Bauer 1997 Plastic Modulus Aisc

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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 .



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.

ENGINEERING JOURNAL /TH IRD QUART ER/ 1997 105



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



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

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Bauer 1997 Plastic Modulus Aisc