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

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395
THE dIFFERENTIAL EQuATION
OF THE ELASTIC CuRVE
Load
A
B
C
P—
2
L—
2
L—
2
P—
2
P—
2
Shear
–
P—
2
— PL
4
Moment
PL
+
2
16EI
Slope
PL
–
2
16EI
Deflection
PL
–
3
48EI
FIGURE 10.6 Relationship among beam diagrams.
supported beam with a concentrated load at midspan. The geometry of the beam was
utilized to locate the points of zero slope and deflection, required as starting points for
the construction. More commonly used methods for calculating beam deflections will be
developed in succeeding sections.
Review of Assumptions
Before proceeding with specific methods for calculating beam deflections, it is helpful to
keep in mind the assumptions used in developing the differential equation of the elastic
curve. All of the limitations that apply to the flexure formula also apply to the calculation
of deflections because the flexure formula was used in the derivation of Equation (10.1). It
is further assumed that
1. The square of the slope of the beam is negligible compared with unity. This assumption
means that beam deflections must be relatively small.
2. Plane cross sections of the beam remain planar as the beam deflects. This assumption
means that beam deflections due to shear stresses are assumed negligible.
3. The values of E and I remain constant for any segment along the beam. If either E or I
varies along the beam span, and if this variation can be expressed as a function of the
distance x along the beam, a solution of Equation (10.1) that considers the variation
may be possible.