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

214
EQuILIbRIuM OF bEAMS
function f (x 1 ) and the M diagram is a second-order function f (x 2 ). As can be seen, the order of
the function increases successively by 1 in going from the w to the V to the M diagrams.
If the V diagram is constant for a beam segment, then the M diagram will be linear,
making the M diagram relatively straightforward to sketch. If the V diagram is linear for a
beam segment, then the M diagram will be quadratic (i.e., a parabola). A parabola can take
one of two shapes: either concave or convex. The proper shape for the M diagram can be
determined from information found on the V diagram, since the slope of the M diagram is
equal to the intensity of the shear force V [Rule 5: Equation (7.2)]. Various shear-force diagram
shapes and their corresponding bending-moment shapes are illustrated in Figure 7.12.
If the shear-force diagram is positive and looks like this . . .
VA
Constant
(zero order)
V B
linear
(first order)
V D
V E
linear
(first order)
V C
V F
V
. . . then the bending-moment diagram looks like this:
linear
(first order)
MB
Quadratic
(second order)
M D
M F
M A
M C
M E
Quadratic
(second order)
Constant slope
(an upward ramp)
Slope becomes
more positive
(a hill that
gets steeper)
(a) Positive shear-force diagrams
If the shear-force diagram is negative and looks like this . . .
Slope becomes
less positive
(an arch)
V
V C
V F
VA
Constant
(zero order)
V B
linear
(first order)
V D
V E
linear
(first order)
. . . then the bending-moment diagram looks like this:
M A
M C
M E
Quadratic
(second order)
linear
(first order)
MB
Quadratic
(second order)
M D
M F
Constant slope
(a downward ramp)
Slope becomes
more negative
(a waterfall)
Slope becomes
less negative
(a valley)
(b) Negative shear-force diagrams
FIGURE 7.12
Relationships between V and M diagram shapes.