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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
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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
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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
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( 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,
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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
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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
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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)
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