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

256
bENdINg
“W12 by 50.” This shape is nominally 12 in. deep, and it weighs 50 lb/ft. W shapes are
manufactured by passing a hot billet of steel through several sets of rollers, arrayed in series,
that incrementally transform the hot steel into the desired shape. By varying the spacing
between rollers, a number of different shapes of the same nominal dimensions can be produced,
giving the engineer a finely graduated selection of shapes. In making W shapes, the
distance between flanges is kept constant while the flange thickness is increased. Consequently,
the actual depth of a W shape is generally not equal to its nominal depth. For
example, the nominal depth of a W12 × 50 shape is 12 in., but its actual depth is 12.2 in.
In SI units, the nominal depth of the W shape is measured in millimeters. Instead of
weight per unit length, the shape designation gives mass per unit length, where mass is measured
in kilograms and length is measured in meters. A typical SI designation is W310 ×
74. This shape is nominally 310 mm deep, and it has a mass of 74 kg/m.
Figure 8.9b shows a tee shape, which consists of a flange and a stem. Figure 8.9c
shows a channel shape, which is similar to a W shape, except that the flanges are truncated
so that the shape has one flat vertical surface. Steel tee shapes are designated by the letters
WT, and channel shapes are designated by the letter C. WT shapes and C shapes are named
in a fashion similar to the way W shapes are named, where the nominal depth and either the
weight per unit length or the mass per unit length is specified. Steel WT shapes are manufactured
by cutting a W shape at middepth; therefore, the nominal depth of a WT shape is
generally not equal to its actual depth. C shapes are rolled so that the actual depth is equal to
the nominal depth. Both WT shapes and C shapes have strong and weak axes for bending.
Figure 8.9d shows a rectangular tube shape called a hollow structural section (HSS).
The designation used for HSS shapes gives the overall depth, followed by the outside width,
followed by the wall thickness. For example, an HSS10 × 6 × 0.50 is 10 in. deep and 6 in.
wide and has a wall thickness of 0.50 in.
Figure 8.9e shows an angle shape, which consists of two legs. Angle shapes are designated
by the letter L followed by the long leg dimension, the short leg dimension, and
the leg thickness (e.g., L6 × 4 × 0.50). Although angle shapes are versatile members that
can be used for many purposes, single L shapes are rarely used as beams because they are
not very strong and they tend to twist about their longitudinal axis as they bend. However,
pairs of angles connected back-to-back are regularly used as flexural members in a configuration
that is called a double-angle shape (2l).
Cross-sectional properties of standard shapes are presented in Appendix B. While one
could calculate the area and moment of inertia of a W shape or a C shape from the specified
flange and web dimensions, the numerical values given in the tables in Appendix B are
preferred, since they take into account specific section details, such as fillets.
ExAmpLE 8.3
A flanged cross section is used to support the loads shown on the beam in the accompanying
diagrams. The dimensions of the shape are given. Consider the entire 20 ft length of the
beam, and determine
(a) the maximum tensile bending stress at any location along the beam, and
(b) the maximum compressive bending stress at any location along the beam.
Plan the Solution
The flexure formula will be used to determine the bending stresses in this beam. However,
the internal bending moments that are produced in the beam and the properties of the
cross section must be determined before the stress calculations can be performed. With
the use of the graphical method presented in Section 7.3, the shear-force and bendingmoment
diagrams for the beam and loading will be constructed. Then, the centroid