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

10.4 Determining Deflections by Integration
of a Moment Equation
v=0
Pin support
v=0
Roller support
v=0
Pin support
Whenever the assumptions of the previous section are satisfied and the bending moment
can be readily expressed as an integrable function of x, Equation (10.1) can be solved for
the deflection v of the elastic curve at any location x along the beam’s span. The procedure
begins with the derivation of a bending-moment function M(x) based on equilibrium considerations.
A single function that is applicable to the entire span may be derived, or it may
be necessary to derive several functions, each applicable only to a specific region of the
beam span. The moment function is substituted into Equation (10.1) to define the differential
equation. This type of differential equation can be solved by integration. Integration of
Equation (10.1) produces an equation that defines the beam slope dv/dx. Integrating again
produces an equation that defines the deflection v of the elastic curve. This approach to
determining the elastic curve equation is called the double-integration method.
Each integration produces a constant of integration, and these constants must be evaluated
from known conditions of slope and deflection. The types of conditions for which
values of v and dv/dx are known can be grouped into three categories: boundary conditions,
continuity conditions, and symmetry conditions.
v=0
Roller support
dvdx=0
v=0
Fixed support
V=0
M=0
Free end
FIGURE 10.7 Boundary
conditions.
Boundary conditions
Boundary conditions are specific values of the deflection v or slope dv/dx that are known at
particular locations along the beam span. As the term implies, boundary conditions are
found at the lower and upper limits of the interval being considered. For example, a bendingmoment
equation M(x) may be derived for a particular beam within a region x 1 ≤ x ≤ x 2 . The
boundary conditions, in this instance, would be found at x = x 1 and x = x 2 .
Boundary conditions are known slopes and deflections at the limits of the bendingmoment
equation M(x). The term “boundary” refers to the bounds of M(x), not necessarily
the bounds of the beam. Although boundary conditions are found at beam
supports, only those supports within the bounds of the bending-moment equation
should be considered.
Figure 10.7 shows several supports and lists the boundary conditions associated with each.
A pin or roller support represents a simple support at which the beam is restrained from
deflecting transversely (either upward or downward for a horizontal beam); consequently,
the beam deflection at either a pin or a roller must be v = 0. Neither a pin nor a roller, however,
restrains a beam against rotation, and consequently, the beam slope at a simple support
cannot be a boundary condition. At a fixed connection, the beam is restrained against both
deflection and rotation; therefore, v = 0 and dv/dx = 0 at such a connection.
While boundary conditions involving a deflection v and a slope dv/dx are normally
equal to zero at supports, there may be instances in which the engineer wishes to analyze
the effects of support displacement on the beam. For instance, a common design concern is
the possibility of support settlement, in which the compression of soil underneath a foundation
causes the support to be displaced downward. To examine possibilities of this sort,
nonzero boundary conditions may sometimes be specified.
One boundary condition can be used to determine one and only one constant of integration.
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