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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 are 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 Loading and Supports:



Points of Reaction and what happens to the Beam under a Load

Beam Loading and Reaction Equations

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:
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:

Decide what material you want to use (Wood or Steel). We are not designing
Concrete Beams in the Section of the Web Site.

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.

If the span is short, it would probably be better to use Wood.

Field conditions will sometimes dictate which would be better to use.

Determine what is going to be the loads imposed on the Beam.

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.

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.

Check the Span (length), and what will support the Beam at each end.

The span is the distance between one support and the other support, at each end
of the Beam.

Once you have all of the above information then you will start the Actual Beam
Design.

The equation Total Load = W x L is to determine the Total Load on the Beam.

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.

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.

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.

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.

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.

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:

