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Deflection in Beams
(Here we will address Structural Beams, Joists and Lintels, not Faux - Fake Beams)

Joists and Lintels are in reality just Beams.

A beam is a structural element that is usually placed horizontally and capable of withstanding loads, primarily by resisting bending. The bending force induced into the material of the beam as a result of loads, which includes its own weight (beam weight) and superimposed loads (other loads called live loads and dead loads, such as, people and furniture). These loads produce what is called bending moments in the beam, and may also have bending moments at each supported end, when the ends are fixed to the end supports. Fixed means that they are attached in such a way that some of the load in the beam is transferred to the end connections (such as the walls or columns).

Beams come in various sizes and shapes. They are usually, either homogeneous or composite. A homogeneous beam is one that is mode of one material, such as, wood or steel. A composite is one made of materials that are not the same, such as, a concrete beam with steel reinforcing.

Some Beam Types:
Beam Bending Loading Beam Bending Loading
Beam Bending Loading Beam Bending Loading

Beam Loading and Supports:
Beam Loading

Points of Reaction and what happens to the Beam under a Load
Beam Bending Loading
Beam Loading and Reaction Equations
Beam Load Values

All of this may appear to be overwhelming, but it is not.

Some experts say that engineering is 80% logic and 20% application. Some may debate this. but here we will provide you with basic engineering information and applications which you may not always find available.

While beams are loaded in all different ways. The simply supported beam is a commonly used beam (as shown above).

Below you will be shown how all of this works and How to select a Beam (wood or steel).

We also touch on the selection of a Concrete Beam in the Beam Section.

Simple Supported Uniformally Distributed Loaded Beam with Equations and Solutions:

Beam Bending Loading

The example above has the steps necessary for selecting and designing a Wood Beam. If you wanted to select and design a Steel Beam, the steps would be the same. There are a few things that change, such as, the Bending Stress in the Material, the Moment of Inertia, the Modulus of Elasticity and the Section Modulus. All of the other equations would be the same, if you had the same loading (W) and span (L).

The usual steps to Designing a Beam are:
  1. Decide what material you want to use (Wood or Steel). We are not designing Concrete Beams in the Section of the Web Site.
    1. If the loading is going to be Heavy, you may want to use Steel, since it will be able to receive a larger load for the same span.
    2. If the span is short, it would probably be better to use Wood.
    3. Field conditions will sometimes dictate which would be better to use.
  2. Determine what is going to be the loads imposed on the Beam.
    1. The loading is usually taken from the Building Code. The Code contains a list of what the minimum loads for most type of uses. In residences, the Code usually requires that a minimum of 40 pounds per square feet be used for what is called Living Spaces. Be careful, because the Code has much larger loading requirements for Balconies and Stairs. A link to the Building Code is provided under the Building Code Section of this Web Site.
    2. Sometimes there are loading conditions that are larger than what is specified in the Code. Please be aware that the Code provided a minimum requirement, and you may exceed the minimum.
  3. Check the Span (length), and what will support the Beam at each end.
    1. The span is the distance between one support and the other support, at each end of the Beam.
  4. Once you have all of the above information then you will start the Actual Beam Design.
    1. The equation Total Load = W x L is to determine the Total Load on the Beam.
    2. Once you have the Total Load on the Beam, it is divided by 2 to determine the load that is transferred to each end of the Beam, which is transposed to either the wall or a column. This is important since you need to verify that the wall or column can carry the loads.
    3. Get the Moment, the Maximum Moment should be obtained, for this reason, the Moments at other points along the Beam have been ignored. We want to Beam to be designed for the Maximum Safety. For the Simply Supported Beam with a Uniformly Distributed Load is M = WL 2 /8.
    4. So far, we have the Loading and the Moment for the Beam. Now we need to know if the Beam is going to be Wood or Steel. If the Beam is Wood, then, depending on the kind of Wood, the typical fb (Bending Stress) may vary from 1000 psi (pound per square inch) to 1200 psi. Typically, a conservative value would be around 1000 psi, if you are using fir or hemlock, this can also be obtained from the Building Code, for various species of Wood. Similarly, if you intend to use Steel, then a value for Fy = 36000 Steel would be fb = 24000 psi (where, fb = 0.66 x Fy). As it can be seen, the Steel is 24000 and Wood is 1000, which indicates that Steel is approximately 24 times stronger than Wood, in bending. Which also indicated that the Steel Beam would be smaller than a Wood Beam. So if you have limited space, a Steel Beam may be a better selection.
    5. Now we need to compute the Sx (Section Modulus) that is required by Code. This is done using the equation Sx = M / fb. We have the M (Moment) from our computations. Simply apply the calculations. This calculation is what is required and shall be a minimum allowed. You can either select a Wood Beam from a Table of Wood Sections, which are available in most Wood Manuals or from our Web Site, or you similarly select a Steel Beam in the same way. Naturally, you could select a Wood Member and then calculate the Section Modulus for that Member, as shown in the example. The Section Modulus must be equal or greater than the computed Section Modulus.
    6. There is one final step, that is to find the Deflection of the Beam caused by the loading. When you place a load on the Beam it will bend downward, and this vertical displacement downward is called the Deflection and is measured in inches (or mm). As can be seen in the example we have computed the Maximum Deflection at the center of the Beam. In the example the Maximum Deflection allowed is controlled the Code. The various allowed Deflections are shown in the example. To compute the Deflection, we need some additional information, which is E (the Modulus of Elasticity) of the material and I (Moment of Inertia) for the selected member. (See Section to Calculate Moment of Inertia on this Web Site)
      The Modulus of Elasticity (E) for Wood varies around 1190000, for these examples the 119000 value have been used. If Steel is used, then the E value would be aroung 29000000, as shown in the examples.
      The Moment of Inertia (I), would be either computed or selected from Tables provided or Computed. (See Section on Calculate Moment of Inertia)
      The Allowed Deflection are: Supporting Floors and Ceilings L/360, Supporting Roofs that have less than a 3 in 12 slope L/240 and Supporting Roofs greater than 3 in 12 slope L/180. L = Spans, for example: 12 feet, multiply 12 feet x 12 inches = 144 inches divided by 360, 240 or 180, which ever applies.
    7. Finally compare the Computed Deflection with the Allowed Deflection. If the Computed Deflection is greater than the Allowed Deflection, then you must select a larger Beam Member, and recalculate.

Simple Supported Point Concentrated Loaded Beam with Equations and Solutions:

Beam Bending Loading