application of aisc design provisions for tapered members

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A p p l i c a t i o n of A I S C D e s i g n P r o v i s i o n s fo r T a p e r e d M e m b e r s

G. C. LEE AND M. L. MORRELL

T H E U S E O F t a p e r e d s t r u c t u r a l m e m b e r s h a v i n g t a p e r e d

d ep th s an d /o r wid th s was f i r s t p ro p o sed b y Am i r ik i an ^

fo r r easo n s of eco n o m y . Ty p ica l l y , t ap ered m em b ers a re

used in s ing le s tory s t ructures of one or more bays (Figs .

l a an d l b ) an d i n can t i l ev ered sec t i o n s o f b u i ld in g s

(F ig . I c ) . So m e lo w- r i se s t ru c tu res a l so co n t a in t ap ered

m em b ers (F ig . Id ) . In v i ew o f t h e l ack o f b as i c u n d er

s t an d in g o f t h e b eh av io r o f t ap ere d m em b ers , a s we l l a s

d e s i g n c r i t e r ia , a j o i n t W R C - C R C s u b c o m m i t t e e w a s

es t ab l i sh ed i n 1 9 6 6 to co o rd in a t e r esea rch s tu d i es . In

a d d i t i o n , a s u b c o m m i t t e e of t h e M B M A T e c h n i c a l

Co m m i t t e e was es t ab l i sh ed i n 19 7 3 to f ac i l i t a t e t h e d e

v e lo p m en t o f d es ig n g u id es fo r t ap e red m e m b ers . T w o

p rev io u s pap ers^ -^ co n t a in in g d es ig n r ec o m m e n d a t io n s

were a r esu l t o f t h e t ech n i ca l g u id an ce o f t h ese su b

co m m i t t ee s . Th e m ajo r t h ru s t of t h ese p ap e r s d ea l w i th

th e fo rm u la t i o n o f a l l o wab le ax i a l s tr es s , a l l o wab le

b en d in g s t res s , an d co m b in e d ax i a l - fl ex u ra l co m p ress io n

f or t a p e r e d m e m b e r s . T h e s e d e s ig n r e c o m m e n d a t i o n s

h a v e b e e n i n c o r p o r a t e d i n t o t h e A I S C S p e c if i c at io n a s

A p p e n d i x D . ^

T h e p h i lo so p h y b eh in d t h e d ev e lo p m en t of t h e a l l o w

ab l e s t r es s fo rm u las fo r t ap ered m e m b ers i s t o u se t h e

fo rm o f t h e p r i sm a t i c fo rm u las i n A IS C S p ec i f i ca ti o n

S e c t s . 1 .5 .1 .3 , 1.5.1.4, and 1 .6 .1 and include a length

m o d i f i ca t i o n f ac to r . Th e ap p ro p r i a t e f ac to r s a r e su m

m a r i ze d i n Ref . 2 for ax i a l b u ck l in g an d l a t e ra l b u ck

l i n g . T h e y were d e t e rm in ed b ased o n t h e fo l l o win g ap

p r o a ch : fi rs t, t h e Ra y le ig h -R i t z Me th o d was u sed t o

d e t e rm i n e t h e c r i t ica l st r es s o f a t ap er ed m em b e r ; t h en ,

th at s t ress was equ ate d to the cr i t ical s t ress for a pr ism at ic

m e m b er h a v in g t h e sam e c ro ss sec t i o n as t h e sm al l e r en d

o f t h e t ap ere d m e m b er , b u t of d i f fe ren t l en g th . Th ese

m o d i f i ca t i o n f ac to r s a r e fu n c t i o n s o f t h e t ap er i n g r a t i o sa n d o t h e r g e o m e t r i c a l p r o p e r t i e s of t h e m e m b e r . T h e

d es ig n fo rm u las o f A IS C Sp ec i f i ca t i o n Ap p e n d ix D a re

f or li n e a r l y w e b - t a p e r e d m e m b e r s o n l y .

G. C. Lee is Professor and Chairman, Dept. of Civil Engineering,

State University of Nevo York at Buffalo.M. L. Morrell is Asst. Professor, Dept. of Civil Engineering, State

University of New York at Buffalo.

Th e fo rm s o f t h e a l l o wab le co m p ress iv e s t r es s fo rm u

l as for t ap ere d m em b ers i n Sec t . D 2 * a re i d en t i ca l t o

t h o s e f or A I S C p r i s m a t i c m e m b e r d e s i g n. T h e t a p e r i n g

inf luence i s inc orp ora ted in the effective leng th factor .

S in ce t h e fo rm u las a re i n t en d ed t o b e u sed fo r web -

t ap ered m em b ers , t h e weak -ax i s e f f ec t i v e l en g th f ac to r

can b e t ak en t h e sam e as t h a t for a p r i sm at i c m e m b er .

Fo r stro ng- axis buc kli ng the ef f̂ ective len gt h fact or differs

a p p r e c i a b l y f r o m a p r i s m a t i c m e m b e r . T h e s t r o n g - a x i s

e f f ec t i v e l en g th f ac to r s fo r t ap ered m em b ers , K^ , h a v e

b een ca l cu l a t ed fo r t y p i ca l t ap ered f r am in g s i t u a t i o n s^

a n d a r e i n c lu d e d i n t h e A d d e n d a t o t h e C o m m e n t a r y o n

th e A IS C Sp ec i f i ca t i o n ( see S u p p lem en t No . 3 ) .*

Th e a l l o wab le b e n d in g st r es s fo rm u las for t ap ered

me mb ers in Sec t . D 3 d if f'er in th re e aspects f rom the A IS C

fo rm u las fo r p r i sm at i c m em b er d es ig n . F i r s t , t h e l en g th

m o d i f i ca t i o n f ac to r s , hy , a n d hs , ap p ea r in t h e fo rm u las

wh ich acco u n t fo r t h e t ap er in g . Th ese fu n c t i o n s were d e

t e rm in ed i n a s im i l a r way as for ax i a l co m p ress io n .^ Th e

ad o p t io n o f t h ese two fu n c t io n s was d i c t a t ed b y t h e co m

p l i ca t ed d ep e n d e n ce o f t h e fu n c t i o n o n t h e wa rp in g an dS t . Ven an t r i g id i t i e s , an d Sy t h e u se o f t h e d o u b le

fo rm u la p ro ced u re fo r c r i t i ca l l a t e r a l b u ck l in g s t r es s .

Seco n d , t h e u se o f t h e t o t a l r es i s t an ce fo rm u la t i o n ^ t o

d e t e rm in e t h e a l l o wab le b en d in g st r ess . Th i rd , co n s id er

a t i o n o f t h e r es t r a in in g e f fec t o f ad j ac en t sp an s wh e n a

m e m b e r i s c o n t i n u o u s o v e r l a t e r a l s u p p o r t s . T h e r e

s t ra in ing effect i s incl ude d in the coefficien t B.* *

C o m b i n e d a x i a l c o m p r e s s io n a n d b e n d i n g fo r t a p e r e d

m e m b ers is h an d led i n t h e sam e m a n n er as for p r i sm at i c

m e m b ers . Th i s case is co v ered i n Sec t . D 4 .

A IS C Sp ec i f i ca t i o n d es ig n p ro v i s io n s ap p l i ca b l e t o

t ap e red m e m b ers can b e g ro u p ed in to t h ree ca t eg o r i es

( see Ta b le 1 ) :

A. P r i sm at i c m e m b er p ro v i s io n s t h a t m a y b e d i r ec t l y

a p p l i e d t o t a p e r e d m e m b e r s , provided maximum

stresses are determined at the correct locations (usually

corresponding to the smallest net cross-sectional area).

* Sections and equations with the prefix "Z)" refer to Appendix D of

the AISC Specification {Supplement No. 3).

** In Ref. 3 the restraint effect is called Ry .

F I R S T Q U A R T E R / 1 9 7 5



^^^^4k^^^^^^^^^^^^^^^^^^^^^^^^^.^^^^^^^^^^^^^^^^^^^^^^^^

^^^^^^^^ ^^^^^^^v^^^^^^v^l^^ vvv\v\vtk>vv,^vvkvv\vvv^v\tv w

v\k>f.vvv\\vvvv\\\v\vt\vs^\\vv\\vvv>\^it>\vxv\\^^^

{b) Multi-bay single story tapered frames

(a) Multiple tapered beam gable frames

X

w(c) Tapered beam used in cantilevered situations

rO l

{d) Use of tapered members in low-rise frames

Figure 1

E N G I N E E R I N G J 0 U R N A L / A M E R I C A N I N S T I T U T E O F S T E E L C O N S T R U C T I O N



B . Pr i sm at i c m em b er p ro v i s io n s t h a t h av e b een

m o d i f i ed in A p p e n d i x D for ap p Hca t io n to t a p

e r e d m e m b e r s .

C . P r i sm at i c m em b er p ro v i s io n s t h a t h av e no t b e e n

a d e q u a t e l y i n v e s t i g a t e d w i t h r e g a r d to t a p e r e d

m e m b e r s . Caution should be exercised in the applica

tion of these provisions to the design of tapered mem

bers.

T a b l e 1

S e c t . 1.5

1.5.1.1

1.5.1.2

1.5.1.3

1.5.1.4

1.5.1.5

S e c t . 1.6

1.6.1

1.6.2

1.6.3S e c t . 1.8

1.8.2

1.8.3

1.8.4

S e c t . 1.9

1.9.1

1.9.2

S e c t . 1.10

S e c t . 1.11

A I S C S p e c i f i c a t i o n S e c t i o n

A l l o w a b l e S t r e s s e s

T e n s i o n

S h e a r

C o m p r e s s i o n

B e n d i n g

B e a r i n g

C o m b i n e d S t r e s s e s

A x i a l C o m p r e s s i o n an d B e n d i n g

A x i a l T e n s i o n and B e n d i n g

S h e a r an d T e n s i o nS t a b i l i t y an d S l e n d e r n e s s R a t i o s

S i d e s w a y P r e v e n t e d

S i d e s w a y No t P r e v e n t e d

M a x i m u m R a t i o s

W i d t h - T h i c k n e s s R a t i o s

U n s ti fi 'e n e d E l e m e n t s U n d e r C o m

pres s ion

S t i ff e ne d E l e m e n t s U n d e r C o m p r e s

s ion

P l a t e G i r d e r s an d R o l l e d B e a m s

C o m p o s i t e C o n s t r u c t i o n

C a t e g o r y

A

A

B

B

A

B

A

A

B

B

C

A

C *

C

C

Fo r web tapered elements only.

FRAME AND STRESS ANALYSIS

Th e in -p l an e an a ly s i s of cer t a in t y p es of s ing le-s tory

t ap ered f r am e assem b l i es has b e e n s h o w n to be s i m p l e r

t h a n the analysis of co n v en t io n a l p r i sm at i c r i g id f r am es .^

A d v a n t a g e is t a k e n of the f a c t t h a t e a c h t a p e r e d m e m b e r

in the f r a m e (see Fig. la) has a p i n n e d end c o n d i t i o n at

i t s smal ler end and a fixed end c o n d i t i o n at its l a r g e r

e n d , i.e., the b e n d i n g m o m e n t d i a g r a m is s im i l a r to t h a t

of a can t i l ev ere d b ea m . Th i s p e rm i t s d e f l ec ti o n co m p u ta

t ions of s im p le f r am es by s u m m i n g the def lect ions of e a c h

c o m p o n e n t m e m b e r . T h i s a p p r o a c h was v e r y a p p e a l i n g

in the ear ly 1950 's before the d e v e l o p m e n t of d ig i t a l

c o m p u t a t i o n a l c a p a b i l i t i e s .O t h e r s t a n d a r d m e t h o d s of s t ru c tu ra l an a ly s i s b ased

o n e i t h e r the flexibility or s t i f fness approach , such as

m o m e n t d i s t r i b u t i o n , can be r e a d i l y a d a p t e d to t a p e r e d

frame assemblies (stiffness and car ry -o v er f ac to r s for

t a p e r e d b e a m - c o l u m n s are g iv en in the A p p e n d i x to th is

p a p e r ) . For h i g h l y r e d u n d a n t f r a m e s , h o w e v e r , a m o r e

a u t o m a t e d a p p r o a c h is d e s i r a b l e , and the f in i te element

( d i s p l a c e m e n t ) m e t h o d has d i s t i n c t a d v a n t a g e s .

Fo r t ap ered m em b er f r am es f r ee f ro m to r s io n , the ele

ment s t i f fness matr ix (based on f i rs t o rder theory) ma y

b e fo u n d in s t a n d a r d m a t r i x m e t h o d b o o k s d e a l i n g w i t h

s t ru c tu ra l an a ly s i s . For p r o b l e m s i n v o l v i n g w a r p i n g

to r s io n an d /o r seco n d -o rd er e f f ec t s , the a p p r o p r i a t e

e l e m e n t a l m a t r i c e s are c o n t a i n e d in Ref. 5 w i t h s o m e

e x a m p l e s . The m em b er s t i f fn ess m at r i x for p r i s m a t i c

m e m b e r s can be sat i sfactor i ly used for t a p e r e d m e m b e r s

if the t a p e r e d m e m b e r is d i sc re t i zed by a suflficient n u m

b e r of e l e m e n t s .

T h e c o m p u t a t i o n of stresses in t a p e r e d m e m b e r s is

c o m p l i c a t e d by the f ac t t h a t at l eas t one new p a r a m e t e r

m u s t be i n t r o d u c e d i n t o the so lu t i o n to d escr ib e th e v a r i a

t ion of cro ss - sec t i o n a l p ro p er t i e s a lo n g the l e n g t h of the

m e m b e r . For m e m b e r s w i t h s m a l l a n g l e s of taper , less

t h a n 15°, the n o r m a l l y a s s u m e d a p p r o x i m a t i o n s of

s t ru c tu ra l m ech an ics s t i l l can be a p p l i e d , w i t h the e x c e p

t i o n t h a t the cross-sect ional p roper t ies vary .^ The de

s ig n er m u s t r ea l i ze t h a t , s i n ce in g en era l b o th s t r es s

resu l t an t s an d cro ss - sec ti o n a l p ro p e r t i e s do v a r y , it is not

a lway s p o ss ib l eto

findthe

p o in t sof

m a x i m u m s t r e s sby

i n sp ec t i o n . By c a l c u l a t i n g the stress at sev era l p o in t s an d

p lo t t i n g the s t r es s v a r i a t i o n , the m a x i m u m s t r e s s can be

d e t e r m i n e d . A l s o , the e q u a t i o n for the s t r es s v a r i a t i o n

c a n be d i f f e ren t i a t ed to find the m a x i m u m s t r e s s .

DESIGN EXAMPLES

T h e p r i m a r y i n t e n t of the p a p e r is to give i l lust rat ions of

t h e ap p l i ca t i o n of A IS C Sp ec i f i ca t io n d es ig n p ro v i s io n s

f o r t a p e r e d m e m b e r s , w i t h p a r t i c u l a r e m p h a s i s on t h o se

p ro v i s io n s t h a t are modif ied by A p p e n d i x D ( c a t e g o r y B

o f T a b l e 1.) S o m e a t t e n t i o n is g iv en to p ro v i s io n s in

C a t e g o r y A. S e v e r a l e x a m p l e s are p r e s e n t e d for t h i s p u r

p o s e , a t t e m p t i n g to i l l u s t r a t e all of the o p t i o n s in S e c t s .D 3 ( B e n d i n g ) and D4 ( C o m b i n e d S t r e s s e s ) of A p p e n

d ix D.

A sp ec i a l n o t e sh o u ld be m a d e w i t h r e g a r d to the

l a t e r a l su p p o r t s . In o r d e r to a p p l y the d es ig n fo rm u las ,

it is r e q u i r e d t h a t t h e r e be no l a t e r a l m o v e m e n t and no

twi s t i n g of the cross sect ion at the p o in t s of l a t e r a l su p

p o r t . Gi r t s an d p u r l i n s are u sed to a t t ach s id in g an d

roof ing to the f r am e; t h ese su p p ly su p p o r t s o n ly to one

flange, e.g., the o u t s id e f lang e. U n d er n o rm al s i t u a t i o n s ,

b rac in g o n ly one flange (when in co m p ress io n ) wi l l be

sufficient to m e e t t h i s r e q u i r e m e n t . If, h o wev er , t h i s

f lange braced by a g i r t or p u r l i n is in t en s io n , t h en an

a d d i t i o n a l s u p p o r t m u s t be su p p l i ed to the o t h e r ( c o m

pression) f lange, ind icated by an '' X" in the figures.

E x a m p l e 1

Given:

D e t e r m i n e the a l l o wab le b en d in g s t r es s for s e g m e n t 2-3

in the t a p e r e d b e a m . Fig. 2, and c o m p a r e it to the

ca l cu l a t ed b en d in g s t r es s . Fy = 42 ksi.

F I R S T Q U A R T E R / 1 9 7 5



q = 0 . 1 3 k / i n .

1 i 1 i i i i i 1 1 i i 1 1 i i i i i i

"̂ T^3 4

4(a 1 0 6 . 6 = 4 2 6 . 3

' - • A

i ^

r

O j5

A

0.25 - p - 1

6 . 0

Mo m ent Dia g ra m , k - i n

S e c t . #

12

3

45

25.0in.

28.532.0

35.539.0

12.00in.12.8813.75

14.6315.50

X

1188in?16102109

26943368

X

95.1in?

112.9131.8

151.7172.7

X

9.95in.

11.1812.39

13.5714.74

y

1.23in.1.181.15

1.111.08

T

1.50in

1.47

1.45

1.431.40

Fig. 2. Design Examples 1 and 2: design of segments 2-3 and 4-5 of propped cantilever; Fy = 42 ksi, dimensions shown in inches

Solution:

Fo r com pres sion on ex t rem e f ibers of tap ere d f lexural

m e m b e r s , F o r m u l a s ( D 3 - 1 ) a n d ( D 3 - 2 ) a p p l y . T h e

fo l l o win g q u an t i t i e s a r e n eed ed t o ca l cu l a t e kg a n d h^:

72-3 = (ds - d2)/d2 = (32 .0 - 28 . 5) /2 8 .5 = 0 .12

Id2/Af = 106.6 X 28 .5 / (0 .5 X 6 .0) = 1013

/ / r^^ = 10 6 .6 /1 .47 = 72 .30

.-. hs = 1.0 + (0 .0230 X 0 .12 X V 1O I3 ) = 1 .090

h^ = 1.0 + (0 .00385 X 0 .12 X V7 2. 30 ) = 1.004

Nex t , t h e two t e rm s F^ ^ a n d F^ ^ a r e c o m p u t e d :

F,^ = 12,000/(1 .090 X 1013) = 10 .87

F^^ = 170,000/(1 .004 X 72 .30)2 = 32 .26

Before F^ ^ i n Fo r m u la (D3 -2 ) can b e ca l cu l a t e d , t h e

t e r m B, w h i c h i nc l u d e s m o m e n t g r a d i e n t a n d / o r e n d

res t r a in t e f fec ts , n eed s t o b e d e t e rm in ed f rom , su b

p arag rap h s D3 a , b , c , o r d . Case D3 a can b e u sed ,

s in ce seg m en t 2- 3 h as a sp an o f eq u a l l en g th o n each

s id e an d t h e m ax im u m m o m en t fo r t h ese t h ree seg

m en t s o ccu r s o n seg m en t 2 -3 :

M2 = 1508 k ip- in . (occurs at 152 in . )

Ml = - 3 1 8 k i p - i n .

M1/M2 = 0 *

.-. ^ = 1.0 + 0.37 (1.0 - 0) + 0.50 (0.1 2)(1 .0 - 0)

= 1.43

' See Spec. Appendix D, footnote to subparagraph D3a.




S u b s e q u e n t l y , f r om F o r m u l a ( D 3 - 2 ) :

F^ ^ = 1 .43 V (1 0 . 87 )2 + (32 .26)2 = 48 .68 ksi

Ho we v er , t h i s v a lu e i s g rea t e r t h an y^^Fy = 14.0 ksi.

T h e r e f o r e , F o r m u l a ( D 3 - 1 ) s h o u l d b e u s e d .

F o r m u l a ( D 3 - 1 ) :

2Fby — 1.0 4 2

6 X 4 8 . 6 8 .4 2

= 24.0 ksi < 0.6Fy = 25 .2 ksi

T h e m a x i m u m c o m p u t e d b e n d i n g s tr e ss f or s e g m e n t

2-3 i s taken as:

M2 1508

= J^ = 12.4 ksi < F,,

.*. segment is o.k.

E x a m p l e 2

Given:

De term in e t h e a l l o wa b le b en d in g st r es s for seg m en t

4 -5 i n t h e t ap ered b eam , F ig . 2 , an d co m p are i t t o t h e

c a l c u l a t e d b e n d i n g s t r e s s . Fy = 42 ksi.

Solution:

Fo r com pres sion on ex t rem e fibers of tap ere d f lexural

m e m b e r s . F o r m u l a s ( D 3 - 1 ) a n d ( D 3 - 2 ) a p p l y . T h e

fo l l o win g q u an t i t i e s a r e n eed ed t o ca l cu l a t e hg a n d h^^:

74-5(^5 - d,)/d4 = (39 .0 - 35 . 5) /3 5 .5 - 0 .10

Id4/Af = 106.6 X 35 .5 / (0 .5 X 6 .0) = 1261

l/rr, = 1 0 6 .6 /1 .4 3 = 7 4 .8

,;h, = 1 .0 + (0 .0230 X 0 .10 V l 2 6 1 ) = 1 .081

h^ = 1.0 + (0.00385 X 0.10 A / 7 4 . 8 ) = 1.003

Nex t , t h e two t e rm s F^ ^ a n d F^ ^ a r e c o m p u t e d :

F,^ = 12 ,000/(1 .081 X 1261) = 8 .80

F^^ = 170,000/(1 .003 X 74 .8)2 = 30 .20

Before F^ y i n Fo r m u la (D3 -2 ) can b e ca l cu l a t ed , t h e

t e r m B, w h i c h i nc l u d e s m o m e n t g r a d i e n t a n d / o r e n d

res t r a in t e f f ec t s , n eed s t o b e d e t e rm in ed f ro m su b

p arag rap h s D3 a , b , c , o r d . E i t h e r case D3 b o r D3 c

may apply , bu t f i rs t the fo l lowing s t resses must be

c a l c u l a t e d :

Mz 1265h = ^ = TTT-Z = 9.60 ksi

JH ~

M,

131.8

3 3 7 8

172.7= 19.56 ksi

S in ce t h e l a rg er s t r es s , i n m ag n i tu d e , o ccu r s a t t h e

l a rg er en d , case D3 b ap p l i es .*

5 = 1.0 + 0.58 ll.O + - ^ 1 -L 19.56J

<0.70 X 0 .10 M.C9 . 6 0 '

1 ^ 5 6

1.76


F^y = 1 .76 V (8 .8 0) 2 + (30 .20)2 = 55 .36 ksi

Ho w ev er , t h i s v a lu e i s g rea t e r t h an ^"^Fy = 14.0 ksi.

T h u s , F o r m u l a ( D 3 - 1 ) s h o u l d b e u s e d .

F o r m u l a ( D 3 - 1 ) :

_ 21.0

4 2

6 X 55 .36 .42

= 24.46 ksi < 0.6Fy = 25.2 ksi

T h e m a x i m u m c o m p u t e d b e n d i n g s tr e ss f or s e g m e n t

4-5 is

^5 = ^5A ^5 = 19 .56 < F^ y

.*. segment is o.k.

E x a m p l e 3

Given:

De term in e t h e ad e q u a cy o f t h e co lu m n 1 -2 i n F ig . 3 .

Th e co lu m n is p in n ed i n b o th weak - an d s t ro n g -

d i r ec t i o n s a t it s b ase , an d b ra ced a t t h e t o p ag a in s t

l a t e r a l m o v e m e n t . T h e r e is n o i n t e r m e d i a t e b r a c i n g .

Th e co lu m n is n o t p rev en t e d f ro m sway in g i n t h e

p l an e . Th e c o lu m n i s A3 6 s t eel (F y = 36 ksi.)

Solution:

Fi r s t , t h e fo l lo win g m a x im u m s t r esses a re co m p u te d :

Ax ia l s t r es s :

f^ = P i 2 / ^ i = 8 7 . 5 / 2 3 . 6 = 3.71 ksi

Sh ear s t r es s :

f^ = Vu/A,i = 20. 1 / (1 0 .9 X 0 .5) = 3 .69 ksi

Ben d in g s t r es s :

/ , = M21/S,,, = 324 X 12 /25 4 = 15 .30 ksi

Nex t , t h e a l l o wab le s t r es ses a re o b t a in ed :

A l l o w a b l e s h e a r s t r e s s ( A I S C S p e c . S e c t . 1.5.1.2):

F, = OAFy = 14.5 ksi > / . = 3 .69 ksi o .k .

* Th e ratio for fb^/fi,5 is positive for reverse curvature.

F I R S T Q U A R T E R / 1 9 7 5



1.875 k/ft.

t i i ^ A i - i l i i i i t ' * ^ ^ ^ ^ ^ ^ i I *—*—i—^

20.1 k^X

-15 ft. >57.5 k

60 ft.

53.5 k

-408

211

535

f

A-

1 l\

V ^oment Diagram

324

324

324

Member:

1-2

2-3

3r4

2-5

12.4

12.4

8.0

in. 24.8 in. 0.50

Tj o n -u- 1 1AW JU X il D

24.8 0.50

24.0 0.42

in. 12.1 in.

12.1

7.0

0.75 in.

0.75

0.60

Fig. 3. Design E xample 3: design of member 7-2 of a frame with tapered columns; Fy = 36 ksi

Al lo w ab le co m p ress io n s t res s (AIS C S p ec . Ap p en d ix

D , S e c t . D 2 ) :

The effect ive length factor for the weak ax is i s ob

t a in ed b y t r ea t i n g t h e weak d i r ec t i o n as p r i sm at i c .

Fo r a p in en d ed co lu m n , Ky = 1 .0 . F ig u re GD I . 5-11,

A I S C C o m m e n t a r y S e c t . D 2 , ^ c a n b e u s e d t o d e

t e r m i n e Kx '

Ti-2 = (^2 - di)/di = (24 .8 - 12 . 4) /1 2 .4 = 1 .0

GT = brh/LlT = 720 X 671/(192 X 4930) = 0 .51

Gi j = 1 0 (AIS C reco m m en d ed v a lu e for p in n e d en d )

F r o m F i g . C D l . 5 - 1 1 :

K, = K^^ 1.35

E N G I N E E R I N G J 0 U R N A L / A IVI E R I C A N I N S T I T U T E O F S T E E L C O N S T R U C T I O N



S l en d ern ess r a t i o s fo r t h e weak an d s t ro n g ax e s :

{Kl/n)y = 1.0 X 192/3 .06 = 62 .7

{Kl/r,)^ = 1 .35 X 192/5 .33 = 48 .6

Th u s , t h e weak ax i s g o v ern s .

Fa^ = 1 7 .1 7 k s i (AI S C S p ec . Ta b l e 1 -3 6)

> fa = 3.71 ksi

A l l o w a b l e b e n d i n g s tr e ss ( A I S C S p e c . A p p e n d i x D ,

S e c t . D 3 ) :

71.2 = 1.0 (see above)

Ido/Af = 192 X 12 .4 / (12 .1 X 0 .75) = 262 .4

/ / rxo = 1 92/ 3 .3 3 = 57 .64

,;h, = 1 .0 + (0 .0 23 0 X I . 0 V 2 6 2 . 4 ) = 1 .3 73

h^ = 1.0 + (0.003 85 X 1.0A/57.64) = 1.029

Fs ^ = 1 2 ,0 0 0 / (1 .3 7 3 X 2 6 2 .4 ) = 3 3 .31

F^^ = 170, 000/ (1 .02 9 X 57 .64)^ = 48 .32

B = 1.75/(1.0 + 0 .2 5 \ / lL0 ) = 1 .4 0

s in ce t h e re a re n o ad j acen t sp an s an d t h e m o m en t a t

t h e sm al l e r en d is ze ro ( see su b p a rag rap h D 2 d ) .

S u b s e q u e n t l y , f r o m F o r m u l a ( D 3 - 2 ) :

F,^ = 1.40\/(33.31)2 + (48.32)2 = 82 .16 ksi

Ho w ev er , t h i s v a lu e i s g rea t e r t h a n ^^^Fy — 12 ksi.

T h u s , Fo rm u la (D3 -1 ) sh o u ld b e u sed .

F. = 1.036

6 X 82

^]

36 = 22.2 5 ksi > 0.6F^

:,Fb^ = 0.6Fy = 22 ksi > A = 15.30 ksi

C o m b i n e d a x i a l c o m p r e s s i o n a n d b e n d i n g ( A I S C S p e c .

A p p e n d i x D , S e c t . D 4 ) :

F r o m p r e v i o u s c a l c u l a t i o n , / a / F ^ ^ = 0 . 2 2 ; t h u s , b o t h

F o r m u l a s ( D 4 - l a ) a n d ( D 4 - l b ) m u s t b e c h e c k e d .

S in ce b en d in g is ab o u t t h e s t ro n g ax i s ,

F\^ = 6 3 .2 k s i (A IS C S p ec . Ta b l e 2 )

Also , s ince the co lum n is al low ed to sway in i t s ow n p la ne,

C^ = 0.85

F i n a l l y , F o r m u l a ( D 4 - l a ) g i v e s :

3.71

17.17 +0.85

1.0 -3.71

X1 5 . 3 0 '

2 2= 0.84 < 1.0 o.k.

63.16

a n d F o r m u l a ( D 4 - l b ) g i v e s :

3.71 15.30

-22 +-22— 0.86 < 1.0 O.k.

E x a m p l e 4

Given:

De term in e t h e a l l o wab le b en d in g s tr es s fo r seg m en t

8-9 in Fig. 4 (A440 steel, Fy = 5 0 k s i ) . Fo r co m p r es

s ion on the ex t re me f ibers of tap ere d f lexural I -sh ape d

m e m b e r s , F o r m u l a s ( D 3 - 1 ) a n d ( D 3 - 2 ) a p p l y .

Solution:Th e fo l lo win g q u a n t i t i e s a r e n eed ed t o ca l cu l a t e hs

a n d hw .

78-9 = (ds — d^/d% = 0.17

Id^jAf = 63.25 X 14 .375/5 X 0 .1875 = 969 .8

l/r^^ = 6 3 .2 5 /1 .2 2 = 5 1 .8

/,h, = 1.0 + (0 .0230 X 0 . 17 \ / 96 9 . 8) = 1 .125

h^ = 1.0 + (0.00385 X OAlVsi.S) = 1.005

Nex t t h e two t e rm s Fs ^ a n d Fw y a r e c o m p u t e d :

Fsy = 1 2 ,0 0 0 / (1 .1 2 5 X 9 6 9 .8 ) = 1 1 .0 0

F^^ = 170,000/(1 .005 X 51 .8)2 = 62 .73

Before F^^ i n Fo rm u la (D3 -2 ) can b e ca l cu l a t ed , t h e

t e r m B, w h i c h in c l u d e s m o m e n t g r a d i e n t a n d / o r e n d

res t r a in t e f fec ts , n eed s t o b e d e t e rm in ed f ro m su b p ara

g rap h D3 a , b , c o r d . E i t h e r D3 b o r D3 c m ig h t ap p ly i n

th is case, bu t f i rs t the fo l lowing bending s t resses are

n eed ed a t p o in t s 7 an d 9 . Case D3 a is n o t ap p l i cab l e b e

cau se t h e t ap e r in g r a t i o i s n o t l i n ea r .

/ , , = M,/S,, = 5.68 ksi

/ ,3 = M,/S,, = 20.1 ksi

S ince the larger s t ress occurs at the smal ler end of a two-

s p a n s e g m e n t , s u b p a r a g r a p h D 3 c a p p l i e s . * T h e n ,

5 .68"

2 0 1^ = 1.0 + 0.55 1.0 +

f 2 X 0.17 X 1.05.6811

20.1 J j

= 1.66


F,^ = 1 . 6 6 A / I 1 . 0 0 2 + 62.732 = 105.7 ksi.

Ho wev er , t h i s v a lu e i s g rea t e r t h a n }yiFy = 16.7 ksi.

Th ere fo re , Fo rm u la (D3 -1 ) sh o u ld b e u sed .

F o r m u l a ( D 3 - 1 ) :

F. = 1.0 -5 0

6 X 105.7 J5 0

= 30.70 ksi > 0.6Fy

:,F. = 0.6Fy = 3 0 .0 k s i > / , , = 2 0 .1 k s i o.k.

* Th e ratio fbi/fbd i^ considered as negative when producing single

curvature.

F I R S T Q U A R T E R / 1 9 7 5



Moment Diagram

k-in

SYM.

L L o a d i n g

0.045 k/in

l U 4 I I U i ̂

0 . 1 8 7 5 i n .

t

0 . 1 5 7 i n ,

Beam :

5

6

7

8

9

Fig. 4.

5 i n .

1 0 . 8 k

H i 2 4 0 i n

7JW< 2 4 0 in . — ^

Column:

Sect.#

12345

d

8.375in.

12.375

16.375

20.375

24.375

A

3.13in?

3.76

4.39

5.02

5.64

IX

38.1in^

92.2

176.4

295.7

455.1

SX

9.1in?

14.9

21.5

29.0

37.3

rX

3.49in.

4.95

6.34

7.68

8.98

r

y

1.12in.

1.02

0.94

0.88

0.83

^T

1.31in

1.25

1.20

1.16

1.12

24.375

21.875

19.375

16.875

14.375

.6 4

.2 5

.8 6

.4 6

.0 7

4 5 5 .

3 5 0 .

2 6 2 .

1 8 9 .

1 3 0 ,

3 7 . 3

3 2 . 0

2 7 . 1

2 2 . 4

1 8 . 1

,1 7

.3 5

.5 1

.6 6

0 . 8 3

0 . 8 6

0 . 9 0

0 . 9 4

0 . 9 8

1 .1 2

1 . 1 4

1 . 1 7

1 . 2 0

1

Design Examples 4 and 5: design of segments 8-9 and 4-5 of a tapered member gable frame; Fy

22

50 ksi

E x a m p l e 5

Given:

Determ in e t h e ad eq u acy o f t h e co lu m n 1 -5 i n F ig . 4 .

Th e c o lu m n i s p in n e d a t it s b ase in b o th weak a n d

s t ro n g d i r ec t i o n s an d l a t e ra l l y b raced a t 5 - f t i n t e rv a l s .

Th e co lu m n is n o t p rev en t e d f ro m s id esway . T h e

co lum n y ield st ress i s 50 ksi . Ea ch seg me nt should be

chec ked to f ind the cr i t ica l one . After hav ing d one

th is , segment 4-5 sat i sf ied the design cr i ter ion most

c lo se ly . Th e ad eq u acy o f o n ly t h i s seg m en t i s p re

sen t ed .

Solution:

Fi r s t , t h e fo l lo win g m a x im u m s t r esses a re co m p u te d :

Axial s t ress :

fa = Pi/Ai = 1 0 . 8 / 3 . 1 3

= 3.45 ksi


A == M,/S,, = 7 0 0 / 3 7 . 3

- 18.8 ksi

ENGINEERING J 0URN AL/A M ERIC AN INSTITUTE OF STEEL CONSTRUC TION



N e x t , the al lowable s t resses are o b t a i n e d :

A l l o w a b l e c o m p r e s s i o n s tr e ss ( A I S C S p e c . A p p e n d i x D ,

S e c t . D 2 ) :

The effect ive length factor for the weak ax is i s ob tained

b y t r e a t i n g the w e a k d i r e c t i o n as p r i sm at i c ; t h e re fo re

Ky — 1.0. T h e c h a r t s in t h e A I S C C o m m e n t a r y , S e c t.

D2,^ can be u sed to d e t e r m i n e K^- S i n c e the c o l u m ni s r es t r a in ed at t h e t o p b y a t a p e r e d m e m b e r , a n e q u i v a

l e n t m o m e n t of i n e r t i a n eed s to be d e t e r m i n e d . T h e

e q u i v a l e n t m o m e n t of i n e r t i a , /g, ca n be def ined by

e q u a t i n g the stiffness coefficient of a p r i s m a t i c b e a m

w i t h a p i n n e d end (Êljt) to t h a t for a t a p e r e d

b e a m and so lv ing for 1^:

?> EAA\X ~ CAB^BA)

where ^^^ is the l e n g t h of b e a m 5-9, KÂ is the stiff

ness coefficient at A ( l a rg er en d) of a t a p e r e d b e a m

w i t h e n d B (smal ler end) f ixed , C^ B is th e car ry o v er

factor f rom A to B, an d CB A is the car ry o v er f ac to rfrom B to A. The s t i f fness and carry over factors are

c o n t a i n e d in A p p e n d i x A. The fo l l o win g q u an t i t i e s

a r e n e e d e d for th e b e a m 5-9:

75-9 = d^/d^ — 1 = 0.7

P s = 7.0 kips

P , , , = ir'ÊIJbT'' = 185.5 k ips

P/P,,, =0.04

: . K A A = 2.3 X 4EI,/b^ (F ig . A l )

CA B = 0.4 (F ig . A3 )

C^ A = 0.65 (F ig . A4 )

. - . / , = 2.3 X 4 X 130.3 (1 - 0.4 X 0 . 6 5 ) / 3

= 296 in.4

U s i n g the c h a r t s in A I S C C o m m e n t a r y S e c t. D 2 :

71-5 = ds/di - 1 = 1.91

G^ = BTII/LI, = 268 X 38.1/(240 X 296)

= 0.14

G B = 10 ( A I S C r e c o m m e n d e d v a l u e for p i n n e d e n d )

.\K, = K ̂ = 0.92

S l en d ern ess r a t i o s for t h e w e a k and s t ro n g ax es a re :

(KL/r4)y = 1.0 X 6 0 / 0 . 8 8 = 67.9

(KL/ri):c = 0.92 X 2 4 0 / 3 . 4 9 - 63.3

Th u s , t h e weak -ax i s g o v ern s .

Ea ^ = 21.3 ksi ( A I S C S p e c . T a b l e 1-50)

> fa = 3.45 ksi

A l l o w a b l e b e n d i n g s t re s s ( A I S C S p e c . A p p e n d i x D,

S e c t . D 3 ) :

Th e fo l l o win g q u an t i t i e s are n e e d e d to c a l c u l a t e kg

a n d h^:

71-5 = 1.91

Id4/Af = 60 X 2 0 . 3 7 5 / 5 X 0.1875 = 1304

l/r^^ = 6 0 / 1 . 1 6 = 51.7

.'.hs = 1.0 + (0 .0230 X 1.91\/l304) = 1.163

h^ = 1.0 + (0.00385 X 1.91 V S T J ) = 1.005

N o w ,

Fs ^ = 1 2 , 0 0 0 / ( 1 . 1 6 3 X 1304) = 7.91

F^ ^ = 1 7 0 , 0 0 0 / ( 1 . 0 0 5 X 51.7)2 = ̂ 2.97

Before Fb ^ in F o r m u l a ( D 3 - 2 ) can be c a l c u l a t e d , the

t e r m B, w h i c h i n c l u d e s m o m e n t g r a d i e n t a n d / o r end

res t rain t ef fects , needs to be d e t e r m i n e d . S i n c e s e g

m en t 4 -5 h as b o th ax i a l an d b en d in g s t r es ses . Sec t . D4wi l l h av e to be ch eck ed l a t e r . When fa/fay > 0.15 the

v a l u e of B sh a l l be u n i t y (see th e l a s t p a r a g r a p h in

S e c t . D 3 ) . F o r s e g m e n t 4 - 5 , / a / F a ^ = 0 .1 6 5 ; t h e re fo re

B = 1.0. S u b s e q u e n t l y f r om F o r m u l a ( D 3 - 2 ) :

F, ^ = 1.0^(7.91)2 + (62.97)2 = 63.46 ksi

Ho wev er , t h is v a lu e is g r e a t e r t h a n ^^'sFy = 16.7 ksi;

t h u s , Fo rm u la (D3 -1 ) ap p l i es .

= ' fl.O - '' )3 \ 6 X 6 3 . 4 6 /

50

= 28.9 ksi < 0.6P^ = 30.0 ksi

C o m b i n e d a x i a l c o m p r e s s i o n and b e n d i n g ( A I S C S p e c .

A p p e n d i x D , S e c t . D 4 ) :

F ro m p rev io u s ca l cu l a t i o n s , / a / i ây = 0 .1 6 5 ; t h u s , b o th

F o r m u l a s ( D 4 - l a ) and ( D 4 - l b ) m u s t be c h e c k e d .

S i n c e b e n d i n g is a b o u t the s t ro n g ax i s ,

F'ey = 37.3 ksi ( A I S C S p e c . T a b l e 2)

Also , s ince the c o l u m n is p e r m i t t e d to s w a y in its

p l a n e ,

Cr . = 0.85

F i n a l l y, F o r m u l a ( D 4 - l a ) g i v e s :

3.45

21.32 +0.85

/ 3.45\

V' - 37:3)

X18.8

2 8 ^= 0.77 < 1.0 o.k.


3.45 18.8 ^^ ^ , ^

1 = 0.77 < 1.0

30 28.9

o . k .

F I R S T Q U A R T E R / 1 9 7 5



0 . 0 6 k / i n .

U i i i U i i i i i i i i i i i i i i i M I i i

1 2 0 i n .

1 9 2 k -

2 4 0 i n .

0 . 5 6 3 i n .

JT

0 . 1 8 7 5 i n .

TT — 0 . 4 7 i n .

0 . 1 5 7 i n .

5 i n .

A-A5 i n .

B-B

::3 V

Column 1-2

continuous

lateral support

8.39 k

30.45 k

Center Column:

Sect.#

1

2-1

2-3

3

8.375in.

24.375

24.375

14.375

9.04in?

16.6

5.64

4.07

lOlinJ

1290

455.1

130.3

24 .1 in?

106

3 7 . 3

1 8 . 1

X

3 . 3 4 i n .

8 . 8 3

8 . 9 8

5 . 6 6

y

1.14in .

0 . 8 5

0 . 8 3

0 . 9 8

T

1.32in .

1 . 1 3

1 .1 2

1 .2 2

E x a m p l e 6

Given:

Fig. 5. Design Example 6: design of the central column of a two-bay gable frame; Fy = 42 ksi

Solution:

D e t e r m i n e t h e a d e q u a c y o f t h e c e n t r a l c o l u i n n

1-2 in the d oub le ba y gab le f rame , Fig . 5 . T he co l

u m n is a s su m e d to b e p in n ed i n t h e s t ro n g d i r e c t i o n

a t it s b ase . Th e f r am e i s n o t b rac ed ag a in s t s i d esway .

Th e we ak d i r ec t i o n h as a co n t in u o u s b race a lo n g t h e

shea r cen ter ax is . T he m ate r ia l y ield s t ress i s 42 ksi .

F i r s t t h e fo l lo win g m ax im u m s t r esses a re co m p u te d :

Axial s t ress :

f^ = P^^/AT, = 30.45/9.04 = 3.37 ks i

Sh ear s t r es s :

/ ^ = Vn/A,^ = 8.39/( 7 .25 X 0 .47) = 2 .46 ksi

10





/ ^ = M 2 i / ^ . : , = 2 0 1 4 /1 0 6 = 1 9 .0 k s i

Nex t , t h e a l l o w ab le s tr es ses a re o b t a in e d :

A l l o w a b l e s h e a r s t r e s s ( A I S C S p e c . S e c t . 1.5.1.2):

F, = OAFy = 17 ksi > / , 2.46 ksi o . k .

A l l o w a b l e c o m p r e s s i o n st re s s ( A I S C S p e c . A p p e n d i x D ,

S e c t . D 2 ) :

S in ce t h e co lu m n i s co n t in u o u s ly su p p o r t ed a lo n g t h e

w e a k - a x i s , Ky = 0 . T h e c h a r t s i n t h e A I S C C o m m e n

t a ry Sec t . D2 , can b e u sed t o d e t e rm in e Kx - N o t i n g

th a t t h e co l u m n is r es t r a in ed a t t h e t o p b y two t ap ered

b ea m s , an eq u iv a l en t m o m e n t o f i n e r t i a n eed s t o b e

d e t e rm in ed i n t h e i d en t i ca l wa y as i n Ex am p le 5 .

No te t h a t b o t h b eam s in F ig . 5 a r e t h e sam e.

72-3 = {d^ — d^/dz = 0 .7 ( for the beam)

P/Pex-i — 0 (ax ial force in beam is smal l )

F ro m th e f i g u res i n t h e Ap p en d ix :

K A A = 2.3 X AEh/br ( F i g . A l )

CAB = 0 .4 (Fig . A3)

Cs A = 0 .65 (Fig . A4 )

.*./ , = 2.3 X 4 X 130 (1 - 0.4 X 0. 65 )/3

= 296 in.4

N o w , u s i n g t h e c h a r t s i n t h e A I S C C o m m e n t a r y , S e c t .

D 2 :

71-2 = {d i —• d\)/d\ — 1.9 ( for the co lumn)

_h^hT _ 101 /2 5 2 252N

~ Z ^ ^ ~ 240 \2 96 296>= 0.72

GB = 1 0 ( A I S C r e c o m m e n d e d v a lu e fo r p i n n e d

e n d )

:,Kx = K^ = 1.25

Th e s l en d ern ess r a t i o s fo r t h e weak an d s t ro n g ax es a re :

{KL/n)y = 0

{KL/ri)x = 1 .25 X 24 0/3 .34 = 89 .8

Th u s , t h e s t ro n g ax i s g o v ern s .

Fa^ 1 5 .5 8 k s i (A IS C S p ec . Ta b l e 1 -4 2)

> /a = 3.37 ksi

A l l o w a b l e b e n d i n g s tr e ss ( A I S C S p e c . A p p e n d i x D , S e c t .

D 3 ) :

Fo r a co n t in u o u s ly l a t e ra l su p p o r t ed b eam th e a l l o w

ab l e b en d in g s t r es s i s :

C o m b i n e d a x i a l c o m p r e s s i o n a n d b e n d i n g ( A I S C S p e c .

A p p e n d i x D , S e c t . D 4 ) :

F r o m p r e v io u s c a l c u l a t i o n s , / a / F a ^ = 0 . 2 2 ; t h u s , b o t h

F o r m u l a s ( D 4 - l a ) a n d ( D 4 - l b ) m u s t b e c h e c k e d .

S in ce b en d in g is ab o u t t h e s t ro n g ax i s ,

F'ey = 1 8 .5 2 k s i (A IS C Sp ec . Ta b l e 2 )

an d for t h e case wh en th e co lu m n i s a l l o wed to sw ay

in i t s p lane,

C^ = 0 .85

F i n a l l y , F o r m u l a ( D 4 - l a ) g i v e s :

19.0^= 1.006 o.k .

18.52y


3.37 19.0

1 = 0.890.6 X 42 25

C O N C L U S I O N S

o . k .

0.6i^, 25 ksi

Th e ex am p les p resen t ed h ere se rv e t o i l l u s t r a t e t h e u se

of the design speci f icat ions for tapered members as re

q u i r e d b y A I S C S p e c i f ic a t io n A p p e n d i x D . T h e r e a r e

s t i l l s ev era l u n an swered q u es t i o n s r eg ard in g t h e r a t i o n a l

d es ig n of t ap er ed m em b e r . Th r ee t o p i cs o f cu r ren t i n

terest are (1) local buckl ing l imi tat ions, (2) tension

f lange sup po rt require me nts ,^ an d (3) the C^̂ factor for

b e a m - c o l u m n s in u n b r a c e d f r a m e s . T h e d e s ig n e r m u s t

s t i l l u se ex p er i en ce an d so u n d en g in eer in g j u d g em en t

r e g a r d i n g d e s i g n p r o c e d u r e s n o t p r o v i d e d b y A p p e n d i x

D of the S peci f icat ion .

A C K N O W L E D G M E N T S

Th i s p a p e r is b ased o n r esea rch r esu l t s o n t ap e red

m e m b er s t u d i es ca r r i ed o u t a t t h e S t a t e U n iv er s i t y o f

Ne w York at Buffalo . T he rese arc h i s f inancial ly sup

p o r t ed b y t h e Am er i c an In s t i t u t e o f S t ee l C o n s t ru c t i o n ,

M e t a l B u i l d i n g M a n u f a c t u r e r s A s s o c i a ti o n a n d t h e

N a v a l F a c i li t ie s E n g i n e e r i n g C o m m a n d . T e c h n i c a l

g u id an ce o f t h e r esea rch was p ro v id ed b y t h e j o in t

C o l u m n R e s e a r c h C o u n c i l - W e l d i n g R e s e a r c h C o u n c i l

S u b c o m m i t t e e o n T a p e r e d M e m b e r s a n d t h e S u b c o m

m i t t e e o n T a p e r e d M e m b e r s of t h e T e c h n i c a l C o m m i t t e e

o f t h e M e t a l B u i ld i n g M a n u f a c t u r e r s A s s o c i a t i o n.

REFERENCES

1. Amirikian, A., Wedge-Beam Framing Trans. ASCE, Vol. 777,p . 596(7952).

2. Lee, G. C, M. L. Mo rrell, and R. L. Ketter Design of TaperedMembers WRC Bulletin No. 773, June 7972.

3. Morrell, M. L. and G. C. Lee Allowable Stress for Web-Tapered Beams with Lateral Restraints WRC Bulletin, No.792, Feb. 7974.

11

F I R S T Q U A R T E R / 1 9 7 5



4 . American Institute of Steel Construction S u p p l e m e n t N o . 3 t o t h e

S peci f i ca t ion for the D es ig n , Fabr ic a t io n & Erec t ion of

S t r u c t u r a l S t e e l f o r B u i l d i n g s June 12, 1974 {including Ad-

denda to the C ommentary on the Specification.)

5. Lee, G. C. and M. L. Morrell S t a b i l i t y o f S p a c e F r a m e s C o m

p o s e d o f T h i n - W a l l e d M e m b e r s Conf. Preprint, ASCE Na

tional Conference-Structural Engineering, April 22-26, 1974, Cin

cinnati, Ohio

6. Lee, G. C. and M. L. M orrell A l l o w a b l e S t r es s f o r W e b -

T a p e r e d B e a m s w i t h T e n s i o n F l a n g e B r a c i n g {manuscript in

preparation).

A P P E N D I X :

S T I F F N E S S F A C T O R S A N D C A R R Y - O V E R F A C T O R S F O R W E B - T A P E R E D M E M B E R S

2 0 2 0

Fig. A1. Stiffness factors: small end fixed Fig. A2. Stiffness factors: large end fixed

12




2 0 1

1 - 5 h

o

o 10

0- 5

[-

1 m^ M

1 zr=o(c/

[• /

L ^^ ^^^

[ y= oa )^ / ' "1 o-sa)—' ,\ 10(T>—'

20(Tr1 1 1 1 1 1 1 1 1 1 1 1 1 1

o-5(a/

l OOxH

22CiJ

• • • • 1

0 0-5 1 0 1-5 2 0

Fz]^. ^ J . Carry-over factors: small end fixed

^00 5 10 1-5P/P

eXo

Fig. A4. Carry-over factors: large end fixed

2 0

13

application of aisc design provisions for tapered members

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