what does the EI in deflection of beams formula stand for?

Question

I know it has something to do with young modulus, but how do you apply it in real life. ( say you have a reinforced concrete beam)

Answer 1

• Young's modulus $E$ times
• the second moment of area $I$ of the cross section.

The caveat is that this formula is simple enough when you have a beam made from one material. In the case of composite beams (ie. reinforced) the treatment is a bit more involved.

In general $EI$ provides a measure of the resistance of the beam to bending, which accounts for the material properties and geometrical aspects of the cross-section

Single material

Take for example this two beam cross-sections:

These two cross-sections have the same area 40x40=1600$mm^2$, however their moments of area are different.

Cross-section	Area	second moment of area $I$
A	$40\cdot40=1600$	$\frac{1}{12}40^4=2.133\cdot 10^5$
B	$60\cdot40 - 20\cdot 40=1600$	$\frac{1}{12}40\cdot 60^3 - \frac{1}{12}20\cdot 40^3 =2.133\cdot 10^5 =6.133\cdot 10^5$

So, despite having the same overall area (and weight), the beam B (hollow section) is more resistant to bending, and will exhibit less deflection.

Note: I'm not going into $I_{xx}$ and $I_{yy}$ calculations because this would be too long for this article.

two material - symmetric

Two or more materials need to take the sum of the EI for the individual material. So for example, if you take cross-section B from above and you fill the hollow part with a material with modulus $E_2$, then the total $EI$ , can be calculated using the following formula:

$$\sum EI = E_1\cdot I_1 + E_2\cdot I_2 $$

where:

• $I_1 = \frac{1}{12}40\cdot 60^3 - \frac{1}{12}20\cdot 40^3 =2.133\cdot 10^5 =6.133\cdot 10^5$
• $I_2 = \frac{1}{12}20\cdot 40^3 =1.067\cdot 10^5$

NOTE: I've picked a symmetric cross-section to avoid additional calculations issues with finding the neutral axis etc. In the generic case, this can get a bit messy, to do it by hand.

Answer 2

$EI$ is the indicator of the stiffness of the structural member - the larger the $EI$ of a member (stiffer), the smaller the deflection. For example, the deflection of a cantilever beam with a concentrated load at the free end, $\Delta_{max} = \dfrac{PL^3}{3EI}$. In which $E$ is the material property namely the "Young's Modulus of Elasticity" or simply "Modulus of Elasticity", or "Elastic Modulus"; and $I$ is the geometric property of the member, namely the "Second Moment of Area" or simply "Moment of Inertia".

For a structural steel element (beam, column..), $E$ and $I$ are easily obtained as $E_s$ (modulus of elasticity of steel) is a well known constant, and $I$ can be calculated using its geometric parameters (width, depth) that are usually remain constant throughout the loading stages and service life, however, it is not the case for the reinforced concrete elements - a composite material of steel and concrete, for which the concrete will crack that effectively reduces the contact depth of the concrete, thus resulting in a reduction in moment of inertia. (Note, the cracked moment of inertia of concrete, $I_{cr}$, is a function of the reinforcing ratio and the strength of the concrete $f'_{c}$.)

Furthermore, in contrast to the structural steel, Young's Modulus of concrete is not a standard constant for all concrete elements but depends on the compressive strength of the concrete, $f'_{c}$, and the weight of the concrete mixture, $w_c$. The Young's Modulus of concrete, $E_c$ is to be calculated using the empirical formula provided by the structural code and can be found in textbooks for the design of reinforced concrete structures.

In the US (for normal weight concrete):