Economical analysis of large framed-tube structures

..i,,,

vo,

Pergamoo Press

rin ed in Oreat.ritain

I

'

.

Economical Analysis of Large Framed=lube Structures P A U L F. AST* JOSEPH S C H W A I G H O F E R t A simplified method is presented .[or the analysis of framed-tube structures when subjected to bending due to lateral loading, which proves to be very economical with large structures. The method consists in replacing the threedimensional tube by an equivalent two-dimensional system. Several additional simplifications are outlined and their influence on the accuracy ~?f the results is demonstrated.

INTRODUCTION

the bending moments in the columns at or near the base. The usual range for employing the framedtube system is between 40-100 storeys. The threedimensional tube then contains several thousand joints and consequently the analysis becomes for most practical cases prohibitive because of high costs. In such instances a plane equivalent frame approach [2] is called for. In connection with very large framed-tubes [3] the approach outlined below [4] is appropriate.

A N U M B E R of tall buildings have recently been erected employing the framed-tube system of construction. In this structural concept the horizontal and vertical elements along the periphery of the building provide the principal resistance against lateral loading. Most actual framed-tube structures have 2 axes of symmetry, they are square or rectangular in plan [figure l(a)], and consist of 2 pairs of rigid frames (or perforated shear walls) orthogonally arranged along the perimeter of the building. The floors act as rigid horizontal diaphragms. Framed-tubes are three-dimensional structures and can be analysed as such. However practical considerations favour less time-consuming and cheaper alternatives. The lateral load on the structure is primarily resisted by frame action of a pair of frames [2] and a pair of frames [1]. The latter are subjected to axial deformations along corner columns, and to some extent to out-of-plane bending. It appears however [I ] that neglecting this latter effect will only slightly change the axial force but to a somewhat greater extent influence

M E T H O D OF ANALYSIS This approach, which is based on the elastic method of analysis, utilizes, when present, symmetry about both the x- and )'-axes. One quarter of the framed-tube is represented by the two half frames (1) and (2). These are disconnected from each other at the corner column A [figure l(b)]. Displacement boundary conditions are prescribed along the axes EE and FF in accordance with the deformation of the actual structure, The appropriate external lateral loading W~ is applied to frame (2). All members of frame (2) will deform and the column line AA will elongate. As a consequence vertical interaction forces I k between the two frames will be activated at every floor level in the actual structure. Obviously the solution to the analysis of the tube structure lies in determining these interaction forces and combining their effect on

*Design Engineer, FARKAS BARRON JABLONSKY, Toronto, Canada. ";'Professor of Civil Engineering, University of Toronto, Toronto, Canada presently: Swiss Federal Institute of Technology, Zurich, Switzerland. 73

74

Paul b, As: and Joseph 5ct, watg/u!/V:*

ly g l

| •

W

t

!

|

l

Wind

u'2 K

~f

E

~A 1

?

Frame (2) ~4cUf

r ra:'flo

(a) I'~,~, I. t:ramed-iube ~lru( tm'e,

frame (2) with the effect which the horizontal external loading W~ produces. Frame (t) is solely subjected to the interaction forces. The displacement compatibility in the vertical direction of frames (1) and (2) along the corner joints is expressed by 1{5i =

25~'-[-2(')i

{l)

where is the vertical displacement of corner joint i of flame (l) when loaded by all interaction forces is the vertical displacement of corner joint i of frame (2) when loaded by the wind forces W~ only, 26 i

= vertical displacement of corner joint i of frame (2) when loaded by the interaction forces only.

In particular is I fi i = 1 6 i l l l

q-t ~Si212 q- .... q - l ~ ) i k l ,

47 . . +

(2) where l(~ik

= vertical displacement of joint caused by a vertical unit force I placed at joint k of frame (1).

lk

= interaction force acting at corner joint k = 1,2 ..... n.

Similarily the vertical displacement of joint i of frame (2) due to the interaction force 1,, can be written : 2i) i =

2 5 i l 1 1 + 2(51212 +

+ 20inl n

(3)

where 2 "

f) i k

= vertical displacement of joint i caused by a vertical unit force placed at joint k of frame (2).

Substituting equations (2) and (3) into equation (1) and noting that the displacement 26~ has a direction opposite to that of 1,5; ~ yields: ll(tdil

Jr- 2 ( ) i 2 ) + 1 2 ( l i 5 i 2

+ 2():2) i .

t !n(l()in-F 2~ji, t) =

-' 6'i'

(4)

'Fhere exists one such compatibility equation for each of the n cornel" joints. This means a total of n equations can be written, and from these can be computed the n unknown interaction lbrces I k, The effect of the interaction forces combined with those of' the external forces will yield tile true forces and moments in frame (2). For the execution of the above outlined analysis a frame program which is oriented toward the analysis of large structures with numerous members and joints should be used. In computing tables tk~r tile analysis of tube structures (3) the authors employed the so-called FR2 program [5] which they appropriately modified to take into account the effect of non-prismatic members. The modified cross-sectional properties and subsequent stiffnesses were determined in accordance with [6]. To obtain representative results in particular For the horizontal displacements of the joints it i~ mandatory to take account of the .joint stiffness in tube structures especially when the column width or spandrel depth exceeds about 15 per cent of the column spacing or the storey height, respectively.

75

Economical Anah,sis ol° Larffe Framed-Tube Structures

SHORT CUTS The computing effort for large framed-tube structures is substantial and for e c o n o m y sake ought to be reduced considerably in particular in the preliminary design, without imparing the accuracy. The replacing of the three-dimensional tube by a plane equivalent frame is the first and dominant step [2, 3]. For very large structures [3] and only for these a second important improvement is the complete separation of frames (1) and (2) and thereby drastically reducing the computer storage requirement. Another device which proved successful is to apply m the above analysis the unit forces along column line A A not at every floor level but at every second or at every fourth ltoor level. In this way the number of loading cases is cut by a factor of two or four, respectively. The effect which applying the interaction forces at e\,ery second and fourth floor, respectively has on the shear forces in the spandrel beams will be demonstrated using a rather small framed-tube structure. Its geometry and loading are shown in iigure 2. In ligures 3(a) and 3(by is plotted in heavy lines the variation o[" the shear forces ,C~ in the spandrel beams of the first bay (between columns I and 2) as function of the height for the cases tha! the interaction forces are applied at the second and fourth floor levels, respectively. Superimposed in these diagrams is (in light lines) the variation of the shear force in the same spandrel beams for the case where the interaction forces are applied at every floor level. Figures 4(a), 4(b), 5(a) and 5(b) represent plots of the shear forces 2 C, and .~C~ in the spandrel beams of the second bay (between column lines 2 and 3) and of the third bay., respectively when the interaction forces are applied at every second and every fourth floor, respectively. Finally in figure 6 are plotted the interaction l\~rces versus height of the building for the cases that the interaction forces are applied at every floor (solid line), at every second floor (circles) and a t e~ery fourth floor (triangles). It should be noted that for the last two cases the magnitudes shown are ~ and ~ of the actual interaction forces. From Iigures 3, 4 and 5 it becomes apparent thal the interaction forces in this particular example can be applied without noticeable loss in accurac) at every second floor level" also the influence of concentrated interaction forces decreases rapidly towards the center of the frame. INFIA~ENCE

OF

THE

JOINT

-'["Corner, IO'-O X

0

--'

7'

3'

b o

IC

2C

=~:~[

3 i

~C

LL j L j i , u - u

LZ -] [Z '~ F_ J i i ; ;i i i[ Ji/i 'ft ~:

i

! I~

5 xlO - 0

P i a ~,

Elevation

Fi,~,. 2, Kvample.

'

k,~,,ery

Z"° floor

iO

"~

kE

3

0

o

very floor

~

4

force

,C

IQ'~.

!~ f l o o r

Every

2

Shear

E.

,C

(bY

,

5 kips

3. k 2 w i a l i o n

Sheor

4

force

qf ~hear lm'('u

floor

C

,

k,ps

,(

(a]

0

Every

t

:::0or

2 nd f l o o r

Every

floor

STIFFNESS

It was mentioned earlier that the computed deflections and the forces in the framed-tube may change substantially if in the analysis the width

2

Shear

force

4

,C

. kips

,

Shear

force

F(~,. 4. Variation q/'Shear Force :('b

I

C

kips

Paul F. Ast and ,loseph SchwaLehoJer

76

,3 ~ 2015-- ~ ~

ol the members is neglected. [ h e structure ~,I figure 2 is used to demonstrate the d{fl'erences

(b}

(a) Every 4 '~

f*oo~

Every 2 n~ f l o o r I0

which may result. In figure 7 i> plotted ~,he horizontal deflection of column line i as a function ,,~f the height, first for the case th:n all members are assumed to be prismatic and then for the case of non-prismatic members, that i~ ~,:~ say ihc end sections of all columns and beams are nominally stilT. In figure 8 is plotted the varialion of the shee~r

5 i

2

C,

4

Shear force

//,

::

2

d

Shear force .C

~C , kips

K~ps

2O

Fig. 5. Variation of shear Jbrce ~C~.

2o\ 16

Interocflon

forces a p p h e d

E v e r y floor

\

IO

......

P at~c

Every 2 ~d floor

members

Every 4 'n floor

>

8

--i---

i

0

o u_

t__

~0

:

20

S h e a r force in s p a n d r e l beams,

0

4

8

Interaction

~2

forces I . ,

J6

Fig 8. Influence of joint stiJJness on the shear force,

zb

kips

Fig, 6. Interaction forces vs hei,qht.

15

o

g

>

,o

30 kips

o

?

o m u_

forces in the spandrel beams versus height lbr both cases, namely when employing prismatic members only and when using stiff ended members, in figure 8 the shear forces for the spandrel beams between columns I and 2 are plotted. Figure 7 indicates clearly that consideration of stiff ended members is mandatory in order to obtain representative deflections. On the other hand figure 8 shows that the shear forces in the spandrel beams are much less effected by the degree of joint stiffness. CONCLUSIONS

5

0.5 Hortzontol

I-0

clisplocernent,

f~

Fig. 7. Influence of joint stiffness on the displacements.

A method has been presented for the economical analysis of large flamed-tube structures. Although the particular case of a tube without a core has been treated, the analysis may readily be extended to include tubes with cores [4]. In particular in the preliminary analysis it should be advantageous to apply the interaction force between the orthogonal frames at every second floor level only.

Economical Analysis of Large Framed-Tube Structures REFERENCES

I. A. RUTENBERG,Discussion of reference [2], Proc. ASCE, J. Struct. Div., 98, 942 (1972). 2. A. COULL and N. K. SUBEDI, Framed-tube structures for high-rise buildings, Proc. ASCE, Strttct. Div., 97, 2097 ( 1971 ). 3. J. S('HWAIGHOVERand P. F. AST, Tables for the Analysis of Framed-tube Buildings, Dept. Civ. Engrg., Univ. of Toronto, Publ. No. 72 01, March 1972. 4. P. F. Asr, The Analysis of Framed-Tube Structures for Tall Buildings M.A.Sc.-Thesis, Dept. Civ. Engrg., Univ. of Toronto, 1972. 5. W. WEAVER. Computer Programs/or Structural Analysis, Van Nostrand, Princeton, N.J. (1967). 6. J. S(THWA~(;HOFERand F. H. MICROYS, Analysis of shear wall structures using standard computer programs, J. Am. Concr. Inst., 66, 1005 (1969).

On prdsente une mdthode simplifide pour l'analyse de tube ou cadre fi ossature pdriphdrique (framed-tubes) d'immeubles de grande hauteur soumis fi la flexion due aux actions lat6rales (p. ex. vent). La m6thode consiste h remplacer le systdme tube 3-dimensions par un systeme 6quivalent ~t 2-dimensions. On indique, en outre, des simplifications compldmentaires et on discute de leurs influences sur l'exactitude des rdsultants. Cette mdthode simplifi6e se rdv61e particuli6rement 6conomique dans le cas de grandes structures.

Eine vereinfachte Methode fuer die Berechnung von Roehren Bauten die horizontalen Kraeften unterworfen sind, ist angegeben. Sie erweist sich als sehr wirtschaftlich fuer groessere Roehrenbauten. Die Methode besteht darin, dass die dreidimensionale Roehre durch ein equivalentes zwei-dimensionales System ersetzt wird. Mehrere zusaetzliche Vereinfachungen sind angegeben und ihr Einfluss auf die Genauigkeit der Resultate ise angezeigt.

77