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

394
bEAM dEFLECTIONS
+ M + M
Positive internal moment:
concave upward
– M
– M
Negative internal moment:
concave downward
FIGURE 10.4 Bendingmoment
sign convention.
v
A
Negative M
Positive
d 2 v
dx 2
B
d 2 v Positive M
Negative
dx 2
FIGURE 10.5 Relationship of
d 2 v/dx 2 to sign of M.
C
x
Sign conventions
The sign convention for bending moments established in Section 7.2 (see Figure 10.4) will
be used for Equation (10.1). Both E and I are always positive; therefore, the signs of the
bending moment and the second derivative must be consistent. With the coordinate axes as
shown in Figure 10.5, the beam slope changes from positive to negative in the segment
from A to B; therefore, the second derivative is negative, which agrees with the sign convention
of Section 7.2. For segment BC, both d 2 v/dx 2 and M are seen to be positive.
Careful study of Figure 10.5 reveals that the signs of the bending moment and the
second derivative are also consistent when the origin is selected at the right, with x positive
to the left and v positive upward. However, the signs are inconsistent when v is positive
downward. Consequently, v will always be chosen as positive upward for horizontal beams
in this book.
Relationship of Derivatives
Before proceeding with the solution of Equation (10.1), it is instructive to associate the
successive derivatives of the elastic curve deflection v with the physical quantities that they
represent in beam action:
Deflection = v
dv
Slope = = θ
dx
M = EI d 2
Moment v
2
dx
(fromEquation 10.1)
dM
V = = EI dv 3
Shear
3
dx dx
(for EI constant)
dV
Load w = = EI d 4
v
4
dx dx
(for EI constant)
The signs are as defined in Sections 7.2 and 7.3.
Starting from the load diagram, a method based on these differential relations was
presented in Section 7.3 for constructing first the shear diagram V and then the moment
diagram M. This method can be readily extended to the construction of the slope diagram
θ and the beam deflection diagram v. From Equation (e), we obtain
dθ =
dx
M
EI
(f)
This equation can be integrated to give
θB
xB
M d θ = ∫
∴θ − θ =
θ EI dx
B A
A xA
∫
∫
x
x
B
A
M
EI dx
The rightmost equation shows that the area under the moment diagram between any two
points along the beam (with the added consideration of EI) gives the change in slope between
the same two points. Likewise, the area under the slope diagram between two points
along the beam gives the change in deflection between the points. These two equations have
been used to construct the complete series of diagrams shown in Figure 10.6 for a simply