Back Matter

REFERENCE LIST AISC (1999) “Load and Resistance Factor Design Specification for Structural

Steel Buildings ” AISC, Inc., Chicago.

Boyer J.P. (1964). “Castellated Beams – New Developments” AISC Engineering Journal, 2nd qtr, pp 104-108.

Chen, W.F. and Lui, E.M. (1987). “Structural Stability: Theory and Implementation” New York : Elsevier.

Clark, J.W. ,and Hill, H.N. (1960). “Lateral Buckling of Beams” AISC Engineering Journal – Structural Division, July, No. ST7, pp 175-196.

Galambos, T. (1993). “Bracing of Trussed Beams” Is Your Structure Suitably

Braced?, Structural Stability Research Council Conference, April, pp 39-49.

Galambos, T. (1998). “Guide to Stability Design Criteria for Metal Structures”

John Wiley and Sons Inc., Fifth Edition, New York, pp 192-213. Halleux, P. (1967). “Limit Analysis of Castellated Beams” Acier-Stahl-Steel,

No. 3, pp 133-144. Hosain, M.U. and Speirs, W.G. (1973). “Experiments on Castellated Steel

Beams” Journal of the American Welding Society, Vol. 52, pp 329-342. Jackson, R. (2002). “Vibration and Flexural Strength Characteristics of

Composite Castellated Beams.” M.S. Thesis, Virginia Tech, Blacksburg, Virginia.

Knowles, P.R. (1991). “Castellated Beams” Proceeding of the Institution of Civil

Engineers, Part 1, No. 90, pp 521-536. Kerdal, D. and Nethercot, D.A. (1982). “Lateral-Torsional Buckling of

Castellated Beams” The Structural Engineer, Part B, No. 3 pp 53-61. Kerdal, D. and Nethercot, D.A. (1983). “Buckling of Laterally Unsupported

Castellated Beams” Structural Stability Research Council Proceedings, 3rd International Colloquium, Stability of Metal Structures, Conference Code:03354, pp 151-171.

Kerdal, D. and Nethercot, D.A. (1984). “Failure Modes for Castellated Beams”

Journal of Constructional Steel Research, 4th qtr., pp 295-315.

56

Murray, T.M. Allen, D.E. and Ungar, E.E. (1997). AISC Steel Design Guide Series 11: Floor Vibrations Due to Human Activity. American Institute of Steel Construction, Chicago.

Pattanayak, U. and Chesson, E. (1974). “Lateral Instability of Castellated Beams”

AISC Engineering Journal, 3rd qtr, pp 73-79. Salmon, C. and Johnson, E. (1996). “Steel Structures – Design and Behavior”

Prentice Hall, 4th Edition, pp 479-559, 1008-1009. SMI Steel Products. (2002). “SMI Steel Products; Smart Beam, The Intelligent

Alternative”. Steel Joist Institute. (1994). “Fortieth Edition Standard Specifications Load

Tables and Weight Tables For Steel Joist and Joist Girders” Toprac, A., Altfillisch, M. and Cooke, B. (1957). “An Investigation of Open-Web

Expanded Beams ” Journal of the American Welding Society, Vol. 29, pp 77-88.

57

Appendix A

CB24x26 Calculations

58

A.1 Measured Dimensions of CB24x26 Specimen

e = 6.25" e = 6.25"

dg = 23.375"

dt = 4.125"

dt = 4.125"

ho = 15.188"

e = 6.25"

tf = 0.344"

tw = 0.251"

b = 4.5" b = 4.5"bf = 4.603"

A.2 “Tee” Section Properties

( )( ) ( )( )( )[ ]( ) 2.75.52 inttdtbA wftffTotal =−+=

( ) 63333

.154.02436

1 inthtbC wff

w =

+=

( ) ( ) 433 .194.0231 inhttbJ wff =

+=

( )( ) ( )( ) 433 .09.102121

121 inttdbtI wftffy =

−+=

4.09.683 inIx =

3.45.58 incIS x

x ==

inAIr y

y 32.1==

59

A.2.1 Classical Lateral-Torsional Buckling Solution

00004.04

15.7232

2

2

1

=

=

==

GJS

ICX

EGJAS

X

x

y

w

x

π

bbbb

cr LLPLwLM 075.000325.048

22

+=+=

( )22

211 28.20189.94221075.000325.02

12 22

bbybb

y

b

y

by

xbcr

LkLkLL

rLkXX

rLkXSCM

φφ+=+⇒

+=

When ky = 1.0 and kφ = 1.0:

ftLb 0.25=


ftLb 0.25=


ftLb 0.27=

A.2.2 Addition of Load Location Term

bbbb

cr LLPLwLM 075.000325.048

22

+=+=

C 50.02 =

( )( )

( )( )

−+=+⇒

−

++=

bbbybb

w

b

w

bby

ybcr

LkLkLkLL

GJEC

LkC

GJCEC

LkLkGJEICM

φφ

φφ

πππ

25.257.287.9189.94221075.000325.0

11

2

22

2

2

2

2

60


ftLb 9.24=


ftLb 9.24=


ftLb 8.26=

A.2.3 Galambos Formula

02)~( 33

=−−−

= ox

chordtopechordbottomx y

IyAydA

β

. 84.10~ inA

dAy

total

echordbottom==

0=+−=y

echordbottomyo

IdIyy

02)(2)(2

2243

)(2)(2

243

164

)(2192)4)(3(2

164

2

3

4

3

4

2

3

42

2

2

3

42222

=

+

−

−

+

−

+

+

−

+−

+++

+

bb

w

b

y

ox

b

yb

xb

yb

KLGJ

KLEC

KLEI

yKL

EIwLwL

aKL

EIwLPP

πππ

βπππ

βπππππ

[ ]

( ) [ ]

0310707392.3)(

93217113568.)(

5.021425822532

0)(

5.0214258225320.26724.99

0)(

5.02142582253224.1730005.156033

33

32

b

3

=

+

−

−

+

−+

bbb

bb

bb

KLKLKL

KLLL

KLL

61

When K = 1.0:

ftLb 2.20=

When K = 0.8:

ftLb 4.23=

When K = 0.5:

ftLb 1.32=

A.3 Full Section Properties

( )( )( ) ( )( )( ) 2.55.9*22 inttdtbA wfgffTotal =−+=

62

.28.13414

inIhC yw ==

( ) 433 .273.0231 inhttbJ wff == +

( )( ) ( )( ) 433 .11.10121

61 intthbtI wfffy =−+=

4.46.755 inIx =

3.64.64 incIS x

x ==

.03.1 inAIr y

y ==

62


24.04

88.10002

2

2

1

=

=

==

GJS

ICX

EGJAS

X

x

y

w

x

π

bbbb

cr LLPLwLM 075.000325.048

22

+=+=

( )22

211 16.125347135.112053075.000325.02

12 22

bbybb

y

b

y

by

xbcr

LkLkLL

rLkXX

rLkXSCM

φφ+=+⇒

+=


ftLb 8.29=


ftLb 6.33=


ftLb 7.35=

When ky = 1.0 and Lb = 37.5ft:

kφ = 0.30


kφ = 0.40


bbbb

cr LLPLwLM 075.000325.048

22

+=+=

63

50.02 =C

( )( )

( )( )

−+=+⇒

−

++=

bbbybb

w

b

w

bby

ybcr

LkLkLkLL

GJEC

LkC

GJCEC

LkLkGJEICM

φφ

φφ

πππ

02.17740.1587587.9135.112053075.000325.0

11

2

22

2

2

2

2


ftLb 8.26=


ftLb 0.30=


ftLb 9.31=


kφ = 0.18


kφ = 0.23


02)~( 33

=−−−

= ox


IyAydA

β

.03.8~ inA

dAy

total

echordbottom==

0=+−=y

echordbottomyo

IdIyy

64

02)(2)(2

2243

)(2)(2

243

164

)(2192)4)(3(2

164

2

3

4

3

4

2

3

42

2

2

3

42222

=

+

−

−

+

−

+

+

−

+−

+++

+

bb

w

b

y

ox

b

yb

xb

yb

KLGJ

KLEC

KLEI

yKL

EIwLwL

aKL

EIwLPP

πππ

βπππ

βπππππ

[ ]

( ) [ ]

067.15113783)(

30.5311894469871)(

33.61428637690

0)(

33.614286376900.26724.99

0)(

33.6142863769024.1730005.156033

33

32

b

3

=

+

−

−

+

−+

bbb

bb

bb

KLKLKL

KLLL

KLL

When K = 1.0:

ftLb 8.24=

When K = 0.8:

ftLb 2.29=

When K = 0.5:

ftLb 1.41=

A.4 Weighted Average Section Properties

%2910022

% =×+

=eb

eTee

%2910022

% =×+

=eb

eSolid

%4210022

2% =×+

=eb

bTransition

65

2.65.7 inATotal =

6.72.670 inCw =

4.234.0 inJ =

4.10.10 inIy =

4.28.719 inIx =

3.54.61 inSx =

.18.1 inry =


14692.04

68.8692

2

2

1

=

=

==

GJS

ICX

EGJAS

X

x

y

w

x

π

bbbb

cr LLPLwLM 075.000325.048

22

+=+=

( )22211 01.76958116.106009075.000325.0

212 2

2

bbybb

y

b

y

by

xbcr

LkLkLL

rLkXX

rLkXSCM

φφ+=+⇒

+=


ftLb 4.28=


ftLb 6.31=

66


ftLb 6.33=


bbbb

cr LLPLwLM 075.000325.048

22

+=+=

C 50.02 =

( )( )

( )( )

−+=+⇒

−

++=

bbbybb

w

b

w

bby

ybcr

LkLkLkLL

GJEC

LkC

GJCEC

LkLkGJEICM

φφ

φφ

πππ

44.143534.929387.9155.103513075.000325.0

11

2

22

2

2

2

2


ftLb 48.25=


ftLb 9.27=


ftLb 7.29=


02)~( 33

=−−−

= ox


IyAydA

β

.43.9~ inA

dAy

total

echordbottom==

67

0=+−=y

echordbottomyo

IdIyy

02)(2)(2

2243

)(2)(2

243

164

)(2192)4)(3(2

164

2

3

4

3

4

2

3

42

2

2

3

42222

=

+

−

−

+

−

+

+

−

+−

+++

+

bb

w

b

y

ox

b

yb

xb

yb

KLGJ

KLEC

KLEI

yKL

EIwLwL

aKL

EIwLPP

πππ

βπππ

βπππππ

[ ]

( ) [ ]

0012910588.0)(

50.119473434925)(

5.681427230111

0)(

5.6814272301110.26724.99

0)(

15.68114272301124.1730005.156033

33

32

b

3

=

+

−

−

+

−+

bbb

bb

bb

KLKLKL

KLLL

KLL

When K = 1.0:

ftLb 3.23=

When K = 0.8:

ftLb 3.27=

When K = 0.5:

ftLb 3.38=

68

Appendix B

CB24x26 Specimen Test Data

69

CASTELLATED BEAM TEST SUMMARY TEST IDENTIFICATION: CB24x26 TEST DESCRIPTION Loading Gravity Point of Load Application Mid-span Span 37'-6" Bracing Points None Number of beams 1 End Condition Web-to-column flange double angle connection FAILURE MODE:

Lateral-Torsional Buckling THEORETICAL CRITICAL UNBRACED LENGTH: (a) Classical Lateral-Torsional Buckling Solution = 35.7 ft

(b) Addition of Load Location Term = 31.9 ft

(c) Galambos Formula = 29.2 ft EXPERIMENTAL CRITICAL UNBRACED LENGTH: Total Applied Load = 300 lb Unbraced Length = 37.5 ft R-VALUE: R(a) = Experimental Length/Theoretical Length = 1.05 R(b) = Experimental Length/Theoretical Length = 1.18 R(c) = Experimental Length/Theoretical Length = 1.28 DISCUSSION:

10 lb weights were loaded on a loading plate clamped to the top flange of the castellated beam at midspan. Catch bracing was installed to stop excessive deflections and help characterize failure.

Concentrated Load (lb) Test LengthEccentricity

48.3ft 44.8ft 41.3ft 37.5ft

e = 0 170 220 260 300 e = 1 1/2" 100 150 190 260

e = 2" 80 120 150 200

70

Photos of CB24x26 Testing

Support Column

Quarter Point Catch Bracing

Midspan Catch Bracing


Photo of CB24x26 Entire Test Set-up

71

Location of Failure

Photo of CB24x26 Specimen at failure

72

Appendix C

CB27x40 Calculations

73

C.1 Measured Dimensions of CB27x40 Specimen

e = 7.5" e = 7.5"

dg = 26.875"

dt = 4.188"

dt = 4.188"

ho = 18.5"

e = 7.5"

tf = 0.524"

tw = 0.320"

b = 6.0" b = 6.0"bf = 6.063"

C.2 “Tee” Section Properties

( )( ) ( )( )( )[ ]( ) 2.70.82 inttdtbA wftffTotal =−+=

( ) 63333

.555.02436

1 inthtbC wff

w =

+=

( ) ( ) 433 .667.0231 inhttbJ wff =

+=

( )( ) ( )( ) ( ) 433 .48.192121

121 inttdbtI wftffy =

−+=

4.66.1393 inIx =

3.71.103 incIS x

x ==

.50.1 inAIr y

y ==

74

C.2.1 Classical Lateral-Torsional Buckling Solution

00002.04

99.9292

2

2

1

=

=

==

GJS

ICX

EGJAS

X

x

y

w

x

π

bbbb

cr LLPLwLM 075.0005.048

22

+=+=

( )22

211 27.21178.242910075.0005.02

12 22

bbybb

y

b

y

by

xbcr

LkLkLL

rLkXX

rLkXSCM

φφ+=+⇒

+=


ftLb 0.30=


ftLb 0.30=


ftLb 4.32=

C.2.2 Addition of Load Location Term

bbbb

cr LLPLwLM 075.0005.048

22

+=+=

C 50.02 =

( )( )

( )( )

−+=+⇒

−

++=

bbbybb

w

b

w

bby

ybcr

LkLkLkLL

GJEC

LkC

GJCEC

LkLkGJEICM

φφ

φφ

πππ

31.269.287.9178.242910075.0005.0

11

2

22

2

2

2

2

75


ftLb 9.29=


ftLb 9.29=


ftLb 2.32=

C.2.3 Galambos Formula

02)~( 33

=−−−

= ox


IyAydA

β

.61.12~ inA

dAy

total

echordbottom==

0=+−=y

echordbottomyo

IdIyy

02)(2)(2

2243

)(2)(2

243

164

)(2192)4)(3(2

164

2

3

4

3

4

2

3

42

2

2

3

42222

=

+

−

−

+

−

+

+

−

+−

+++

+

bb

w

b

y

ox

b

yb

xb

yb

KLGJ

KLEC

KLEI

yKL

EIwLwL

aKL

EIwLPP

πππ

βπππ

βπππππ

[ ]

( ) [ ]

0936879923.5)(

30784498707.)(

9.942751370895

0)(

9.9427513708950.401715.95

0)(

9.94275137089537.1930005.156033

33

32

b

3b

=

+

−

−

+

−+

bbb

bb

b

KLKLKL

KLLL

KLL

76

When K = 1.0:

ftLb 1.24=

When K = 0.8:

ftLb 9.27=

When K = 0.5:

ftLb 2.38=

C.3 Full Section Properties

( )( )( ) ( )( )( ) 2.62.14*22 inttdtbA wfgffTotal =−+=

62

.32.33904

inIhC yw ==

( ) 433 .869.0231 inhttbJ wff == +

( )( ) ( )( ) 433 .53.19121

61 intthbtI wfffy =−+=

4.47.1562 inIx =

3.28.116 incIS x

x ==

.16.1 inAIr y

y ==

77


10.04

40.12272

2

2

1

=

=

==

GJS

ICX

EGJAS

X

x

y

w

x

π

bbbb

cr LLPLwLM 075.0005.048

22

+=+=

( )22

211 46.99663167.277619075.0005.02

12 22

bbybb

y

b

y

by

xbcr

LkLkLL

rLkXX

rLkXSCM

φφ+=+⇒

+=


ftLb 0.34=


ftLb 7.37=


ftLb 1.40=


kφ = 0.28


kφ = 0.37


bbbb

cr LLPLwLM 075.0005.048

22

+=+=

78

50.02 =C

( )( )

( )( )

−+=+⇒

−

++=

bbbybb

w

b

w

bby

ybcr

LkLkLkLL

GJEC

LkC

GJCEC

LkLkGJEICM

φφ

φφ

πππ

85.15752.1262287.9167.277619075.0005.0

11

2

22

2

2

2

2


ftLb 9.30=


ftLb 7.33=


ftLb 9.35=


kφ = 0.16


kφ = 0.231


02)~( 33

=−−−

= ox


IyAydA

β

. 38.9~ inA

dAy

total

echordbottom==

0=+−=y

echordbottomyo

IdIyy

79

02)(2)(2

2243

)(2)(2

243

164

)(2192)4)(3(2

164

2

3

4

3

4

2

3

42

2

2

3

42222

=

+

−

−

+

−

+

+

−

+−

+++

+

bb

w

b

y

ox

b

yb

xb

yb

KLGJ

KLEC

KLEI

yKL

EIwLwL

aKL

EIwLPP

πππ

βπππ

βπππππ

[ ]

( ) [ ]

0348047648.4)(

811.774788594942)(

2.122758505739

0)(

2.1227585057390.401715.95

0)(

2.12275850573937.1930005.156033

33

32

b

3b

=

+

−

−

+

−+

bbb

bb

b

KLKLKL

KLLL

KLL

When K = 1.0:

ftLb 0.28=

When K = 0.8:

ftLb 8.32=

When K = 0.5:

ftLb 0.46=

C.4 Weighted Average Section Properties

%2810022

% =×+

=eb

eTee

%2810022

% =×+

=eb

eSolid

%4410022

2% =×+

=eb

bTransition

80

2.66.11 inATotal =

6.44.1695 inCw = 4.768.0 inJ =

4.50.19 inIy =

4.07.1478 inIx =

3.00.110 inSx =

.33.1 inry =


05681.04

31.10892

2

2

1

=

=

==

GJS

ICX

EGJAS

X

x

y

w

x

π

bbbb

cr LLPLwLM 075.0005.048

22

+=+=

( )22211 61.59281191.267415075.0005.0

212 2

2

bbybb

y

b

y

by

xbcr

LkLkLL

rLkXX

rLkXSCM

φφ+=+⇒

+=


ftLb 7.32=


ftLb 7.35=


ftLb 0.38=

81


bbbb

cr LLPLwLM 075.0005.048

22

+=+=

C 50.02 =

( )( )

( )( )

−+=+⇒

−

++=

bbbybb

w

b

w

bby

ybcr

LkLkLkLL

GJEC

LkC

GJCEC

LkLkGJEICM

φφ

φφ

πππ

74.11832.714287.9197.260820075.0005.0

11

2

22

2

2

2

2


ftLb 8.29=


ftLb 7.31=


ftLb 8.33=


02)~( 33

=−−−

= ox


IyAydA

β

. 99.10~ inA

dAy

total

echordbottom==

0=+−=y

echordbottomyo

IdIyy

82

02)(2)(2

2243

)(2)(2

243

164

)(2192)4)(3(2

164

2

3

4

3

4

2

3

42

2

2

3

42222

=

+

−

−

+

−

+

+

−

+−

+++

+

bb

w

b

y

ox

b

yb

xb

yb

KLGJ

KLEC

KLEI

yKL

EIwLwL

aKL

EIwLPP

πππ

βπππ

βπππππ

[ ]

( ) [ ]

0142463786.0)(

759.532394689720)(

6.032754938317

0)(

6.0327549383170.401715.95

0)(

6.03275493831737.1930005.156033

33

32

b

3b

=

+

−

−

+

−+

bbb

bb

b

KLKLKL

KLLL

KLL

When K = 1.0:

ftLb 5.26=

When K = 0.8:

ftLb 0.31=

When K = 0.5:

ftLb 3.43=

83

Appendix D

CB24x26 Specimen Test Data

84

CASTELLATED BEAM TEST SUMMARY TEST IDENTIFICATION: CB27x40 TEST DESCRIPTION Loading Gravity Point of Load Application Mid-span Span 42.5" Bracing Points None Number of beams 1 End Condition Web to column flange double angle connection FAILURE MODE:

Lateral Torsional Buckling THEORETICAL CRITICAL UNBRACED LENGTH: (a) Classical Lateral-Torsional Buckling Solution = 40.1 ft

(b) Addition of Load Location Term = 35.9 ft

(c) Galambos Formula = 32.8 ft EXPERIMENTAL CRITICAL UNBRACED LENGTH: Total Applied Load = 300 lb Unbraced Length = 42.5 ft R-VALUE: R(a) = Experimental Length/Theoretical Length = 1.06 R(b) = Experimental Length/Theoretical Length = 1.18 R(c) = Experimental Length/Theoretical Length = 1.30 DISCUSSION:

10 lb weights were loaded on a loading plate clamped to the top flange of the castellated beam at midspan. Catch bracing was installed to stop excessive deflections and help characterize failure.

Concentrated Load (lb) Test LengthEccentricity

51.8ft 47.3ft 44.5ft 42.5ft

e = 0 self wt. 120 270 300 e = 1 1/2" self wt. 60 210 250

e = 2" self wt. 40 160 190

85

Photos of CB27x40 Testing

Support Column


Midspan Catch Bracing


Photo of CB27x40 Entire Test Set-up

86

Location of Failure

Photo of CB27x40 Specimen at failure

87

88

VITA

T. Patrick Bradley was born on July 14, 1977 in Clemmons, North

Carolina. He graduated from West Forsyth high school in Lewisville, North

Carolina. He received his Associate in Applied Science in Architectural

Technology from Guilford Technical Community College in May 1998. He

received his Bachelor of Science in Civil Engineering from North Carolina

Agricultural and Technical State University in Greensboro, North Carolina in

May of 2001. He enrolled in the graduate program at Virginia Tech in the fall of

2001 and plans to work for a metal building manufacturer in North Carolina after

completion.

_______________________ T. Patrick Bradley

Back Matter

Documents

Transcript of Back Matter