design example for beams with web openings

8/10/2019 Design Example for Beams With Web Openings

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Design Example forBeam s wi th W eb Openings

R I C H A R D L . K U S S M A N A N D P E T E R B . C O O P E R

D U E

T O the increasing cost of energy an d the d i f f icuhy of

obtain ing raw mater ials , economy has a h igh pr ior i ty in al l

aspects of design. In the design of multistory steel buildings,

sav ings can be real ized by passing ductwork for heat ing ,

ventilation, and air conditioning systems

through

steel floor

beams, rather than under them. Not on ly does th is p ract ice

save in the overall height of the structure, along with all the

related benefits in material savings, but i t also saves in the

cost of heating and air conditioning by enclosing less volume

to be heated or cooled . Eccentr ic openings are of special

in terest because not al l f loor beams and g i rders are of the

same depth at any g iven level . I t i s desi rab le to keep the

duct wo rk on a relat ively level p lane to cu t down on the cost

of fabricating bends. It is doubtful that i t costs any more to

fabr icate an eccentr ic opening in a s teel beam than i t does

to fabr icate a concentr ic opening . Thus, even more sav ings

can be real ized by using the eccentr ic opening . Th e addi t ion

of reinfo rcem ent i s also som et ime s desi rab le so tha t a

heavier sect ion i s no t required because of the opening .

Des ig n fo rm u las h av e b een d ev e lo p ed fo r b eam s wi th

co n cen t r i c an d eccen t r i c web o p en in g s , b o th u n re in fo rced

and reinforced;^ the purpose of this paper is to i l lustrate the

appl icat ion of these formulas in a typ ical design prob

lem.

P R O B L E M S T A T E M E N T

A port ion of the f loor system support ing a concourse in a

mu l t i s to ry bui ld ing is show n in Fig . 1 . T h e f loor system

co n s i st s of g i rd er s sp an n in g f ro m co lu m n to co lu m n , wi th

f loor beams supported by the g i rders . The f loor beams and

girders supp ort a 4 - in . concrete slab , which in tu rn provides

cont inuous lateral support to the top f langes of the f loor

m em b ers . Mo m en t co n n ec t io n s a re p ro v id ed b e tween th e

columns and g i rders ; therefore, the g i rder ends are assumed

to be f ixed . I t i s fur ther assumed that the co lumns are W14

sect ions and that the g i rder span is taken f rom column face

to co lumn face. The f loor beams are at tached to the g i rders

wi th sh ear co n n ec t io n s ; s im p le su p p o r t s a r e t h e re fo re as

su m ed . Fo r a rch i t ec tu ra l r easo n s t h e f l o o r b eam s a re l im

i ted to a 21- in . dep th .

Th e h ea t i n g , v en t i l a t i o n , an d a i r co n d i t i o n in g (HVAC)

ducts run paral lel to the g i rders , wi th serv ice ducts

Richard L. Kussman is Graduate Research Assistant, Dept. of

Civil Engineering, Kansas State University, Manhattan, Kans.

Peter B. Cooper is Professor of Civil Engineering, Dept. of Civil

Engineering, Kansas State University, Manhattan, Kans.

b ran ch in g o u t a t r i g h t an g l es . Fo r b o th a rch i t ec tu ra l an d

aesthet ic reasons i t i s undesi rab le to run the H V A C system

b elo w th e f lo o r m em b ers , so t h ey m u s t p e n e t r a t e t h em . I t

i s a lso desi rab le to keep the H V A C system on a level p lane,

thereby reducing instal lat ion costs by decreasing the

number of bends in the duct mater ial . I t i s necessary to

provide a duct area of 144 sq. in. with l-in.-thick insulation

on al l s ides . Ver t ical posi t ion ing of the ducts used in th is

ex am p le a re sh o wn in F ig . 2 . Th e co rn er r ad iu s was d e

termined f rom recommendat ions for members subjected to

fa t i g u e l o ad in g s .^ Wh i l e s t ru c tu res d es ig n ed b y p l as t i c

methods are no t subject to fat igue s i tuat ions, these gu ide

l ines were used to provide a reasonable basis for deter

mining the corner rad i i o f web openings. I t i s a lso possib le

that a s l igh t ly smal ler opening could have been used to

accommodate the duct . The use of a smal ler opening might ,

however , cause problems in the instal lat ion of the duct

i n su l a t i o n a n d t h e reb y i n c rease co s ts . A l i b e ra l c l ea ran ce

was t h e re fo re p ro v id ed i n t h i s ex am p le .

T h e f loor system is to be designe d to carry a l ive load of

100 psf^ and a dead load of 80 psf (50 psf for the concrete

slab and 30 psf for o ther dead loads) . Using A36 s teel and

the AISC Speci f icat ion"^ for p last ic design , along wi th

curren t research resu l ts , the locat ions where openings can

be p laced along the length of the f loor beams and g i rders

wi l l b e ex p lo red wi th t h e o p en in g r e in fo rcem en t v a ry in g

as fo l lows: (1) no opening reinforcement i s p rovided , (2)

GIRDER-

B E A M S -

G I R D E R z J

BEAMS

k-

3 5 ' - 0

ink

1 ^

d

II

cJ

rol

Ui

Fig.

7.

Plan view of loor system

48

ENGINEERING JOURNAL / AMERICAN INSTITUTE OF STEEL CONSTRUCTION


2/9

s

INSULATION

DUCT

t

W 2 I X 8 2

S

(a) Floor beam elevation at opening

i ^ l N S U L A T I O N

DUCT

W 2 7 X 8 4

r?

Figure 2

(b) Girder elevation at opening

enough reinforcement is supplied to resist the maximum

shear force in the beam, and (3) the minimum reinforce

ment requ ired to develop the shea r strength of the section

is furnished.

FLOOR BEAMS

Select ion of Beam Sect ion

Uniformly Distributed Load:

w = (1.7)(12)(0.10 + 0.08) = 3.67 kips/ft

Design Moment:

M = (3.67)(35)V8 = 562 kip-ft

Required Plastic Section Modulus:

Z = (562)(12)/36 = 187in.3

Try W21 X 82 (Z = 192 in.^ > 187

in.^):

Design Shear:

V = (3.67)(35)/2 = 64.2 kips

Plastic Shear Force:

Vp = (0.55)(36)(20.86)(0.499)

= 206 kips > 64.2 kips

Use W21 X 82

Proper t ies for Inves t igat ing Local S t rength

at the Opening

Opening Parameters:

h = 6

in.;

a

= 9.5 in.;

e = 0

Cross Section Properties:

Af

= (8.962)(0.795) = 7.12in.2

A^ = (20.86 )(0.499 ) = 10.4 in.2

Reference Values:

M p= (192 )(36 )/12 = 576 kip-ft

Vp =

206 kips (see above)

Calcula t ion of In ternal Beam ForcesThe in ternal

beam forces are described by the two expressions below,

based on Fig. 3. These forces are plotted in Fig. 4 along

with the interaction diagrams for the floor beams.

V

V,

M

M .

64.2 - 3.67A:

206

_ 64.2^ - 1.84A:^

576

Opening Locat ions for A^ . = 0Calculate the interaction

diagram coordinates, using formulas presented in the

Appendix, with

Ar = e = 0:

\ 1 6 /

V 9.5 / V 20. 86/

10.4 [/ ^ 12 \ 2 1 1i /2

1 +

104 n _ / 1 yl

7 .1 2 [ 4 \ 2 0 . 8 6 / J

1 +

10.4

= 0.911

(4)(7.12)

/M_

-

1 - 0.288

1 +

10.4

= 0.522

(4)(7.12)

\VpJ\ LV2 20.86 / VlO.4/ ^ J

= 0.158

49

SECOND QUARTER / 1976


3/9

Vm,

3. 67 k/ f t

\l /

\L \b \U s i/ \ 1 / \ ] / \ 1 / \ ] /

35'

77m,

64.2 k

- 6 4 . 2 k

Fig. 3. Floor beam loading, shear, and moment diagrams

NO OPENI NGS ALLOWED

17.5'

(a) Ar = 0

nz] nu

T

t

10.5

10.2'

10.5

'-1

zz\

^

(b)Ar= 1.26 in.^

16,6;

i

1

y\

{c)Ar = 2.81 in.^

Fig. 5. Permissible opening locations for floor beams

LU

0 . 9

0 . 8

0.7

0 . 6

Mp

0.4

0.3

0.2

0.1

I NTERNAL BEAM FORCES

- \ \ \ \

\ \ \ \

\ \ \ \

-

-

Ar =

0 2 ^

-

1

1 i \

^ A r = 1 . 2

1

^ A r

>

in.^

1

2.81 in.2

1

0.1 0.2 0.3 0.4 0.5

V

Fig. 4. Floor beam interaction diagrams

0.6

Th e ab o v e co o rd in a t es a r e p lo t t ed t o fo rm th e ap p ro x i

m ate i n t e rac t i o n d i ag ram sh o wn in F ig . 4 fo r

A^ =

0. As

sh o w n sch em at i ca l l y i n F ig . 5 an d g rap h ica l l y i n F ig . 4 ,

t h e re a re n o p o s i t i o n s a lo n g t h e b eam wh ere t h e o p en in g

can be p laced for A^ .

=

0 . This i s because al l the po in ts on

the internal beam force curve lie outside the failure envelope

fo r

Ar

= 0.

O p e n i n g L o c a t i o n s f o r

A^.

L a r g e E n o u g h t o A c c o m

m o d a t e M a x i m u m B e a m S h e a r T h e f o l l o w i n g q u a

d ra t i c eq u a t io n i n

Ar

i s o b t a in ed f ro m th e g en era l i zed

equat ion for

{V/Vp)\

p resen t ed i n t h e Ap p en d ix , wi th

Ar

\^r Jmin

AAr^ -{- BAr-^ C=0

w h e r e

A =

16a

AJ{\ +a)

( l + a M ^ LV d) \Vp) J

\~~d) Vl-Va) ~ \V~

1 /2

Ca lcu lat ion of v4 :

VmaJVp

= 0 .312

(1 6 ) (0 .1 6 3 )

A =

(1 0 .4 )2 (1 .1 6 3 )

= 0 . 0 2 0 7

50



4/9

( 8) (0 .1 63 ) [ -/ 12 \2 , , . _ ] i / 2

^ (1.163X10.4) L O ^ -(0.312)2J

= 0.0311

V 20.86/ 1.163)

0721

Ar =

-B

VB^

- 4AC

2A

-0 .0311 \/( 0.031 1)2 - (4)(0.020 7)(-0.072 T)

(2)(0.0207)

= 1.26 in.2

Us e l-Bar 3 X Vigabove and bel ow the opening on

one side only.

2 br/tr = 6.9 < 8.5)

Ar= 1.31 in. 2> 1.26 in.

Calculation of interaction diagram coordinates using for

mulas given in the Appendix with A^ = 1.26 in.2,


5/9

Properties for Investigating Local Strength

at the Opening

Opening Parameters :

/z = 6 in.; a = 9.5 in.; ^ = 3 in.

Cross Section Properties:

Af = (9.963)(0.636) = 6.34 in.2

A^, = (0.463)(26.69) = 12.4 in.2

Reference Values:

M p = (244)(36)/12 = 732 kip-ft

Vp

= 245 kips (see above)

Calculation of Internal Beam ForcesThe in te rna l

beam forces form a discontinuous curve described by the

expres sions below^, based on Fig . 6, w ith the app rop riat e

limits. This curve is shown in Fig. 7.

128.4 k 128.4 k

M

V

128

245

= 0.522

- 7 3 0 + 128 x1

732

Limits: 0 < x < 11.4 ft

M

M .

= 0

= 0.997

]Limits: 11.4ft < x < 17.4ft

wrherexis taken to be positive from the column face toward

midspan.

Opening Locations for

A^.=

0

Calculation of interac

tion diagram coordinates using formulas presented in the

Appendix with

Ar =

0^

e = 3

in.:

3 /26 .69 \2 /

12

26.69 26.69

r

157

0T =

13B

=

3 /2_6.69\2

12.4

(2)(6.34)

12.4

12 6

26.69 26

6 _ \ 2 _

.69/

0.889

[ / , 12 + 6\ 2 1 11/2

( 1 ) = 0 . 2 9 6

LV 26 .6 9/ 1.157 J

f / i 1 2 - 6 \ 2 1 11/2

2) 6.34)LV 26 .6 9/ 1.889 J ~

t +

12.41 -1 (6)2 + (2)( 6)( 3)]

/ M\ 6.34L4 (26.69)2 J

VMJO

/_M

r

1 +

1 - 0 . 5 5 2

1 +

12.4

12.4

(4)(6.34)

= 0.301

= 0.867

(4)(6.34)

11.4

12'

11.4'

128.4 k

-128.4 k

- 7 7 0 k f t

Fig. 6. Girder loading, shear, and mom ent diagrams

Fig. 7. Girder interaction diagrams

52

ENGINEERING JOURNAL / AMERICAN INSTITUTE O F STEEL CONSTRUCTION


6/9


7/9


8/9

N O M E N C L A T U R E

R E F E R E N C E S

Ar

\-^r)min

Ayj

M

Mp

\M/Mpl

{M/Mp)o,

{M/Mp),

P

V

I

V/Vpl

(y/Vph

(Vs/Vph

(Vr/Vph

br

c

d

e

h

U

w

X

Area of one f lange

{b j

X

tj)

Are a of r e in fo rcem en t ab o v e o p en in g ,

also area of reinforcement below opening

M i n i m u m a r e a of r e i n fo r c e m e n t r e

qui red to reach shear capaci ty of sect ion

Area o f web

(d X t^)

Mo m en t a t cen t e r l i n e o f o p en in g

Plast ic moment of sect ion

Ab so lu t e v a lu e o f m o m en t t o p l as t i c m o

m en t r a t i o a lo n g b eam s an d g i rd er s

Co o rd in a t es o f p o in t s o n ap

p ro x im ate i n t e rac t i o n d i ag ram

Co n cen t r a t ed l o ad

Shear force at cen ter l ine of opening

Plast ic shear force of sect ion

Absolu te value of shear force to p last ic

sh ear fo rce a lo n g b eam s an d g i rd er s

Co o rd in a t e o f p o in t o n ap p ro x im ate i n

t e r ac t i o n d i ag ram

Rat io of shear force in bo t tom tee sect ion

(VB)

to plas tic she ar force for poin t 1 on

a p p r o x i m a t e i n t e r a c t i o n d i a g r a m

Ratio of shear force in top tee section

(

VT)

to pla stic she ar force for point 1 on

a p p r o x i m a t e i n t e r a c t i o n d i a g r a m

Plast ic sect ion modulus

On e-h a l f o p en in g l en g th

F l a n g e w i d t h

Wid th o f r e in fo rcem en t

W eb th i ck n ess p lu s wid th of r e in fo rce

m e n t

(br ~^ t^)

Beam d ep th

Eccen t r i c i t y (d i s t an ce b e tween m id -

depth of sect ion and mid-depth of open

ing)

On e-h a l f o p en in g d ep th

F lan g e t h i ck n ess

Th ick n ess o f r e in fo rcem en t

Web th i ck n ess

Distance f rom edge of opening to face of

r e in fo rcem en t

Un i fo rm ly d i s t r i b u t ed l o ad

Coordinate def in ing posi t ions along

b eam s an d g i rd er s

Coeff icien ts used in ap pro xim ate design

fo rm u las wi th su b scr ip t s

T

a n d

B

to de

note top and bot tom tee sect ions, respec

tively

1.

Wang, T. M., R . R. Snell, P. B. Cooper

S t r ength of Beams

w i th Eccent r i c Reinforced H oles Journal of the Structural Di

vision, ASCE, Vol. 101, No. ST9, Proc. Paper 11540, Sept.

1975.

2. Subcommittee on Beams w ith W eb Openings of the Structural

Division

Sugges ted D es ign G uides for Beams w i th Web H oles

Journal of the Structural Division, ASC E, Vol. 97, No. ST11,

Proc. Paper 8536, Nov. 1971.

3.

U ni form Bui ld ing Code

1973 Edition, International Confer

ence of Building Officials, W hittier,

Calif.

1973.

4. M an ua l of S tee l Cons t ruc t ion

7th Edition, American Institute

of Steel Construction, New York, 1970.

5.

Redwood, R. G.

Tables fo r P las t i c D es ign of Beams w i th Rec

t a n g u l a r H o l e s

Engineering Journal, AISC , Vol. 9, No. 1, Jan.

1972.

A P P E N D I X

A P P R O X I M A T E I N T E R A C T I O N D I A G R A M

E Q U A T I O N S

T h e eq u a t io n s p resen t e d b e lo w a re u sed t o co n s t ru c t an

ap p ro x im ate i n t e rac t i o n d i ag ram s im i l a r t o F ig . 9 .^

P o i n t 0 o n t h e d i a g r a m h a s t h e c o o r d i n a t e s [ 0 , ( M / M ^ ) -

o] ,

P o in t 1 is d e s c ri b e d b y [ ( F / F ^ ) i , ( M / M p ) i ] , a n d t h e

poin t on the

V/Vp-axis

is

(V/Vp)\

f rom the or ig in .

These equat ions are for the general case of a reinforced

eccen t r i c web o p en in g . To o b t a in eq u a t io n s fo r o th er ,

less general cases ,

Ar, e,

or bo th

Ar

a n d

e,

a re t ak en

eq u a l t o ze ro as ap p ro p r i a t e .

F o r

e < w.

1 + ^

V M J O

Ar (Ih

2h \ ,4w/l__ h^

+

2he\

\d) Af\A d^ )

1 +

AAf

F o r

u


9/9

Figure 9

F o r

Ar < {Ar)min

4Af

\VpJ\ \VpJ\ \Vp)\

F or

Ar > (Ar)

; )

A.

1 -

\Mp/i

_

A

/M_\ ^ Af^

where the

A^

value is {Ar)r)

1 +

4Af

2h

d

Vp)\

^ 1 + a r V^ / / 2^ /

r / 2/? + 2g \2 1 16 a r

/ ^ r y ] '

LV d / 1 + a r (1 + a r ) H ^ J / J

)

I "/ 2 A - 2 e \ 2 1 1 6a g (^rVA

V\ d

/ 1 + B ( H - a B ) 2 U / J

^

1 + aB\Af/

2Aj

2h-2eV

1

/ 2

3 /a f \2 / 2 / i 2e \

2A 2\2

^

d)

56


design example for beams with web openings

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