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Calculation of
the
Plastic Section Modulus
sing the Computer
DOMINIQUE BERNARD BAUER
ABSTRACT
A simple spreadsheet is presented which calculates the plas
tic section modulus of structural mem bers. The method con
sists in dividing the cross section into rectangles and arrang
ing all calculations conveniently into a spreadsheet program.
The basic algorithm and the required spreadsheet formulas
are given as well as a numerical ex ample.
INTRODUCTION
With the increasing use of the limit states design of steel
structures, engineers often have to calculate the plastic bend
ing resistance,
M
r
, of structural members, which is a function
of the plastic modulus, Z, of the cross section, that is,
M= ZF
y
(1)
where
<) = performa nce factor
F
y
= yield strength of steel.
Although the calculation of the plastic section modulus can
be done easily by hand, it can also be done quickly and
reliably using the computer. The following technical note
presen ts a simple spreadsh eet for the calculation of the plastic
modulus. It is restricted to cross sections that can be approxi
mated by a series of rectangles, which should cover most
situations that structural engineers encounter in the design
office.
SPREADSHEET ALGORITHM
The proposed algorithm
is
described
below.
The cross section
to be analyzed must first be divided into
N
rectangles (Figure
la) .
Each rectangle must comprise the entire width of the
cross section at any particular
height.
Hence, the arrangement
shown in Figure l a is valid, while the one shown in Figure lb
is not valid.
The w idth and the height of each rectangle will be entered
into the spreadsheet, going consecutively from top to b ottom
of the cross section. These values are the only required input
Dominique Bernard Bauer, P. Eng., MASCE, M CSC E, B. Eng.,
M. Eng., Ph.D ., structural engineering consultant n Montreal.
104 ENGINEERING JOURNAL/THIRD QUAR TER/ 1997
data. All the calculations p resented below are arranged
so
that
the equations can
be
expressed
as
spreadsheet formulas which
will be evaluated automatically by the spreadsheet program.
With a datum line placed at the top of the cross section, the
vertical distance from the datum line to the centroid, y
n
, of the
nth rectangle is equal to (Figure 2)
where
h
n
=
height of the
n th
rectangle
Datum line
I n
...n...
.N
(a) valid arrangement
Datum line
(b) invalid arrangement
Figure 1 .
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yo=K = o.
The cross-sectional area,
A
n
,
of the
nth
rectangle is equal to
K
=
b
n
h
n
(3)
where
b
n
=
width of the nth rectangle.
The total area, A
tot
^ of
the
cross section is equal to
A
tntn
i —
SA.
4)
The vertical distance, f, from the datum line to the neutral
axis, which divides the cross section into two portions of
equal areas, is determined by noting that if the neutral axis
passes through the
nth
rectangle, we must have
-^ =
y
£A
i
b
n
fi
n 5)
or
H - l
-5>
k=-
—
( 6 )
where
h
n
positions the neutral axis as shown is Figure 2. The
rectangle through which the neutral axis passes is determined
from the fact that it is the only one for which
h
n
> 0 and h
n
< h
n
(7)
For all other rectangles, Equation 7 is not verified. H ence, the
vertical distance from the datum line to the neutral axis,
Y
, is
equal to
m - l
(8)
where the subscript
m
identifies the single rectangle for w hich
Equation 7 is verified. For the other rectangles through which
the neutral axis does not pass, the values
?
n
= ^ h
i +
h
n
(9)
are meaningless and therefore discarded.
The contribution to the plastic m odulus,
Z
n
,
of each rectan
gle through which the neutral axis does
n ot
pass is equal to
Z „ = A
n
abs (5 j
(10)
where the distance from the neutral axis to the centroid of the
nth
rectangle,
<3
n
,
is equal to Figure 3a.
a
n
f-y
n
(11)
The contribution to the section modulus, Z
m
, of the rectangle
through which the neutral axis
does
pass is equal to Figure
3b.
Datum line
Neutral axis
(a)
Datum line
'
i
K n
1
(
r
r
| i...
l - n - l
Centroid of
^
ectangle n
u
n
n
* -
1 '
1
Neutral axis
i
L
y
n
Figure
2.
Datum line
h
'
i
j
1
1
——
,
2 :
h
m
- f i
m
2
k
'
b„
m
1
1
Neutral axis ,
k
?
'
i
(b)
Figure
3.
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z =
bjii bjh
m
-nj
(12)
2 2
Finally, the plastic modulus of the cross section,
Z,
is equal to
Z =
Z
m
J
j
Z
n
(13)
n= 1 toNexceptro
SPREADSHEET FORMULAS
The spreadsheet formulas required for the calculation of the
plastic section m odulus correspond
to
Equations 2 to
13
given
above. There m ust be as many sets of formulas, arranged in
rows in the spreadsheet, as there are rectangles into which the
cross section is divided. Assum ing that the cross section to be
analyzed is composed of 3 rectangles (see the example be
low),
there would be 3 sets of formulas, arranged in 3 rows,
say rows 9 to 11 in the spreadsheet. The formulas for the first
rectangle, in row 9, would be as shown in Table 1.
Note that the syntax used with the formulas given in Table
1 is that of Microsoft Excel. The formulas can be easily
Ub
mm
6 mm
75
mm
^ "
1
'
i
i
i
*
i
i
8 mm
1 mm
8 mm
(a)
(b)
Table 1.
Spreadsheet Formulas
Expression
b„
h
n
y
n
A
n
/=1
A 7 1
/ = 1
hn
Y
n
d
n
Z
n
or Z
m
Atotal
Atotal/2
Y=Ym
Z
Equation No.
2)
3)
6)
7) and 9)
11)
10) a nd 12)
4)
7) a nd 8)
13)
Cell
C9
D9
E9
F9
G9
H9
19
J9
K9
L9
F5
F4
J5
L5
Formula
input data
input data
D8/2+E8+D9/2
C9*D9
D8+G8
F8+H8
( F 4-H9)/C9
IF(AND(I9>0;I9<=D9);G9+I9;0)
IF(J9=0; J 5-E9;'Neutral Axis')
IF(J9=0;ABS(K9)*F9;C9*l9
A
2/2
+C9*(D9-l9)
A
2/2)
SUM(F9:F11)
F5/2
SUM(J9:J11)
SUM(L9:L11)
Figure 4.
modified to meet the syntax rules of other spreadsheet pro
grams such as Lotus
1-2-3,
Quattro Pro, etc.
EXAMPLE
Calculate the plastic modulus of the cross section shown in
Figure 4a.
The cross section is divided into 3 rectangles as shown in
Figure 4b. With the width and height of each rectangle entered
as input data in cells C9, D9, CIO, D10, C ll and D ll , the
value of the plastic modulus is calculated by the spreadsheet
as Z = 94,733 mm
3
and displayed in cell L5 (Figure 5).
CONCLUSION
A simple spreadsheet is presented which can be used to
calculate the plastic section modulus of structural members.
It should be useful to design engineers, espec ially when there
are many values to be calculated.
APPENDIX I. NOTATION
A
n
= area of the
nth
rectangle
o u l
=
a r e a
°f
m e
entire cross section
b
n
= width of the nth rectangle
F
y
= yield strength of steel
106 ENGINEERING JOURNAL yTHIRD QUARTER / 1997
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K
K
M
r
?
Y
m
t
= height
of
the nth rectangle
= distance from the top of the
nth
rectangle to the
neutral axis
of
the cross section
= plastic bending mom ent resistance
= distance from
the
datum line
to the
neutral axis
of
the cross section
= correct value
of
Y obtained with rectangle m
(Y
m
=
?)
=
value
of
Y obtained with
the
nth rectangle
(Y
n
* Y
for all rectangles except rectangle m)
y
n
z
z
z
n
< >
= distance from the datum line
to the
centroid
of
the
nth rectangle
= plastic section modulus
= contribution
of
rectangle m
to the
plastic section
modulus
= contribution
of the
nth rectangle
to the
plastic
section modulus
= performance factor
Column—>
U'Row
1
3
1
4
5
6
7
1 8
1
9
1
10
11
Note:
n
1
2
3
• •
C
bn
D E
Atotaj/2 =
K
125
6 '100
75
*|
iii
Atota]
y
n
4
58
112
Indicates input d
F
1100
2200
A,
1000
600
600
ata.
G
n - l
V
t
1=1
0
8
108
H
n - l
2 > i
0
1000
1600
I
Y =
K
8.8
16.667
-6.667
J
24.667
X
0
24.667
0
K
Z =
1
20.667
N . Axis
-87.33
1
L
94733.3|
Zn
20666.7|
21666.7|
52400J
Figure 5.
ENGINEERING JOURNAL
THIRD QUARTER
1997
107