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PTC_CE_BSD_2.1_us_mp.mcdx

Mathcad ® Enabled Content Copyright © 2011 Knovel Corp.

Building Structural Design: Composite Beam Section Properties Thomas P. Magner, P.E. © 2011 Parametric Technology Corp.

Chapter 2: Structural Steel Beams

2.1 Composite Beam Section Properties

DisclaimerWhile Knovel and PTC have made every effort to ensure that the calculations, engineering solutions, diagrams and other information (collectively “Solution”) presented in this Mathcad worksheet are sound from the engineering standpoint and accurately represent the content of the book on which the Solution is based, Knovel and PTC do not give any warranties or representations, express or implied,including with respect to fitness, intended purpose, use or merchantability and/or correctness or accuracy of this Solution.

Array origin:

≔ORIGIN 1

Description

This application calculates the horizontal shear and section properties for composite steel beam and concrete slab sections with solid slabs, composite steel decks, or haunches. Computations are made for beams and slabs over a complete usable range of composite action from 25% to 100%.

The user must identify the steel section that is used. The user must also enter the dimensions and section properties of the steel section, the dimensions of the slab section and "haunch", the compressive strength of the concrete, the unit weight of concrete and the yield strength of the steel section.Composite steel beams consisting of rolled structural beams and either solid slabs or slabs of composite steel deck and concrete in-fill are commonly used, especially in office construction. In composite construction the slab and beam are connected together and made to act as one unit by field welding steel shear studs to the beam prior to placing the concrete. Some added economy may be achieved by using the AISC Specifications provisions of Section I1 for partial composite action. These provisions permit the use of fewer studs than required for full composite behavior when the strength and stiffness of a given beam section is adequate with partial composite behavior.

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Reference: AISC "Specification for Structural Steel Buildings -- Allowable Stress Design and Plastic Design with Commentary." June 1, 1989

Input

Input Variables

The user should enter the steel section designation and plate size in text.

Steel Section: W16x26 Plate: None

Depth of the steel beam: ≔d 15.69 in

Dimension from bottom of steel section to neutral axis of steel section:

≔ybs ―d

2

Cross-sectional area of the steel section:

≔As 7.68 in2

Moment of inertia of steel section:

≔Is 298.10 in4


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Thickness of solid slab or thickness of concrete above top of steel deck:

≔t 3.5 in

Effective concrete flange width: ≔b 77.5 in

Width of concrete haunch or equivalent width of the concrete filled ribs of a steel deck parallel to the beam span of steel deck ribs:

≔xh 0 in

Depth of concrete haunch or depthof steel deck parallel to beam span:

≔yh 2 in

Notes

For sections with a composite steel deck parallel to the span, xh is equal to the equivalent width of concrete in the ribs of the deck and yh is equal to the depth of the steel deck.

For sections with a composite steel deck transverse to the beam span, xh is equal to 0 inches and yhis equal to the depth of steel deck.

Computed Variables

h total depth of composite section

yt dimension from top of slab to neutral axis of composite section

yb dimension from bottom of steel section to neutral axis of composite section

Ac cross-sectional area of concrete

Itr moment of inertia of composite section with 100% composite action

Ieff effective moment of inertia of composite section with partial composite action


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Ss section modulus of steel section referred to bottom of section

Str section modulus of fully composite section to bottom of steel section

St section modulus of composite section to top of slab

Seff effective section modulus to bottom of steel beam for section with partial composite action

Vh total horizontal shear between point of maximum positive moment and pointsof zero moment for full composite action

V'h total horizontal shear between point of maximum positive moment and points of zero moment for partial composite action

N.A. neutral axis of composite section

Material Properties

Enter the compressive strength of concrete, yield strength of steel section and the unit weight of concrete.

Specified compressive strength of concrete:

≔f'c 4 ksi

Specified yield strength of steel section:

≔fy 36 ksi

Weight of concrete (minimum weight of 90 pcf):

≔wc 145 pcf


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Modulus of elasticity of steel:

≔Es 29000 ksi

The following variables are computed from the entered material properties.

Modulus of elasticity of concrete:

≔Ec ⋅⋅⋅⎛⎜⎝――wc

pcf

⎞⎟⎠

1.5

33‾‾‾――f'c

psipsi =Ec 3644 ksi

Modular ratio: ≔n ―Es

Ec

=n 7.958

For lightweight concrete the modular ratio for normal weight is used for stress calculations, and the modular ratio for lightweight concrete is used for deflection calculations.

Solution

Cross-sectional area of concrete:

≔Ac +⋅b t ⋅xh yh =Ac 271.25 in2

Total horizontal shear for 100% composite action, using AISC Specification, Eqs. (14-1) and (14-2),combined:

≔Vh if⎛⎜⎝

,,≥――⋅As fy

2――――

⋅⋅0.85 f'c Ac

2――――

⋅⋅0.85 f'c Ac

2――

⋅As fy

2

⎞⎟⎠

=Vh 138.24 kip


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Section modulus of the steel section referred to the bottom flange:

≔Ss ――Is

ybs

=Ss 37.999 in3

Dimension from the neutral axis of the steel section to the top of the steel section:

≔yts −d ybs =yts 7.845 in

Total depth of composite section:

≔h ++t yh d =h 21.19 in

Test for Location of Neutral Axis

This section determines if the neutral axis for the 100% composite section lies within the steel beam, within the haunch or the ribs of the steel deck parallel to the beam span, between the slab and the steel beam, or within the slab.

These conditions are summarized by the following four cases:

Case 1: The neutral axis lies within the steel beam.

Case 2: The neutral axis lies within the haunch or the ribs of the steel deck parallel to the beam span.

Case 3: The neutral axis lies between the slab and the steel beam with the steel deck transverse to the beam.

Case 4: The neutral axis lies within the slab.

≔Case if⎛⎜⎝

,,≥⋅⋅n As yts +⋅⋅b t⎛⎜⎝

+―t

2yh

⎞⎟⎠

―――⋅xh yh

2

21 if

⎛⎜⎝

,,≥⋅⋅n As ⎛⎝ +yts yh⎞⎠ ――⋅b t2

2if ⎛⎝ ,,>xh ⋅0 in 2 3⎞⎠ 4

⎞⎟⎠

⎞⎟⎠


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=Case 3

Case 1 The neutral axis lies within the steel beam.

Dimension from top of slab to neutral axis of composite section:

≔yt1if

⎛⎜⎜⎜⎝

,,＝Case 1 ――――――――――――

++――⋅b t2

2⋅⋅xh yh

⎛⎜⎝

+t ―yh

2

⎞⎟⎠

⋅⋅n As ⎛⎝ −h ybs⎞⎠

+Ac ⋅n As

⋅0 in

⎞⎟⎟⎟⎠

=yt10 in

Dimension from bottom of steel section to the neutral axis of composite section:

≔yb1if ⎛

⎜⎝,,＝Case 1 −h yt1

⋅0 in⎞⎟⎠

=yb10 in

Moment of inertia of composite section with 100% composite action:

≔Itr1if

⎛⎜⎜⎝

,,＝Case 1 ⋅

⎛⎜⎜⎝

+++++――⋅b t3

12⋅⋅b t

⎛⎜⎝

−yt1―t

2

⎞⎟⎠

2

⋅n Is ⋅⋅n As ⎛⎜⎝

−yb1ybs⎞

⎟⎠

2

―――⋅xh yh

3

12⋅⋅xh yh

⎛⎜⎝

−−yt1t ―

yh

2

⎞⎟⎠

2⎞⎟⎟⎠

―1

n⋅0 in4

⎞⎟⎟⎠

=Itr10 in4


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Case 2 The neutral axis lies within the haunch or the ribs of a steel deck parallel to beam span.

Dimension from top of slab to the neutral axis of composite section:

≔yt2if

⎛⎜⎜⎜⎝

,,＝Case 2 ――――――――――――――――――――――――――――

+−⎛⎝ ++⋅−xh t ⋅b t ⋅n As⎞⎠‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾

−⎛⎝ ++⋅−xh t ⋅b t ⋅n As⎞⎠2

⋅⋅4 ―xh

2

⎛⎜⎝

+−――⋅−b t2

2⋅⋅n As ⎛⎝ −h ybs⎞⎠ ――

⋅xh t2

2

⎞⎟⎠

xh

⋅0 in

⎞⎟⎟⎟⎠

=yt20 in


≔yb2if ⎛

⎜⎝,,＝Case 2 −h yt2

⋅0 in⎞⎟⎠

=yb20 in


≔Itr2if

⎛⎜⎜⎝

,,＝Case 2 ⋅

⎛⎜⎜⎝

+

⎛⎜⎜⎝

++――⋅b t3

12⋅⋅b t

⎛⎜⎝

−yt2―t

2

⎞⎟⎠

2

⋅n Is

⎞⎟⎟⎠

⎛⎜⎝

+⋅⋅n As ⎛⎜⎝

−yb2ybs⎞

⎟⎠

2

⋅⋅―1

3xh ⎛

⎜⎝−yt2

t⎞⎟⎠

3⎞⎟⎠

⎞⎟⎟⎠

―1

n⋅0 in4

⎞⎟⎟⎠

=Itr20 in4

Case 3 The neutral axis lies between the slab and the steel beam.

(Note: The steel deck is transverse to the beam span)


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≔yt3if

⎛⎜⎜⎜⎝

,,＝Case 3 ――――――――

+⋅⋅―1

2b t2 ⋅⋅n As ⎛⎝ −h ybs⎞⎠

+⋅b t ⋅n As

⋅0 in

⎞⎟⎟⎟⎠

=yt33.882 in


≔yb3if ⎛

⎜⎝,,＝Case 3 −h yt3

⋅0 in⎞⎟⎠

=yb317.308 in


≔Itr3if

⎛⎜⎜⎝

,,＝Case 3 ⋅

⎛⎜⎜⎝

+++――⋅b t3

12⋅⋅b t

⎛⎜⎝

−yt3―t

2

⎞⎟⎠

2

⋅n Is ⋅⋅n As ⎛⎜⎝

−yb3ybs⎞

⎟⎠

2⎞⎟⎟⎠

―1

n⋅0 in4

⎞⎟⎟⎠

=Itr31175.559 in4

Case 4 The neutral axis lies within the slab.

Dimension from top of slab to the neutral axis of composite section:

≔yt4if

⎛⎜⎜⎝

,,＝Case 4 ―――――――――――――+⋅−n As

‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾+⎛⎝ ⋅n As⎞⎠

2

⋅⋅⋅⋅2 b n As ⎛⎝ −h ybs⎞⎠b

⋅0 in

⎞⎟⎟⎠


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=yt40 in


≔yb4if ⎛

⎜⎝,,＝Case 4 −h yt4

⋅0 in⎞⎟⎠

=yb40 in


≔Itr4if

⎛⎜⎜⎝

,,＝Case 4 ++Is ⋅As ⎛⎜⎝

−yb4ybs⎞

⎟⎠

2

―――

⋅b ⎛⎜⎝yt

4⎞⎟⎠

3

⋅3 n⋅0 in4

⎞⎟⎟⎠

=Itr40 in4

Section Properties for 100% Composite Action


≔yt ∑ yt =yt 3.882 in

Dimension from bottom of steel section to neutral axis of composite section:

≔yb ∑ yb =yb 17.308 in


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≔Itr ∑ Itr =Itr 1175.559 in4

Transformed section modulus of composite section with 100% composite action referred to the bottom flange of the steel section:

≔Str ―Itr

yb

=Str 67.921 in3

Transformed section modulus of composite section with 100% composite action referred to the bottom flange of the steel section:

≔St ―Itr

yt

=St 302.812 in3

Seff and V'h as Functions of the Percent of Composite Action (CA) from 25% to 100%

Values for the horizontal shear, effective section modulus and effective moment of inertia are displayed in the Summary section:

≔V'h ((CA)) ―――⋅CA Vh

100≔CA , ‥25 30 100

Effective section modulus computed using a AISC Specification, Eq. (I2-1) with percent of compositeaction substituted for Vh and V'h:

≔Seff ((CA)) +Ss ⋅‾‾‾‾――CA

100⎛⎝ −Str Ss⎞⎠

Effective moment of inertia computed using a AISC Specification, Eq. (I4-4) with percent of composite action substituted for Vh and V'h:


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≔Ieff ((CA)) +Is ⋅‾‾‾‾――CA

100⎛⎝ −Itr Is⎞⎠

Summary

Modulus of elasticity of concrete:

=Ec 3644 ksi

Modular ratio: =n 7.958

Total depth of composite section:

=h 21.19 in


=yt 3.882 in

Dimension from bottom of steel section to neutral axis of composite section:

=yb 17.308 in

Cross section area of concrete:

=Ac 271.25 in2


=Itr 1175.6 in4

Section modulus of steel section referred to bottom of section:

=Ss 37.999 in3


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Section modulus of fully composite section to bottom of steel section:

=Str 67.921 in3

Section modulus of fully composite section to top of slab:

=St 302.812 in3

Total horizontal shear between point of maximumpositive moment and points of zero moment for full composite action:

=Vh 138.240 kip

Plot V'h in kip for (25% to 100% Composite Action) versus Seff in in3:

50

60

70

80

90

100

110

120

130

30

40

140

55.5 57 58.5 60 61.5 63 64.5 66 67.552.5 54 69


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=CA

253035404550556065707580859095

100

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=V'h ((CA))

34.5641.47248.38455.29662.20869.1276.03282.94489.85696.768

103.68110.592117.504124.416131.328138.24

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

kip =Seff ((CA))

52.9654.38855.70156.92358.07159.15760.18961.17662.12263.03363.91264.76265.58566.38567.16367.921

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

in3 =Ieff ((CA))

0.7370.7790.8170.8530.8870.9190.9490.9781.0061.0321.0581.0831.1071.1311.1531.176

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⋅103 in4

User Notices

Equations and numeric solutions presented in this Mathcad worksheet are applicable to the specific example, boundary condition or case presented in the book. Although a reasonable effort was made to generalize these equations, changing variables such as loads, geometries and spans, materials and other input parameters beyond the intended range may make some equations no longer applicable. Modify the equations as appropriate if your parameters fall outside of the intended range.For this Mathcad worksheet, the global variable defining the beginning index identifier for vectorsand arrays, ORIGIN, is set as specified in the beginning of the worksheet, to either 1 or 0. If ORIGIN is set to 1 and you copy any of the formulae from this worksheet into your own, you need to ensure that your worksheet is using the same ORIGIN.


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