LIST OF REFERENCES Wilson, E. L. and Habibullah, A. (1992). “SAP90--A Series of Computer Programs for the Finite Element Analysis of Structures—Structural Analysis User’s Manual,”

76

LIST OF REFERENCES

Band, B. S. Jr. (1996). “Several Vibration Performance Aspects of Joist and Joist-GirderSupported Floors.” M. S. Thesis, Virginia Polytechnic Institute and StateUniversity, Blacksburg, Virginia.

Gibbings, D. R., Easterling, W. S., and Murray, T. M. (1991). “Strength of CompositeLong Span Joists.” Report No. CE/VPI 91/02. Virginia Polytechnic Institute andState University, Blacksburg, Virginia

Gibbings, D. R., Easterling, W. S., and Murray, T. M. (1993). “Analysis of CompositeJoists.” Report No. CE/VPI 93/12. Virginia Polytechnic Institute and StateUniversity, Blacksburg, Virginia.

Hanagan, L. M. (1994). “Active Control of Floor Vibrations.” Ph.D. Dissertation,Virginia Polytechnic Institute and State University, Blacksburg, Virginia.

Kitterman, S. S. (1994). “Investigation of Several Aspects of the VibrationCharacteristics of Steel Member-Supported Floors.” M. S. Thesis, VirginiaPolytechnic Institute and State University, Blacksburg, Virginia.

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

Ohmart, R. D. and Lenzen, K. H. (1968). “An Approximate Method for the Response ofStiffened Plates to Aperiodic Motion.” Studies in Engineering Mechanics, ReportNo. 30, The University of Kansas Center for Research in Engineering Science,Lawrence, Kansas

Rahman, R. H. and Murray, T. M. (1974). “Measurement and Analysis of FloorVibrations.” School of Civil Engineering, University of Oklahoma, Norman,Oklahoma.

Rottmann, C. (1996). “The Use of Tuned Mass Dampers to Control Floor Vibrations.”M. S. Thesis, Virginia Polytechnic Institute and State University, Blacksburg,Virginia.

Shamblin, C. L. (1989). “Floor System Response to a Heel-Drop Impact.” M. S. Thesis,Virginia Polytechnic Institute and State University, Blacksburg, Virginia.

Steel Joists and Joist Girders (1995). Vulcraft Corporation, Norfolk, Nebraska.

77

Wilson, E. L. and Habibullah, A. (1992). “SAP90--A Series of Computer Programs forthe Finite Element Analysis of Structures—Structural Analysis User’s Manual,”Computers and Structures, Inc., Berkeley, California.

Wilson, E. L. and Habibullah, A. (1996). “SAP2000--A Series of Computer Programsfor the Finite Element Analysis of Structures—Structural Analysis User’sManual,” Computers and Structures, Inc., Berkeley, California.

78

APPENDIX A

DERIVATION OF THE PROPOSED EQUATIONS FOR IEFF

79

Derivation of Ieff Equation

The following is a derivation of Equation (1.7), and listed again as Equation (3.1). It isderived from the combination of the bending and shear deflection caused by a midspanpoint load.

The total deflection of the composite system due to a midspan point load is broken upinto the shear deflection and the flexural deflection:

∆ ∆ ∆p s bnc c

PlEI

PlEI

= + =

+γ β β

3 3

48 48

The term βPl

E

3

48 is factored out of the above equation, and the collected terms in the

brackets are renamed:

∆ pnc c p

PlE I I

PlE I

= +

=

β

γβ

3 3

481

481

The term βPl

E

3

48 can be canceled out, which leaves the following relationship:

γI I Inc c p

+ =1 1

The equation is now rearranged to solve for Ip, which is then renamed Ieff:

I

I I

I

C

p

nc c

eff

r

=+

=

= −

11

11

γ

γ

where γ = portion of non-composite deflection that is due to shear deflection, and Cr isdefined in Equation (1.6).

80

APPENDIX B

DESCRIPTION OF JOIST DESIGNS USED IN

CHAPTER 3 STUDY

81

The description of the joist geometry variables, L1, L2, L3, and L4 in Table B.1 aredefined in Figure 4.1. The scheme for naming the web members is also defined in Figure4.1.

Table B.1 Joist GeometryCase Total Nominal L1 L2 L3 L4 P Vertical

Length Depth (in) (in) (in) (in) # Interior Members(in) (in) Panels

J1 240 14 34.5 25 9.5 19 8 NJ2 240 16 36 24 12 24 6 NJ3 240 20 36 24 12 24 6 NJ4 360 20 48 36 12 24 10 NJ5 360 24 36 24 12 24 11 NJ6 360 30 42 30 18 48 5 YJ7 480 26 65 43 7 48 7 YJ8 480 32 79 48 2 53 6 YJ9 480 40 70 43 2.5 67 5 Y

J10 600 32 80 52 8 53 8 YJ11 600 40 95 59 4 67 6 YJ12 600 52 73 63 32 78 5 YJ13 720 40 100 75 25.5 67 7 YJ14 720 48 96 64 12 72 7 YJ15 720 60 84 54 6 90 6 YJ16 240 10 25 15.5 9.5 19 9 NJ17 360 15 28 18.5 9.5 19 15 NJ18 480 20 50 43 22 48 7 YJ19 600 25 63 50 21 48 9 YJ20 720 30 75 57 21 48 11 YG1 240 24 37 24 11 48 3 YG2 360 36 0 0 0 90 4 YG3 480 48 69 40 11 80 4 YG4 600 60 67 43 18.75 85.75 5 YG5 720 72 121 90 59 180 2 Y

82

The following tables give the member size information for each joist design usedin Chapter 3. The dimensions given are design, or nominal, values.

Table B.2 Member Sizes for Case J1



Member Size (in.)

Top Chord 2L-1.25x1.25x0.133Bottom Chord 2L-1.00x1.00x0.109

Web RB 0.50 Dia.

Member Size (in.)


W2 RB 0.536 Dia.W3 RB 0.50 Dia.W4 RB 0.50 Dia.W5 RB 0.50 Dia.W6 RB 0.50 Dia.W7 RB 0.50 Dia.W8 RB 0.50 Dia.W9 RB 0.50 Dia.

W10 RB 0.50 Dia.

Member Size (in.)



W10 RB 0.50 Dia.

83



Member Size (in.)



W10 RB 0.594 Dia.W11 RB 0.594 Dia.W12 RB 0.594 Dia.W13 RB 0.594 Dia.W14 RB 0.594 Dia.

Member Size (in.)



W10 RB 0.656 Dia.W11 RB 0.656 Dia.W12 RB 0.656 Dia.W13 RB 0.656 Dia.W14 RB 0.656 Dia.W15 RB 0.656 Dia.

84



Member Size (in.)


Verticals 1L-1.00x1.00x0.109W2 RB 0.875 Dia.W3 1L-1.00x1.00x0.109W4 1L-1.75x1.75x0.155W5 1L-1.00x1.00x0.109W6 1L-1.50x1.50x0.123W7 1L-1.00x1.00x0.109W8 1L-1.25x1.25x0.109W9 1L-1.00x1.00x0.109

Member Size (in.)


Verticals 1L-1.00x1.00x0.109W2 2L-2.00x2.00x0.205W3 1L-1.50x1.50x0.143W4 2L-1.25x1.25x0.133W5 2L-1.25x1.25x0.133W6 1L-2.00x2.00x0.176W7 1L-1.50x1.50x0.143W8 1L-1.75x1.75x0.170W9 1L-1.25x1.25x0.109

W10 1L-1.50x1.50x0.138W11 1L-1.25x1.25x0.109

85



Member Size (in.)



W10 1L-1.50x1.50x0.170

Member Size (in.)



86



Member Size (in.)



W10 1L-1.75x1.75x0.155W11 1L-1.25x1.25x0.133

Member Size (in.)



W10 1L-2.00x2.00x0.163

87



Member Size (in.)


First & Last Vert. 1L-1.50x1.50x0.143Rest of Verticals 1L-1.50x1.50x0.138

W2 2L-2.00x2.00x0.232W3 1L-1.75x1.75x0.155W4 2L-2.00x2.00x0.179W5 2L-1.25x1.25x0.133W6 1L-1.75x1.75x0.155W7 1L-1.50x1.50x0.143W8 1L-2.00x2.00x0.205W9 1L-1.50x1.50x0.123

Member Size (in.)



W10 1L-2.00x2.00x0.205W11 1L-1.50x1.50x0.170

88




Member Size (in.)



W10 2L-1.50x1.50x0.138W11 1L-1.50x1.50x0.170

Member Size (in.)



W10 2L-1.75x1.75x0.155

Member Size (in.)


Web RB 0.50 Dia.

89



Member Size (in.)



W10 RB 0.625 Dia.W11 RB 0.50 Dia.W12 RB 0.50 Dia.W13 RB 0.50 Dia.W14 RB 0.50 Dia.W15 RB 0.50 Dia.W16 RB 0.50 Dia.W17 RB 0.50 Dia.W18 RB 0.50 Dia.W19 RB 0.50 Dia.

Member Size (in.)



W10 1L-1.50x1.50x0.138W11 1L-1.25x1.25x0.109

90



Member Size (in.)



W10 1L-1.75x1.75x0.155W11 1L-1.25x1.25x0.133W12 1L-1.75x1.75x0.155W13 1L-1.50x1.50x0.125

Member Size (in.)



W10 2L-1.50x1.50x0.138W11 1L-2.00x2.00x0.187W12 1L-2.00x2.00x0.163W13 1L-1.50x1.50x0.170W14 1L-2.00x2.00x0.163W15 1L-1.50x1.50x0.170

91

Table B.22 Member Sizes for Case G1



Member Size (in.)


Verticals 1L-1.50x1.50x0.155W2 2L-1.50x1.50x0.170W3 1L-1.50x1.50x0.155W4 1L-2.00x2.00x0.187W5 1L-1.50x1.50x0.170W6 1L-1.75x1.75x0.155W7 1L-1.50x1.50x0.155

Member Size (in.)


Verticals 1L-2.00x2.00x0.163W2 2L-2.00x2.00x0.250W3 2L-2.50x2.50x0.250W4 2L-1.50x1.50x0.155W5 2L-1.75x1.75x0.155

Member Size (in.)


Verticals 1L-1.75x1.75x0.155W2 2L-3.50x3.50x0.375W3 1L-1.75x1.75x0.170W4 2L-3.00x3.00x0.250W5 2L-2.00x2.00x0.232W6 2L-2.50x2.50x0.250W7 2L-1.50x1.50x0.155W8 2L-2.00x2.00x0.163

92



Member Size (in.)



Member Size (in.)


Verticals 2L-2.50x2.50x0.230W2 2L-5.00x5.00x0.625W3 2L-3.00x3.00x0.227W4 2L-4.00x4.00x0.438W5 2L-3.00x3.00x0.281W6 2L-3.50x3.50x0.313

93

APPENDIX C

DATA USED TO VERIFY IEFF

94

Table C.1 Data for Round Bar Joists

Case D S L Ieff Ieff Ratio Freq. Freq. RatioSAP hand SAP/hand SAP hand SAP/hand

(in) (in) (in) (in^4) (in^4) (Hz.) (Hz.)

Kit_J1A 14 24 240 64.0 67.4 0.950 10.5 10.6 0.991Kit_J1B 14 24 240 78.0 74.4 1.048 9.80 9.65 1.016Kit_J1C 14 24 240 102.2 87.5 1.168 9.51 8.87 1.072







95

Table C.2 Data for Joists J6 through J13


(in) (in) (in) (in^4) (in^4) (Hz.) (Hz.)

Kit_J6B1 30 24 360 440.0 424.3 1.037 10.30 10.10 1.020Kit_J6B2 30 36 360 451.3 432.5 1.043 8.62 8.43 1.023Kit_J6C1 30 24 360 494.9 457.8 1.081 9.28 8.92 1.040Kit_J6C2 30 36 360 509.2 466.3 1.092 7.76 7.43 1.044

Kit_J7B1 26 24 480 1438 1585 0.907 9.40 9.93 0.947Kit_J7B2 26 36 480 1538 1689 0.911 8.30 8.74 0.950Kit_J7C1 26 24 480 1652 1775 0.931 8.80 9.18 0.959Kit_J7C2 26 36 480 1769 1886 0.938 7.70 7.98 0.965

Kit_J8C1 32 24 480 1580 1709 0.925 8.85 9.24 0.958Kit_J8C2 32 36 480 1664 1782 0.934 7.62 7.90 0.965Kit_J8D1 32 24 480 1701 1806 0.942 8.56 8.85 0.967Kit_J8D2 32 36 480 1797 1883 0.954 7.35 7.54 0.975

Kit_J9C1 40 36 480 1998 2048 0.976 8.43 8.52 0.989Kit_J9C2 40 48 480 2047 2085 0.982 7.48 7.53 0.993Kit_J9D1 40 36 480 2128 2135 0.997 8.06 8.07 0.999Kit_J9D2 40 48 480 2182 2174 1.004 7.15 7.12 1.004

Kit_10C1 32 24 600 3113 3509 0.887 7.39 7.89 0.937Kit_10C2 32 36 600 3340 3748 0.891 6.55 6.97 0.940Kit_10D1 32 24 600 3336 3713 0.898 7.19 7.63 0.942Kit_10D2 32 36 600 3594 3975 0.904 6.35 6.71 0.946


Kit_12C1 52 48 600 4522 4883 0.926 6.98 7.24 0.964Kit_12C2 52 60 600 4599 4653 0.988 6.36 6.59 0.965Kit_12C3 52 120 600 4794 5167 0.928 4.70 4.84 0.971Kit_12D1 52 48 600 4742 5046 0.940 6.63 6.84 0.969Kit_12D2 52 60 600 4827 5117 0.943 6.04 6.21 0.973Kit_12D3 52 120 600 5042 5283 0.954 4.45 4.55 0.978


96

Table C.3 Data for Joists J14, J15, J18 through J20


(in) (in) (in) (in^4) (in^4) (Hz.) (Hz.)



Kit_18B1 20 24 480 1363 1535 0.888 8.36 8.97 0.932Kit_18B2 20 36 480 1484 1672 0.888 7.63 8.15 0.936Kit_18C1 20 24 480 1610 1770 0.910 8.10 8.57 0.945Kit_18C2 20 36 480 1767 1931 0.915 7.31 7.69 0.951



97

Table C.4 Data for Joists G1 through G5


(in) (in) (in) (in^4) (in^4) (Hz.) (Hz.)

Kit_G1B1 24 24 240 536 541.3 0.990 24.30 24.50 0.992Kit_G1B2 24 36 240 560.3 558.7 1.003 20.90 20.90 1.000Kit_G1C1 24 24 240 623.2 590.1 1.056 22.60 22.10 1.023Kit_G1C2 24 36 240 653.6 607.5 1.076 19.30 18.70 1.032

Kit_G2C1 36 36 360 2631 2677 0.983 16.70 16.70 1.000Kit_G2C2 36 48 360 2716 2744 0.990 15.00 14.90 1.007Kit_G2D1 36 36 360 2789 2785 1.001 16.00 15.90 1.006Kit_G2D2 36 48 360 2885 2854 1.011 14.30 14.10 1.014

Kit_G3C1 48 48 480 8142 8219 0.991 13.90 13.90 1.000Kit_G3C2 48 60 480 8381 8410 0.997 12.90 12.80 1.008Kit_G3C3 48 120 480 8992 8879 1.013 9.84 9.72 1.012Kit_G3D1 48 48 480 8553 8488 1.008 13.30 13.20 1.008Kit_G3D2 48 60 480 8806 8681 1.014 12.30 12.50 0.984Kit_G3D3 48 120 480 9473 9151 1.035 9.37 9.15 1.024

Kit_G4C1 60 48 600 17347 18102 0.958 12.20 12.40 0.984Kit_G4C2 60 60 600 17863 18574 0.962 11.4 11.6 0.983Kit_G4C3 60 120 600 19302 19839 0.973 8.99 9.03 0.996Kit_G4D1 60 48 600 18034 18603 0.969 11.7 11.8 0.992Kit_G4D2 60 60 600 18593 19092 0.974 11 11 1.000Kit_G4D3 60 120 600 20145 20379 0.989 8.55 8.52 1.004

Kit_G5C1 72 48 720 34892 36483 0.956 11.3 11.6 0.974Kit_G5C2 72 60 720 35955 37522 0.958 10.7 11 0.973Kit_G5C3 72 120 720 39119 40531 0.965 8.62 8.75 0.985Kit_G5D1 72 48 720 35824 37346 0.959 10.9 11.1 0.982Kit_G5D2 72 60 720 36942 38431 0.961 10.2 10.5 0.971Kit_G5D3 72 120 720 40266 41513 0.970 8.17 8.27 0.988

98

APPENDIX D

SAMPLE CALCULATION OF THE FIRST NATURAL

FREQUENCY OF A SINGLE BAY FLOOR

99

The following is a sample calculation to determine the first natural frequency of a joistand joist-girder floor system. These calculations use Equations (1.1), (1.2) and (1.7).This sample calculation is for Floor 4 defined in Section 4.3.4. This floor has round barweb joists which are supported by angle web joist girders.

CALCULATION OF JOIST FREQUENCY

Description of Joist Tee-Beam

A cross section of the tee-beam which is used to model the joist behavior is shown inFigure D.1. The physical properties of the system are:

Length of joist, L: 360 in.Depth of joist, D: 16 in.Weight of joist, wj: 5.42 plf

Weight of concrete, wc: 110 pcfCompressive strength, fc’: 4000 psiWeight of steel deck, wd: 1 psf

Area of top chord, Atc: 0.763 in2

Area of bottom chord, Abc: 0.539 in2

Area of joist, Aj: 1.302 in2

Depth of slab, ds: 1.9 in.Height of ribs, dr: 0.6 in.Width of slab, S: 30 in.

Centroid of top chord, ytc: 0.424 in.Centroid of bottom chord, ybc: 0.354 in.Centroid of joist, yj: 6.73 in.

Moment of inertia of top chord, Itc: 0.164 in4

Moment of inertia of bottom chord, Ibc: 0.081 in4

100

30”

6.73”0.424”

0.354” 2L 1.25x1.25x0.113

2L 1.5x1.5x0.133

1.9”

0.6”

16”

Figure D.1 Cross Section of Joist Tee-Beam

Calculation of Ieff

Calculate Ichords

I I A d I A d

I

I in

chords tc tc bc bc

chords

chords

= + × + + ×

= + × + + ×

=

2 2

2 2

4

0164 0 763 6 31 0 081 0539 8 92

735

. ( . . ) . ( . . )

.

Calculate Icomp

First , calculate the transformed area of the deck.

Dynamic Ec = 1.35 [33(wc)1.5 fc' ]

Dynamic Ec = 1.35 [33(110)1.5 3000 ] = 2815 ksi

bS

EsEc

intr = = =30

290002815

2 9. .

Ad = btr * ds = 2.9 * 1.9 = 5.51 in2

101

Next calculate the centroidal axis of the composite section. This is given by yc asmeasured from the top of the slab.

( )

yA

dA y d d

A A

y

y in

c

ds

j j r s

d j

c

c

=+ + +

+

=+ + +

=

2

553192

13 6 73 0 6 19

68352 53

( )

..

. . . .

.. .

Next, Icomp can be calculated

I I A d b d A d

I

I in

comp chords j tr r d

comp

comp

= + × + + ×

= + × + × + ×

=

2 3 2

2 3 2

4

112

735 13 6 71

122 9 19 53 158

147 4

. ( . . ) . . (5. . )

.

Calculate Cr

For Round Bar Joists:C L DC

r

r

= += + =

0 721 0 007250 721 0 00724 360 16 08841. . ( / ). . ( / ) .

Calculate Ieff

I

I I

C

I

eff

chords comp

r

eff

=+

= − = − =

=+

11

11

108841

1 0131

10131735

1147 4

γ

γ.

.

.. .

Ieff = 116.7 in4

102

Calculate fn

Calculate System Weight

Weight of one Joist

5.42 plf * 30 ft = 162.6 lbs.

Weight of Slab Width:

30 2 2 3601728

110 15133 3

in in inin ft

pcf lbs. . . .

/.

× ×× =

Weight of Steel Deck:

30 360144

1 752 2

in inin ft

psf lbs. .

/.

×× =

Total System Weight:

W = 162.6 + 1513 + 75 = 1,751 lbs., or 1.75 kips

Calculate fn of joist tee-beam

fgEIWL

f

neff

n

=

=× ×

×

=

157

157386 4 29000 116 7

175 360

3

3

.

.. .

.

f 6.28 Hzn .

CALCULATION OF GIRDER FREQUENCY

Description of Girder Tee-Beam

A cross section of the tee-beam which is used to model the joist-girder behavior is shownin Figure D.2. The physical properties of the system are given below.

Length of joist, L: 300 in.Depth of joist, D: 20 in.Weight of joist, wj: 32.3 plf

Weight of concrete, wc: 110 pcfCompressive strength, fc’: 4000 psi

103

Weight of steel deck, wd: 1 psf

Area of top chord, Atc: 4.186 in2

Area of bottom chord, Abc: 4.186 in2

Area of joist, Aj: 8.372 in2

Depth of slab, ds: 1.9 in.Height of ribs, dr: 0.6 in.Joist seat height, b: 1.0 in.Width of slab, S: 60 in.

Centroid of top chord, ytc: 0.991 in.Centroid of bottom chord, ybc: 0.991 in.Centroid of joist, yj: 10.0 in.

Moment of inertia of top chord, Itc: 4.904 in4

Moment of inertia of bottom chord, Ibc: 4.904 in4

60”

10.0”0.991”

0.991” 2L 3.5x3.5x0.313

2L 3.5x3.5x0.313

1.9”

1.6”

20”

Figure D.2 Cross Section of Girder Tee-Beam

Calculation of Ieff

Calculate Ichords

I I A d I A d

I

I in

chords tc tc bc bc

chords

chords

= + × + + ×

= + × + + ×

=

2 2

2 2

4

4 9 419 9 01 4 9 419 9 01

690

. ( . . ) . ( . . )

104

Calculate Icomp

Dynamic Ec = 2815 ksi

bS

EsEc

intr = = =60

290002815

58. .

Ad = btr * ds = 5.8 * 1.9 = 11.1 in2

Next calculate the centroidal axis of the composite section. This is given by yc asmeasured from the top of the slab.

( )

yA

dA y d d b

A A

y

y in

c

ds

j j r s

d j

c

c

=+ + + +

+

=+ + + +

=

2

111192

8 37 10 0 0 6 19 10

19 46 36

( )

..

. . . . .

.. .

Next, Icomp can be calculated

I I A d b d A d

I

I in

comp chords j tr r d

comp

comp

= + × + + ×

= + × + × + ×

=

2 3 2

2 3 2

4

112

690 37 7151

1258 19 111 541

1444

(8. . ) . . ( . . )

Calculate Cr

For Angle Web Joists:

C e

C er

L D

r

= −

= − =

−

−

0 9 1

0 9 1 08627

0 28 2 8

0 28 300 20 2 8

. ( )

. ( ) .

. ( / ) .

. ( / ) .

105

Calculate Ieff as a Joist

I

I I

C

I

I in

eff j

chords comp

r

eff j

eff j

,

,

,

..

.

=+

= − = − =

=+

=

11

11

10 8627

1 0 159

10 159690

11444

1083 4

γ

γ

Calculate Girder Ieff

I C I I I C I I

I

I in

eff r chords deck eff j r chords deck

eff

eff

= × + + − × +

= × + + − × +

=

(( ) ( )) /

(( . . ) ( . . )) /

.

, 4

08627 690 333 1083 08627 690 333 4

719 2 4

Calculate fn

Calculate System Weight (½ of the total floor weight)

Weight of one Girder

32.2 plf * 25 ft = 805 lbs

Weight of 5.5 Joists

5.5 * 5.42 plf * 30 ft = 894.3 lbs.

Weight of ½ Floor Slab:

300 2 2 1801728

110 75603 3

in in inin ft

pcf lbs. . . .

/.

× ×× =

Weight of ½ Floor Steel Deck:

300 180144

1 3752 2

in inin ft

psf lbs. .

/.

×× =

Total System Weight:

W = 805 + 894.3 + 7560 + 375 = 9,630 lbs., or 9.63 kips

106

Calculate fn of Girder Tee-Beam

fgEIWL

f

neff

n

=

=× ×

×

=

157

157386 4 29000 719 2

9 63 300

3

3

.

.. .

.

f 8.74 Hzn .

CALCULATION OF SYSTEM FREQUENCY

The two values of fn obtained for the joist tee-beam and the girder tee-beam are combinedusing Dunkerly’s relationship.

1 1 1

1 16 28

18 74

2 2 2

2 2 2

f f f

f

s j g

s

= +

= +. .

fs = 5.10 Hz.

107

Timothy Allan Beavers

Timothy Allan Beavers was born July 13, 1974 in Hampton, Virginia. He soon

moved to Newport News, Virginia, where he spent his childhood. He graduated from

Ferguson High School in June, 1992. In May 1996 he received his Bachelor of Science

degree in Civil Engineering from Virginia Polytechnic Institute and State University. In

August 1996, he entered the graduate program in the Structures Division of Civil

Engineering at the same university.

Documents

LIST OF REFERENCES Wilson, E. L. and Habibullah, A. (1992). “SAP90--A Series of Computer Programs for the Finite Element Analysis of Structures—Structural Analysis User’s Manual,”