Lecture 40:
Beam Deflection
A loaded beam deflects by an amount that depends on several factors including:
- the magnitude and type of loading
- the span of the beam
- the material properties of the beam (Modulus of Elasticity)
- the properties of the shape of the beam (Moment of Inertia)
- the beam type (simple, cantilever, overhanging, continuous)
Deflection may or may not be critical. Excessive deflection may result in cracked plaster, objectionable appearance, or sags in flat roofs which then "pond". It may transfer loads to non- bearing members under the beam such as partitions, doors, windows, etc., which in turn may cause partitions to crack or doors and windows to stick. A floor beam which deflects excessively is apt to be "springy", creating an undesirable walking surface, even if it is in no danger of failing. A springy floor is especially unsatisfactory for a room housing sensitive instruments.
A general rule for limiting deflection of simple spans for floor construction, or for plastered ceilings, is that the deflection should not exceed the span (in inches) divided by 360 (max D= L/360). The deflection for exposed ceiling beams at the roof is often allowed to be 50% to 100% greater (l/240 or l/180). Codes usually specify that these deflections are based on live load only, but experience shows that this is sometimes excessive. A conservative approach is to limit the deflection to these values for total load in lightweight construction (such as wood and steel). These guidelines are general and apply in most cases, but certainly not all. For example, the springiness of a floor is influenced more by the mass of the floor than by the total deflection; more dead load could cause more deflection, but probably less springiness.
Formulas given in tables will be used to compute deflection for some loading conditions; these will be expanded to approximate deflections for other conditions. The deflection formulas will not be derived. In deflection formulas, "w" refers to pounds per inch of length of loading (not pounds per foot). "W" refers to the total distributed load. The beam length is in inches. Most mistakes in computing deflections are caused by using the length in feet instead of inches and/or using "w" to mean pounds per foot instead of pounds per inch. Also, convert any distributed load "w" to "W".
There are four parts to deflection formulas:
1. COEFFICIENT, which takes into account:
- Type of beam
simple, cantilever, overhanging, etc.
- Loading condition
2. LOAD FACTOR
- Distributed loads "W" (total weight)
- Distributed loads "w" (weight per inch of length)
- Concentrated loads "P" (weight of one concentrated load)
3. LENGTH FACTOR
- L^3 (inches) usually for load factor "P" and "W"
- L^4 (inches) usually for load factor "w"
- La, Lb (inches) used to locate some types of loads
(There is some confusion about the symbol "L" and "l";. Lower case "l" is often used to indicate length in inches and upper case "L" for length in feet. However, lower case "l" can be confused with the number "1" which causes more serious
problems. Therefore, in this class, "L" refers to inches in these deflection formulas.)
4. STIFFNESS FACTOR
1/EI
E = material stiffness (modulus of elasticity)
I = stiffness of the section based on the geometry of the section (moment of inertia).
CAUTION: The most common mistake in computing deflection is caused by using "w" as load per foot instead of load per inch. The derivation of the deflection formulas uses the unit of inches for all the factors in the formula. Uniformly distributed beam loads "w" are normally described as load per foot which must be converted to the proper load per inch value before insertion into the deflection formula. If "W" is used for the total distributed load instead of the load per unit of length ("w") this conversion is not necessary.
Note that W = wL; will simplify many of the formulas.
The following are equations for finding the deflection of the more common beam types and their associated loading. Note how the co-efficient reflects the stiffness of the system and the loading.
Beam
Deflection
Deflection Approximation
Copyright © 1995 by Chris H. Luebkeman & Donald Peting
22V95