# Normal Stress, Normal Strain and Young's Modulus

#### Normal Stress

The previous topics have discussed what a Force and a Moment is. However, there is clearly a question of scale that needs looking at, applying a force to a large object will surely have a different effect than if the same force was applied to a small object?

Imagine, pushing on a finger with the thumb of your other hand. In this situation we can usually push as hard as we want and very little will happen. Now (don’t try this bit at home) if we applied the same force, but this time pressed on the finger with a pin, rather than your thumb, we would clearly do significant damage. And yet, the same force has been applied in both instances, how can the differences be so drastic for the same force?

The key is that although the same force has been applied, the force has been applied over much different areas. We define a force applied over an area as a stress:

$$\sigma = \frac {F}{A}$$

Where, $$\sigma$$ is the “normal” stress that would arise due to a Force, $$F$$, applied in a perpendicular direction to an area, $$A$$. For information on units of stress, click For useful information relating to the difference between stress, pressure and patch loading click .

In the metric system, the typical units for force is the Newton, N, and the typical units for area is m², this gives the units of stress equal to N/m², this is sometimes called the Pascal, or 1Pa. However, as a Newton is around 100g a stress equal to 1 N/m² is a very low stress. In Engineering we often instead deal with units of stress of N/mm² (1N/mm² = 1MPa), this is around 100g of mass applied to an area 1mm x 1mm and is much easier to handle in Engineering applications.
Stress and pressure are both defined as a force over an area, both having units of N/mm². The difference is that a pressure is usually seen as an APPLIED load over an area, while a stress is what is CAUSED in the materials by a load. To add to the confusion, in Civil and Structural Engineering loading can be defined as a “Patch Load” this is again a force distributed over an area, typical patch loads are 5kN/m² to represent pedestrian crowd loading

The key word in the above description is "perpendicular", "normal stress" is the stress that arises due to a force acting "perpendicularly" or "normal" to a plane. Forces that are applied at any angle other than perpendicular to a plane will give rise to Shear Forces, these are discussed in a separate article.

#### Normal Strain

When an object undergoes a stress it will deform by stretching or compressing, depending on the direction of the load. Some object will deflect more than others, to quantify the amount of deformation, we use the variable “strain”:

$$\varepsilon = \frac {\Delta l}{l}$$

Where $$\varepsilon$$ is the value of “normal” strain, $$\Delta l$$ is the change in length of an object and $$l$$ is the original length of the object. For information on units of strain, click

In the metric system, the typical units for both length, l and change in length, δl are metres. As change in length is divided by length the units cancel out, so a value of strain is unit less. Typically, large strains, such as ultimate rupture strain of a ductile material are given in percentages, “% strain”. However, we typically want to stay well below these % limits, so for more typical working loads of Structural Engineering we will use “microstrain”, με, 1micro strain = 1x10⁻⁶ strain.

#### Young's Modulus / Elastic Modulus

When a load is applied to an object it will deform, as the load increases, the deformation will also increase, so there must be some relationship between stress and strain. You would also expect that less stiff materials will deflect more than stiffer objects under the same load. The parameter that links stress and strain is the “Elastic Modulus”, which is a fundamental property of a given material and is defined as:

$$E = \frac{\sigma}{\epsilon} = \frac {P/A}{\Delta l / l}$$

For most materials the behaviour between stress and strain is approximately linear, doubling the stress on an object will double the stress. E, the elastic modulus is often also referred to as the “Young’s Modulus” of the material and is a property of a given material, for a list of typical values for Elastic Modulus (Young’s Modulus) click here. The higher the value of elastic modulus (Young’s modulus) the stiffer a material is, with a higher load required to produce a given strain. For information on the units of Young's modulus click

As stress has units of Pascals (or N/m²) and strain is unitless then the units of elastic modulus (Young’s modulus) is also the Pascal (or N/m²)

By plotting values of stress against strain for a material it is possible to see immediately the difference between a stiff or flexible material. Figure 3: Stress vs strain graph showing a stiff material (high Young's modulus) and a flexible material (low Young's modulus)