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12
Beams
The straight beam represented by engineering beam theory has been used as an example
throughout this work. This theory takes into account the effects of extension, bending, and
shear deformation. The local and global forms of the straight beam equations were derived
in Chapters 1 and 2, respectively. It is the intention here to present rather complete equations
for straight beams. The derivations are based on the reduction of three-dimensional elas-
ticity equations to the appropriate beam theory. The governing equations can be integrated
and solved, using the displacement method, to find the displacements, including rotations,
and the corresponding forces (stress resultants) along the member. Furthermore, analy-
ses are presented for the cross-sectional properties needed for the study of beams. Also,
analytical expressions for the distribution of normal and shear stresses on the cross-section
are discussed. Computational methods for calculating cross-sectional properties and stress
distributions for arbitrary cross-sectional shapes are presented.
12.1
Displacements and Forces in Straight Bars
We begin with the derivation of a linear theory in which the three-dimensional continu-
um is reduced to a combination of a two-dimensional cross-sectional problem and a one-
dimensional longitudinal analysis, see Wunderlich (1977). Both the cross-sectional and
the longitudinal analyses are substantially simpler to carry out than the original three-
dimensional problem.
The coordinate system remains the same as for simple beams (Fig. 12.1a). Positive forces
and moments are shown in Figs. 12.1a and b, respectively. Note that these definitions are
an extension of Sign Convention 1. The displacements corresponding to these forces are
shown in Fig. 12.2.
12.1.1
Virtual Work
By definition, the dimensions of a bar normal to the axial (longitudinal) coordinate are very
small relative to the length of the bar. The influence of the strains associated with these
directions will be neglected. This means that the principle of virtual work relation
δ
=
W
0
727
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