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

General procedure for constructing Shear-Force
and Bending-moment Diagrams
The method for constructing V and M diagrams presented here is called the graphical
method because the load diagram is used to construct the shear-force diagram and then the
shear-force diagram is used to construct the bending-moment diagram. The six rules just
outlined are used to make these constructions. The graphical method is much less time
consuming than the process of deriving V(x) and M(x) functions for the entire beam, and it
provides the information necessary to analyze and design beams. The general procedure
can be summarized by the following steps:
Step 1 — complete the Load Diagram: Sketch the beam, including the supports,
loads, and key dimensions. Calculate the external reaction forces, and if the beam is a
cantilever, find the external reaction moment. Show these reaction forces and moments
on the load diagram, using arrows to indicate the direction in which they act.
Step 2 — construct the Shear-Force Diagram: The shear-force diagram will be constructed
directly beneath the load diagram. For that reason, it is convenient to draw a
series of vertical lines beneath significant locations on the beam in order to help align
the diagrams. Begin the shear-force diagram by drawing a horizontal axis, which will
serve as the x axis for the V diagram. The shear-force diagram should always start and
end on the value V = 0. Construct the V diagram from the leftmost end of the beam toward
the rightmost end, using the rules outlined on p. 209. Rules 1 and 2 will be the
rules most frequently used to determine shear-force values at important points. Rule 3 is
useful for sketching the proper shape of the diagram between these key points. Label all
points where the shear force changes abruptly and all locations where maximum or
minimum (i.e., maximum negative values) shear forces occur.
Step 3 — Locate Key points on the Shear-Force Diagram: Special attention should be
paid to locating points where the V diagram crosses the V = 0 axis, because these points indicate
locations where the bending moment will be either a maximum or a minimum value.
For beams with distributed loadings, Rule 3 will be essential for this task.
Step 4 — construct the Bending-moment Diagram: The bending-moment diagram
will be constructed directly beneath the shear-force diagram. Begin the bendingmoment
diagram by drawing a horizontal axis, which will serve as the x axis for the M
diagram. The bending-moment diagram should always start and end on the value M = 0.
Construct the M diagram from the leftmost end of the beam toward the rightmost end,
using the rules outlined on p. 209. Rules 4 and 6 will be the rules most frequently used
to determine bending-moment values at important points. Rule 5 is useful for sketching
the proper diagram shape between these key points. Label all points where the bending
moment changes abruptly and all locations where maximum or minimum (i.e., maximum
negative values) bending moments occur.
211
gRAPHICAL METHOd FOR
CONSTRuCTINg SHEAR ANd
MOMENT dIAgRAMS
The graphical method is most
useful when the areas associated
with Equations (7.3) and (7.4)
are simple rectangles or
triangles. These types of areas
exist when beam loadings are
concentrated loads or uniformly
distributed loads.
The idea of starting and ending
at V = 0 is related to the beam
equilibrium equation ΣF y = 0.
A shear-force diagram that
does not return to V = 0 at the
rightmost end of the beam
indicates that equilibrium has
not been satisfied. The most
common cause of this error in
the V diagram is a mistake in the
calculated beam reaction forces.
The idea of starting and ending
at M = 0 is related to the beam
equilibrium equation ΣM = 0. A
bending-moment diagram that
does not return to M = 0 at the
rightmost end of the beam
indicates that equilibrium has
not been satisfied. The most
common cause of this error in
the M diagram is a mistake in the
calculated beam reaction forces.
If the applied loads included
concentrated moments, another
common error is “jumping”
the wrong direction at the
discontinuities.
In the example problems that follow, a special notation is used to denote values at discontinuities
on the V and M diagrams. To illustrate this notation, suppose that a discontinuity
occurs at x = 15 on the shear-force diagram. Then the shear value on the −x side
of the discontinuity will be denoted V(15 − ), and the value on the +x side will be denoted
V(15 + ). Similarly, if a bending-moment discontinuity occurs at x = 0, then the moment
values at the discontinuity will be denoted M(0 − ) and M(0 + ).