©COMPUTERS AND STRUCTURES, INC., BERKELEY, CALIFORNIA DECEMBER 2001
COMPOSITE BEAM DESIGN AISC-ASD89
Technical Note
Transformed Section Moment of Inertia
This Technical Note describes in general terms how the program calculates
the transformed moment of inertia for a composite section, Itr. The calculated
transformed moment of inertia applies for full (100%) composite connection.
See Technical Note Elastic Stresses with Partial Composite Connection Composite Beam AISC-ASD89 for a description of partial composite connection.
The Technical Note also describes in detail a method that can be used to calculate the transformed section moment of inertia by hand that will yield the
same result as the program. The exact methodology used by the program is
optimized for computer-based calculations and is unsuitable for hand calculations and for presentation in this Technical Note.
Note that for the AISC-ASD89 specification, the transformed section properties used for stress calculations for a beam may be different from those used
for deflection calculations for the same beam. For AISC-ASD89 composite
beam design stress calculations, the value of Ec is always calculated from
Equation 1, assuming that the unit weight of concrete, wc, is 150 pounds per
cubic foot, regardless of its actual specified weight.
(
)
E c = w1.5
33 fc'
c
Eqn. 1
In Equation 1, Ec is in pounds per square inch (psi), wc is in pounds per cubic
foot (pcf) and f c' is in pounds per square inch (psi).
For AISC-ASD89 composite beam design deflection calculations, the value of
Ec is taken from the material property specified for the concrete slab.
Background
Page 1 of 21
Composite Beam Design AISC-ASD89
Transformed Section Moment of Inertia
Background
Figure 1 shows a typical rolled steel composite floor beam with the metal deck
ribs running parallel to the beam. Figure 2 shows a typical composite userdefined steel beam with the metal deck ribs running parallel to the beam.
Note that the user-defined beam may have a different top and bottom flange
size, and that no fillets are assumed in this beam.
For each of these configurations the following items may or may not be
included when calculating the transformed section moment of inertia:
Concrete in the metal deck ribs: The concrete in the metal deck ribs is
included in the calculation when the deck ribs are oriented parallel to the
beam (typically the case for girders). It is not included when the deck ribs are
oriented perpendicular to the beam (typically the case for infill beams).
•
Cover plate: The cover plate is only included if one is specified by you in
the composite beam overwrites.
Note that the deck type and deck orientation may be different on the two
sides of the beam as described in "Multiple Deck Types or Directions Along
the Beam Length" of Technical Note Effective Width of the Concrete Slab
Composite Beam Design.
Because composite behavior is only considered for positive bending, the
transformed section moment of inertia is only calculated for positive bending
(top of composite section in compression). Calculation of the transformed
section moment of inertia is greatly complicated by the requirement that the
concrete resist no tension.
The first task in calculating the transformed section moment of inertia of the
composite section is to compute properties for the steel beam alone (plus the
cover plate, if it exists). The properties required are the total area, Abare; the
location of the ENA, ybare; and the moment of inertia, Is.
Background
Page 2 of 21
Composite Beam Design AISC-ASD89
Transformed Section Moment of Inertia
hr
tc
Concrete slab
d
Metal deck
Bottom cover plate
tcp
Rolled steel beam
bcp
Concrete slab
hr
tf-top
tc
Figure 1: Composite Rolled Steel Beam Shown With Metal Deck Ribs
Running Parallel To Beam
Beam web
d
tw
tf-bot
Beam top flange
h = d - tf-top - tf-bot
bf-top
Metal deck
Bottom cover plate
bcp
tcp
Beam bottom flange
bf-bot
Figure 2: Composite User-Defined Steel Beam Shown With Metal Deck
Ribs Running Parallel To Beam
Background
Page 3 of 21
Transformed Section Moment of Inertia
ybare
Elastic neutral axis of steel beam
plus cover plate if applicable.
Ibare is taken about this axis.
y1 for top flange
Composite Beam Design AISC-ASD89
Bottom of bottom flange of steel
beam. Ybare and y1 are
measured from here
Figure 3: Illustration of ybare and y1
Properties of Steel Beam (Plus Cover Plate) Alone
The location of the ENA for the steel beam alone (plus cover plate if applicable) is defined by the distance ybare, where ybare is the distance from the bottom of the bottom flange of the beam to the ENA, as shown in Figure 3. If
there is a cover plate, ybare is still measured from the bottom of the bottom
flange of the beam, not the bottom of the cover plate.
Figure 3 also illustrates an example of the dimension y1 that is used in Tables
1 and 2. For a given element of a steel section, the dimension y1 is equal to
the distance from the bottom of the beam bottom flange to the centroid of
the element. Figure 3 illustrates the distance y1 for the beam top flange.
If the beam section is a rolled steel beam or channel chosen from the program section database, Abare, ybare and Ibare are calculated as shown in Table 1
and Equations 1 and 2. If the beam section is a user-defined (welded) beam,
they are calculated using Table 2 and Equations 1 and 2.
Properties of Steel Beam (Plus Cover Plate) Alone
Page 4 of 21
Composite Beam Design AISC-ASD89
Table 1:
Transformed Section Moment of Inertia
Section Properties for Rolled Steel Beam Plus Cover Plate
Item
Area, A
Steel beam
As
Cover plate
bcptcp
Sums
Table 2:
y1
−
d
2
t cp
2
ΣA
2
IO
Ay1
2
Is
Ay1
Ay1
2
Σ(Ay1)
Σ(Ay1 )
Ay1
Ay1
Ay1
b cp t 3cp
12
2
ΣIO
Section Properties for User-Defined (Welded) Steel Beam Plus
Cover Plate
Item
Area, A
Top flange
bf-toptf-top
Web
Bottom flange
Cover plate
Sums
y1
d−
t f − top
2
2
Ay1
Ay1
Ay1
Ay1
2
htw
d
2
Ay1
Ay1
2
bf-bottf-bot
t f − bot
2
Ay1
Ay1
2
Ay1
Ay1
2
Σ(Ay1)
2
Σ(Ay1 )
bcptcp
−
ΣA
t cp
2
IO
b f − top t 3f − top
12
t wh3
12
b f − bot t 3f − bot
12
b cp t 3cp
12
ΣIO
The area of the steel section (including the cover plate if it exists), Abare, is
given by Equation 1.
Abare = ΣA
Eqn. 1
The ENA of the steel section is located a distance ybare from the bottom of the
bottom flange of the steel beam section (not bottom of cover plate) where
ybare is determined from Equation 2.
Properties of Steel Beam (Plus Cover Plate) Alone
Page 5 of 21
Composite Beam Design AISC-ASD89
y bare =
Transformed Section Moment of Inertia
∑ (Ay )
∑A
1
Eqn. 2
The moment of inertia of the steel section (plus cover plate, if one exists)
about its ENA, Ibare, is given by Equation 3.
I bare =
∑ (Ay ) + ∑ I
2
1
O
−
(∑ A ) y
Eqn. 3
2
bare
Following is the notation used in Tables 1 and 2 and Equations 1 through 3:
Abare
= Area of the steel beam (plus cover plate, if one exists),
in2.
As
= Area of rolled steel section alone (without the cover plate
even if one exists), in2.
Ibare
= Moment of inertia of the steel beam (plus cover plate if
one exists), in4.
IO
= The moment of inertia of an element of the beam section
taken about the ENA of the element, in4.
Is
= Moment of inertia of the steel beam alone (without the
cover plate even if one exists), in4.
bcp
= Width of steel cover plate, in.
bf-bot
= Width of bottom flange of a user-defined steel beam, in.
bf-top
= Width of top flange of a user-defined steel beam, in.
d
= Depth of steel beam from outside face of top flange to
outside face of bottom flange, in.
h
= Clear distance between flanges for user-defined (welded)
sections, in.
tcp
= Thickness of cover plate, in.
tf-bot
= Thickness of bottom flange of a user-defined (welded)
section, in.
Properties of Steel Beam (Plus Cover Plate) Alone
Page 6 of 21
Composite Beam Design AISC-ASD89
Transformed Section Moment of Inertia
tf-top
= Thickness of top flange of a user-defined (welded) section,
in.
tw
= Thickness of web of user-defined (welded) section, in.
ybare
= Distance from the bottom of the bottom flange of the steel
section to the ENA of the steel beam (plus cover plate if it
exists), in.
y1
= Distance from the bottom of the bottom flange of the steel
beam section to the centroid of an element of the beam
section, in.
ΣA
= Sum of the areas of all of the elements of the steel beam
section, in2.
Σ(Ay1)
= Sum of the product A times y1 for all of the elements of
the steel beam section, in3.
Σ(A y12 ) = Sum of the product A times y12 for all of the elements of
the steel beam section, in4.
ΣIO
=
Sum of the moments of inertia of each element of the
beam section taken about the ENA of the element, in4.
Properties of the Composite Section General
Calculation Method
The first step, and potentially most calculation-intensive step in the process of
determining the composite properties is to calculate the distance from the
ENA of the steel beam (plus cover plate if it exists) to the ENA of the full
composite section. This distance is designated ye in Figure 4.
Recall that concrete in tension is ignored when calculating the composite
properties. Because of the possibility that some of the concrete may be in
tension, and because the amount of concrete that is in tension is initially unknown (if any), the process for calculating the distance ye is iterative. After
the distance ye has been determined, the other calculations to determine the
composite properties are relatively straight-forward.
Properties of the Composite Section General Calculation Method
Page 7 of 21
Composite Beam Design AISC-ASD89
Transformed Section Moment of Inertia
Elastic neutral axis of steel beam
alone, including cover plate if it
exists
ye
z
Elastic neutral axis of composite
beam
Figure 4: Illustration of ye and z
The program uses the following method to calculate the properties of the
composite section.
1. The location of the ENA of the composite section, defined by ye (see Figure 4), is calculated using the following iterative process:
a. The program assumes (guesses) that the ENA of the composite section
is within the height of the steel beam and uses Equation 4 to calculate
the distance ye that defines the location of the ENA for the composite
section. Note that with this assumption, all of the concrete is above
the ENA of the composite section and thus it is all in compression and
can be considered.
ye =
Σ(A element delement )
ΣA element
Eqn. 4
where,
Aelement
= Area of an element in the composite section, ignoring any
area of concrete that is in tension and ignoring any concrete in the metal deck ribs when the metal deck span is
perpendicular to the beam span, in2.
Properties of the Composite Section General Calculation Method
Page 8 of 21
Composite Beam Design AISC-ASD89
delement
Transformed Section Moment of Inertia
= Distance from the ENA of the element considered to the
ENA of the steel beam alone (including cover plate, if it
exists), in. Signs are considered for this distance. Elements located below the ENA of the steel beam alone (including cover plate, if it exists) have a negative distance
and those above have a positive distance.
If the ENA as calculated is within the height of the steel beam, as assumed, the assumed location of the ENA is correct and the calculation
for ye is complete.
b. If the calculated ENA is not within the height of the steel beam, as assumed in Step a, the assumed location of the ENA is incorrect and calculation for ye continues.
i
Using the incorrect location of the ENA calculated in Step a, the
program calculates the location of ye again using Equation 4, ignoring any concrete that is in tension.
ii
If the newly calculated location of the ENA is the same as the previously calculated location (Step i), the assumed location of the
ENA has been identified and the calculation for ye is complete.
c. If the newly calculated location of the ENA is not the same as the previously calculated location (Step i), the most recent assumed location
of the ENA is incorrect and another iteration is made.
The program repeats the iterations until the location of the ENA has
been determined. After the location of the ENA is known, the rest of
the process for calculating the composite properties is non-iterative.
2. Given that the ENA has been located, the program determines if any concrete is below the ENA. If so, the program ignores it in the remaining calculations.
3. The program sums the product of the area of each element of the composite section (except concrete in tension) times its distance to a convenient axis (such as the bottom of the beam bottom flange).
4. The program divides the sum calculated in step 3 by the sum of the areas
of each element of the composite section (except concrete in tension).
Properties of the Composite Section General Calculation Method
Page 9 of 21
Composite Beam Design AISC-ASD89
Transformed Section Moment of Inertia
This calculation yields the distance from the convenient axis to the ENA of
the composite section.
5. After the ENA of the composite section has been determined, the section
properties of the composite section are quickly calculated using standard
methods.
A hand calculation method for determining the distance ye described in steps
1a through 1c above is presented in the next section entitled "Equivalent
Hand Calculation Method to Calculate the Distance ye." A hand calculation
method for the calculation of the composite properties described in steps 2
through 5 above is presented in the section entitled "Equivalent Hand Calculation Method to Calculate the Composite Properties" later in this Technical
Note.
Equivalent Hand Calculation Method to Calculate the
Distance ye
The following hand calculation method for determining the distance ye is
similar to and provides the same result as the calculations performed by the
program.
After ybare has been calculated, ye is calculated by equating the forces above
and below the ENA using either Equation 5a or Equation 5b. Recall that ye is
the distance from the ENA of the steel beam alone, plus cover plate if it exits,
to the ENA of the fully composite section, as illustrated in Figure 4.
ye =
ye =
X1 + X 2 + X 3 + X 4
A bare + X 5 + X 6 + X 7 + X 8
2
- X10 ± X10
− 4X 9 (X1 + X 2 + X 3 + X 4 )
2X 9
Eqn. 5a
Eqn. 5b
Equations for use in calculating values for the variables X1 through X10 in
Equations 5a and 5b are presented in the following subsection entitled "Background Equations." The actual process to calculate ye is described in the subsection of this Technical Note entitled "Hand Calculation Process for ye."
Equivalent Hand Calculation Method to Calculate the Distance ye
Page 10 of 21
Composite Beam Design AISC-ASD89
Transformed Section Moment of Inertia
Background Equations
This subsection presents the equations for the variables X1 through X10 in
Equations 5a and 5b. The exact equation to use for the variables X1 through
X10 depends on the assumed location of the ENA.
For the purposes of determining the ye distance, there are nine possible locations for the ENA. Those locations are as follows:
1. The ENA is located within the height of the steel section (including
cover plate, if it exists).
2. The ENA is located within the height of the metal deck on both the left
and the right sides of the beam.
3. The ENA is located within the height of the metal deck on the left side
of the beam and within the height of the concrete above the metal
deck (or within a solid slab) on the right side of the beam.
Note: Recall that you can have different deck properties on the two
sides of the beam.
4. The ENA is located within the height of the metal deck on the left side
of the beam and above the concrete on the right side of the beam.
5. The ENA is located within the height of the concrete above the metal
deck (or within a solid slab) on the left side of the beam and within the
height of the metal deck on the right side of the beam.
6. The ENA is located within the height of the concrete above the metal
deck (or within a solid slab) on both sides of the beam.
7. The ENA is located within the height of the concrete above the metal
deck (or within a solid slab) on the left side of the beam and above the
concrete on the right side of the beam.
8. The ENA is located above the concrete on the left side of the beam and
within the height of the metal deck on the right side of the beam.
9. The ENA is located above the concrete on the left side of the beam and
within the height of the concrete above the metal deck (or within a
solid slab) on the right side of the beam.
Equivalent Hand Calculation Method to Calculate the Distance ye
Page 11 of 21
Composite Beam Design AISC-ASD89
Transformed Section Moment of Inertia
The first two columns in Table 3 list the nine possible locations of the ENA of
the composite section. The columns labeled Left Side and Right Side indicate
the location of the ENA relative to the left and right sides of the beam, respectively. The third column of Table 3, labeled "ye Eqn" specifies whether
Equation 5a or 5b should be used to calculate ye. Columns 4 through 13 of
Table 3 list the equation numbers to be used to determine the value of the
variables X1 through X10 for the location of the ENA specified in the first two
columns of the table.
When using Table 3, the location of the ENA of the composite section and the
location of the ENA of the composite section relative to the elements that
make up the composite section are initially unknown. Thus, begin by assuming a location of the ENA. It works best if you assume that the ENA of the
composite section is within the steel section. Then, calculate the actual location of the ENA and check the validity of the assumption. This process is described in the subsection entitled "Hand Calculation Process for ye."
Equations 7 through 16 define the terms X1 through X10 in Table 3 and Equations 5a and 5b. A term that is repeatedly used in Equations 7 through 16 is
z. As previously illustrated in Figure 4, z is the distance from the ENA of the
steel beam alone (plus cover plate, if it exists) to the top of the concrete slab.
The distance z, which can be different on the left and right sides of the beam,
is defined by Equations 6a and 6b.
zleft = d + hr left + tc left - ybare
Eqn. 6a
zright = d + hr right + tc right - ybare
Eqn. 6b
The equations for the variables X1 through X10 in Equations 5a and 5b and Table 3 follow. In most cases, there are multiple equations for each variable.
See Table 3 for specification of which equation to use for any assumed location of the ENA.
Equivalent Hand Calculation Method to Calculate the Distance ye
Page 12 of 21
Composite Beam Design AISC-ASD89
Transformed Section Moment of Inertia
Table 3: Table Identifying Circumstances for Using Equations 5a and 5b and
Identifying Appropriate Equations to Use to Calculate the Values of
Variables X1 through X10 that Appear in Equations 5a and 5b
Left
Side
Right
Side
ye
Eqn
X1
Eqn
X2
Eqn
X3
Eqn
X4
Eqn
X5
Eqn
X6
Eqn
X7
Eqn
X8
Eqn
X9
Eqn
X10
Eqn
Steel section
hr
hr
hr
tc
hr
>tc
tc
hr
tc
tc
tc
>tc
>tc
hr
>tc
tc
5a
5b
5b
5b
5b
5b
5b
5b
5b
7a
7a
7a
7a
7b
7b
7b
0
0
8a
8b
8b
8b
0
0
0
0
0
9a
9a
9b
0
9a
9b
0
9a
9b
10a
10b
0
0
10b
0
0
10b
0
11a
11a
11a
11a
11b
11b
11b
0
0
12a
12b
12b
12b
12c
12c
12c
0
0
13a
13a
13b
0
13a
13b
0
13a
13b
14a
14b
14c
0
14b
14c
0
14b
14c
N.A.
15a
15a
15a
15a
15a
15a
15a
15a
N.A.
16a
16c
16a
16d
16b
16b
16a
16b
Table Descriptive Notes:
1.
The columns labeled Left Side and Right Side indicate the assumed location of the
ENA of the composite section relative to the left and right sides of the beam. Steel
section means that the ENA falls within the height of the steel section (including the
cover plate, if it exists). The designation hr means that the ENA is within the height of
the metal deck. The designation tc means that the ENA is within the height of the
concrete slab above metal deck or within the height of a solid slab. The designation
>tc means that the ENA is above the concrete slab.
2. The column labeled "ye Eqn" tells you whether to use Equation 5a or Equation 5b to
calculate ye for the assumed location of the ENA listed in the first two columns of the
table.
3. The columns labeled "X1 Eqn" through "X10 Eqn" indicate the equation numbers that
should be used to calculate the value of the variables X1 through X10 for use in Equations 5a and 5b. If one of the cells for X1 through X8 contains a "0," the value of Xn is
zero for that location of the ENA.
4. The variables X9 and X10 are not used if the ENA falls within the height of the steel
beam.
5. The variables X2, X4, X6 and X8 are always taken as zero if the deck span is oriented
perpendicular to the beam span.
6. Using this table requires a trial and error process. You must assume a location for the
ENA and then check if the assumption is correct. See the subsection entitled "Hand
Calculation Process for ye" later in this chapter for more information.
Equivalent Hand Calculation Method to Calculate the Distance ye
Page 13 of 21
Composite Beam Design AISC-ASD89
Transformed Section Moment of Inertia
Important note: The terms X2, X4, X6 and X8 are always taken as zero if the
deck span is oriented perpendicular to the beam span; otherwise they are
taken as given in the equations below.
t c left

X1 = X 5  z left −
2

z
X1 = X 5  left
 2




Eqn. 7a



Eqn. 7b
X2 is taken as zero if the deck span is oriented perpendicular to the beam
span; if the deck span is oriented parallel to the beam span, X2 is as specified
in the equations below.
hr left

X 2 = X 6  z left − t c left −
2

(
X 2 = X 6 zleft − t c left
Eqn. 8a
)2
Eqn. 8b
t c right

X 3 = X 7  z right −
2

 zright
X 3 = X 7 
 2








Eqn. 9a




Eqn. 9b
X4 is taken as zero if the deck span is oriented perpendicular to the beam
span; if the deck span is oriented parallel to the beam span, X4 is as specified
in the equations below.
hr right

X 4 = X 8  z right − t c right −
2

(
X 4 = X 8 zright − t c right
X5 =




)2
b eff left E c left t c left
Es
Equivalent Hand Calculation Method to Calculate the Distance ye
Eqn. 10a
Eqn. 10b
Eqn. 11a
Page 14 of 21
Composite Beam Design AISC-ASD89
X5 =
b eff left E c left z left
Es
Transformed Section Moment of Inertia
Eqn. 11b
X6 is taken as zero if the deck span is oriented perpendicular to the beam
span; if the deck span is oriented parallel to the beam span, X6 is as specified
in the equations below.
X6 =
X6 =
X6 =
X7 =
b eff left E c left wr left hr left
E s Sr left
b eff left E c left wr left
2E s Sr left
b eff left E c left
Eqn. 12b
Eqn. 12c
2E s
b eff right E c right t c right
Es
X7 =
Eqn. 12a
b eff right E c right z right
Es
Eqn. 13a
Eqn. 13b
X8 is taken as zero if the deck span is oriented perpendicular to the beam
span; if the deck span is oriented parallel to the beam span, X8 is as specified
in the equations below.
X8 =
X8 =
b eff right E c right wr right hr right
E s Sr right
b eff right E c right wr right
X8 =
2E s Sr right
b eff right E c right
2E s
Eqn. 14a
Eqn. 14b
Eqn. 14c
X9 = X6 + X8
Eqn. 15a
X9 = X8
Eqn. 15b
Equivalent Hand Calculation Method to Calculate the Distance ye
Page 15 of 21
Composite Beam Design AISC-ASD89
Transformed Section Moment of Inertia
X9 = X6
Eqn. 15c
(
)
X 10 = − A bare − X 5 − 2X 6 z left − t c left −
(
X 7 − 2X 8 z right − t c right
Eqn. 16a
)
Eqn. 16b
X10 = − A bare − X 5 − X 7
(
)
X10 = − A bare − X 5 − X 6 zleft − t c left − X 7
(
X10 = − A bare − X 5 − X 7 − X 8 zright − t c right
Eqn. 16c
)
Eqn. 16d
The notation used in equations 5a through 16d are as follows:
Abare
= Area of the steel beam (plus cover plate), in2. This area
does not include the concrete area.
Ec
= Modulus of elasticity of concrete slab, ksi. Note that this
could be different on the left and right sides of the beam.
Also note that this it may be different for stress calculations and deflection calculations.
Es
= Modulus of elasticity of steel, ksi.
Sr
= Center-to-center spacing of metal deck ribs, in. Note that
this may be different on the left and right sides of the
beam.
beff
= Effective width of the concrete flange of the composite
beam, in. This width is code dependent. Note that this
width may be different on the left and right sides of the
beam. See Technical Note Effective Width of the Concrete
Slab Composite Beam Design for additional information.
d
= Depth of steel beam from outside face of top flange to
outside face of bottom flange, in.
hr
= Height of metal deck rib, in. Note that this may be different on the left and right sides of the beam.
tc
= Thickness of concrete slab, in. If there is metal deck, this
is the thickness of the concrete slab above the metal
Equivalent Hand Calculation Method to Calculate the Distance ye
Page 16 of 21
Composite Beam Design AISC-ASD89
Transformed Section Moment of Inertia
deck. Note that this may be different on the left and right
sides of the beam.
wr
= Average width of a metal deck rib, in. Note that this may
be different on the left and right sides of the beam.
ybare
= Distance from the bottom of the bottom flange of the
steel beam to the ENA of the steel beam (plus cover
plate, if it exists) alone, in.
ye
= The distance from the ENA of the steel beam (plus cover
plate, if it exists) alone to the ENA of the fully composite
beam, in.
z
= Distance from the ENA of the steel beam (plus cover
plate, if it exists) alone to the top of the concrete slab, in.
Note that this distance may be different on the left and
right sides of the beam.
Hand Calculation Process for ye
The location of the ENA of the composite section, defined by ye, is calculated
using the following process:
1. Assume the ENA is within the height of the steel beam. Use Equation 5a to
calculate the location of the ENA. Table 3 identifies the equations to use to
determine values for the variables X1 through X8 in Equation 5a.
2. If the location of the ENA calculated in step 1 is within the height of the
steel beam, as initially assumed, the location of the ENA is correct and the
calculation for ye is complete.
3. If the calculated ENA is not within the height of the steel beam, as initially
assumed, the location is incorrect and a new assumption for the location
of the neutral axis is made. The new assumption for the location of the
ENA is wherever it was calculated to be in step 1 and is one of the choices
defined in the first two columns of Table 3.
4. Use Equation 5b to calculate the location of the ENA. Note that Table 3
identifies the equations to use to determine values for the variables X1
through X10 for use in solving Equation 5b.
Hand Calculation Process for ye
Page 17 of 21
Composite Beam Design AISC-ASD89
Transformed Section Moment of Inertia
5. If the calculated location of the ENA is the same as the new location assumed in step 3, then the assumption is correct and the calculation for ye
is complete.
6. If the calculated location of the ENA is not the same as the location assumed in step 3, the location is incorrect and another iteration is made.
The new assumption for the location of the ENA is wherever it was calculated to be in step 4 and is one of the choices defined in the first two columns of Table 3.
7. Repeat steps 4 through 7 as many times as required until the assumed
location of the ENA (based on the choices in the first two columns of Table
3) and the calculated location of the ENA match.
Equivalent Hand Calculation Method to Calculate the
Composite Properties
After the location of the ENA has been calculated, the other calculations to
determine the composite section moment of inertia are non-iterative and
relatively straightforward. The other calculation steps are as follow.
8. Calculate the transformed section properties for full composite connection
as illustrated in Table 4. When reviewing Table 4 note:
a. If the deck spans perpendicular to the beam span, the concrete in the
metal deck ribs is ignored. If the deck spans parallel to the beam span,
the concrete in the metal deck ribs is considered.
b.
The cover plate may or may not be present.
c. The concrete slab and metal deck may not exist on one side of the
beam or the other.
Equivalent Hand Calculation Method to Calculate the Composite Properties
Page 18 of 21
Composite Beam Design AISC-ASD89
Transformed Section Moment of Inertia
Table 4: Transformed Section Properties for a Fully Composite Beam
Item
Transformed
Area, Atr
Concrete
slab, left side
b eff t *c E c
Es
d + hr + tc −
b eff t *c E c
Es
t*
d + hr + tc − c
2
Concrete
slab, right
side
Concrete in
metal deck
ribs, left side
Concrete in
metal deck
ribs, right side
Steel beam
plus cover
plate
Sums
y1
t *c
2
2
Atry1
Atry1
Atry1
Atry1
2
b eff E c t *c 3
12E s
Atry1
Atry1
2
b eff E c t *c 3
12E s
Atry1
Atry1
2
b eff w r E c h *r 3
12S r E s
b eff w r E c h *r 3
12S r E s
b eff h *r w r E c
Sr E s
d + hr −
b eff h *r w r E c
Sr E s
h*
d + hr − r
2
Atry1
Atry1
2
Abare
ybare
Atry1
Atry1
2
Σ(Atry1)
Σ(Atry1 )
ΣAtr
h *r
2
IO
Ibare
2
ΣIO
d. The top of the concrete slab may be at a different elevation on the two
sides of the beam.
e. Any concrete that is below the ENA of the composite section is not included in the calculation.
Following is a list of the variables introduced in Table 4 that have not been
mentioned previously in this Technical Note.
Atr
= Area of an element of the composite steel beam section, in2.
h *r
= Height of the metal deck ribs above the ENA (i.e., that is in
compression) used for calculating the transformed section
properties, in. Note that this could be different on the left and
right sides of the beam.
Equivalent Hand Calculation Method to Calculate the Composite Properties
Page 19 of 21
Composite Beam Design AISC-ASD89
Transformed Section Moment of Inertia
If the deck ribs are oriented perpendicular to the beam span,
h *r = 0.
If the deck ribs are oriented parallel to the beam span, one of
the following three items applies:
1. If the ENA is below the metal deck, h r = hr.
*
2. If the ENA is within the metal deck, h r equals the height of
*
the metal deck above the ENA.
3. If the ENA is above the metal deck, h r = 0.
*
t *c
= Height of the concrete slab above the metal deck
(or solid
slab) that lies above the ENA (i.e., is in compression) that is
used for calculating the transformed section properties, in. Note
that this could be different on the left and right sides of the
beam.
One of the following three items applies:
1. If the ENA is below the top of the metal deck (bottom of the
concrete slab), t c = tc.
*
2. If the ENA is within the concrete slab, t c equals the height
*
of the concrete slab above the ENA.
3. If the ENA is above the concrete slab, t c = 0
*
ΣAtr
= Sum of the areas of all of the elements of the composite steel
beam section, in2.
Σ(Atry1) =Sum of the product Atr times y1 for all of the elements of the
composite steel beam section, in3.
Σ(Atry12) =Sum of the product Atr times y12 for all of the elements of the
composite steel beam section, in4.
Equivalent Hand Calculation Method to Calculate the Composite Properties
Page 20 of 21
Composite Beam Design AISC-ASD89
Transformed Section Moment of Inertia
The neutral axis of the transformed composite section is located
a distance y from the bottom of the bottom flange of the steel
beam section (not bottom of cover plate). The distance
y can
be determined from either Equation 17a or from Equation 17b.
They both give the same result.
y=
∑ (A y )
∑A
tr
1
Eqn. 17a
tr
y = ybare + ye
The distance
Eqn. 17b
y is illustrated in Figure 5.
The transformed section moment of inertia about the ENA of
the composite beam, Itr, is calculated using Equation 18.
I tr =
∑A
2
tr y1
+
∑I
O
−
(∑ A ) y
Eqn. 18
2
tr
Bottom of bottom flange of steel
beam. The dimensions y, ybare
and y1 are measured from here.
y1 for top flange
y
ye
Elastic neutral axis (ENA) of steel
beam alone, including cover plate
if it exists
ybare
Elastic neutral axis (ENA) of
composite beam. Itr is taken
about this axis.
z
Figure 5 illustrates the axis about which Itr is taken.
Figure 5: Illustration of y
Equivalent Hand Calculation Method to Calculate the Composite Properties
Page 21 of 21