Design To Minimize Deflection
by John F Mann, PE
Structural elements must be designed to satisfy requirements for strength, to prevent failure or collapse, and stiffness, to prevent excessive movement or deflection.
This discussion is focused primarily on deflection of beam elements.
Design of many beam elements, such as floor joists, is often governed by deflection, not strength. This means that, for the required or proposed configuration, the limit for deflection is reached before the limit for strength.
Determination of the appropriate limit for deflection is one of the first issues that must be addressed during the design process. See "Wood Framing For Tile Flooring" for additional discussion of deflection limits.
Of course, there is usually a trade-off between more conservative design and cost. Greater strength and stiffness generally costs more. However, there are often ways to improve a design that can actually minimize deflection while reducing cost.
Obviously, a conservative deflection limit can be specified to minimize deflection, assuming design and construction is then performed correctly.
However, it is more useful to find ways to minimize deflection by more efficient design.
For a given design load, the following ways (if feasible) are most often used to minimize deflection, in general order of maximum effect or practicality;
(1) Decrease length of beam
(2) Move one or both supports inward from end of beam
(3) Use moment joints at ends of beam
(4) Increase beam moment of inertia
(5) Increase beam modulus of elasticity
(6) Decrease load on beam
(7) Share load with other beams
(8) Prestressing & camber
Deflection is highly dependent on length of beam element. For a given total load and distribution, deflection varies with the cube (third power) of span length. Therefore, if length of beam is doubled, deflection increases by a factor of 8, which is 2 cubed (2^3).
Even if beam length is increased by only 10 percent, deflection increases by 33 percent.
For a simple beam with uniform load (w), deflection varies with the fourth power of span length. However, total load (wL) is increased. If beam length is doubled, deflection increases by factor of 16 (for the same beam).
One way to reduce effective span length is to move supports inward from each end if feasible. Not only is maximum span length reduced, but load on the cantilever actually causes upward deflection in the main span that offsets deflection due to load on the main span. However, a load case without live load on the cantilever should always be considered. See below for further discussion of this method for a line of multiple beams.
Deflection is reduced when one or both ends of a beam resist moment, instead of being completely free to rotate ("hinged"). This method is generally available for steel or reinforced concrete construction, not wood.
Deflection is directly proportional to beam moment of inertia, modulus of elasticity and, for a given load distribution, total load.
For the same amount of material, some shapes (such as I-beam) have greater moment of inertia. However, increasing depth of a beam is the most practical way to increase moment of inertia.
Modulus of steel is essentially the same (29,000 psi) for all the various grades and alloys.
Modulus of wood varies significantly between species and grade. Modulus can be increased substantially by using LVL (laminated veneer lumber).
Modulus of concrete varies with compressive strength, but is not especially sensitive.
Standard code provisions include generous safety factors for variable ("live") loads and material properties. Code recommendations for weight of permanent materials ("dead load") are also conservative, but not as conservative as for live loads.
For a specific member, rearrangement of framing members is one way to reduce design loads. Use of lighter-weight materials (within the building) should also be considered if feasible.
For repetitive members such as floor joists, design load can be reduced by using reduced spacing. This may allow for reduction of joist depth. However, net cost for all the joists remains effectively the same or greater, due to increased labor costs.
For beam elements such as floor joists, one relatively simple way to reduce deflection is to use continuous elements that span over one or more interior supports, in addition to the usual support at (or near) each end. However, continuous beams result in conditions that must be carefully evaluated.
Deflection Of Simply Supported Beam
A beam with two supports (one at each end) is typically described as a single-span or "simply supported" beam. For uniform load (w) along the entire beam, maximum deflection (at midspan) is calculated by the standard formula;
Deflection, midspan (inches) = 5 w L^4 / 384EI
w = Uniform load (lbs per inch)
L = Span length (inches)
E = Modulus of elasticity for beam (lbs per square inch; psi)
I = Moment of inertia for beam (in^4)
This deflection is taken as a baseline index value of 1.00 for comparison purposes.
The baseline index value is applicable for uniform dead load (by itself), uniform live load (by itself), and total uniform load.
Deflection value is most often calculated for live load to compare with code limits for live load. However, dead load deflection should also be checked.
Load applied to each support is half of the total load on the beam, or (wL / 2).
Deflection of Two-Span Continuous Beam
Now consider a beam with three supports, forming two separate spans, with each span having the same length L as for the simply-supported beam in the previous example.
This beam is continuous over the center support, such that there is no joint in the beam as there is when two separate beams are used.
For the same uniform load w (used for single-span, simply supported beam) applied to both spans of the two-span continuous beam, maximum deflection for each span is only 0.42 (index value), which is 42-percent of the baseline value (1.00). Just as for the simply supported beam, this value is applicable (separately) for uniform dead load, uniform live load, and total load.
However, we must consider a separate load case, when live load is applied to only one of the two spans. For this case, dead load deflection remains the same while maximum live load deflection (in the one span) increases to 0.70 (70-percent of live load deflection for simply supported beam).
Other loading conditions (such as point loads) must also be considered. However, a continuous beam tends to reduce maximum deflections significantly, using the same beam size as for two single-span beams.
Therefore, use of two-span continuous beam may allow for reduction in beam moment of inertia and beam size.
The following results of using a two-span beam must be considered;
** Greater reaction force at center support (compared to two single-span beams). For two-span beam, with equal spans, reaction force is 25-percent greater than total (combined) reaction force for two single-span beams (of same total length). This affects design of whatever element may be providing the center support, such as a separate girder, as well as foundation walls and footings.
** For greatly unequal span lengths (one span much longer than the other), uplift reaction force may occur at outer end of shorter span for live load on the longer span only. If such uplift can not be resisted, deflection of the longer span would be increased greatly. Also, upward movement will usually be unacceptable, especially for a building.
Cantilever Method for Multi-Span Beam Line
For a line of beams with multiple supports, deflection can be reduced (compared to simply-supported beams spanning between supports) almost as much as for continuous beams by using the cantilever method. This method eliminates the need for moment-resisting connections which are often required for continuous beams.
Consider the simple case of two spans on three supports, which are often columns (C1, C2, C3). One beam (B1) can be extended (cantilevered) over the interior support (C2). The second beam (B2) then spans from inside end of B1 to the third support (C3). Connection between B2 and B1 is a simple hinge-type that resists shear only.
Inside end of B1, at end of the cantilever segment, is designated P1 for further discussion. Point P1 deflects downward due to load from B2. However, P1 deflects upward due to load on B1 between supports C1 and C2. Net deflection may therefore be upward, depending on loading conditions.
As long as the net cantilever segment of B1 is not excessive, the following advantages result, compared to using two beams that span between supports;
(1) Moment and deflection of B2 is less since span length of B2 has been reduced. However, deflection of any point on B2, relative to each support (C2 & C3) must include net deflection of point P1 (at end of B1). For live loads on B2 only, live load deflection of point P1 will be downward.
(2) Deflection of B1, between supports C1 and C2, is reduced due to point load from B2 at point P1, which causes upward movement of B1 between supports.
This basic design concept can of course be used for multiple beams with more than three supports.