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

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ENERgy METHOdS
the real external loads. (Review Section 7.2 for a discussion on deriving bendingmoment
equations.) Note that the same x coordinate must be used in both equations.
The origin for the x coordinate may be located anywhere on the beam and should be
chosen so that the number of terms in the equation is minimized. Use the standard
convention for bending-moment signs, illustrated in Figures 7.6 and 7.7, for both the
virtual and real internal-moment equations.
5. Virtual-Work Equation: Determine the desired beam deflection by applying Equation
(17.35), or compute the desired beam slope by applying Equation (17.36). If the
beam has been divided into segments, then you can evaluate the integral on the righthand
side of Equation (17.35) or Equation (17.36) by algebraically adding the integrals
for all segments of the beam. It is, of course, important to retain the algebraic sign of
each integral calculated within a segment.
If the algebraic sum of all of the integrals for the beam is positive, then D or θ is in
the same direction as the virtual unit load or virtual unit moment. If a negative value is
obtained, then the deflection or slope acts opposite to the direction of the virtual unit
load or virtual unit moment.
Examples 17.13–17.15 illustrate use of the virtual-work method to determine
beam deflections and beam slopes.
ExAmpLE 17.13
w
Calculate (a) the deflection and (b) the slope at end A of the cantilever
beam shown. Assume that EI is constant.
A
L
Plan the Solution
B
The deflection at end A can be determined through the use of a virtual
unit load acting downward at A. Consider the beam with the real load w
removed and a virtual load applied at A. An equation for the variation of
the virtual internal moment m can be derived, and this equation will be
continuous over the entire length of the span. Next, consider the beam without the virtual
load, but with the real load w reapplied. Derive an equation for the variation of the real internal
moment M. This equation will also be continuous over the entire span. Therefore, the
beam need not be subdivided in calculating its deflection and slope. Once equations for m
and M are obtained, apply Equation (17.35) to compute the beam deflection D at A. To determine
the slope of the beam at A, the virtual load will be a concentrated moment applied at
that point. After deriving a new equation for m, use Equation (17.36) to calculate θ.
A
1
x
L
B
SolutioN
(a) Virtual Moment m for Calculating the Beam Deflection: To
determine the downward deflection of the cantilever beam, first remove
the real load w from the beam and apply a virtual unit load
downward at A.
1
A
x
v
m
Next, draw a free-body diagram around end A of the beam. Place the origin of the x
coordinate system at A. From the free-body diagram, derive the following equation for the
virtual internal moment m:
m =−1x 0 ≤ x ≤ L