1
THE CAPACITY DESIGN OF REINFORCED CONCRETE HYBRID STRUCTURES FOR MULTISTOREY BUILDINGS
T. Paulay
1and W. J. Goodsir
2SYNOPSIS
T o c o m p l e m e n t existing capacity d e s i g n procedures used in N e w Zealand for reinforced concrete b u i l d i n g s in which earthquake resistance is provided by d u c t i l e frames or ductile structural w a l l s , an analogous m e t h o d o l o g y is presented for the design of d u c t i l e hybrid structures. M o d e l l i n g and types of structures in w h i c h the m o d e of wall c o n t r i b u t i o n is different are briefly d e s c r i b e d . A step by step d e s c r i p t i o n of a capacity design p r o c e d u r e for a structural system in w h i c h fixed base ductile frames and w a l l s , both of identical h e i g h t , interact, is presented. T h e rationale for each step is outlined and, where n e c e s s a r y , evidence is offered for the selection of particular d e s i g n p a r a m e t e r s and their m a g n i t u d e s . A number of issues w h i c h require further study are b r i e f l y outlined. These relate to irregularity in layout, torsional e f f e c t s , diaphragm
flexibility, shortcomings in the p r e d i c t i o n s for dynamic shear d e m a n d s in w a l l s , and to limitations of the proposed design p r o c e d u r e . It is believed that the m e t h o d o l o g y is l o g i c a l , r e l a t i v e l y simple and that it should e n s u r e , when combined with appropriate d e t a i l i n g , excellent seismic structural response.
INTRODUCTION
W h e n lateral load resistance is provided by the c o m b i n e d c o n t r i b u t i o n s of ductile m u l t i - storey frames and structural w a l l s , the system is o f t e n referred to as a "hybrid s t r u c t u r e " . In North A m e r i c a , the term
"dual system" is used. These structures combine the a d v a n t a g e s of their constituent c o m p o n e n t s . Because of the large stiffness of w a l l s which are provided with adequate r e s t r a i n t s at the f o u n d a t i o n s , e x c e l l e n t storey d r i f t control may be obtained. M o r e - o v e r , suitably designed w a l l s can ensure that storey m e c h a n i s m s (soft storeys) w i l l not develop in any e v e n t . Interacting ductile frames o n the o t h e r h a n d , w h i l e carrying the m a j o r part of the gravity load, can p r o v i d e , w h e n r e q u i r e d , significant energy d i s s i p a t i o n , p a r t i c u l a r l y in the upper s t o r e y s .
D e s p i t e the a t t r a c t i v e n e s s and indeed e x i s t e n c e of m a n y such structures in N e w Zealand, comparatively little research e f f o r t h a s been directed to them. T h e N e w Zealand C o d e of P r a c t i c e for D e s i g n of C o n c r e t e S t r u c t u r e s1 d r a w s d e s i g n e r s ' a t t e n t i o n to the need for "special studies"
w h e n d e s i g n i n g "ductile hybrid s t r u c t u r e s " . N o specific g u i d a n c e i s , h o w e v e r , provided.
K n o w n studies refer primarily to elastic r e s p o n s e , d e s p i t e the o b v i o u s importance of the features of inelastic behaviour.
A study w a s initiated w i t h the aim of u l t i m a t e l y formulating a design procedure for h y b r i d s t r u c t u r e s w h i c h w o u l d be
a n a l o g o u s to those developed in New Zealand for ductile frames and ductile structural w a l l s1. It w a s h o p e d that a scheme could b e formulated w h i c h w o u l d provide a smooth t r a n s i t i o n b e t w e e n design a p p r o a c h e s for
space f r a m e s *1 and those for b u i l d i n g s in w h i c h seismic resistance is p r o v i d e d by structural w a l l s o n l y *1' 2 . To this end n u m e r o u s analytical studies of p r o t o t y p e b u i l d i n g structures were c o n d u c t e d3'4 to p r o v i d e appropriate calibration of the p r i n c i p a l design p a r a m e t e r s . This paper reports on the findings and conclusions as they relate to design procedures rather than on details of features of structural b e h a v i o u r .
The traditional procedure of d e s i g n i n g for e a r t h q u a k e r e s i s t a n c e , utilizing e l a s t i c a n a l y s i s techniques and equivalent lateral static l o a d s , is well e s t a b l i s h e d . The r e s u l t i n g distribution of lateral load r e s i s t a n c e over the h e i g h t of b u i l d i n g s w i t h ductile f r a m e s , or structural w a l l s , is generally accepted as meeting satisfact- orily actual earthquake load d e m a n d s . T h e r e w a s little evidence to indicate that this w o u l d be the case also w i t h hybrid
s t r u c t u r e s . One source of concern for p o s s i b l y drastic differences b e t w e e n
"elastic static" and "elasto-plastic d y n a m i c " responses of hybrid structures stems from the recognition of fundamental differences in the behaviour of
Details of the "capacity d e s i g n " of reinforced concrete structures are given in N Z S 3101:1982 and the b a c k g r o u n d to this design philosophy is o u t l i n e d in some detail in the commentary of the code of p r a c t i c e .
1 Professor o f Civil E n g i n e e r i n g , U n i - versity of C a n t e r b u r y , New Zealand.
2 E n g i n e e r , Ove A r u p P a r t n e r s h i p , L o n d o n , England.
BULLETIN OF THE NEW ZEALAND NATIONAL SOCIETY FOR EARTHQUAKE ENGINEERING, Vol. 19, No. 1, March 1986
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b e a m - c o l u m n frames and structural w a l l s . T h e s e d i f f e r e n c e s stem from dissimilar d e f o r m a t i o n p a t t e r n s w h e n subjected to the same lateral load, as shown in Fig. 1.
F r a m e s and w a l l s , w h i l e sharing in the r e s i s t a n c e of shear forces in the lower s t o r e y s , o p p o s e each other in the storeys n e a r the top of the building. It w a s of major interest to examine the load sharing between these two types of interacting elements during inelastic d y n a m i c response to a m a j o r seismic e v e n t .
Lateral Load
Fig. 1
Frame Element Walt Element Coupled Frame-Wall (Shear Model (Bending Mode) Building
Deformation Patterns of L a t e r a l l y Loaded F r a m e , W a l l s and Coupled W a l l - F r a m e E l e m e n t s .
A step by step d e s i g n m e t h o d o l o g y is
proposed^ to m e e t the intent of the "capacity d e s i g n " p h i l o s o p h y . The p r e s e n t a t i o n
concentrates solely o n issues relevant to the largest expected seismic event envisaged by the c o d e5. The emphasis is therefore o n issues of ductility and the prevention of collapse. Existing procedures , to satisfy d e s i g n criteria for stiffness and m i n i m u m strength, both r e l e v a n t primarily to damage c o n t r o l , and considered to be equally a p p l i c a b l e to hybrid s t r u c t u r e s , are not referred to in this p a p e r .
T h e d o m i n a n t feature of the capacity d e s i g n strategy is the a p r i o r i establishment of a rational h i e r a r c h y in strength b e t w e e n the components of the entire structural system.
A c c o r d i n g l y , the approach to the d e s i g n of e a c h primary lateral load resisting compon- ent w h i c h is to be protected against yield- ing or brittle f a i l u r e , such as due to shear, can be described by the simple general
e x p r e s s i o n for the ideal strength S., thus
S. l o code (1)
w h e r e S , is the required dependable s t r e n g t h o r the m e m b e r selected for energy d i s s i p a t i o n , as determined by elastic analysis techniques for a c o d e5 specified lateral static load o n the structure;
<J> is the ratio of the m a x i m u m strength, SQ, w h i c h can b e developed in the selected inelastic c o m p o n e n t (as built) by large d i s p l a c e m e n t s during a severe seismic e v e n t , to the strength required, s C OHe' for the same m e m b e r by the code specified lateral loading; and w is a dynamic m a g n i f i c a t i o n factor w h i c h q u a n t i f i e s d e v i a t i o n s in strength d e m a n d s on the m e m b e r to be p r o t e c t e d , from demands
indicated by elastic a n a l y s i s . E x t r e m e d e m a n d s are expected to occur during the inelastic dynamic r e s p o n s e of the structure.
For the sake of c o m p l e t e n e s s , certain aspects of the "capacity d e s i g n " of ductile frames are restated.
2. TYPES OF HYBRID STRUCTURES A N D THEIR MODELLING
In the following, some distinct and common types of hybrid s t r u c t u r e s , in which w a l l s and frames interact in a particular m a n n e r , are described. N o attempt is m a d e , h o w e v e r , to categorize all p o s s i b l e
combinations in which these two systems m a y be u t i l i z e d . C o n v e n t i o n a l m o d e l l i n g t e c h - n i q u e s , to b e used for the purposes of a n a l y s i s , are briefly reviewed and suggest- ions m a d e for choices of suitable energy dissipating systems in hybrid structures.
2.1 Interacting Ductile F r a m e s and D u c t i l e Cantilever Walls
In the m a j o r i t y of reinforced c o n c r e t e m u l t i s t o r e y b u i l d i n g s , lateral load
resistance is assigned to both d u c t i l e space frames and cantilever structural w a l l s . F i g u r e 2(a) shows in plan the somewhat idealized symmetrical disposition of frames and walls in a 12 storey example b u i l d i n g . T h e p r o p e r t i e s of these two d i s t i n c t structural elements m a y b e conveniently lumped into a single frame and a single cantilever w a l l , as shown in Fig. 2 ( b ) . Instead of individual w a l l s , shown in F i g . 2 ( a ) , tubular c o r e s , or coupled structural w a l l s , are also used frequently.
Direction of 1 • — - 1 - h — H —
IV' 1
m! -, . I ^ i r
j\ i,„ j
1 I
• r ^ 1 1 I T 1 j
1
. . 1 ,
ALL
Structural •
wall K Main frame
X
(a) PLAN Transverse
frames
Secondary beams
Total Shear
Krode, total'
(hi STRUCTURAL MODELLING (c) STOREY SHEAR FORCES F i g . 2 Modelling of and L a t e r a l Load
Sharing in a Typical W a l l - F r a m e System.
It is customary to assume that floor slabs at all levels have infinite inplane rigidity.
Such diaphragms will then ensure that storey d i s p l a c e m e n t s for frames and w a l l s are the same or that in the case of storey t o r s i o n , a simple linear relationship exists b e t w e e n the storey displacements of vertical
e l e m e n t s . W h e n diaphragms are relatively slender and w h e n large concentrated lateral storey forces need to be introduced to
3 r e l a t i v e l y stiff w a l l s , particularly w h e n
t h e s e are spaced far a p a r t , the flexibility of floor d i a p h r a g m s m a y need to b e taken into a c c o u n t . This issue is briefly reviewed in Section 4.3.
The e x t e n s i o n a l l y infinitely rigid h o r i z o n - tal connection b e t w e e n lumped frames and w a l l s at each f l o o r , shown in Fig. 2 ( b ) , enables the analysis of such laterally loaded e l a s t i c s t r u c t u r e s to be carried out speedily. Typical results are shown in Fig. 2 ( c ) . Here the sharing between w a l l s and frames of the total storey shear forces is illustrated. The relative p a r t i c i p a t i o n s w h i c h reflect the b e h a v i o u r of the two d i f f e r e n t s y s t e m s , as shown in F i g . 1, indicate a rapid decline w i t h h e i g h t of the c o n t r i b u t i o n of the w a l l s to shear r e s i s t - ance . Figure 2(c) also shows h o w the two systems o p p o s e e a c h o t h e r in the top
s t o r e y s . The d i s t r i b u t i o n of m a g n i t u d e s of shear forces w i t h h e i g h t for each system w i l l depend p r i m a r i l y on the relative stiff- n e s s e s of the w a l l s and frames. This example s t r u c t u r e , s h o w n in Fig. 2,will be s u b s e - quently used to illustrate typical d i s t r i b u t i o n s of forces and moments for b o t h w a l l s and f r a m e s , as a consequence of inelastic dynamic response to seismic e x c i t a t i o n s .
As the flexural response of w a l l s is intend- ed to control d e f l e c t i o n s in hybrid s t r u c t - ures , the d a n g e r of developing "soft s t o r e y s " should n o t a r i s e . The designer may therefore freely choose those members or localities in frames w h e r e energy d i s s i - p a t i o n should take place w h e n required. A p r e f e r a b l e and p r a c t i c a l m e c h a n i s m for the frame of F i g . 2 is shown in Fig. 4 ( a ) . In this frame, p l a s t i c h i n g e s , w h e n required d u r i n g a large expected seismic e v e n t , are made to develop in all the beams and at the b a s e of all v e r t i c a l e l e m e n t s . A t roof
level, p l a s t i c h i n g e s may form in either the b e a m s or the c o l u m n s . The main advantage of this s y s t e m is in the detailing of the p o t e n t i a l p l a s t i c h i n g e s . Generally it is e a s i e r to detail b e a m rather than column e n d s for p l a s t i c r o t a t i o n . M o r e o v e r , the avoidance of p l a s t i c hinges in columns a l l o w s lapped splices to be constructed at the b o t t o m end rather than a t m i d h e i g h t of c o l u m n s in each upper storey.
The design p r o c e d u r e described in consider- able detail in Section 3, is relevant to this type o f s t r u c t u r a l system and its p r e f e r r e d energy dissipating m e c h a n i s m s .
W h e n long span beams are used, and in p a r t i c u l a r w h e n gravity rather than e a r t h - quake loading governs the strength of b e a m s , it m a y be preferable to admit the d e v e l o p - m e n t o f plastic h i n g e s at both ends of all columns over the full h e i g h t of the s t r u c t - ure , as shown i n Fig. 4 ( c ) .
2.2 Ductile Frames and Walls C o u p l e d b y B e a m s .
Structural w a l l s , instead of b e i n g isolated as free standing c a n t i l e v e r s , may be
connected by continuous beams in their p l a n e to a d j a c e n t frames. The m o d e l of such a system is shown in Fig. 3 ( a ) . Beams w i t h span lengths ij_ and £2 are rigidly c o n n e c t - ed to the w a l l s . A possible m e c h a n i s m that c a n b e utilized in this type of system is shown in Fig. 4 ( b ) . Beam hinges at or close to the wall edges m u s t develop. H o w e v e r , at c o l u m n s , the designer may decide to allow plastic hinges to form in either the beams or the columns, above and below each floor, ,as shown in Fig. 4(c)
Plastic hinges
Wall
(a) lb) let
Fig. 4 Complete Energy Dissipating M e c h a n i s m s Associated with Different Hybrid Structural S y s t e m s .
This type of system could be utilized also in the building of Fig. 2(a) if the w a l l s were to be connected to the adjacent columns by primary lateral load resisting b e a m s . In
that case the entire structural system w o u l d consist of 7 ductile f r a m e s , shown in Fig.
2(b) and two coupled frame-walls of the type given in Fig. 4 (b) .
(a) (b) (c) (d)
F i g . 3 Modelling of D i f f e r e n t T y p e s of Hybrid S y s t e m s .
S i m p l i f i e d a n a l y s i s t e c h n i q u e s , useful at least for p r e l i m i n a r y d e s i g n , have been developed for such a m i x t u r e of interacting frames and w a l l s ^ . T h e structural
i d e a l i z a t i o n used in shown in Fig. 3 (b). T h e stiffness of all w a l l s , as quantified by the second m o m e n t of a r e a , are lumped into a s t i f f n e s s , I , of a single w a l l . W h e r e a p p r o p r i a t e Allowance for shear d i s t o r t i o n s in the w a l l s should also be m a d e2. T h e b e a m s framing d i r e c t l y into wall e d g e s , i.e.
t h o s e with span lengths and in Fig.
3 ( a ) , are lumped at each floor into a single b e a m having a m e a n second m o m e n t of area, I , and span I , as shown in Fig. 3 ( b ) . A l l o t h e r b e a m s , suSh as span in Fig. 3(a) and the b e a m s of the frames in Fig. 2 ( a ) , are also lumped and replaced at each floor w i t h beams having the m e a n properties of 1^
and &b shown in Fig. 3 ( b ) . The aim is to o b t a i n r e p r e s e n t a t i v e m e a n I/£ ratios for the b e a m s . Finally all the columns of the b u i l d i n g s are lumped into two identical c o l u m n s , each having one half of the m o m e n t i n e r t i a , 0.51 , of the sum of the m o m e n t of inertia of al5 columns of the real
s t r u c t u r e . Standard solutions for a r a n g e of r e l a t i v e stiffnesses h a v e b e e n presented for this type (Fig. 3(b)) of structure^.
A n o t h e r u s e f u l t e c h n i q u e replaces all frames w i t h a single e q u i v a l e n t "shear" cantilever and connects this continuously over the h e i g h t of the building to a single equivalent
"bending" cantilever w a l l7. The method is similar to that used in the "laminar
a n a l y s i s " of coupled structural w a l l s . It is limited to regular structures w i t h v e r t i c a l l y constant g e o m e t r i c p r o p e r t i e s .
B e f o r e the design of individual m e m b e r s can b e finalised, it is necessary to identify clearly the locations in beams and columns at w h i c h p l a s t i c h i n g e s are intended in order to enable the capacity design p r o c e d u r e to be applied.
2.3 F r a m e s Interacting with Walls of P a r t i a l H e i g h t
A l t h o u g h in m o s t b u i l d i n g s structural w a l l s extend over the full h e i g h t , there are c a s e s w h e n for architectural or other r e a s o n s , w a l l s are terminated below the level of the top floor. A model of such a structure is shown in Fig. 3 ( c ) .
B e c a u s e of the abrupt discontinuity in total stiffnesses at the level where w a l l s
t e r m i n a t e , the seismic response of these s t r u c t u r e s is viewed w i t h some concern.
G r o s s d i s c o n t i n u i t i e s are expected to r e s u l t in p o s s i b l y critical features of dynamic r e s p o n s e w h i c h are d i f f i c u l t to p r e d i c t . It is suspected that the regions of d i s c o n t i n - u i t y may suffer p r e m a t u r e damage and that local d u c t i l i t y d e m a n d s during the largest expected seismic events m i g h t exceed the a b i l i t y of affected components to deform in the plastic range w i t h o u t significant loss of r e s i s t a n c e .
O n the other h a n d , elastic analyses for lateral static loads show that structural w a l l s in the u p p e r storeys m a y serve no useful structural p u r p o s e . Figure 2
suggests t h a t the t e r m i n a t i o n of w a l l s b e l o w the top floor m a y beneficially affect
o v e r a l l b e h a v i o u r .
T h e response of such structures h a s also been studied r e c e n t l y1! . A limited number of case studies, using the 1940 El Centro earthquake record, suggested no features that could not be readily accommodated in currently used d e s i g n p r o c e d u r e s . T h e findings of this study, together w i t h the appropriate application of the capacity design approach, will be reported separately.
2.4 Hybrid Structures w i t h W a l l s on D e f o r m a b l e F o u n d a t i o n s
It is customary to assume that cantilever w a l l s are fully restrained against rotations at the b a s e . It is r e c o g n i s e d , h o w e v e r , that full base fixity for such large structural elements is very d i f f i c u l t , if not impossible, to a c h i e v e . F o u n d a t i o n compliance may result from soil deformations below footings and/or from deformations occurring w i t h i n the foundation s t r u c t u r e , such as p i l e s . B a s e rotation is a v i t a l component of w a l l d e f o r m a t i o n s . T h e r e f o r e it m a y significantly affect the stiffness of cantilever walls and h e n c e p o s s i b l y their share in the lateral load resistance w i t h i n elastic hybrid s t r u c t u r e s . T h e reluctance to address the problem m a y b e attributed to our limitations in being able to estimate reliably stiffness p r o p e r t i e s of soils.
M o r e o v e r , soil stiffness is g e n e r a l l y very d i f f e r e n t for static and dynamic loading.
F o r the latter, frequency and amplitude are also parameters which affect soil r e s p o n s e . T o g a u g e the sensitivity of hybrid structures of the type shown in Fig. 2(b) w i t h respect to foundation compliance of the wall elements only, parametric studies w e r e c o n d u c t e d1 1. T h e major variables in the structures chosen for analyses w e r e :
(1) V a r i a t i o n of w a l l r e s t r a i n t b e t w e e n the extreme limits of full rotational fixity and a hinge at the b a s e .
(2) V a r i a t i o n in the number of storeys in a building. Predominantly 6 and 12 storey structures w e r e studied.
(3) The relative c o n t r i b u t i o n of w a l l s to total lateral load resistance w i t h i n the structure w e r e v a r i e d . • This w a s
achieved with appropriate v a r i a t i o n of wall l e n g t h s , shown in Fig. 2 ( a ) .
(4) Elastic response to code specified lateral static load w a s compared with the elasto-plastic dynamic r e s p o n s e of the structure to the 1940 EI C e n t r o earthquake record.
Details of this study are to b e reported.
T h o s e aspects of the conclusions w h i c h are particularly relevant to the issues
examined in this paper are as follows:
(a) A b o v e the first floor, the static response of the structure with w a l l s w i t h m o d e r a t e stiffness is not significantly affected by the d e g r e e of base restraint.
A s a corollary, the stiffer a w a l l the m o r e profound is the influence of foundation compliance.
(b) In pinned base w a l l s , as expected, very large and reversed b a s e shear forces are predicted by elastic analyses for static lateral load. This points to the need for studying the transfer of these large forces to the diaphragm at
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first floor l e v e l . The w a l l shear reversals in the first storey necessitated dramatic i n c r e a s e s in column shear forces in that storey. In the first storey the sum of the c o l u m n shear forces exceed therefore the total static b a s e shear for the entire s t r u c t u r e .
(c) T h e single m o s t important parameter affecting the seismic dynamic response of such h y b r i d structures w a s found to be the period shift brought about by the r e d u c t i o n of w a l l stiffnesses w h e n complete loss of r o t a t i o n a l r e s t r a i n t a t the base w a s assumed.
(d) E x t r e m e levels of shear f o r c e s , p r e d i c t e d by elastic analyses for c o l u m n s and w a l l s , did n o t eventuate during t h e time h i s t o r y analysis for the E l C e n t r o e v e n t .
(e) In the u p p e r storeys i m p o r t a n t d e s i g n q u a n t i t i e s for the example hybrid s t r u c t u r e s , such as d r i f t s , column and w a l l m o m e n t s , and rotational ductility d e m a n d s in p l a s t i c h i n g e s of b e a m s , w e r e only
i n s i g n i c i a n t l y affected w h e n walls w e r e m o d e l l e d w i t h pinned b a s e s .
(f) F u l l w a l l base fixity is normally assumed in d e s i g n , although it is k n o w n to b e generally u n a v a i l a b l e . T h e s e p a r a m e t r i c studies indicated, h o w e v e r , that errors d u e to quite significant relaxation in b a s e r e s t r a i n t , are not likely to s e r i o u s l y a f f e c t elasto-plastic dynamic r e s p o n s e .
B r i e f c o m p a r i s o n s of a few features of the a n a l y t i c a l l y p r e d i c t e d response of p r o t o - t y p e hybrid structures with fixed or pinned b a s e d w a l l s , are m a d e in Section 3.
3. D E T A I L S OF A CAPACITY DESIGN P R O C E D U R E
In the following sections a capacity design approach for hybrid structures is described in a s t e p - b y - s t e p m a n n e r . T h e presentation f o l l o w s the p a t t e r n o f , and is similar t o , the d e s i g n procedure suggested for
r e i n f o r c e d concrete d u c t i l e frames in the C o m m e n t a r y of N Z S 3 1 G 11. W h e r e necessary, t h e p r e s e n t a t i o n of a design step is followed by c o m m e n t s , sometimes e x t e n s i v e . T h e s e are relevant to the p u r p o s e of and intend to explain the justification for that p a r t i c u l a r step. F r e q u e n t reference is m a d e to Fig. 2, w h i c h shows a p r o t o t y p e frame- w a l l s t r u c t u r e .
T h e p r o c e d u r e outlined in the following 19 steps is r e l e v a n t to the types of structures shown in F i g s . 4(a) and ( b ) . In these c o l u m n s in upper storeys are intended to b e p r o t e c t e d a g a i n s t significant plastic d e f o r m a t i o n s . T h e r e b y various concessions w i t h r e s p e c t to their detailing for
d u c t i l i t y m a y be utilised.
Step 1 - D e r i v e the b e n d i n g m o m e n t s and shear forces for all m e m b e r s of the f r a m e - s h e a r w a l l system subjected to the c o d e specified e q u i v a l e n t lateral static e a r t h q u a k e load only. T h e s e actions are s u b s c r i p t e d "code".
In the analysis for the elastically
responding structure, d u e a l l o w a n c e should be m a d e for the effects of cracking on the stiffness of both frame m e m b e r s and w a l l s . B o t h frames and w a l l s m a y g e n e r a l l y b e assumed to be fully restrained at their b a s e . Load effects are referred to as E .
Step 2 - Superimpose the b e a m bending m o m e n t s obtained in Step 1 u p o n corresponding beam m o m e n t s which are derived for
appropriately factored gravity loading o n the structure..
T h i s superposition corresponds w i t h the combinations of factored loads U = D + 1.3L
± E , w h e r e D is the dead load and LR is the live load reduced as the tributary area increases^.
Step 3 - If a d v a n t a g e o u s ,
r e d i s t r i b u t e design m o m e n t s obtained in Step 2 horizontally at a floor b e t w e e n any or all beams in each b e n t , and v e r t i c a l l y b e t w e e n beams of the same span at d i f f e r e n t f l o o r s .
In t h e process of m o m e n t r e d i s t r i b u t i o n , the peak values of beam m o m e n t s , resulting from the load combinations U = D + 1.3L
± E , m a y b e reduced by up to 3 0 %1. H o w e v e r , the curtailment of the beam flexural
reinforcement along a beam m u s t b e such that at least 7 0 % of the m o m e n t obtained from elastic analyses in Step 2 can be r e s i s t e d1. F i n a l moments obtained after r e d i s t r i b u t i o n should be checked to ensure that no loss in the total lateral load resistance of the structure r e s u l t s8. A l s o combinations for g r a v i t y load a l o n e , U = 1.4D + 1.7L , m u s t b e examined before the proportioning of b e a m s commences.
T h e principles of redistribution of m o m e n t s at a level among d i f f e r e n t spans of b e a m s w i t h i n frames are well e s t a b l i s h e d1>8. O n e of the advantages which m a y result is the reduction of the peak beam n e g a t i v e m o m e n t at an exterior column which is associated^_with the load combination U = D + 1 . 3 LR + E . T h e reduction is achieved at the expense of increasing the (usually non-critical) p o s i t i v e m o m e n t at the same section associated w i t h the combination U = D + 1. 3 LD + E . In the latter case the
K
o Factored gravity moments only
• Gravity and earthquake moments from elastic analysis
x After horizontal redistribution I After vertical redistribution,
-800 -400 0 400 800 (kNm) F i g . 5 T h e Redistribution of D e s i g n M o m e n t s
Among Beams of a Hybrid S t r u c t u r e .
g r a v i t y and earthquake m o m e n t s , superimposed in Step 2, oppose each o t h e r . A n example in F i g . 5 shows m a g n i t u d e s of b e a m d e s i g n m o m e n t s at each floor at an exterior c o l u m n at v a r i o u s stages of the a n a l y s i s . T h e gravity m o m e n t s (always n e g a t i v e ) ,
shown by c i r c l e s , are changed b y the_^
a d d i t i o n of earthquake moments E or E , to v a l u e s shown b y solid c i r c l e s .
B e c a u s e near their b a s e , w a l l s m a k e very s i g n i f i c a n t contributions to the resistance of both horizontal shear (Fig. 2) and overturning m o m e n t , the flexural demands o n the beams of hybrid structures are
r e l a t i v e l y small in the lower s t o r e y s . T h e d i s t r i b u t i o n of beam and wall m o m e n t d e m a n d s w i t h the h e i g h t of the elastic structure d e p e n d s on the relative stiffness of w a l l s and f r a m e s ^ . In the example f r a m e s , the b e a m m o m e n t s at the exterior column could b e redistributed so as to result in m a g n i t u d e s shown by crosses in F i g . 5. It is seen that the negative and p o s i t i v e m o m e n t demands are now comparable in m a g n i t u d e s .
T o o p t i m i z e the practicality of b e a m d e s i g n , w h e r e b y beams of identical strength are p r e f e r r e d o v e r the l a r g e s t p o s s i b l e number of a d j a c e n t f l o o r s , some vertical
r e d i s t r i b u t i o n of beam d e s i g n m o m e n t s should also b e considered. In the e x a m p l e of F i g . 5, the design m o m e n t s shown b y c r o s s e s m a y b e redistributed up and down the frames so as to result in m a g n i t u d e s shown b y the continuous stepped lines. It is seen that b e a m s of the same flexural strength could be used over 6 floors. :The' stepped line has been chosen in such a w a y that the area enclosed by it is approximately the same as that w i t h i n the curve formed by the c r o s s e s . T h i s choice m e a n s that the
c o n t r i b u t i o n of the frames to the r e s i s t a n c e of overturning m o m e n t s is only insignificantly altered by vertical m o m e n t r e d i s t r i b u t i o n . It m a y be noted that horizontal r e d i s t r i b u t - ion of beam m o m e n t s at a particular level w i l l change the m o m e n t input to individual c o l u m n s . H e n c e the shear demand across individual columns w i l l also change w i t h r e s p e c t to that indicated by the elastic a n a l y s i s used in Step 1. H o w e v e r , the total shear demand on columns of a b e n t m u s t not c h a n g e . This is referred to as
r e d i s t r i b u t i o n of design shear forces b e t w e e n c o l u m n s .
W h e n v e r t i c a l r e d i s t r i b u t i o n of beam m o m e n t s is carried o u t , the total m o m e n t input to some or all columns at a floor w i l l a l s o c h a n g e . H e n c e the total shear demand on columns of a particular storey m a y d e c r e a s e
(the 5th storey in F i g . 5 ) , w h i l e in other storey (the 2nd storey in F i g . 5) it w i l l i n c r e a s e . T o ensure that there is no d e c r e a s e in t h e total storey shear r e s i s t - ance intended by the code specified
lateral loading, there m u s t be a h o r i z o n t a l r e d i s t r i b u t i o n of shear forces b e t w e e n the v e r t i c a l elements of the s t r u c t u r e , i.e.
columns and w a l l s . It w i l l be shown
subsequently that the upper regions of w a l l s w i l l b e provided w i t h sufficient shear and flexural strength to accommodate a d d i t i o n a l shear forces shed by upper storey c o l u m n s . The principles involved in v e r t i c a l load
r e d i s t r i b u t i o n discussed h e r e are similar to those used in the d e s i g n of coupling beams of coupled structural w a l l s2.
To safeguard against p r e m a t u r e yielding in beams during small e a r t h q u a k e s , the
reduction of beam m o m e n t s resulting from combined h o r i z o n t a l and v e r t i c a l m o m e n t redistribution should n o t exceed 3 0 % .
Step 4 - Design all critical beam sections so as to p r o v i d e the required dependable flexural s t r e n g t h s , and detail the reinforcement for all b e a m s in all frames.
These routine steps r e q u i r e the d e t e r m i n - ation of the size and number of reinforcing b a r s to b e used to resist m o m e n t s along all beams in accordance w i t h the demands of m o m e n t envelopes obtained after m o m e n t redistribution. It is important at this stage to locate the two potential plastic h i n g e s in each span (Fig. 4(a)) for each direction of e a r t h q u a k e attack. In locating plastic h i n g e s w h i c h require the bottom
(positive) flexural r e i n f o r c e m e n t to yield in t e n s i o n , both load combinations U = D + 1 . 3 LR + E and U = 0.9D + E ,5 should be considered, as each c o m b i n a t i o n may indicate a d i f f e r e n t h i n g e p o s i t i o n . Detailing of the beams should then be carried out in conformity with the r e l e v a n t c o d e1 p r o v i s i o n s .
Step 5 - In each b e a m d e t e r m i n e the flexural overstrehgth of each of the two potential plastic h i n g e s corresponding with each of the two directions of earthquake attack.
The p r o c e d u r e , incorporating allowances for strain hardening of the steel and the
possible p a r t i c i p a t i o n in flexural resistance of all reinforcement p r e s e n t in the structure as b u i l t , is the same as that used in the design of beams of d u c t i l e f r a m e s1. The primary aim is to estimate the m a x i m u m m o m e n t input from b e a m s to adjacent columns
associated with the largest seismic event.
Step 6 - Determine the lateral displacement induced shear f o r c e , V , associated with the d e v e l o p m e n t of ° flexural overstrength at the two plastic hinges in each beam span for each d i r e c t i o n of earthquake attack.
These shear forces are readily obtained from the flexural o v e r s t r e n g t h s of potential plastic h i n g e s , determined in Step 5, w h i c h were located in Step 4. W h e n combined w i t h gravity induced shear f o r c e s , the design shear envelope for each beam span is obtained, and the required shear r e i n f o r c e - m e n t can then b e d e t e r m i n e d1. T h e displacement induced m a x i m u m beam shear f o r c e s , V , are used subsequently to determine°She m a x i m u m lateral displacement induced axial column load input at each floor.
Step 7 - D e t e r m i n e the beam flexural overstrength factor, <j> , at the centre line of each column at each floor for both directions of earthquake attack. Fixed v a l u e s of <f) are:
(a) A t groun8 level <f> = 1.4 (b) A t roof level <f>° = 1.1.
T h i s factor is subsequently used to estimate the m a x i m u m m o m e n t which could be
introduced to columns by fully plastified b e a m s . T h e beam overstrength factor, $ , at a c o l u m n , is the ratio of the sum of the flexural o v e r s t r e n g t h s developed by adjacent b e a m s , as d e t a i l e d , to the sum of the flexural strengths required in the g i v e n d i r e c t i o n by the code specified lateral earthquake loading alone, both sets of values being taken at the centre line of the relevant column .
T h e beam m o m e n t s at column centre lines can b e readily obtained graphically from the d e s i g n bending m o m e n t e n v e l o p e s , after the flexural overstrength m o m e n t s at the exact locations of the two plastic hinges along the beam h a v e been plotted.
Step 8 - E v a l u a t e the column design shear forces in each storey from
col (D <J) V _
cTo code (2)
w h e r e c o l u m n dynamic shear m a g n i f i c a t i o n factor, wc, is 2.5, 1.3 and 2.0 for the b o t t o m , intermediate and top storeys respectively. T h e d e s i g n shear force in the bottom storey columns should not be less than
col + 1.3$ M - )
yo code
col U + 0. 5 hb) (3)
w h e r e M col = the flexural overstrength of the column base section consistent with the axial load and shear which are associated with the direction d i r e c t i o n of earthquake attack.
Mc o d e toy t^i e v alu e o£ Mc ^ for the ' ^ column at the re line
of the first floor b e a m s . I = the clear h e i g h t of the
n = 2
column
hk ~ t h e depth of the first floor beam
w h e r e = (1 - n/67) > 0.7 (5) is a reduction factor which takes the
number of floors, n, above the storey under c o n s i d e r a t i o n , into account.
T h e m a g n i t u d e s of the m a x i m u m lateral d i s p l a c e m e n t induced beam shear f o r c e s , V e, at each floor, w e r e obtained in Step 6.
T n e probability of all beams above a p a r t i c u l a r level developing simultaneously p l a s t i c hinges at flexural overstrength diminishes with the number of floors above that level. T h e reduction f a c t o r , R , m a k e s an approximate allowance for tXis.
Step 10 - Determine the total design axial load on each column for each of the two directions of earthquake attack from
e,max P + P + P *D LR eq and P . = 0.9P^ - P^ e , m m D eq
(6) 17) w h e r e P and P are axial forces due to dead ana reduced live loads•respectively.
Step 11 - Obtain the design m o m e n t s for columns above and below each floor from
col m Yo code ° -3 hbVc o l > ( 8 ) w h e r e OJ = the dynamic m o m e n t m a g n i f i c a t i o n
factor, the value of w h i c h is given in Fig. 6.
the beam overstrength factor a p p l i - cable to the floor and the direction of lateral loading under c o n s i d e r - ation
the depth of the beam which frames into the column
and R 1 + 0.55(a) - 1) (10 f' A
c g
1) < 1 (9) T h e procedure for the evaluation of column
d e s i g n shear forces is very similar to that u s e d in the capacity d e s i g n of ductile frames-1-. It reflects a higher degree of conservation b e c a u s e of the intent to avoid a column shear failure in any event. Case studies show that in spite of the apparent severity of E q s . (2) and ( 3 ) , shear require- m e n t s very seldom g o v e r n the amount of transverse reinforcement to be used in c o l u m n s .
is a design moment reduction factor appli- able w h e n
0.15 < f1 A c g
< 0.10
w h e r e Pe is to be taken negative w h e n c a u s i n g axial t e n s i o n .
These requirements are very similar to those recommended for columns of ductile f r a m e s1. Dynamic analyses of example hybrid
s t r u c t u r e s1 1 indicated that shear forces induced in the b o t t o m and top storey columns m a y exceed by a large m a r g i n the magnitudes predicted by elastic (Step 1) analyses, V a ^ - It m a y b e noted, h o w e v e r , that in hybrid s t r u c t u r e s , as Fig. 2 suggests, the computed static column shear f o r c e s , V ^ are often very small in these two
storeys.
Step 9 - Estimate in each storey the m a x i m u m likely lateral d i s p l a c e m e n t induced axial load on each column from
P = R EV eq v oe (4)
—
"i
5
- 1-0 J
F i g . 6 Dynamic M o m e n t M a g n i f i c a t i o n F a c t o r for Columns in Hybrid S t r u c t u r e s .
T h e steps in the d e r i v a t i o n of t h e column d e s i g n m o m e n t , M ^, are given in F i g . 7 as follows. T h e v a r i a t i o n of c o l u m n m o m e n t s d u e to code loading (Step 1 ) , M , , above and below a floor is shown w i t hc§ n i d e d l i n e s . T h e s e m o m e n t s are m a g n i f i e d t h r o u g h o u t the h e i g h t of the column to § M , w h e n b e a m s adjacent to the column d e v e J o p flexural o v e r s t r e n g t h s at their p l a s t i c h i n g e s . It is assumed that the m a x i m u m m o m e n t input from the two b e a m s , ^ , as shown in F i g . 7, cannot b e exceeded during a n e a r t h q u a k e . H o w e v e r , the d i s t r i b u t i o n of this t o t a l m o m e n t imput b e t w e e n the columns above and b e l o w the f l o o r , during the dynamic r e s p o n s e , is u n c e r t a i n . A l l o w a n c e for d i s p r o p o r t i o n a t e d i s t r i b u t i o n is made b y the dynamic m a g n i - fication factor OJ ^ 1.2. For e x a m p l e the e s t i m a t e d m a x i m u m m o m e n t for the u p p e r
column in F i g . 7, m e a s u r e d at t h e b e a m centre l i n e , is thus w <J>0 MC O ( i e. At the top of the b e a m , at the critical section o f t h i s c o l u m n , t h e m o m e n t is l e s s . The reduction d epe nds on the m a g n i t u d e of the column s h e a r force generated s i m u l t a n e o u s l y . For this purpose a c o n s e r v a t i v e a s s u m p t i o n is m a d e , w h e r e b y V n = ° *6 Vm a x = ° '6 vc o l <S t eP 8> • Hence the m o m e n t r e d u c t i o n a t the top of the b e a m b e c o m e s 0.5 h>
in F i g . 7. 2b vm i n = 0 -3 hb vc o l as shown W h e n the a x i a l load on the column p r o d u c e s small compression / i » e . pe < o . 1 f£ Aa, or results ifn net axial t e n s i o n , some y i e l d i n g of the column is not u n a c c e p t a b l e . Such columns should e x h i b i t suf fic ient ductility even w i t h o u t special confining r e i n f o r c e m e n t in the end r e g i o n s . Hence for this situation the design m o m e n t s are reduced by the factor Rjp given in Eq. (9) . This e x p r e s s i o n will give the same values that have b e e n r e c o m - m e n d e d1 for columns of ductile f r a m e s , p r o - vided that the value o f w is not taken larger than 1.2. The m i n i m u m value of 1 ^ is 0.72.
This will enable the a m o u n t of required tension r e i n f o r c e m e n t in exterior c o l u m n s , w h e r e this situation a r i s e s , to be reduced.
Because the v a l u e of the dynamic m o m e n t m a g n i f i c a t i o n factor for columns in hybrid
structures is r e l a t i v e l y s m a l l , i.e.ui £ 1.2, the above reduction of column design m o m e n t s w i l l seldom exceed 2 0 % . T o simplify c o m p u t a t - ions , the designer m a y p r e f e r to u s e = 1.0.
When the reduction f a c t o r , Rm, is used in determining the a m o u n t of column r e i n f o r c e - m e n t , the design shear Vc o l, obtained in Step 8, may also be reduced p r o p o r t i o n a l l y . Having obtained the critical design q u a n t - ities for each c o l u m n , i.e. Mc oj _ from Step 11 and Vc oi from Step 8, the r e q u i r e d flexural and shear r e i n f o r c e m e n t at each critical section can be found. B e c a u s e the design q u a n t i t i e s h a v e b e e n derived from b e a m o v e r s t r e n g t h s i m p u t , the a p p r o p r i a t e strength r e d u c t i o n factor for these columns is cj) = 1 . 01. End r e g i o n s of c o l u m n s n e e d further be checked to e n s u r e that t h e t r a n s - verse r e i n f o r c e m e n t p r o v i d e d satisfies the c o d e1 r e q u i r e m e n t s for c o n f i n e m e n t , s t a b - ility of v e r t i c a l r e i n f o r c i n g b a r s and lapped s p l i c e s .
Moment gradient with 0.6Vcof y y with Vcoly
' MCode
*o Mcode
Fig.
two beams at overstrength
™beam
7 The Derivation of Design M o m e n t s for C o l u m n s .
The design of columns at the b a s e , w h e r e the d e v e l o p m e n t of a p l a s t i c hinge in each column m u s t be e x p e c t e d , is the same as for c o lu m ns of ductile f r a m e s1.
Step 12 - Determine the appropriate gravity and e a r t h q u a k e induced axial forces on w a l l s .
In the example structure (Fig. 2 ) , it w a s implicitly assumed that lateral load on the b u i l d i n g does not introduce axial forces to the cantilever w a l l s . F o r this situation the design axial forces on the walls are
P§ = P p + l s 3 pL R or pe = 0,9 pn - G e n e r - ally the l a t t e r , w h e n considered together w i t h lateral load induced m o m e n t s , governs the amount of ve rt ic al wall r e i n f o r c e m e n t to be used.
If w a l l s are connected to columns via rigid- ly connected b e a m s , as shown for example in F i g . 3 ( a ) , the lateral load induced axial forces on the w a l l s are o b t a i n e d from the initial elastic analysis of the structure (Step 1 ) . Similarly this applies w h e n , instead o f ^ c a n t i l e v e r w a l l s , c o u p l e d s t r u c t - ural w a l l sz share with frames in lateral load r e s i s t a n c e .
Step 13 - Determine the m a x i m u m b e n d i n g m o m e n t at the base of each w a l l and design the necessary flexural r e i n f o r c e m e n t , taking into a c c o u n t the m o s t a d v e r s e
c o m b i n a t i o n w i t h axial forces on the w a l l . This simply implies that the r e q u i r e m e n t s of strength design be satisfied. The appropriate combination of actions is Mu =
^ o d e a n <3 Pu = Pe • Because the w a l l should m e e t the a d d i t i o n a l seismic r e q u i r e m e n t s specified by the c o d e1, the appropriate strength reduction factor to be used is cj) = 0 . 91, irrespective o f the level of axial c o m p r e s s i o n . The e x a c t a r r a n g e m e n t of b a r s w i t h i n the w a l l section a t the b a s e , as b u i l t , is to b e determined to allow the flexural o v e r s t r e n g t h of t h e section to b e e s t i m a t e d .
Ideal moment strength to be provided
Elastic moment pattern
• provided at base
Fig. 8 Design Moment Envelope for Walls of Hybrid S t r u c t u r e s .
Step 14 - W h e n curtailing the
vertical r e i n f o r c e m e n t in the upper storeys of w a l l s , p r o v i d e flexural resistance not less than g i v e n by the m o m e n t envelope in Fig. 8.
The envelope shown is similar to but not the same as that recommended for cantilever w a l l s1. It specifies slightly larger flexural r e s i s t a n c e in the top s t o r e y s . Its construction from the initial m o m e n t d i a g r a m , ^ o b t a i n e d from the elastic analysis in Step 1, may be readily followed in Fig. 8.
It is important to note that the e n v e l o p e is related to the ideal flexural strength of a w a l l at its b a s e , as b u i l t , rather than the m o m e n t required at that section by the analysis for lateral load. The envelope refers to e f f e c t i v e ideal flexural strength.
H e n c e v e r t i c a l b a r s in the w a l l m u s t extend by at least full development length beyond levels indicated by the e n v e l o p e .
The aim of this apparently conservative approach is to ensure that significant yielding w i l l not occur beyond the assumed h e i g h t , \] t of the plastic hinge at the b a s e . Thereby the shear strength of the w a l l in the upper storeys is also i n c r e a s - e d1 , and h e n c e reduced amounts of h o r i z o n - tal shear reinforcement may be used.
F i g u r e 9 compares w a l l m o m e n t d e m a n d s , encountered during the analysis for the 1940 E l C e n t r o and the 1971 Pacoima Dam
earthquake r e c o r d s , w i t h the m o m e n t
envelopes given in Fig. 8. T h r e e 12 storey b u i l d i n g s , in plan as shown in Fig. 2, w i t h w a l l s of 3.0, 3.6 and 7.0 m e t r e s l e n g t h , w e r e studied. All buildings w e r e designed in accordance with this capacity design p r o c e d u r e . While the e n v e l o p e s appear to provide considerable reserve flexural strength in the upper storeys d u r i n g the El Centro record, at various instants of the extreme (and unrealistic) Pacoima Dam e v e n t the analysis predicted the a t t a i n m e n t of the ideal flexural strength in m o s t s t o r e y s . Analyses showed, h o w e v e r , that curvature ductility d e m a n d s , e v e n d u r i n g this extreme e v e n t , were very small in the upper storeys. As part of the study of the effects of foundation c o m p l i a n c e , discussed in Section 2.4, these s t r u c t u r e s , w i t h p i n - ned base w a l l s , but otherwise identical w i t h the prototype s t r u c t u r e s , were also analysed for the El Centro record. It is seen in Fig. 9, that wall m o m e n t demands for the El Centro event in the u p p e r storeys are very similar to those experienced with fixed base w a l l s .
Step 15 - Determine the m a g n i t u d e of the flexural overstrength factor <j>n w, for each w a l l . This is the ratio of the flex- ural overstrength of the w a l l , Mu, as d e t a i l e d , to the moment required to resist the code specified lateral l o a d i n g , MP n H Q; b o t h moments taken at the base section of' a w a l l .
The m e a n i n g and puroose of this factor,
<f>o.w = ( M ° / Mc o d e,
b a s e \ls the same as that evaluated for beams in Step 7. Strictly, for walls there are two limiting values of o v e r s t r e n g t h , M ° , which could be considered.
These are the moments developed in the presence of two different axial load inten- s i t i e s , i.e. P ma x and Pe m ii r H o w e v e r , it is considered to be sufficient for the intended purpose to evaluate flexural o v e r - strength developed with axial compression on cantilever walls due to dead load a l o n e2.
Step 16 - Compute the wall shear r a t i o , This is the ratio of the sum of the shear forces at the base of all w a l l s , I V ,-j , , predicted by the analysis for
, w a l l , code ± - r - r—r—i—s = — r - ^ ~ r d e s i g n load, to the total design b a s e shear for the entire structure, V , . . .
- code,total The relative contribution of all w a l l s to the required total lateral load resistance is expressed as a matter of convenience by
5 w 15 20 MNm 0 fO 20 30 iOMNm 0 20 W 60 80 MNm WALL BENDING MOMENT
F i g . 9 Wall M o m e n t Demands Encountered During Earthquake R e c o r d s .
10
the shear ratio n
i> = a v ,
i=l i ,wall, code code,total base (10) It a p p l i e s strictly to the base of the s t r u c t u r e . A s Fig. 2(c) s h o w s , such a shear r a t i o w o u l d rapidly reduce with h e i g h t , and n e a r the top it could become n e g a t i v e . This indicates t h a t the parameter ^ is a c o n v e n - ient b u t not unique m e a s u r e to quantify the share of w a l l s in the total lateral load r e s i s t a n c e .
Step 17 - Evaluate for each w a l l the d e s i g n shear force at the base from
Vw a l l , b a s e = K *o,w Vw a l l , c o d e
and 1 + U
1H-
(12)w h e r e u\T is the dyanmic shear m a g n i f i c a t i o n factor relevant to cantilever w a l l s , obtained from
O Jv = 0.9 + n/10 when n < 6 (13a) or a) = 1 . 3 + n/30 < 1.8 w h e n n > 6
V (13b)
1.7, = 0.6 and w* = 1.42, the"ideal shear strength w i l l need to be Design criteria for shear strength will often be found to be c r i t i c a l . At the base the thickness of w a l l s may need to be increased on account of Eq. ( 1 1 ) , and because of the maximum shear stress limit- ations of NZS 3101. Typically w h e n using Grade 380 vertical w a l l r e i n f o r c e m e n t in a 12 storey hybrid s t r u c t u r e , w h e r e the w a l l s have b e e n assigned 6 0 % of the total base shear r e s i s t a n c e , it w i l l be found that w i t h
*p,w." X-6' wv the
i "
^wall = 2.27 VC Q d e. In c o m p a r i s o n , the ideal shear strength of a w a l l , proportioned w i t h strength rather than capacity design p r o c e d u r e s , would be Vwall = vc o d e / ° *8 5 = 1.18 VC Q d e. Thus Eq. (11) implies very large apparent reserve strength in shear.
Analyses in cases s t u d i e d1 1 consistently predicted, h o w e v e r , shear forces which are often 3 0 % larger than those required by Eqn. ( 1 1 ) . Of all the aspects of this
proposed design strategy, the estimation of w a l l shear forces w a s found to be the least satisfactory. Some relevant issues are discussed subsequently in Section 4.4.
Step 18 - In each storey of each w a l l , provide shear resistance not less than that given by the shear design envelope of Fig. 1 0 . w h e r e n is the number of storeys above the
b a s e .
The approach developed for the shear design of w a l l s in hybrid structures is an e x t e n - sion of the two stage methodology used for cantilever w a l l s1'9.
In the first s t a g e , the design shear force is increased from the initial (Step 1) value to that corresponding with the development of a plastic hinge at flexural overstrength at the b a s e of the w a l l . This is achieved w i t h the introduction of the flexural overstrength factor, <j>Q w, obtained in Step 1 5 . In the next s t a g e a l l o w a n c e is made for the
amplification of the base shear force during the inelastic dynamic response of the
structure. While a plastic hinge develops at the base of a w a l l , due to the c o n t r i - b u t i o n of h i g h e r modes of v i b r a t i o n , the centroid of inertia forces over the height of the b u i l d i n g m a y be in a significantly lower position than that predicted by the c o n v e n - tional analysis for lateral l o a d s9. The larger the number of s t o r e y s , the more important is the participation of h i g h e r m o d e s . The dynamic shear magnification for c a n t i - lever w a l l s , oov, given in Eq. ( 1 3 ) , m a k e s allowance for this p h e n o m e n o n9. The values so obtained agree w i t h those recommended in NZS 3101.
It h a s also b e e n f o u n d3, that for a given earthquake r e c o r d , the dynamically induced base shear forces in w a l l s of h y b r i d s t r u c t - ures increased w i t h an increased p a r t i c i - pation of such w a l l s in the resistance of the total base shear for the entire s t r u c t - ure . Wall participation is quantified by the "shear r a t i o " , obtained in Step 1 6 . The e f f e c t of the "shear r a t i o " upon the m a g n i f i c a t i o n the m a x i m u m wall shear force is estimated by Eq. ( 1 2 ) . It is seen that w h e n i|i = 1, = wv.
O'SKvoli.bose
Fig. 10 Envelope for Design Shear Forces for Walls of Hybrid Structures
As Fig. 2(c) s h o w s , shear demands predicted by analyses for static load may be quite small in the upper half of w a l l s . As can be expected, during the response of the building to vigorous seismic e x c i t a t i o n s , much larger shear forces may be generated at these upper levels. A linear scaling up of the shear force diagram drawn for static load, in accordance with Eq. ( 1 1 ) , would give an erronous prediction of shear demands in the upper s t o r e y s . Therefore from case studies the shear design envelope shown in F i g . 10 was developed. It is seen that the envelope gives the required shear strength in terms of the base shear for the w a l l , which w a s obtained in Step 1 7 .
Figure 11 presents some results of the relevant s t u d y1 1 of a 12 storey building.
It is seen that the shear design envelope is satisfactory w h e n structures w i t h relatively slender w a l l s , w i t h ip < 0.57, were subjected
11
- 8 - 5 - 4 - 2 0 2 i 6 MN WALL SHEAR FORCE
WALL SHEAR FORCE
Fig. 11 Predicted Shear Demands for Different Walls in a 12 Storey Hybrid S t r u c t u r e . to the El Centro e x c i t a t i o n . The shear r e s -
p o n s e of the structure with 7 m w a l l s is less satisfactory in the lower s t o r e y s . A s may be e x p e c t e d , the predicted demand for shear in pin b a s e d w a l l s * is l e s s , particularly as the length of the w a l l s , £w, increases . Shear loads p r e d i c t e d for the Pacoima event w e r e found to consistently exceed the suggested design v a l u e s .
W i t h the aid of the shear design e n v e l o p e , the r e q u i r e d amount of horizontal (shear) w a l l r e i n f o r c e m e n t at any level may be readily found. In t h i s , attention m u s t be p a i d to the different approaches u s e d1 to e s t i m a t e the contribution of the concrete to shear s t r e n g t h , vc, in the potential p l a s t i c h i n g e and the elastic regions of a w a l l . In the potential plastic hinge r e g i o n , e x t e n d - ing above the b a s e , as shown in Fig. 1 0 , the m a j o r p a r t of the design shear, Vw a^ ^ , w i l l need to be assigned to shear r e i n f o r c e - m e n t . In the upper (elastic) parts of the w a l l , h o w e v e r , the concrete may be relied o n1 to contribute significantly to shear r e s i s t a n c e , allowing considerable reduction in the d e m a n d for shear reinforcement.
Step 19 - In the end regions of each w a l l , over the assumed length of the p o t e n t - ial p l a s t i c h i n g e , provide adequate t r a n s -
*Note that the shear r a t i o , K given by Eq.
"(10), is n o t applicable to pin based w a l l s .
verse reinforcement to supply the required confinement to parts of the flexural com- pression zone and to prevent premature b u c k l i n g of vertical b a r s .
These detailing requirements for ductility are the s a m e1 as those recommended for c a n t i - lever and coupled structural w a l l s . Recent experimental s t u d i e s1 1i n d i c a t e d , h o w e v e r , that current code r e q u i r e m e n t s , r e l e v a n t to the region of confinement w i t h i n wall s e c t i o n s , should be amended. For this r e a s o n , although presented e l s e w h e r e1 0, the suggested improvement in the procedure is restated h e r e .
The proposed approach to the confinement of w a l l sections rests on the p r e c e p t that concrete should be laterally confined w h e r e - wherever compression s t r a i n s , corresponding with the expected curvature ductility demand on the relevant section, exceed 0.004. The
strain profile shown shaded in F i g . 12 indicates the ultimate c u r v a t u r e , <j>u, which m i g h t be necessary to enable the estimated displacement ductility, y/\f for a p a r t i c u l a r hybrid structure to be sustained, w h e n the concrete strain in the extreme compression fibre theoretically reaches the magnitude of 0 . 004 . This strain profile is associated w i t h a neutral axis d e p t h , cc. An estimate for this critical neutral axis d e p t h , cc, may be m a d e1 with
cc = 0.10 4>* S £w (14)
12
a c
F i g . 12 Strain Profiles for Wall S e c t i o n s . w h e r e &w = length of w a l l ,
S = structural type f a c t o r2 , and (p* = global overstrength f a c t o r , which is the ratio of the total resistance of the hybrid structure to overturning m o m e n t , including the contributions of axial forces in columns and w a l l s and those of plastic h i n g e s at the base of all columns and w a l l s , evaluated at levels of flexural o v e r -
strength , to the corresponding overturning m o m e n t due to code specified lateral static l o a d i n g / When the total strength provided by yielding regions of the structure m a t c h e s very closely that required by the lateral code loading, the value of <J>* w i l l n o t be less than 1.4.
To achieve in a w a l l the same ultimate curvature w h e n the computed neutral axis d e p t h , c, is larger than the critical v a l u e , cc, as Fig. 12 s h o w s , the length of w a l l
section subjected to compression strains larger than 0.004 , b e c o m e s ac. It is this length over w h i c h the compressed concrete needs to be confined. From the geometry shown in Fig. 1 2 , a = 1 - cc/ c .
Because it h a s been found in t e s t s1 0 , 1 1 that, after reversed cyclic loading, o b s e r v e d neutral axis depths tend to be larger than those predicted by conventional section a n a l y s e s , it is suggested that the length of confinement, ac, be derived from
a = 1 - 0.7 cc/ c > 0.5 (15) w h e n e v e r cc/ c < 1. W h e n c is only a little
larger than cc, a very small and impractical value of a w o u l d be obtained. In line w i t h current r e q u i r e m e n t s1, it is suggested that in such cases at least one half of the theoretical compression zone be c o n f i n e d . The tests q u o t e d1 0 also indicated that the amount of confining reinforcement specified in the c o d e1 is likely to be a d e q u a t e .
4. ISSUES REQUIRING FURTHER STUDY The proposed capacity design p r o c e d u r e and the accompanying discussion of the
behaviour of hybrid s t r u c t u r e s , p r e s e n t e d in the previous section, are by necessity restricted to simple and regular structural
systems. The variety of w a y s in w h i c h w a l l s and frames may be combined may present problems to w h i c h a satisfactory solution w i l l r e q u i r e , as in many other s t r u c t u r e s , the application of engineering judgement.
This may necessitate some rational a d j u s t - ments in the o u t l i n e d 19 step p r o c e d u r e .
In the following, a few situations are mentioned w h e r e such j udgement in the application of the p r o p o s e d design method- ology w i l l be necessary. Some directions for premising approaches are also suggested.
4.1 Gross Irregularities in the Lateral Load Resisting System.
It is generally recognised that the larger the departure from symmetry and regularity in the arrangement of lateral load r e s i s t - ing substructures w i t h i n a b u i l d i n g , the less confidence should the designer have in predicting likely seismic response.
Examples of irregularity are w h e n w a l l dimensions change drastically over the height of the building or w h e n walls term- inate at different h e i g h t s , and w h e n s e t - backs occur. Symmetrical positioning of walls in plan may lead to gross e c c e n t r i - cities of applied lateral load with respect to centres of rigidity.
4.2 Torsional Effects
Codes make simple and rational provisions for torsional e f f e c t s . The severity of torsion is commonly quantified by the distance between the centre of rigidity
(or stiffness) of the lateral load resisting structural system and the centre of m a s s . In reasonably regular and symmetrical buildings this distance (horizontal e c c e n t - ricity) , does not significantly change from storey to storey. Errors due to inevitable variations of eccentricity over b u i l d i n g h e i g h t are thought to be compensated for by code specified amplifications of the
computed (static) e c c e n t r i c i t i e s . The corresponding assignment of additional lateral load to resisting e l e m e n t s , p a r t i c - ularly those situated at greater distances from the centre of rigidity (centre of horizontal t w i s t ) , are intended to compen- sate for torsional e f f e c t s . Because minimum and maximum e c c e n t r i c i t i e s , at least with respect to the two principal directions of earthquake attack, need to be considered, the structural system, as designed, will possess increased translational rather than torsional r e s i s t a n c e .
It w a s emphasised that the contributions of w a l l s to lateral load resistance in hybrid structures usually change dramatically over the height of the b u i l d i n g . An example w a s shown in Fig. 2 ( c ) . For this reason the position of the centre of rigidity may also change significantly from floor to floor.
For the purpose of illustrating the v a r i - ation of eccentricity w i t h h e i g h t , consider the example structure shown in Fig. 2 ( a ) , but slightly m o d i f i e d . Because of symmetry, torsion due to variation in the position of the centre of r i g i d i t y , does n o t a r i s e . A s s u m e , h o w e v e r , that instead of the two symmetrically positioned w a l l s shown in Fig. 2 ( a ) , two 6 m long w a l l s are placed side by side at 9.2 m from the left hand end
