How to Calculate Beam Deflection

Knowing how to calculate beam deflection is something that comes handy when dealing with various stresses that surpass those that can be dealt with by simple common sense. Aside from the foundation, beams and columns are the two core components in a structure. Of these two, beams are more susceptible to deflection due to their orientation.

Columns however, are subject to buckling when loaded beyond their capacity. But as logic will dictate, they are capable of withstanding much greater vertical loading when compared to similar horizontal beams. This is largely due to the fact that virtually all common building materials are stronger in the compression or axial direction than the tension, due to the mechanical advantage of the collective bending moments within a beam’s cross section. (See Buckling and Elasticity for details on buckling and the Euler formula.)

Many DIY-type construction projects can be done without any fancy calculations, such as standard floors, walls, and roofs, that don’t have any unique functions or that aren’t required to carry extreme loads. Necessity arises however, when large loads must be shouldered, and when you don’t have the time, budget, or desire to overengineer.

Besides knowing how to calculate deflection, one must also know how to visualize and determine loads on given structural members so the values that are inserted into the formula are accurate. Also, depending on the material and length of member, a given deflection can range from acceptable to unthinkable. As a rule, a member should never deflect more than the span divided by 250 (even this number is discouraged in most applications).

Aside from the obvious structural limitations and dangers, large deflections create movement within the structure that can cause motion sickness and unnecessary discomfort. And although a given material may initially be capable of rigorous deflections without yielding, repeated cyclic-loading of this material will eventually cause it to fail due to fatigue.

Beam Deflection Formula

Before being able to calculate a beam’s deflection, you’ll need a few numbers. Let’s take a look at the formula, and I’ll explain the individual values:

Maximum vertical deflectionwhere

W = weight in newtons (N)
l = total unsupported length (m)
E = modulus of elasticity (N/m2)
I = second moment of area (m4)

To explain further, here’s an example. Say you have a .09 by .09 by 3 meter pine beam supported simply on its two ends. First of all, you’re gonna need the elastic or Young’s modulus of pine – which is roughly 8.962 Gpa. Then, you’ll need the second moment of area for a cross section of 9 by 9 centimeters – which in meters is roughly 0.0483 m4. Now you apply a weight of one ton, which when converted to newtons, equals 9,800N. Here’s how it looks:

deflection calculation exampleThis is more or less the maximum deflection one ton of weight would exact on the center point of a (homogeneous) .09 by .09 by 3 meter pine beam. Of course, if the above load bears down anywhere but the 1.5 meter mark, the deflection would be that much smaller. Also, if the load is spread out across the span of the beam, the deflection will also be less.

The idea is that with a bit of information, you can calculate the effect of various loads on beams and thereby design structures accordingly. In case there was any doubt, a vertical deflection of 113 millimeters on most anything in the 3 meter length range would be quite unacceptable structurally! – This violates the “span-over-250” rule by a factor of 10, not to mention that this rule only applies when the deflection is instantaneous, not ongoing!

*Here’s a pretty cool site, engineering calculator, that has various calculators where you can simply input values and get your answers on the spot.

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