Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

8.5 Introductory Beam Design for Strength
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INTROduCTORy bEAM dESIgN
FOR STRENgTH
At a minimum, a beam must be designed so that it is capable of supporting the loads acting
on it without exceeding allowable bending stresses. A successful design involves the determination
of an economical cross section for the beam—one that performs the intended
function but does not waste materials. Elementary design generally involves either
(a) the determination of appropriate dimensions for basic shapes, such as rectangular or
circular cross sections, or
(b) the selection of satisfactory standard manufactured shapes that are available for the
preferred material.
A complete beam design requires attention to many concerns. This discussion, however,
will be limited to the task of proportioning cross sections so that allowable bending stresses
are satisfied, thus ensuring that a beam has sufficient strength to support the loads that act
upon it.
The section modulus S is a particularly convenient property for beam strength
design. One form of the flexure formula given by Equation (8.10) for doubly symmetric
shapes was
Mc M
σ max = = where S =
I S
If an allowable bending stress is specified for the beam material, then the flexure formula
can be solved for the minimum required section modulus S min :
I
c
S
min
M
≥ (8.11)
σ allow
Using Equation (8.11), the engineer may either
(a) determine the cross-sectional dimensions necessary to attain the minimum section
modulus or
(b) select a standard shape that offers a section modulus equal to or greater than S min .
The maximum bending moment in the beam is found from a bending-moment
diagram. If the cross section to be used for the beam is doubly symmetric, then the maximum
bending-moment magnitude (i.e., either positive or negative M) should be used in
Equation (8.11). In some instances, it may be necessary to investigate both the maximum
positive bending moment and the maximum negative bending moment. One such situation
arises when different allowable tensile and compressive bending stresses are specified for
a cross section that is not doubly symmetric, such as a tee shape.
If a beam has a simple cross-sectional shape, such as a circle, a square, or a rectangle
of specified height-to-width proportions, then its dimensions can be determined directly
from S min , since, by definition, S = I/c. If a more complex shape (e.g., a W shape) is to be
used for the beam, then tables of cross-sectional properties such as those included in
Appendix B are utilized. The general process for selecting an economical standard steel
shape from a table of section properties is outlined in Table 8.1.
The ratio of one dimension to
another is called an aspect ratio.
For a rectangular cross section,
the ratio of height h to width b is
the aspect ratio of the beam.