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

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dEFLECTIONS OF bEAMS by
THE VIRTuAL-wORk METHOd
A B C D E
x 1 x 3
A B C D E
x 1 x 3
x 2
x 4
x 2
x 4
(a) Real loads
(b) Virtual loads
FIGURE 17.24 Choice of x coordinates for integration of expressions in M and m.
distributed loadings spread across only a portion of the span, will cause discontinuities in
the moment equation for a beam. For example, suppose that the deflection at point D is to
be determined for the beam shown in Figure 17.24a. The real internal bending moments M
could be expressed in equations written for segments AB, BC, and CE of the beam. From
Figure 17.24b, however, we observe that the virtual internal moment m could be expressed
with only two equations: one for segment AD and the other for segment DE. Note, though,
that Equations (17.35) and (17.36) must be continuous functions throughout the length of
the segment in order for a single integration to be carried out for each of them. Since, however,
they are both discontinuous at D, segment CE of the beam must be further subdivided
into segments CD and DE.
Typically, several x coordinates must be employed in order to express the moment equation
for various regions of the beam span. To evaluate the integral in Equation (17.35), equations
for the real internal bending moment M and the virtual internal bending moment m in
each of segments AB, BC, CD, and DE of the beam must be derived. Separate x coordinates
may be chosen to facilitate the formulation of moment equations for each of these segments. It
is not necessary that these x coordinates all have the same origin; however, it is necessary that
the same x coordinate be used for both the real-moment and the virtual-moment equations that
are derived for any specific segment of the beam. For example, coordinate x 1 , with origin at A,
may be used with both Equation (17.35) and Equation (17.36) for segment AB of the beam.
Then, a separate coordinate x 2 , also with origin at A, may be used for the moment equations
applicable to segment BC. Next, a third coordinate, x 3 , with origin at E, may be used for segment
DE of the beam, and a fourth coordinate, x 4 , could be used to formulate the expressions
for segment CD. In any case, each x coordinate should be chosen to facilitate the formulation
of equations describing both the real internal moment M and the virtual internal moment m.
procedure for Analysis
The following procedure is recommended for calculating beam deflections and slopes by
the virtual-work method:
1. Real System: Draw a beam diagram showing all real loads.
2. Virtual System: Draw a diagram of the beam with all real loads removed. If a beam
deflection is to be determined, apply a unit load at the location desired for the deflection.
If a beam slope is to be determined, apply a unit moment at the desired location.
3. Subdivide the Beam: Examine both the real and virtual load systems. Also, consider
any variations of the flexural rigidity EI that may exist in the beam. Divide the
beam into segments so that the equations for the real and virtual loadings, as well as the
flexural rigidity EI, are continuous in each segment.
4. Derive moment Equations: For each segment of the beam, formulate an equation
for the bending moment m produced by the virtual external load. Formulate a second
equation expressing the variation in the bending moment M produced in the beam by