to this elastic limit.

So we repeat to emphasize:

Elastic : No permanent deformation. Material returns to original completely, when load is removed.

Plastic : Some or a lot of permanent deformation remains, even after load is removed.

Not only structural materials, even soils can behave as elastic material sometimes, such as sands, which means the settlement would be totally recovered and the soil would bounce back, if you had removed the loading.

Modulus of Elasticity, E, is one of the most important things to know about a material. Not only in civil engineering, but in many other disciplines as well, such as mechanical or metallurgical engineering or materials science. If we rearrange and rewrite the equation we have seen in previous pages:

E = stress / strain = σ / ε

Now remember, we said that the strain has no units, such as inch / inch or cm / cm and so on… So, then the only remaining unit is the unit of stress. So, the unit of E is same as stress, which is,  

Force / Area

or expressed by units,

Newtons / mm2   = Megapascal or shortly MPa (metric units)

or

Pounds / inch2 = psi (English units)

Poisson’s Ratio (μ): One last term to introduce is Poisson’s ratio, which is a property of material that shows how much vertical stress a material can convert to horizontal stress and vice versa. Like modulus of elasticity, this is also applicable not only in structural but in soil mechanics too.

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