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Background Page 1 of 21 ©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, I tr . The calculated transformed moment of inertia applies for full (100%) composite connection. See Technical Note Elastic Stresses with Partial Composite Connection Com- posite Beam AISC-ASD89 for a description of partial composite connection. The Technical Note also describes in detail a method that can be used to cal- culate 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 calcula- tions and for presentation in this Technical Note. Note that for the AISC-ASD89 specification, the transformed section proper- ties 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 E c is always calculated from Equation 1, assuming that the unit weight of concrete, w c , is 150 pounds per cubic foot, regardless of its actual specified weight. ( ) ' c 1.5 c c f 33 w E = Eqn. 1 In Equation 1, E c is in pounds per square inch (psi), w c is in pounds per cubic foot (pcf) and ' c f is in pounds per square inch (psi). For AISC-ASD89 composite beam design deflection calculations, the value of E c is taken from the material property specified for the concrete slab.

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Page 1: D 2001 COMPOSITE BEAM DESIGN AISC-ASD89 …extras.springer.com/2003/978-3-322-80050-3/INFORMATION and POOL... · COMPOSITE BEAM DESIGN AISC-ASD89 Technical Note Transformed Section

Background Page 1 of 21

©COMPUTERS AND STRUCTURES, INC., BERKELEY, CALIFORNIA DECEMBER 2001

COMPOSITE BEAM DESIGN AISC-ASD89Technical Note

Transformed Section Moment of Inertia

This Technical Note describes in general terms how the program calculatesthe transformed moment of inertia for a composite section, Itr. The calculatedtransformed moment of inertia applies for full (100%) composite connection.See Technical Note Elastic Stresses with Partial Composite Connection Com-posite Beam AISC-ASD89 for a description of partial composite connection.

The Technical Note also describes in detail a method that can be used to cal-culate the transformed section moment of inertia by hand that will yield thesame result as the program. The exact methodology used by the program isoptimized for computer-based calculations and is unsuitable for hand calcula-tions and for presentation in this Technical Note.

Note that for the AISC-ASD89 specification, the transformed section proper-ties used for stress calculations for a beam may be different from those usedfor deflection calculations for the same beam. For AISC-ASD89 compositebeam design stress calculations, the value of Ec is always calculated fromEquation 1, assuming that the unit weight of concrete, wc, is 150 pounds percubic foot, regardless of its actual specified weight.

( ) 'c

1.5cc f33wE = Eqn. 1

In Equation 1, Ec is in pounds per square inch (psi), wc is in pounds per cubic

foot (pcf) and 'cf is in pounds per square inch (psi).

For AISC-ASD89 composite beam design deflection calculations, the value ofEc is taken from the material property specified for the concrete slab.

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Composite Beam Design AISC-ASD89 Transformed Section Moment of Inertia

Background Page 2 of 21

BackgroundFigure 1 shows a typical rolled steel composite floor beam with the metal deckribs running parallel to the beam. Figure 2 shows a typical composite user-defined 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 flangesize, and that no fillets are assumed in this beam.

For each of these configurations the following items may or may not beincluded when calculating the transformed section moment of inertia:

Concrete in the metal deck ribs: The concrete in the metal deck ribs isincluded in the calculation when the deck ribs are oriented parallel to thebeam (typically the case for girders). It is not included when the deck ribs areoriented perpendicular to the beam (typically the case for infill beams).

• Cover plate: The cover plate is only included if one is specified by you inthe composite beam overwrites.

Note that the deck type and deck orientation may be different on the twosides of the beam as described in "Multiple Deck Types or Directions Alongthe Beam Length" of Technical Note Effective Width of the Concrete SlabComposite Beam Design.

Because composite behavior is only considered for positive bending, thetransformed section moment of inertia is only calculated for positive bending(top of composite section in compression). Calculation of the transformedsection moment of inertia is greatly complicated by the requirement that theconcrete resist no tension.

The first task in calculating the transformed section moment of inertia of thecomposite section is to compute properties for the steel beam alone (plus thecover plate, if it exists). The properties required are the total area, Abare; thelocation of the ENA, ybare; and the moment of inertia, Is.

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Composite Beam Design AISC-ASD89 Transformed Section Moment of Inertia

Background Page 3 of 21

Figure 1: Composite Rolled Steel Beam Shown With Metal Deck RibsRunning Parallel To Beam

Figure 2: Composite User-Defined Steel Beam Shown With Metal DeckRibs Running Parallel To Beam

��������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

bcp

h rt c

dt cp

Concrete slab

Metal deck

Rolled steel beam

Bottom cover plate

������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

bcp

bf-bot

tw

bf-top

h rt c

t f-top

d

h = d

- tf-t

op -

t f-bot

t cp

t f-bot

Metal deck

Beam top flange

Beam web

Beam bottom flange

Bottom cover plate

Concrete slab

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Composite Beam Design AISC-ASD89 Transformed Section Moment of Inertia

Properties of Steel Beam (Plus Cover Plate) Alone Page 4 of 21

Properties of Steel Beam (Plus Cover Plate) AloneThe location of the ENA for the steel beam alone (plus cover plate if applica-ble) is defined by the distance ybare, where ybare is the distance from the bot-tom of the bottom flange of the beam to the ENA, as shown in Figure 3. Ifthere is a cover plate, ybare is still measured from the bottom of the bottomflange of the beam, not the bottom of the cover plate.

Figure 3 also illustrates an example of the dimension y1 that is used in Tables1 and 2. For a given element of a steel section, the dimension y1 is equal tothe distance from the bottom of the beam bottom flange to the centroid ofthe 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 pro-gram section database, Abare, ybare and Ibare are calculated as shown in Table 1and 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.

Elastic neutral axis of steel beamplus cover plate if applicable.Ibare is taken about this axis.

y bar

e y 1 fo

r top

flang

e

Bottom of bottom flange of steelbeam. Ybare and y1 aremeasured from here

Figure 3: Illustration of ybare and y1

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Composite Beam Design AISC-ASD89 Transformed Section Moment of Inertia

Properties of Steel Beam (Plus Cover Plate) Alone Page 5 of 21

Table 1: Section Properties for Rolled Steel Beam Plus Cover Plate

Item Area, A y1 Ay1 Ay12 IO

Steel beam As 2d

Ay1 Ay12 Is

Cover plate bcptcp 2t cp− Ay1 Ay1

2

12tb 3

cpcp

Sums ΣΣΣΣA ΣΣΣΣ(Ay1) ΣΣΣΣ(Ay12) ΣΣΣΣIO

Table 2: Section Properties for User-Defined (Welded) Steel Beam PlusCover Plate

Item Area, A y1 Ay1 Ay12 IO

Top flange bf-toptf-top 2t

d topf −− Ay1 Ay12

12tb 3

topftopf −−

Web htw 2d

Ay1 Ay12

12ht 3

w

Bottom flange bf-bottf-bot 2t botf − Ay1 Ay1

2

12tb 3

botfbotf −−

Cover plate bcptcp 2t cp− Ay1 Ay1

2

12tb 3

cpcp

Sums ΣΣΣΣA ΣΣΣΣ(Ay1) ΣΣΣΣ(Ay12) ΣΣΣΣIO

The area of the steel section (including the cover plate if it exists), Abare, isgiven by Equation 1.

Abare = ΣA Eqn. 1

The ENA of the steel section is located a distance ybare from the bottom of thebottom flange of the steel beam section (not bottom of cover plate) whereybare is determined from Equation 2.

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Composite Beam Design AISC-ASD89 Transformed Section Moment of Inertia

Properties of Steel Beam (Plus Cover Plate) Alone Page 6 of 21

∑∑=

A

)(Ayy 1

bare 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.

( ) ( ) 2bareO

21bare yAIAyI ∑∑∑ −+= Eqn. 3

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 plateeven if one exists), in2.

Ibare = Moment of inertia of the steel beam (plus cover plate ifone exists), in4.

IO = The moment of inertia of an element of the beam sectiontaken about the ENA of the element, in4.

Is = Moment of inertia of the steel beam alone (without thecover 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 tooutside 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.

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Composite Beam Design AISC-ASD89 Transformed Section Moment of Inertia

Properties of the Composite Section General Calculation Method Page 7 of 21

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 steelsection to the ENA of the steel beam (plus cover plate if itexists), in.

y1 = Distance from the bottom of the bottom flange of the steelbeam section to the centroid of an element of the beamsection, in.

ΣA = Sum of the areas of all of the elements of the steel beamsection, in2.

Σ(Ay1) = Sum of the product A times y1 for all of the elements ofthe steel beam section, in3.

Σ(A 21y ) = Sum of the product A times 2

1y for all of the elements of

the steel beam section, in4.

ΣIO = Sum of the moments of inertia of each element of thebeam section taken about the ENA of the element, in4.

Properties of the Composite Section GeneralCalculation MethodThe first step, and potentially most calculation-intensive step in the process ofdetermining the composite properties is to calculate the distance from theENA of the steel beam (plus cover plate if it exists) to the ENA of the fullcomposite section. This distance is designated ye in Figure 4.

Recall that concrete in tension is ignored when calculating the compositeproperties. Because of the possibility that some of the concrete may be intension, and because the amount of concrete that is in tension is initially un-known (if any), the process for calculating the distance ye is iterative. Afterthe distance ye has been determined, the other calculations to determine thecomposite properties are relatively straight-forward.

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Composite Beam Design AISC-ASD89 Transformed Section Moment of Inertia

Properties of the Composite Section General Calculation Method Page 8 of 21

The program uses the following method to calculate the properties of thecomposite section.

1. The location of the ENA of the composite section, defined by ye (see Fig-ure 4), is calculated using the following iterative process:

a. The program assumes (guesses) that the ENA of the composite sectionis within the height of the steel beam and uses Equation 4 to calculatethe distance ye that defines the location of the ENA for the compositesection. Note that with this assumption, all of the concrete is abovethe ENA of the composite section and thus it is all in compression andcan be considered.

( )element

elementelemente A

dAy

ΣΣ

= Eqn. 4

where,

Aelement = Area of an element in the composite section, ignoring anyarea of concrete that is in tension and ignoring any con-crete in the metal deck ribs when the metal deck span isperpendicular to the beam span, in2.

����������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������������

Elastic neutral axis of compositebeam

y e

Elastic neutral axis of steel beamalone, including cover plate if itexists

z

Figure 4: Illustration of ye and z

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Composite Beam Design AISC-ASD89 Transformed Section Moment of Inertia

Properties of the Composite Section General Calculation Method Page 9 of 21

delement = Distance from the ENA of the element considered to theENA of the steel beam alone (including cover plate, if itexists), in. Signs are considered for this distance. Ele-ments located below the ENA of the steel beam alone (in-cluding cover plate, if it exists) have a negative distanceand those above have a positive distance.

If the ENA as calculated is within the height of the steel beam, as as-sumed, the assumed location of the ENA is correct and the calculationfor ye is complete.

b. If the calculated ENA is not within the height of the steel beam, as as-sumed in Step a, the assumed location of the ENA is incorrect and cal-culation for ye continues.

i Using the incorrect location of the ENA calculated in Step a, theprogram calculates the location of ye again using Equation 4, ig-noring any concrete that is in tension.

ii If the newly calculated location of the ENA is the same as the pre-viously calculated location (Step i), the assumed location of theENA 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 pre-viously calculated location (Step i), the most recent assumed locationof the ENA is incorrect and another iteration is made.

The program repeats the iterations until the location of the ENA hasbeen determined. After the location of the ENA is known, the rest ofthe process for calculating the composite properties is non-iterative.

2. Given that the ENA has been located, the program determines if any con-crete is below the ENA. If so, the program ignores it in the remaining cal-culations.

3. The program sums the product of the area of each element of the com-posite section (except concrete in tension) times its distance to a conven-ient 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 areasof each element of the composite section (except concrete in tension).

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Composite Beam Design AISC-ASD89 Transformed Section Moment of Inertia

Equivalent Hand Calculation Method to Calculate the Distance ye Page 10 of 21

This calculation yields the distance from the convenient axis to the ENA ofthe composite section.

5. After the ENA of the composite section has been determined, the sectionproperties of the composite section are quickly calculated using standardmethods.

A hand calculation method for determining the distance ye described in steps1a through 1c above is presented in the next section entitled "EquivalentHand Calculation Method to Calculate the Distance ye." A hand calculationmethod for the calculation of the composite properties described in steps 2through 5 above is presented in the section entitled "Equivalent Hand Calcu-lation Method to Calculate the Composite Properties" later in this TechnicalNote.

Equivalent Hand Calculation Method to Calculate theDistance ye

The following hand calculation method for determining the distance ye issimilar to and provides the same result as the calculations performed by theprogram.

After ybare has been calculated, ye is calculated by equating the forces aboveand below the ENA using either Equation 5a or Equation 5b. Recall that ye isthe 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.

8765bare

4321e XXXXA

XXXXy

+++++++

= Eqn. 5a

( )9

4321921010

e 2XXXXX4XXX-

y+++−±

= Eqn. 5b

Equations for use in calculating values for the variables X1 through X10 inEquations 5a and 5b are presented in the following subsection entitled "Back-ground Equations." The actual process to calculate ye is described in the sub-section of this Technical Note entitled "Hand Calculation Process for ye."

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Composite Beam Design AISC-ASD89 Transformed Section Moment of Inertia

Equivalent Hand Calculation Method to Calculate the Distance ye Page 11 of 21

Background EquationsThis subsection presents the equations for the variables X1 through X10 inEquations 5a and 5b. The exact equation to use for the variables X1 throughX10 depends on the assumed location of the ENA.

For the purposes of determining the ye distance, there are nine possible loca-tions for the ENA. Those locations are as follows:

1. The ENA is located within the height of the steel section (includingcover plate, if it exists).

2. The ENA is located within the height of the metal deck on both the leftand the right sides of the beam.

3. The ENA is located within the height of the metal deck on the left sideof the beam and within the height of the concrete above the metaldeck (or within a solid slab) on the right side of the beam.

Note: Recall that you can have different deck properties on the twosides of the beam.

4. The ENA is located within the height of the metal deck on the left sideof 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 metaldeck (or within a solid slab) on the left side of the beam and within theheight of the metal deck on the right side of the beam.

6. The ENA is located within the height of the concrete above the metaldeck (or within a solid slab) on both sides of the beam.

7. The ENA is located within the height of the concrete above the metaldeck (or within a solid slab) on the left side of the beam and above theconcrete on the right side of the beam.

8. The ENA is located above the concrete on the left side of the beam andwithin 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 andwithin the height of the concrete above the metal deck (or within asolid slab) on the right side of the beam.

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Composite Beam Design AISC-ASD89 Transformed Section Moment of Inertia

Equivalent Hand Calculation Method to Calculate the Distance ye Page 12 of 21

The first two columns in Table 3 list the nine possible locations of the ENA ofthe composite section. The columns labeled Left Side and Right Side indicatethe location of the ENA relative to the left and right sides of the beam, re-spectively. The third column of Table 3, labeled "ye Eqn" specifies whetherEquation 5a or 5b should be used to calculate ye. Columns 4 through 13 ofTable 3 list the equation numbers to be used to determine the value of thevariables X1 through X10 for the location of the ENA specified in the first twocolumns of the table.

When using Table 3, the location of the ENA of the composite section and thelocation of the ENA of the composite section relative to the elements thatmake up the composite section are initially unknown. Thus, begin by assum-ing a location of the ENA. It works best if you assume that the ENA of thecomposite section is within the steel section. Then, calculate the actual loca-tion of the ENA and check the validity of the assumption. This process is de-scribed in the subsection entitled "Hand Calculation Process for ye."

Equations 7 through 16 define the terms X1 through X10 in Table 3 and Equa-tions 5a and 5b. A term that is repeatedly used in Equations 7 through 16 isz. As previously illustrated in Figure 4, z is the distance from the ENA of thesteel 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 Ta-ble 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 loca-tion of the ENA.

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Composite Beam Design AISC-ASD89 Transformed Section Moment of Inertia

Equivalent Hand Calculation Method to Calculate the Distance ye Page 13 of 21

Table 3: Table Identifying Circumstances for Using Equations 5a and 5b andIdentifying Appropriate Equations to Use to Calculate the Values ofVariables X1 through X10 that Appear in Equations 5a and 5b

LeftSide

RightSide

ye

EqnX1

EqnX2

EqnX3

EqnX4

EqnX5

EqnX6

EqnX7

EqnX8

EqnX9

EqnX10

EqnSteel section 5a 7a 8a 9a 10a 11a 12a 13a 14a N.A. N.A.hr hr 5b 7a 8b 9a 10b 11a 12b 13a 14b 15a 16ahr tc 5b 7a 8b 9b 0 11a 12b 13b 14c 15a 16chr >tc 5b 7a 8b 0 0 11a 12b 0 0 15a 16atc hr 5b 7b 0 9a 10b 11b 12c 13a 14b 15a 16dtc tc 5b 7b 0 9b 0 11b 12c 13b 14c 15a 16btc >tc 5b 7b 0 0 0 11b 12c 0 0 15a 16b

>tc hr 5b 0 0 9a 10b 0 0 13a 14b 15a 16a>tc tc 5b 0 0 9b 0 0 0 13b 14c 15a 16b

Table Descriptive Notes:

�� The columns labeled Left Side and Right Side indicate the assumed location of theENA of the composite section relative to the left and right sides of the beam. Steelsection means that the ENA falls within the height of the steel section (including thecover plate, if it exists). The designation hr means that the ENA is within the height ofthe metal deck. The designation tc means that the ENA is within the height of theconcrete 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 tocalculate ye for the assumed location of the ENA listed in the first two columns of thetable.

3. The columns labeled "X1 Eqn" through "X10 Eqn" indicate the equation numbers thatshould be used to calculate the value of the variables X1 through X10 for use in Equa-tions 5a and 5b. If one of the cells for X1 through X8 contains a "0," the value of Xn iszero for that location of the ENA.

4. The variables X9 and X10 are not used if the ENA falls within the height of the steelbeam.

5. The variables X2, X4, X6 and X8 are always taken as zero if the deck span is orientedperpendicular to the beam span.

6. Using this table requires a trial and error process. You must assume a location for theENA and then check if the assumption is correct. See the subsection entitled "HandCalculation Process for ye" later in this chapter for more information.

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Composite Beam Design AISC-ASD89 Transformed Section Moment of Inertia

Equivalent Hand Calculation Method to Calculate the Distance ye Page 14 of 21

Important note: The terms X2, X4, X6 and X8 are always taken as zero if thedeck span is oriented perpendicular to the beam span; otherwise they aretaken as given in the equations below.

−=

2

tzXX leftc

left51 Eqn. 7a

=2

zXX left

51 Eqn. 7b

X2 is taken as zero if the deck span is oriented perpendicular to the beamspan; if the deck span is oriented parallel to the beam span, X2 is as specifiedin the equations below.

−−=

2

htzXX leftr

leftcleft62 Eqn. 8a

( )2leftcleft62 tzXX −= Eqn. 8b

−=

2

tzXX rightc

right73 Eqn. 9a

=

2

zXX right

73 Eqn. 9b

X4 is taken as zero if the deck span is oriented perpendicular to the beamspan; if the deck span is oriented parallel to the beam span, X4 is as specifiedin the equations below.

−−=

2

htzXX rightr

rightcright84 Eqn. 10a

( )2rightcright84 tzXX −= Eqn. 10b

s

leftcleftclefteff5 E

tEbX = Eqn. 11a

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Composite Beam Design AISC-ASD89 Transformed Section Moment of Inertia

Equivalent Hand Calculation Method to Calculate the Distance ye Page 15 of 21

s

leftleftclefteff5 E

zEbX = Eqn. 11b

X6 is taken as zero if the deck span is oriented perpendicular to the beamspan; if the deck span is oriented parallel to the beam span, X6 is as specifiedin the equations below.

leftrs

leftrleftrleftclefteff6 SE

hwEbX = Eqn. 12a

leftrs

leftrleftclefteff6 S2E

wEbX = Eqn. 12b

s

leftclefteff6 2E

EbX = Eqn. 12c

s

rightcrightcrighteff7 E

tEbX = Eqn. 13a

s

rightrightcrighteff7 E

zEbX = Eqn. 13b

X8 is taken as zero if the deck span is oriented perpendicular to the beamspan; if the deck span is oriented parallel to the beam span, X8 is as specifiedin the equations below.

rightrs

rightrrightrrightcrighteff8 SE

hwEbX = Eqn. 14a

rightrs

rightrrightcrighteff8 S2E

wEbX = Eqn. 14b

s

rightcrighteff8 2E

EbX = Eqn. 14c

869 XXX += Eqn. 15a

89 XX = Eqn. 15b

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Equivalent Hand Calculation Method to Calculate the Distance ye Page 16 of 21

69 XX = Eqn. 15c

( )( )rightcright87

leftcleft65bare10

tz2XX

tz2XXAX

−−

−−−−−=Eqn. 16a

75bare10 XXAX −−−= Eqn. 16b

( ) 7leftcleft65bare10 XtzXXAX −−−−−= Eqn. 16c

( )rightcright875bare10 tzXXXAX −−−−−= Eqn. 16d

The notation used in equations 5a through 16d are as follows:

Abare = Area of the steel beam (plus cover plate), in2. This areadoes not include the concrete area.

Ec = Modulus of elasticity of concrete slab, ksi. Note that thiscould be different on the left and right sides of the beam.Also note that this it may be different for stress calcula-tions and deflection calculations.

Es = Modulus of elasticity of steel, ksi.

Sr = Center-to-center spacing of metal deck ribs, in. Note thatthis may be different on the left and right sides of thebeam.

beff = Effective width of the concrete flange of the compositebeam, in. This width is code dependent. Note that thiswidth may be different on the left and right sides of thebeam. See Technical Note Effective Width of the ConcreteSlab Composite Beam Design for additional information.

d = Depth of steel beam from outside face of top flange tooutside face of bottom flange, in.

hr = Height of metal deck rib, in. Note that this may be differ-ent on the left and right sides of the beam.

tc = Thickness of concrete slab, in. If there is metal deck, thisis the thickness of the concrete slab above the metal

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Hand Calculation Process for ye Page 17 of 21

deck. Note that this may be different on the left and rightsides of the beam.

wr = Average width of a metal deck rib, in. Note that this maybe different on the left and right sides of the beam.

ybare = Distance from the bottom of the bottom flange of thesteel beam to the ENA of the steel beam (plus coverplate, if it exists) alone, in.

ye = The distance from the ENA of the steel beam (plus coverplate, if it exists) alone to the ENA of the fully compositebeam, in.

z = Distance from the ENA of the steel beam (plus coverplate, if it exists) alone to the top of the concrete slab, in.Note that this distance may be different on the left andright sides of the beam.

Hand Calculation Process for ye

The location of the ENA of the composite section, defined by ye, is calculatedusing the following process:

1. Assume the ENA is within the height of the steel beam. Use Equation 5a tocalculate the location of the ENA. Table 3 identifies the equations to use todetermine 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 thesteel beam, as initially assumed, the location of the ENA is correct and thecalculation for ye is complete.

3. If the calculated ENA is not within the height of the steel beam, as initiallyassumed, the location is incorrect and a new assumption for the locationof the neutral axis is made. The new assumption for the location of theENA is wherever it was calculated to be in step 1 and is one of the choicesdefined in the first two columns of Table 3.

4. Use Equation 5b to calculate the location of the ENA. Note that Table 3identifies the equations to use to determine values for the variables X1

through X10 for use in solving Equation 5b.

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Equivalent Hand Calculation Method to Calculate the Composite Properties Page 18 of 21

5. If the calculated location of the ENA is the same as the new location as-sumed 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 as-sumed 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 calcu-lated to be in step 4 and is one of the choices defined in the first two col-umns of Table 3.

7. Repeat steps 4 through 7 as many times as required until the assumedlocation of the ENA (based on the choices in the first two columns of Table3) and the calculated location of the ENA match.

Equivalent Hand Calculation Method to Calculate theComposite PropertiesAfter the location of the ENA has been calculated, the other calculations todetermine the composite section moment of inertia are non-iterative andrelatively straightforward. The other calculation steps are as follow.

8. Calculate the transformed section properties for full composite connectionas illustrated in Table 4. When reviewing Table 4 note:

a. If the deck spans perpendicular to the beam span, the concrete in themetal 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 thebeam or the other.

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Equivalent Hand Calculation Method to Calculate the Composite Properties Page 19 of 21

Table 4: Transformed Section Properties for a Fully Composite Beam

ItemTransformed

Area, Atr y1 Atry1 Atry12 IO

Concreteslab, left side s

c*ceff

EEtb

2t

thd*c

cr −++ Atry1 Atry12

s

3*cceff

12EtEb

Concreteslab, rightside

s

c*ceff

EEtb

2t

thd*c

cr −++ Atry1 Atry12

s

3*cceff

12EtEb

Concrete inmetal deckribs, left side sr

cr*reff

ESEwhb

2h

hd*r

r −+ Atry1 Atry12

sr

3*rcreff

E12ShEwb

Concrete inmetal deckribs, right side sr

cr*reff

ESEwhb

2h

hd*r

r −+ Atry1 Atry12

sr

3*rcreff

E12ShEwb

Steel beamplus coverplate

Abare ybare Atry1 Atry12 Ibare

Sums ΣΣΣΣAtr ΣΣΣΣ(Atry1) ΣΣΣΣ(Atry12) ΣΣΣΣIO

d. The top of the concrete slab may be at a different elevation on the twosides of the beam.

e. Any concrete that is below the ENA of the composite section is not in-cluded in the calculation.

Following is a list of the variables introduced in Table 4 that have not beenmentioned previously in this Technical Note.

Atr = Area of an element of the composite steel beam section, in2.

*rh = Height of the metal deck ribs above the ENA (i.e., that is in

compression) used for calculating the transformed sectionproperties, in. Note that this could be different on the left andright sides of the beam.

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Equivalent Hand Calculation Method to Calculate the Composite Properties Page 20 of 21

If the deck ribs are oriented perpendicular to the beam span,*rh = 0.

If the deck ribs are oriented parallel to the beam span, one ofthe following three items applies:

1. If the ENA is below the metal deck, *rh = hr.

2. If the ENA is within the metal deck, *rh equals the height of

the metal deck above the ENA.

3. If the ENA is above the metal deck, *rh = 0.

*ct = Height of the concrete slab above the metal deck (or solid

slab) that lies above the ENA (i.e., is in compression) that isused for calculating the transformed section properties, in. Notethat this could be different on the left and right sides of thebeam.

One of the following three items applies:

1. If the ENA is below the top of the metal deck (bottom of the

concrete slab), *ct = tc.

2. If the ENA is within the concrete slab, *ct equals the height

of the concrete slab above the ENA.

3. If the ENA is above the concrete slab, *ct = 0

ΣAtr = Sum of the areas of all of the elements of the composite steelbeam section, in2.

Σ(Atry1) =Sum of the product Atr times y1 for all of the elements of thecomposite steel beam section, in3.

Σ(Atry12) =Sum of the product Atr times y1

2 for all of the elements of thecomposite steel beam section, in4.

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Composite Beam Design AISC-ASD89 Transformed Section Moment of Inertia

Equivalent Hand Calculation Method to Calculate the Composite Properties Page 21 of 21

The neutral axis of the transformed composite section is locateda 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.

∑∑=

tr

1tr

A

)y(Ay Eqn. 17a

y = ybare + ye Eqn. 17b

The distance y is illustrated in Figure 5.

The transformed section moment of inertia about the ENA ofthe composite beam, Itr, is calculated using Equation 18.

( ) 2trO

21trtr yAIyAI ∑∑∑ −+= Eqn. 18

Figure 5 illustrates the axis about which Itr is taken.

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Elastic neutral axis (ENA) ofcomposite beam. Itr is takenabout this axis.

y y 1 fo

r top

flang

e

Bottom of bottom flange of steelbeam. The dimensions y, ybareand y1 are measured from here.

y ey b

are

z

Elastic neutral axis (ENA) of steelbeam alone, including cover plateif it exists

Figure 5: Illustration of y