Shear and Bending Moment Diagrams Made Easy Using the

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ASEE Southeast Section Conference
2004

Shear and Bending Moment Diagrams Made Easy Using the
“Continuous Segment Method”

Timothy Wayne Mays, Ph.D., PE
1

Abstract

This paper presents a novel approach for developing shear and bending moment diagrams for beams and frames that
is different than meth
ods typically presented in structural analysis and mechanics of materials textbooks. This
nontraditional approach called the “Continuous Segment Method” utilizes an organized bookkeeping methodology
that results in a simplified means for solving structura
l analysis problems by direct integration. Irrespective of
problem complexity, free body diagrams are not needed to develop shear and bending moment diagrams using the
Continuous Segment Method. A student survey has been performed and students at The Cit
adel strongly prefer this
nontraditional method to traditional methods used in the same and other courses. More importantly, the
methodology is directly compatible with advanced topics in structural analysis such as deflections using virtual work
and the
analysis of indeterminate structures. This paper provides a derivation of the method, a simple outline of
rules and procedures, and various application examples for beams and frames.

Introduction

Typical structural analysis and mechanics of materials text
books introduce shear and bending moment diagrams by
first defining the differential relationship among external loading, shear, and bending moment. Using the
aforementioned relationship, such textbooks illustrate common loading cases graphically so that
students can
develop shear and moment diagrams by recognizing the change in slope of the loading and shear diagrams,
respectively. For examples, see [Hibbeler, 1], [Gere, 2], [Nelson et al., 3], and [Hibbeler, 4]. This traditional
approach to shear and b
ending moment diagrams conflicts with advanced structural topics such a deflection by
virtual work, a process that requires students to define shear and bending moment diagrams as a function of x (i.e.,
along the length of a beam) in order to solve the pro
blem. In addition, traditional methods do not provide self
-
checking mechanisms that can be employed as part of the analysis process.

Other approaches are also used to help students learn structural analysis. For example, [Das, 5] presents how
computer so
ftware can be used to help students learn structural analysis. [Rojiani and Schottler, 6] and [Holzer and
Andruet, 7] provide web based approaches to learning structural analysis. [Lesko et al., 8] presents a more practical
and experimental approach to l
earning structural analysis by having students load actual small
-
scale structures. The
literature suggests that such approaches have quantifiable merit.

This paper presents a novel approach to shear and bending moment diagrams called the “Continuous Segme
nt
Method.” The Continuous Segment (CS) Method is based on a manipulation of the differential relationship among
external loading, shear, and bending moment, whereby constants of integration are accounted for in tabular format.
The CS Method has numerous

advantages over traditional methods to include all of the following.



Free body diagrams are not needed to develop shear and bending moment diagrams.




1

Assistant Professor, The Cita
del, Department of Civil & Environmental Engineering, 171 Moultrie Street,
Charleston, SC, 29409.

ASEE Southeast Section Conference
2004



The method is directly compatible with advanced topics in structural analysis such as deflections using
virtual work and the analysis of indeterminate structures.



The method applies to all beams/frames and for any loading (e.g., students are not required to memorize
ways to solve “special case” problems).



Special tables are not required to locate the centroi
d of abnormal curves (e.g., loading or shear diagrams).

The only disadvantage to the CS Method is also that students are not required to develop free body diagrams of the
structure. Understanding free body diagrams is a major civil engineering concept tha
t students must learn prior to
structural analysis and mechanics of materials courses. Developing and application of free body diagrams should be
enforced during structural analysis and mechanics of materials courses and unless the professor requires the
diagrams, students using the CS Method most often choose not to draw them. Hence, professors choosing to adopt
the CS Method are encouraged to assess their student’s ability to develop and use free body diagrams by requiring
students to show CS Method tab
le values on free body diagrams of the structures.

Development of the Continuous Segment Method

Figure 1 shows a simply supported beam loaded by a continuous load w(x), a point load P
0
, and a point moment M
0
.
To derive the differential relationship among

external, loading, shear, and bending moment for the various loadings,
first consider a differential element of length dx taken somewhere along the region of continuous loading as shown in
Figure 2(a).

Figure 1. Simply supported beam with typical conti
nuous and point loadings shown.

Summing forces vertically, the relationship between continuous loading and shear can be obtained,

Note that the loading w(x) is defined in Figure 1 as positive downward. Hence, a net upward beam loading must be
treated as

a negative loading w(x). Examining the differential element shown in Figure 2(a), it is apparent that the
force w(x)dx does not act directly in the center of the element (i.e., at dx/2 from the right face), but at some distance
kdx from the right face.
Keeping this in mind, the relationship among continuous loading, shear, and bending
moment can be obtained by summing moments about the right face of the differential element.

w(x)
P
0
M
0
x
x
)
x
(
w
dx
dV
or
dx
)
x
(
w
dV
0
)
dV
V
(
dx
)
x
(
w
V
F
y










Vdx
dM
0
M
dM
M
)
kdx
(
dx
)
x
(
w
Vdx
M
Face
Right









ASEE Southeast Section Conference
2004

The

term w(x)dx(kdx) above is the product of two differential values and posse
sses a negligible magnitude relative
to the other terms in the derivation.







(a)





(b)






(c)

Figure 2. Differential beam element taken at (a) any location of continuous loading w(x), (b) location of applied
point load P, and (c) locat
ion of applied point moment M
0
..

Next, consider a differential element of length dx taken at the location of the applied point load P
0

as shown in
Figure 2(b). Summing forces vertically, the relationship between an applied point load and change in shear
can be
obtained.

Note that the change in shear follows the applied load. In other words, if a downward point load acts at some
location on a beam, the shear just to the right of the point load is equal to the magnitude of the shear just to the left of
the

point load minus the magnitude of the point load. Conversely, if an upward point load acts on a beam (e.g., a
reaction), the shear just to the right of the point load is equal to the magnitude of the shear just to the left of the point

load plus the magn
itude of the point load.

The relationship among applied point load, shear, and change in bending moment can be obtained by summing
moments about the right face of the differential element shown in Figure 2(b).

When the terms V and

P
0
/2 are multiplied by
the differential length dx, they are considered to generate a negligible
change in moment

M over the length dx
.

Finally, consider a differential element of length dx taken at the location of the applied point moment M
0

as shown in
Figure 2(c). Summing f
orces vertically, the relationship between an applied point moment and change in shear can
be obtained.

V
dx
dM
or

0
0
y
P
V
0
)
V
V
(
P
V
F










0
2
dx
P
Vdx
M
0
M
M
M
2
dx
P
Vdx
M
0
0
Face
Right













0
V
0
)
V
V
(
V
F
y








w(x)
dx
V+
dV
M+
dM
V
M
dx
V+

V
M+

M
V
M
P
0
dx
V+

V
M+

M
V
M
M
0
ASEE Southeast Section Conference
2004

The relationship among applied point moment, shear, and change in bending moment can be obtained by summing
moments about the right face of the differen
tial element shown in Figure 2(c).

As mentioned in the derivation for point loads, the term Vdx is considered to generate a negligible change in moment
over the length dx.

Application of the Continuous Segment Method for Beams

As the name implies, the CS
Method requires that the beam be visually identified as a series of continuous segments.
For statically determinate structures, a continuous segment is defined as a segment of a beam where the applied
loading (including reactions) is continuous. For exam
ple the beam in Figure 1 has four continuous segments. The
first segment is from the left support to the end of the continuous load. The continuous loading over this portion of
the beam is satisfied by the loading’s description w(x) which means that the
loading is a continuous function of x.
The end of the continuous load creates a discontinuity that requires a new beam segment. The second beam segment
is from the end of the continuous load until the point load is encountered. The nonexistent loading o
ver this range
satisfies the continuous loading requirement by the equation w(x)=0. The point load discontinuity requires a third
beam segment that begins at the location of the point load and ends at the location of the point moment. The
nonexistent loa
ding over this range also satisfies the continuous loading requirement by the equation w(x)=0. The
point moment discontinuity requires a final beam segment from the point moment location to the end of the beam.
The final beam segment also has no loading
and w(x)=0.

It should be noted that if the CS method is used to analyze statically indeterminate structures or to generate equations
used to calculate deflections, the definition of continuous segment must be slightly altered. For such applications, a
con
tinuous segment shall be defined as a segment of a beam where the applied loading, the beam’s modulus of
elasticity, and the beam’s moment of inertia in the direction of bending are all continuous. If any of the three beam
properties are not continuous o
ver a segment, new segments must be tagged and used for CS Method analysis. Since
the focus of this paper is simplified shear and bending moment diagrams for statically determinate beams, the
previous definition of a continuous segment shall be used herei
n.

The steps used in the CS Method are listed below. Following each step, an application example is provided for the
simple beam shown in Figure 3.

Figure 3. Example problem used to illustrate steps of the CS Method.

STEP 1: Determine the Minimum Number

of Continuous Segments

The first step for CS Method analysis is to identify the minimum number of continuous segments for the beam.
Based on the aforementioned definition of a continuous segment, the minimum number of continuous segments for
the beam sho
wn in Figure 3 is two. The left half of the beam will be labeled segment one and the right half of the
beam will be labeled segment two. The point load in the middle of the beam is a discontinuity in the otherwise
nonexistent beam loading. Hence for the

two segments, the continuous loading is w(x)=0.

0
0
0
Face
Right
M
M
Vdx
M
0
M
M
M
M
Vdx
M














10 ft
10 ft
12 k
6 k
6 k
ASEE Southeast Section Conference
2004

STEP 2: Create an Analysis Table for Each Segment

The second step is to create a table consisting of two rows and three columns for each segment. The label for
column one should identify the range of value
s over which the shear and bending moment for the segment are to be
determined. The range shall be in terms of
x

and not x, where
x

is relative to the left end of the segment and x is
relative to the left end of

the entire beam.
x

and x are identical for the first segment only. The second and third
column labels shall provide the location for calculation of shear and bending moment using equations yet to be
obtained. The general labels ar
e
x
=0 and
x
=end where end denotes the end of the segment. These values are
obtained to aid in graphing and to use for subsequent segments of the beam. For the first segment of the beam in
Figure 3, the table he
adings are provided in Figure 4. For the second segment of the beam, the table headings are
provided in Figure 5. Other equations and values listed in the table are presented and discussed in steps 3 and 4.

Figure 4. CS Method table values and appropri
ate headings for segment one.

Figure 5. CS Method table values and appropriate headings for segment two.

STEP 3: Calculate Shear and Bending Moments for Each Segment

The third step is to determine the shear and bending moment as a function of
x

along the length of each segment.
The shear and bending moment equations should be obtained in order beginning with the leftmost segment and
ending with the rightmost segment. For each segment, the shear V as a function of
x
is obtained from the following
expression.

The term V
0

consists of any point load or reaction applied at the leftmost end of the segment and the final shear value
for the previous segment as obtained from the previous table (if one exists). The term

w(
x
) is used in lieu of the
term w(x) to denote that the loading is continuous only over the segment and should be defined as a function of
x
.

For the example beam of Figure 3, the shear along segment one is a c
onstant value of 6 as obtained from the
equation above and shown in Figure 4. Note that since the loading is nonexistent, the term with the integral is 0 and
V
0

consists only of the leftmost support reaction. However, for segment two, the loading is als
o nonexistent but the
term V
0

is comprised of the point load value of

12 and the previous table value of 6 resulting in a net V
0

=
-
6 as
illustrated in Figure 5.

For each segment, the bending moment M as a function of
x
is obtained f
rom the following expression.

60
0
x
6
x
6
0
)
x
(
V
M
M
6
6
6
0
6
)
x
(
w
V
V
10
x
@
0
x
@
ft
10
x
0
0
0
















0
60
x
6
60
)
x
(
V
M
M
6
6
6
12
6
)
x
(
w
V
V
10
x
@
0
x
@
ft
10
x
0
0
0





















)
x
(
w
V
V
0



)
x
(
V
M
M
0
ASEE Southeast Section Conference
2004

The term M
0

consists of any point moment applied at the leftmost end of the segment and the final bending moment
value for the previous segment as obtained from the previous table (if one exists). It is important to note that

as
derived in a previous section, a clockwise applied point moment provides a positive contribution to M
0
. The term
V(
x
) is identical to the expression above for V since V has been obtained as a function of
x
.

For the example beam of Figure 3, the bending moment along segment one is found as a linear function of
x

as
shown in Figure 4. Note that there is no applied point moment at the left end of segment one and that there is no
previous t
able to the one used for segment one. Hence, for segment one, the value for M
0

is zero. However, for
segment two, the term M
0

is the previous table value of 60 as illustrated in Figure 5.

STEP 4: Plot Shear and Bending Moment Functions for Each Segment

The final step in the CS Method is to plot shear and bending moment diagrams as a function of x along the length of
the beam. These plots can be generated readily using table values from the CS Method and equations for V and M
that are generated as a func
tion of
x

as part of the process. The simplest way to generate the diagrams is to first plot
the calculated table values (end reactions) directly on the graph. These points can then be connected by plotting the
shear and moment func
tions directly from point to point. Straight vertical lines are used to connect points with the
same assigned value of x. These vertical lines indicate discontinuities in the beam loading. Shear and bending
moment diagrams for the beam in Figure 3 are s
hown in Figure 6.

Application of the Continuous Segment Method for Frames

For frame analysis, a preliminary step is used to simplify the process. The beam and beam
-
column segments making
up the frame are isolated so that the end reactions of all members c
an be obtained from statics. After obtaining all
end reactions, the frame members are treated as individual beams and the methodology used for beam and frame
analysis is identical. An example problem illustrating the CS Method for frames is provided at t
he end of this paper.

-8
-6
-4
-2
0
2
4
6
8
0
5
10
15
20
25
Location (ft)
Shear (k)

Figure 6. Shear and bending moment diagrams for the two
-
segment example.

Results of Limited Student Survey

To gauge the student’s opinion of CS Method, anonymous survey results have been obtained from St
udents taking
CIVL 309


Structural Analysis course at The Citadel. Given the small sample size, nine students, statistically
presenting the results (all in favor of the CS Method) could be misleading. Rather, it is important to list the general
statemen
ts made by the students regarding the use of the CS Method.



The CS Method provides a convenient check of results and is easy to follow, understand, and apply.



The CS Method is easy to teach to others.

0
10
20
30
40
50
60
70
0
5
10
15
20
25
Location (ft)
Moment (k-ft)
ASEE Southeast Section Conference
2004



Calculations using the CS Method require less time an
d are more straightforward.



There is not enough documentation on the CS Method (several students commented on this).



I would have a much harder time analyzing structures without the CS Method.

Future literature will present the CS Method as applied to adva
nced structural analysis concepts such as
indeterminate beams and virtual work. Also, the CS Method will be taught again in the spring of 2004 and an
expected sample size of approximately 25 will be used in the next survey.

Examples Utilizing the Continuo
us Segment Method

Following the list of references below are examples that illustrate the CS Method for statically determinate beams
and frames. Example problems are provided to show common applications in structural engineering. A commentary
is not prov
ided for each problem. Rather, the reader is encouraged to review previous sections of this paper prior to
working the provided example problems.

References

1.

Hibbeler, Russell C. (2001)
Structural Analysis
, Prentice Hall, Englewood Cliffs, NJ.

2.

Gere, Jam
es M. (2002)
Mechanics of Materials
, Brooks Cole, New York.

3.

Nelson, James K and McCormac, Jack C. (2002)
Structural Analysis: Using Classical and Matrix Methods
, John
Wiley & Sons, New York.

4.

Hibbeler, Russell C. (2002)
Mechanics of Materials
, Prentice Hall
, Englewood Cliffs, NJ.

5.

Das, Nirmal K. (2001) “Teaching Structural Analysis Using MathCAD Software,”
Proceedings of the 2001
American Society for Engineering Education Annual Conference & Exposition
.

6.

Rojiani, Kamal B. and Schottler, Robert (2000) “Java Ap
plets for Structural Analysis,”
Proceedings of the 2000
American Society for Engineering Education Annual Conference & Exposition
.

7.

Holzer, Seigfried M., and Andruet, Raul H. (1999) “Learning Statics with Multimedia and Other Tools,”
Proceedings of the 1999

American Society for Engineering Education Annual Conference & Exposition
.

8.

Lesko, Jack, Duke, Jack Holzer, Seigfried, and Auchey, Flynn (1999) “Hands
-
on
-
Statics Integration into an
Engineering Mechanics
-
Statics Course: Development and Scaling,”
Proceeding
s of the 1999 American Society for
Engineering Education Annual Conference & Exposition
.

ASEE Southeast Section Conference
2004

Example Problem 1


Example Problem 2




7 ft
6 ft
2 k/ft
12 k
50 k
-
ft
7 ft
154 k
-
ft
0
36
x
x
12
36
)
x
(
V
M
M
0
12
x
2
12
)
x
(
w
V
V
6
x
@
0
x
@
ft
6
x
0
36
120
x
12
120
x
12
50
70
)
x
(
V
M
M
12
12
12
0
12
)
x
(
w
V
V
7
x
@
0
x
@
ft
7
x
0
70
154
x
12
154
)
x
(
V
M
M
12
12
12
0
12
)
x
(
w
V
V
7
x
@
0
x
@
ft
7
x
0
2
0
0
0
0
0
0

























































10 ft
10 ft
2 k/ft
10 k
10 k
x
5
1
)
x
(
w

x
5
1
2
)
x
(
w


0
67
.
66
x
30
1
x
67
.
66
)
x
(
V
M
M
10
0
x
10
1
x
2
x
10
1
x
2
0
)
x
(
w
V
V
10
x
@
0
x
@
ft
10
x
0
67
.
66
0
x
30
1
x
10
x
30
1
x
10
0
)
x
(
V
M
M
0
10
x
10
1
10
)
x
(
w
V
V
10
x
@
0
x
@
ft
10
x
0
3
2
0
2
2
0
3
3
0
2
0





































0
2
4
6
8
10
12
14
0
5
10
15
20
25
Location (ft)
Shear (k)
-200
-150
-100
-50
0
0
5
10
15
20
25
Location (ft)
Moment (k-ft)
-15
-10
-5
0
5
10
15
0
5
10
15
20
25
Location (ft)
Shear (k)
0
20
40
60
80
0
5
10
15
20
25
Location (ft)
Moment (k-ft)
ASEE Southeast Section Conference
2004

Example Problem 3 (Preliminary Step of Isolating Members for Frames)











20 ft
2 k/ft
100 k
6 ft
6 ft
100 k
50 k
10 k
20 ft
2 k/ft
100 k
6 ft
6 ft
100 k
50 k
10 k
600 k
-
ft
10 k
10 k
600 k
-
ft
50 k
50 k
ASEE Southeast Section Conference
2004

Example Problem 3 (CS Method for L
eft Column)


Example Problem 3 (CS Method for Beam)







100 k
6 ft
6 ft
100 k
10 k
600 k
-
ft
10 k
x
600
600
600
0
600
)
x
(
V
M
M
0
0
0
100
100
)
x
(
w
V
V
6
x
@
0
x
@
ft
6
x
0
600
0
x
100
x
100
0
)
x
(
V
M
M
100
100
100
0
100
)
x
(
w
V
V
6
x
@
0
x
@
ft
6
x
0
0
0
0
0
































20 ft
2 k/ft
10 k
600 k
-
ft
50 k
0
600
x
x
10
600
)
x
(
V
M
M
50
10
x
2
10
)
x
(
w
V
V
20
x
@
0
x
@
ft
20
x
0
2
0
0


















0
20
40
60
80
100
120
0
5
10
15
Location (ft)
Shear (k)
0
200
400
600
800
0
5
10
15
Location (ft)
Moment (k-ft)
-60
-50
-40
-30
-20
-10
0
0
5
10
15
20
25
Location (ft)
Shear (k)
0
200
400
600
800
0
5
10
15
20
25
Location (ft)
Moment (k-ft)
ASEE Southeast Section Conference
2004

Timothy Wayne Mays

Dr. Mays came to The Citadel in August of 2002. Prior to his arrival at The Citadel, he worked as an associate
structural engineer for Lindbergh & Associates in Charleston,

SC while teaching at The Citadel as an adjunct
professor. He received a Ph.D. in Civil Engineering from Virginia Polytechnic Institute and State University and
specialized in structural/seismic engineering. During his time at Virginia Polytechnic Instit
ute and State University,
Dr. Mays received numerous regional and national awards such as a National Science Foundation Graduate Research
Fellowship, the Earthquake Engineering Research Institute’s most outstanding student paper award, and the
University’s

most outstanding engineering research award. He has also been a guest speaker at numerous regional
and national organizational meetings discussing current topics such as antiterrorism, code development, and seismic
design.