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SHEAR WALL STRUCTURES—DESIGN AND CONSTRUCTION PROBLEMS
SHEAR WALL STRUCTURES-DESIGN AND CONSTRUCTION PROBLEMS WILEM W. FRISCHMANN and SUDHAKAR S. PRABHU Partner and Associate, C. J. Pell and Partners , Consulting Engineers, London. SUMMARY An approximate and an exact method for the analysis of multi-storey interconnected shear walls are described and compared. The results of some of the investigations carried out to determine the effects of perforations on the stiffness and strength of shear walls are given. Various modern techniques of construction are briefly discussed together with their effects on the basic design assumptions. The effects of temperature deformation and of differential stressing of shear walls on the rest of the buildings are also considered.
lNTRODUCTlON
In the design of tall buildings special consideration must be given to providing sufficient stability in all directions against lateral forces, i.e. wind, possible earth tremors and blasts. These forces can produce critical stresses in the structure, set up vibrations in the structure and the cladding and , in addition, can cause lateral sway of the building, which could reach a point of discomfort to the occupants. As discussed in the article on "Planning Concepts using Shear Walls", this restraint against lateral forces is best provided by developing the inherent stiffness of the enclosures to service areas, both from the considerations of cost and the method of construction. Such enclosures, when constructed in concrete, should be located so as to carry as much of the vertical dead weight of the floor system as possible. The vertical loads have the effect of pre-loading the core, which should be so proportioned that (a) the dead weight overcomes the tension induced in the core by horizontal loads, and (b) the increase in compressive stress, solely due to wind forces, is approximately 25 %. (This being the permissible increase as per c.P. 114.) With this optimum arrangement the wind stability can be achieved very economically. When the enclosures are constructed with braced shear walls in steel, the pre-loading is helpful only from considerations of anchorage at the foundations. With these optimum arrangements the lateral stability is achieved economically in contrast with the traditional structural frame where the materials 83
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W. FRISCHMANN AND S. PRABHU
are used uneconomically. Fig. I illustrates the structural efficiencies of traditional frames and various shear wall systems , using the same quantity of materials. (a) (b) (c) (d) (e) (f)
Shows the traditional frame . Shear wall the full depth of the building. Shear wall with storey-high opening. Shear wall with door high opening. Shear wall reduced in thickness and the material placed to form flanges. As in (e) above but with door high openings.
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I. Comparison of d ifferent forms of structure for lateral stiffness.
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85
DESIGN AND CONSTRUCTION PROBLEMS
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(g) As in (e) above but the flanges connected with the web walls each of half the thickness of that in (e) to form a core. (h) As in (f) above but flanges connected with two web walls, each of half the thickness of that in (f) above to form (j), (k), Facade wall with window openings.
acore:----
From a structural stability point of view the architectural planning should aim at providing arrangement (b), (e) or (g). (g) has the advantage of increased stiffness in all directions including torsional stiffness. In practice, however, the structural engineer has to accept a compromise arrangement of (d), (f) or (j). The floors in shear wall structures are made to act as deep horizontal beams, transmitting the lateral loads to the vertical stiff shear walls, which , in turn transmit them, to the foundation. The foundations are designed to distribute highly concentrated loads over a sufficient area to prevent local overstressing of the soil.
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W. FRISCHMANN AND S. PRABHU
ANALYSIS OF SHEAR WALLS
The stiffnessof the floor slabs in the horizontal direction is large compared to the stiffness of various shear walls or a group of walls forming a core; and the simplified assumption is made that the horizontal wind loading is carried in proportion to the stiffnesses of the shear walls. The walls can then be designed as cantilevers (or Vierendeelgirders if interconnected shear walls) fixed at the foundation level. The yielding of the soil should also be allowed for when designing a statically indeterminate (e.g. Vierendeel-like system) interconnected shear wall, as it could alter considerably the pattern of stresses throughout the wall. The interconnected shear walls should be designed taking into account the stiffnessof the connection and the connecting member and not as two separate cantilevers hinged together by the floor, since the connecting member has a considerable effect in reducing the stresses in the wall. The following are two of the methods which the authors have used for the analysis of interconnected shear walls and have found to be quick and reasonably accurate.
1. The Equivalent Column Method The method consists in replacing the frame by an equivalent column, whose stiffnessequals the sum of all the column stiffnesses, and by restraints, applied at each floor level, equivalent to the sum of the beam stiffnesses. The condition of equilibrium of all external and internal forces acting on the equivalent column is expressed as a second-degree differential equation. The nature of the mathematical function is such that with increasing height, that is, with a greater number of restraints, the accuracy of the solution also increases. The general solution may be applied to any number of bays and storeys. The following assumptions are made. (a) All frames are fully fixed at the bases of the columns. (b) All columns deflect, due to the horizontal loads, and remain parallel to each other, that is, it is assumed that there is no elastic shortening of the beams. (c) The sectional properties of columns and beams in each bay are constant for the full height of the structure. However, variations may occur from bay to bay in a multi-bay structure. If the sectional properties of the beams (in each bay) or columns vary with the height of the building, an average value may be taken. This should yield a reasonably accurate answer and it is left to the designer to decide whether further analysis is necessary.
DESIGN AND CONSTRUCTION PROBLEMS
87
(d) The storey height is constant throughout the height of the building. If this is not the case, an average may again give reasonably accurate results. (e) Axial shortening of members and deflection due to shear are neglected. (f) The modulus of elasticity E of the concrete is assumed to be constant throughout the height of the building.
2. The Influence of Coefficient Method This is essentially the method of strain-energy and consists in reducing the structure to a statically determinate system by the introduction of releases at suitable points. Unit constraints are then applied at these points, resulting in a number of simultaneous equations equalling the order of statical indeterminacy. The method permits account to be taken of the variation of inertia of the beams and columns, and of the modulus of elasticity due to the use of mixtures of concrete differing throughout the height of the building. The following assumptions were made in the analysis. (a) The strain-energy due to axial and shearing forces in the structure is neglected. (If thought necessary this could be taken into account.) (b) All horizontal loads are assumed to act at panel points. The methods of analysis are fully described in Ref. 1. The comparison of moments in _the wall analysed by the two methods is shown in Fig. 2 which has been reproduced from the above paper. It should be noted that although for illustration a simple example was chosen, both these methods can be used to analyse several shear wall sections of varying depths and connections. The results of some of the investigations carried out to determine the effects of perforations on the stiffness and strength of shear walls are given in Fig . 3. The variables considered are the width of the opening and the depth of the connecting beams. The results also show the benefits of the homogeneous connection as against the pinned connection between the floor and walls. It appears that the stress in the shear walls is not unduly affected if the width of the openings is limited to approximately 15 % of the total width of both shear walls and the depth of the connecting beam is greater than 20 % of the storey height. For this reason if the perforations are restricted within the limits suggested the shear wall may be considered as fully homogeneous. The adequacy of the connecting members should, however, be assessed from the points of view of both moment and shear capacity. More development work is required before precise recommendation can be stated.
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W. FRISCHMANN AND S, PRABHU
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2. Comparison of moments of shear walls analysed by the method of influence coefficients and the equivalent column method.
DESIGN AND CONSTRUCTION PROBLEMS
89
WIND AND VIBRAnON EFFECTS
Wind tunnel studies have shown that air flow distribution produces a complex three-dimensional flow field in the region of a tall block. The deceleration of the air stream against the building creates a downward flow, frequently producing strong eddies at pavement level. At the same time, on the leeward face of the building the air flow close to the surface of the building is upward, creating problems of rain and snow penetration through cladding. Wind Loading
The wind load forces depend on the mean hourly wind speed, the estimation of an appropriate gust factor, shape and pressure coefficients and the effects of local topography. The wind forces are normally applied on the building as an equivalent uniformly distributed load for its full height. Natural wind has the potential to cause vibration of cladding or sway of the whole building, depending on its dynamic properties. These effects can be caused by either the response of the building or an element of the building to the just spectrum of the wind, or a self-induced aerodynamic instability which occurs in steady wind conditions. Tall buildings built in the last decade, with modern techniques, have a much lower frequency and much reduced damping than previous constructions and require careful checking as they are liable to gust induced sway. The observed effect of aerodynamic instability-which occurs say, on iced transmission lines-is unlikely except in cases of very low damping, since normally wind speeds required to excite this type of instability are outside the wind speeds encountered. The vortex-shedding type of instability is of great significanceto the design of tall buildings, as a vortex-shedding frequency can coincide with the natural frequency of the tower at normal wind speeds, with the possibility of resonant vibration, the amplitude of the sway depending on the structural damping of the tower. If the energy imparted by the exciting forces equals the losses due to internal damping, the amplitude of vibration would increase until failure occurs due to stresses. Slender towsrs have to be checked for such instability, which can be adjusted in the design by altering the natural frequency, the damping or the sectional properties of the building. Increase in frequency has the effect of raising the critical wind speed for onset oscillation. The tower is normally regarded as a cantilever encastre at the foundation, and the building could be excited in the lateral, torsional and longitudinal modes. Effects of Vibration
These have to be considered in relation to the effect of the stresses induced on the fatigue life of the structure, and the effect of vibration on people in the building.
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W. FRISCHMANN AND S. PRABHU
The human body is very susceptible to vibration, and the amplitude which can be tolerated is considerably less than would be permitted purely from stress considerations. A number of research workers have published limit values for intensity of vibration, depending on amplitude and frequency.!" The accepted amplitude of transverse lateral vibration is considerably less than that anticipated from static wind loading on the building, and this, therefore, must be the criterion for the design. Much of our understanding of the effects of wind on buildings has come from scaled models tested under controlled conditions in wind tunnels, reproducing such effects as pressures on cladding, air flow studies, aerodynamic stability and minimum structural damping. Effects of Earthquake Earthquake forces result from the vibratory motion of the grounds on which the structure is supported. The ground vibrates both vertically and horizontally, but the vertical component has a negligible effect on the structure due to its inherent vertical strength. The effectof earthquake forces on design of structures at the present time are empirical, bas-ed on the performance of structures in earthquake zones. In some countries building .regulations require that, in addition to the normal wind forces, the structure be designed to withstand a minimum total lateral seismic force which is assumed to act non-concurrently in the direction of each of the main axes of the building, equivalent to a given percentage of the weight of the building. The percentage generally varies from I to 13 %, depending on the location in relation to the earthquake zones. In addition, walls and partitions are anchored to the rest of the building to resist a force of 20 % of their weight and members such as parapets, ornamentation and cladding are usually required to be anchored for 100% of their weight. DEFLECTION
The maximum deflection due to lateral loads must be limited so that it does not cause discomfort to the occupants especially near the top of the building (as discussed previously) and prevents damage to the remaining structural elements and finishes, e.g. curtain walling, partitions, etc. The calculation of deflection should also take into account the effect of the rotation of the foundations due to compressibility of the soil. The following formula may be used for evaluating the maximum deflection at the top of the building of the example shown in Fig. 2 : wh4 h2 n Ii = maximum deflection = 8E I - 2E l 2 .2 aMi2n - a), d d n a=l
DESIGN AND CONSTRUCTION PROBLEMS
91
where w is the uniformly distributed horizontal loading, h is the total height of the shear wall, n is the number of storeys, I is the moment of inertia of each shear wall, M; is the moment a1 the ath storey due to the shear force Xa, and Ed is the dynamic modulus of elasticity (see Ref. 3). In the above formula the shear deformations have been ignored. The allowable deflection figure that seems generally in use in the United States for braced steel frames is due to H. V. Spurr's' and varies between h/500 and h/750, where h is the height to the building. The report on The Testing of Structures's' states that there is very little basis for any proposals and suggests a limit of lateral movement of 1/1200. In Hong Kong, where typhoons are frequent, a great deal of experience is available for building over 20 storeys. (61 It appears that limiting the height, depth ratio to 10 for a single shear wall and to 15 for a core (shear box), has proved satisfactory. This can also be expressed as a function of the height of the building as follows: ~
wh4 8E dl
wh2 h2 . -2 4Edl
= -- = -
M=L I y
where
Mh 2 f h2 = -- = -.-, 4Edl
Y 4Ed
Assuming concrete 4000 psi, works cube strength at 28 days,
Ed = 4 j(permissible)
Then
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,
u
=
=
X
106 lb/sq, in.
1470 x 1-25Ib/sq. in.,
h2 1470 x 1-25 h/(2 x 10) 4 x 4 X 106
h = 434Thus the allowable deflection for single shear walls is approximately h/430. For cores (or shear boxes) it can be shown that the corresponding deflection is approximately h/300. There are no specific bye-law requirements for maximum permissible deflections and if conservative assumptions are made this could significantly increase the structural cost. In our opinion an approximate value of h/350 to h/400 appears to be reasonable, if the deflection calculations are based on bending (shear deformations if significant) and the effect of rotation of foundation, and provided it has been checked against the tolerable amplitude acceptable to the occupants.
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W. FRISCHMANN AND S. PRABHU
METHODS OF CONSTRUCTION
The overall speed of construction for a building, and hence the cost, largely depends on the time taken for the construction of the shear or core walls. These should therefore be made as simple and uniform as possible so that a slipform or a mandrel type internal form of shuttering (to be described later) may be used. These methods have been found time and labour saving in practice. With in situ techniques the wall thickness should preferably be such as would carry all the loads with nominal reinforcement so that complicated arrangements of horizontal links to prevent lateral buckling are not required. In practice, however, it is very rarely that this ideal is achieved and the engineers are forced to accept high percentages of reinforcement, in many cases as high as 8 %, which is the maximum permissible according to the Code of Practice. The use of prefabricated welded reinforcement ladders has been found of value in such cases. The elements of the shear walls are normally found to carry the highest work load of the structure, and the construction cycle is almost entirely controlled by the speed at which the core can be built. Too high a concentration of workload within a small area will delay the cycle, so that in general a singlemassivecore should be avoided in favour of two smaller core elements. Accepting the premise that the cores are to be of in situ concrete construction, the formwork and fixingof reinforcement will control the cycle. Simplification of the core shape will greatly assist in raising output, and the same attentions should be given to this point where normal formwork is to be used as is invariably given where slipform is to be used. Any shape with a number of small closed cells should be avoided where possible, the fourth wall of ducts, for example, being subsequently constructed in brickwork. Where it is necessary to design lift shafts with four load-bearing walls, construction time may be saved by the use of mandrel type internal forms. Reinforcement should be limited to single storey lifts to avoid erection and dismantling of scaffolding during fixing. Time may also be saved by the use of prefabricated reinforcement panels for core and column construction. Cages should be restricted to 7 or 8 ft width for transport or handling purposes, and adequate bracing rods must be added to provide rigidity. The panels should be jig-assembled to give accurate bar spacing and neat lapping of vertical rods. The following construction methods can be used with advantage on tall buildings. Sliding Formwork
The core can be erected first by the use of continuously sliding shutters operated by hydraulic (oil) jacks. The shutters rise at the rate of about I ft
DESIGN AND CONSTRUCTION PROBLEMS
93
per hour. The reinforcement is fixed first in advance of the shutter. Concrete is deposited in the shutter from a hopper which is lifted by the main crane. Erection of the complete core takes only a few weeks. When the concrete has hardened an auxiliary crane may be erected on the top of the core by means of the main crane. The following points should be considered carefully. (i) The quantity of reinforcement in some cases may be so high that it may not be possible to keep the formwork moving. (ii) Difficulties may arise in supporting the climbing crane on recently poured concrete unless the crane is very light. (iii) The initial erection and subsequent dismantling of the sliding shutter may be time-consuming. (iv) Remedial work tends to be higher than for construction with normal formwork. It is very difficult to ensure that inserts do not become displaced. (v) Finishing trades may be introduced later than with normal construction.
Mandrel Form of Construction A constant internal shaft size is designed for the full height of the building, the variation in thickness of the walls taking place on the outside. The shafts are rectangular in plan with splays along the' corners. The splays in addition to their structural significance facilitate the shrinking of the forms for striking. The mandrel forms are of fully framed box units with an accurately made steel framework lined with high quality plywood reproducing the internal configuration of the shafts. The mandrel formers are constructed to the full storey height plus a projection above the general floor level (see Fig. 4). The mandrel formers can be climbed by the tower crane in one piece without the necessity of dismantling and can be used for the full height of the building with something like 25-30 re-uses, requiring only cleaning down and re-oiling after each successive floor cycle. PRECAST CONSTRUCTION
The success of the precast systems will very much depend on the design of the joint between panels forming the shear wall. The joints must be designed: (a) To provide adequate stability during erection, possibly in conjunction with temporary propping. (b) To reduce crane hook times to a minimum. (c) To enable the jointed members to accept stresses at an early date. (d) To enable the dimensional inaccuracies and tolerances of the units in manufacture and erection to be absorbed.
94
w.
FIG.
FRISCHMANN AND S. PRABHU
4. Mandrel former in position before casting the floor.
Where the dead loads are sufficient to counterbalance the stresses induced in the walls due to wind, etc., simple bedding joints may be used. If, however, the dead loads are insufficient, the precast units must be connected with full tension strength joints such as welding projecting reinforcement bars after erection, prestressing plated joints, grouted joints, coupler joints or some form of sleeved joint illustrated in some detail by W. W. Frischmann elsewhere.(7) When stability depends on the co-action between panels, the vertical joints must be designed to transfer the shear forces. In addition, since the vertical joints are invariably filled with concrete of high water-eement ratio, the joint material shrinks, which results in very high deflections under wind loading as, at the commencement of taking up the load, the panels tend to deflect as independent units. To ensure satisfactory co-action the joints are often corrugated, toothed or reinforced with stirrups to ensure mechanical connection.
DESIGN AND CONSTRUCTION PROBLEMS
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FOUNDATIONS
With the knowledge and experience of soil mechanics large buildings can now be economically founded on relatively poor soils. Sites can, therefore, be better utilized than was possible in the past when foundation difficulties were often a major restriction. To determine the characteristics of the soil and predict its possible behaviour under the final loading, samples of soil are tested in the laboratory. From its mechanical properties it is then possible to estimate the probable distribution of stress in the soil, the settlement, and the sub-structure most suited to support the building. In proportioning foundations the allowable bearing pressure must be chosen to provide an adequate factor of safety against shear failure, and to ensure that both the overall and the relative settlements are within tolerable limits. On non-cohesive soils, e.g. sand and gravels, since the water can move freely the settlements are largely completed by the end of the construction. On cohesive soils, e.g. clays, the spaces between the particles are very small and the movement of water under pressure can only take place slowly. Theoretically consolidation can take place almost indefinitely, although perceptibly for some years only. It is this long, continued settlement which causes so much trouble with structures and finishes. A good deal of experience has been gained in big cities like London in founding tall blocks, in forming deep basements andin stabilizing neighbouring buildings against possible lateral and vertical movements. The London area is underlain by a great depth of clay which increases in strength and decreases in compressibility with increased depth. The choice of foundation for tall blocks therefore lies mainly between a raft, a buoyant foundation, a piled foundation or a combination of these. The principle of the buoyant foundation which has been applied to towers in London consists of excavating sufficient earth for the sub-structure to balance, as far as possible, the weight of the buildings and thus reduce the average settlement to a minimum. The practical difficulties of forming excavations deep enough to make a full buoyant foundation are expensive and normally, therefore, a compromise has to be accepted. This consists of either using a raft at, say, the subbasement level, extending well beyond the outline in plan of the tower block, or a raft placed at a shallower depth used in conjunction with piles which transfer the loads into the stiffer and less compressible soil. Large diameter augered piles transferring loads, partly in friction, partly in end bearing, have been found to be very economic. The bases of these piles can be enlarged by underreaming, if required. The function of a raft is to spread the load over a large area in order to reduce the bearing pressure to the allowable limit. Normally it is made stiff
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W . FRISCHMANN AND S. PRABHU
enough either by itself or in combination with the sub-structure, so as to distribute the load and to reduce the differential settlements in the structure to acceptable values. MODEL ANALYSIS
The behaviour of shear wall structures for different loading conditions can be studied by using advanced theoretical analysis supplemented by model testing. This would help to determine the most suitable structural form and also to predict the behaviour of the structure at working and failure load. Thus the engineer can assess the true factor of safety. Much of the tedious work associated with theoretical analysis is simplified by the use of electronic digital and analogue computers. Less time is spent on the actual computingwork-thus leaving more time for the proper detailing of the structure to simplify site- operations. Testing can be carried out on models varying from Ijl5th to Ij50th scale, constructed from perspex, cement mortar, steel, celluloid, aluminium, etc., depending on the type of material to be used in the final structure and the accuracy of the results required. Examples where model testing has been used are the Pirelli Building in Milan and the Co-operative Insurance building in Manchester. The savings resulting from carrying out such tests more than offset the cost of model testing, since significant reductions in structural costs can be made by using a more appropriate factor of safety.
SECONDAR Y STRESSES
Iflarge massesof concrete are cast in members such as rafts, beams, columns, etc., care should be taken that the heat of hydration is not so high as to overheat the concrete. This can be achieved by careful positioning of construction joints, using low heat cements or cooling the water used in the mix. One of the major secondary stresses that arises in tall blocks is caused by the temperature differential between the exposed outside and the heated or cooled inside parts of the structure. This is specially so in air-conditioned buildings. The relative expansion with respect to the internal columns or cores due to temperature gradient can be considerable for very tall structures. The differential displacement is maximum at the top and decreases towards the bottom. If the displacement is beyond the allowable limits, cracking of finishes in the upper floors may occur. This displacement further induces moments and shears in the floor slab, and in addition a considerable increase in axial load and moments in the columns. In addition, the effect of differential stressing of various vertical members resulting in unequal strains and warpage of the floor slab should be investigated.
DESIGN AND CONSTRUCTION PROBLEMS
97
In particular, this can be of great significance when highly stressed perimeter columns are used in conjunction with shear walls which may not be so highly stressed under normal vertical loads, due to reserve required for wind loading. Reference 8 states how this problem was overcome in the design of the World Trade Centre which will be one of the world's tallest buildings. In a building where shear walls or cores are designed to resist the lateral forces the effect of their horizontal deflection on the remaining structure must be carefully investigated. REFERENCES
w. W., PRABHU, S. S., and TOPPLER, J. F., Multi-storey frames and interconnected shear walls subjected to lateral loads, Concrete and Constructional Engineering, June and July 1963. STEFFENS, R. J., The assessment of vibration intensity and its application to the study of building vibrations. National building studies. Special Report No. 19, D.S.tR., Building Research Station. H.M.S.O. 1952. B.S. Code of Practice C.P.115: 1959. The Structural Use of Prestressed Concrete in Buildings. SPURR, H. V., Wind Bracing' in High Towers, McGraw-Hili, 1939. Report of a committee on The Testing of Structures, The Institution of Structural Engineers, September 1964. PHILCOX, K. T., Some recent developments in the design of high buildings in Hong Kong, Structural Engineer, October 1962. Vol. 40, No. 10, pp. 303-323. FRISCHMANN, W. W., A Review of Structural Form-Structural Design of Tall BUildings, Prestressed Concrete Development Group, University College of South Wales and Monmouthshire. December 1964. Engineering News Record, April 1964.
1. FRIScHMANN,
2. 3. 4. 5. 6. 7.
8.
DISCUSSION J. E. CROFTS : Mr. Crofts (partner Nachshen, Crofts and Leggatt, Consulting Engineers) referring to design and construction problems, said that when tension occurred at the bottom of shear walls, the actual load factor was not obvious. Tension arose from subtraction of wind load stresses from dead load stresses and was generally the small differences of two large numbers. A modest increase in assumed wind speed or a small change in the distribution of wind load between shear walls would cause a large change in tension. This large tension increase would not generally be covered by the difference between working and ultimate stresses in the tension members. Special accuracy in defining dead and wind loads on the shear wall is justified in such a case so as to define the tension requirements with economy. Tall buildings founded in rafts or provided with a grillage type of structure at first floor levelare frequently sufficiently stiff to cope with local variations in ground bearing capacity which arise either from the natural ground condition or the method of construction. Conversely, flexible buildings founded on separate foundations with widely spaced shear walls rely to a greater extent on foundation assumptions being realized. The adequacy of the site investigation data and of construction supervision should be considered in the light of the proposed superstructure design. The specialization in superstructure design and foundation engineering which is common in research, tends to prevent this approach. Uniform wind load is not in all cases the critical wind condition when the main shear member is central, partial wind load resulting in twisting causing heavier loading on the off-centre shear walls. There seems no justification for relying on wind loading on a building being uniform. It is suggested that CP3 loading should be modified to allow for this. Quite moderate winds are a source of delay when cranes are being used for erection. This is a point which requires greater attention than is normally given in the initial design stage. Precast concrete walls are frequently used without reinforcement to form elements of shear walls, the justification being that shrinkage cracks do not appear in practice. This is an attractive concept and an analytical consideration of the practical limits of nonreinforced structural members might lead to their greater use and consequent savings.
P. R. BARNARD : I should like to ask a question concerning the part of the paper dealing with deflection. Could the authors please give the derivation of the formula for maximum deflection of a system of coupled shear walls given on page 90 of their paper. Not having their reference to hand, I assume that the formula applies to a system of two similar walls connected by a row of beams down the middle. I wonder if the same type of formula would be derived for walls of different stiffness and for the case where a system contains more than one row of beams. To be able to calculate the maximum deflection of each of a series of parallel but dissimilar wall systems in a building under unit horizontal loads will give an indication of the relative stiffnesses. If the floor system is stiff enough to act as a rigid diaphragm, then the stiffness of a wall system will be a measure of the amount of load carried by it. This partition of the total wind force to the various wall systems becomes particularly important in the "shear wall-slab" type of apartment structure over 20 or 25 storeys in height. W. W. FRISCHMANN :
The authors would like to thank both Dr. Barnard and Mr. Crofts for their contributions. 98
99
DISCUSSION
The formula for maximum deflection for a system of coupled shear walls may be derived as follows: wh' The deflection of the free cantilever = 8 . Ed . I The effect of BMs due to Xa's is to reduce the deflection by the following:
Mn.
o LJ] M2
~ .h ,
x
n-I
n '
a;.
h
h
h. '-----'
lYP-iW I In~~rarion
Qj@raM.
8.M. wilh unit loaa
at Top-
FlG. A
Hence deflection due to Xa's. I
ah
= - l L - . Ma /I
£ d " =1611
=£
I II
[11 _ a --h 11
~ ah Ma
I £., - 6 - (11 - a)h a =t n
I = £ I d
h2
L -2II • a . Ma(2/1 /I
a ~1
{(II ~, a)h + h} + "] + 4..:.-_-=--_-'-
+ 2h(2n -
2
a)
+ IIh]
.
aJ
Hence total deflection, wh.
= 8£d I
I - 2£ I d
n 2
La. Ma(211 - a)
11 a =1
It is true that this derivation is for a system of two similar walls connected by a row of beams but, as is evident from the Equivalent Column Method, this could give an approximate solution to a number of walls with different openings. The sketch above shows the combined effect of the Equivalent Column (which is the summation of individual wall stiffnesses) and the restraints applied at each floor level. The stiffness of various parallel but dissimilar wall systems can be found out from their deflections under unit load and the wind loading can be distributed accordingly. The authors agree with most of the points raised by Mr. Crofts. The problem of exact wind pressure is difficult and one can only go by the recommendations made by the Code of Practice or on the advice of the National Physical Laboratory. The effect of nonuniform loading may be considered only approximately and it is normally left to the d iscretion of the Designer. Some recommendations in the Code of Pract ice may, however, be very useful and would , moreover, standardize the design data over the whole country.
lNTRODUCTlON
In the design of tall buildings special consideration must be given to providing sufficient stability in all directions against lateral forces, i.e. wind, possible earth tremors and blasts. These forces can produce critical stresses in the structure, set up vibrations in the structure and the cladding and , in addition, can cause lateral sway of the building, which could reach a point of discomfort to the occupants. As discussed in the article on "Planning Concepts using Shear Walls", this restraint against lateral forces is best provided by developing the inherent stiffness of the enclosures to service areas, both from the considerations of cost and the method of construction. Such enclosures, when constructed in concrete, should be located so as to carry as much of the vertical dead weight of the floor system as possible. The vertical loads have the effect of pre-loading the core, which should be so proportioned that (a) the dead weight overcomes the tension induced in the core by horizontal loads, and (b) the increase in compressive stress, solely due to wind forces, is approximately 25 %. (This being the permissible increase as per c.P. 114.) With this optimum arrangement the wind stability can be achieved very economically. When the enclosures are constructed with braced shear walls in steel, the pre-loading is helpful only from considerations of anchorage at the foundations. With these optimum arrangements the lateral stability is achieved economically in contrast with the traditional structural frame where the materials 83
S4
W. FRISCHMANN AND S. PRABHU
are used uneconomically. Fig. I illustrates the structural efficiencies of traditional frames and various shear wall systems , using the same quantity of materials. (a) (b) (c) (d) (e) (f)
Shows the traditional frame . Shear wall the full depth of the building. Shear wall with storey-high opening. Shear wall with door high opening. Shear wall reduced in thickness and the material placed to form flanges. As in (e) above but with door high openings.
ELEVATION
-==---.-------- -=-
PL.AN
PLAN
o.
b.
TAAOITI~ FRAME. APPRaJ: STI'F"€5S _ I
SHUR WALL OF 'Ul..L. WIDTH OF IUIL.DING.
APPRQX STI,'NE.SS -
'''0
EL(V.TION
PL.AN
PLAN
<1
C
SH[AA WALLS CONN!CT£D BY FlOOR aEA'"
SHEAR ""LLS CONNECTED .,.,. TH IN FLCX>R SLAB APPAOX . STIFFNESS - 28
APPROX . ST IFFNESS-U2 .
__ II II
~i II-
-
-
-
-
-
-
-
-
....-J
(L(VAT ION
ELEVATION
II
~
II
r
OPlHIHG
EJ
"
PLAN
•
WALL. WITH FLANGES Iil EXTAEMTIES APPAOX. STIFFN£55- 2&0
FIG .
.LAN
I
11
II
~
Ii II
WAU.S WITH FLANGES AT UTAEWITIES ~ BY FI.DOA 8EAM .'PROX . STIFFHESS_22S
I. Comparison of d ifferent forms of structure for lateral stiffness.
-
85
DESIGN AND CONSTRUCTION PROBLEMS
-oPENING
o
e.,
~I~.
ELEVATION
m
PLAN
PLAN
--==:.....-
VA': 1,
1
_
___ · h.
9 WALL WITH FLANGU AT EXTREMITIES TO FOAM A CORE APPROX. ST IFFNESS. ,210
'--- POOOOOODI ,
WALL WITH 'LANGES AT DTA(MlTtES
TO FORM A CORE
APP"'OX , STI,.,N€SS • US
~~
ELEVAT ION
'Lf:VATION~ I I \ />; , \ /\ 1\ / \
1
I
I
PLAN
J. FA~"DE
SHIA"
k. WALL
'~"'Ol
APPAOI . STI"HW_lJolU
A~..cl .
FIG
SHE""
WI.LL STI"NESS - 11 0
J. (contd)
(g) As in (e) above but the flanges connected with the web walls each of half the thickness of that in (e) to form a core. (h) As in (f) above but flanges connected with two web walls, each of half the thickness of that in (f) above to form (j), (k), Facade wall with window openings.
acore:----
From a structural stability point of view the architectural planning should aim at providing arrangement (b), (e) or (g). (g) has the advantage of increased stiffness in all directions including torsional stiffness. In practice, however, the structural engineer has to accept a compromise arrangement of (d), (f) or (j). The floors in shear wall structures are made to act as deep horizontal beams, transmitting the lateral loads to the vertical stiff shear walls, which , in turn transmit them, to the foundation. The foundations are designed to distribute highly concentrated loads over a sufficient area to prevent local overstressing of the soil.
S6
W. FRISCHMANN AND S. PRABHU
ANALYSIS OF SHEAR WALLS
The stiffnessof the floor slabs in the horizontal direction is large compared to the stiffness of various shear walls or a group of walls forming a core; and the simplified assumption is made that the horizontal wind loading is carried in proportion to the stiffnesses of the shear walls. The walls can then be designed as cantilevers (or Vierendeelgirders if interconnected shear walls) fixed at the foundation level. The yielding of the soil should also be allowed for when designing a statically indeterminate (e.g. Vierendeel-like system) interconnected shear wall, as it could alter considerably the pattern of stresses throughout the wall. The interconnected shear walls should be designed taking into account the stiffnessof the connection and the connecting member and not as two separate cantilevers hinged together by the floor, since the connecting member has a considerable effect in reducing the stresses in the wall. The following are two of the methods which the authors have used for the analysis of interconnected shear walls and have found to be quick and reasonably accurate.
1. The Equivalent Column Method The method consists in replacing the frame by an equivalent column, whose stiffnessequals the sum of all the column stiffnesses, and by restraints, applied at each floor level, equivalent to the sum of the beam stiffnesses. The condition of equilibrium of all external and internal forces acting on the equivalent column is expressed as a second-degree differential equation. The nature of the mathematical function is such that with increasing height, that is, with a greater number of restraints, the accuracy of the solution also increases. The general solution may be applied to any number of bays and storeys. The following assumptions are made. (a) All frames are fully fixed at the bases of the columns. (b) All columns deflect, due to the horizontal loads, and remain parallel to each other, that is, it is assumed that there is no elastic shortening of the beams. (c) The sectional properties of columns and beams in each bay are constant for the full height of the structure. However, variations may occur from bay to bay in a multi-bay structure. If the sectional properties of the beams (in each bay) or columns vary with the height of the building, an average value may be taken. This should yield a reasonably accurate answer and it is left to the designer to decide whether further analysis is necessary.
DESIGN AND CONSTRUCTION PROBLEMS
87
(d) The storey height is constant throughout the height of the building. If this is not the case, an average may again give reasonably accurate results. (e) Axial shortening of members and deflection due to shear are neglected. (f) The modulus of elasticity E of the concrete is assumed to be constant throughout the height of the building.
2. The Influence of Coefficient Method This is essentially the method of strain-energy and consists in reducing the structure to a statically determinate system by the introduction of releases at suitable points. Unit constraints are then applied at these points, resulting in a number of simultaneous equations equalling the order of statical indeterminacy. The method permits account to be taken of the variation of inertia of the beams and columns, and of the modulus of elasticity due to the use of mixtures of concrete differing throughout the height of the building. The following assumptions were made in the analysis. (a) The strain-energy due to axial and shearing forces in the structure is neglected. (If thought necessary this could be taken into account.) (b) All horizontal loads are assumed to act at panel points. The methods of analysis are fully described in Ref. 1. The comparison of moments in _the wall analysed by the two methods is shown in Fig. 2 which has been reproduced from the above paper. It should be noted that although for illustration a simple example was chosen, both these methods can be used to analyse several shear wall sections of varying depths and connections. The results of some of the investigations carried out to determine the effects of perforations on the stiffness and strength of shear walls are given in Fig . 3. The variables considered are the width of the opening and the depth of the connecting beams. The results also show the benefits of the homogeneous connection as against the pinned connection between the floor and walls. It appears that the stress in the shear walls is not unduly affected if the width of the openings is limited to approximately 15 % of the total width of both shear walls and the depth of the connecting beam is greater than 20 % of the storey height. For this reason if the perforations are restricted within the limits suggested the shear wall may be considered as fully homogeneous. The adequacy of the connecting members should, however, be assessed from the points of view of both moment and shear capacity. More development work is required before precise recommendation can be stated.
88
W. FRISCHMANN AND S, PRABHU
~
18'9
16·4
36'0
- 5. '0
- 18'0
(ml :::-·144-51
- 21 '0
- 10 '0
Inll :::::- 15511 .
- 13 '0
-
lHI ::-. tltil
27'5 :::
49 '6
37'1 :-:-. 60 '1
42'6 (n·ll 45-4 (\l '11 45 '7
3·71
:::::' ::::-.' .:-: . :::::-. :::::.
67'9 fill 73'2
tn ·§!
76 '2
(IHI ::::::-'(7I-51 437 :::::-.
71\'9
14N):-::::-' l7HI
39·4 :-:.:- 75'2 (11'1) :::::-: tn'5I
32-7 ::::.' 70'8 !!l'11 :::::-. lit') 23-3 .:.' 63'5 IIBI ::.:-. (IlI-II n-o ;., 52-3 fI 'll .:.' In 'll 4·7 . 3&3
5'37
5'59 5 '73
11-1' ..' Ill'"
24 '2
5'77
IlUI
48 'S
151'7)
19'0
1Il·11
. 5'9
.' ftHl
78 '• . ..:::: 37'5 fatel ..::' : lUil 115-0 :·.. . 77-5
5'00
11 lUI
160'0
1171 '11
215-3
1121'0 283·2
(lU '
:.::..:: 1IIi!
-:'.':- -: 127'8
. .... -:: 1141'01 :-:-:-::::.::.::- 190·7
. . . .;:;: :::-:: OMil
. . :::: 269-1
1214051
f:.t~I""-;';";'"'-'-'-'-'-'-'-'''-'--'-'-"'-'-''';''''''''' SHEARING FORCE
BENDING MOMENT DIAGRAM
Raulta obtained by Equlvolcnt in c_ctlng !>cOIlS. cokI. . Method gl.... ln brackets . DIAGRAM
FIG.
~ - 47 '0
. nO'11
-
-
-
8'2 5 '3 3 '7 3 '0
· ·
6 '0 3'5 2 '3 1 '7
-
I '8
4 '9
-
8 '6
- 4'
3·3
2' I 2 ·7 I
- 24'0
6 '11
·74 '0
12·0
· ·
· ·
20 '0
- 32'0
13 '0
• 112'0
9 '7
25 '0
11 '2
15'0
7 '0
10 0
5 '9
7 '2
5 '2 4 '2
·
5 ·7
PERCENTAGE PlffERENCE BETVtU:N RESUiiS .
2. Comparison of moments of shear walls analysed by the method of influence coefficients and the equivalent column method.
DESIGN AND CONSTRUCTION PROBLEMS
89
WIND AND VIBRAnON EFFECTS
Wind tunnel studies have shown that air flow distribution produces a complex three-dimensional flow field in the region of a tall block. The deceleration of the air stream against the building creates a downward flow, frequently producing strong eddies at pavement level. At the same time, on the leeward face of the building the air flow close to the surface of the building is upward, creating problems of rain and snow penetration through cladding. Wind Loading
The wind load forces depend on the mean hourly wind speed, the estimation of an appropriate gust factor, shape and pressure coefficients and the effects of local topography. The wind forces are normally applied on the building as an equivalent uniformly distributed load for its full height. Natural wind has the potential to cause vibration of cladding or sway of the whole building, depending on its dynamic properties. These effects can be caused by either the response of the building or an element of the building to the just spectrum of the wind, or a self-induced aerodynamic instability which occurs in steady wind conditions. Tall buildings built in the last decade, with modern techniques, have a much lower frequency and much reduced damping than previous constructions and require careful checking as they are liable to gust induced sway. The observed effect of aerodynamic instability-which occurs say, on iced transmission lines-is unlikely except in cases of very low damping, since normally wind speeds required to excite this type of instability are outside the wind speeds encountered. The vortex-shedding type of instability is of great significanceto the design of tall buildings, as a vortex-shedding frequency can coincide with the natural frequency of the tower at normal wind speeds, with the possibility of resonant vibration, the amplitude of the sway depending on the structural damping of the tower. If the energy imparted by the exciting forces equals the losses due to internal damping, the amplitude of vibration would increase until failure occurs due to stresses. Slender towsrs have to be checked for such instability, which can be adjusted in the design by altering the natural frequency, the damping or the sectional properties of the building. Increase in frequency has the effect of raising the critical wind speed for onset oscillation. The tower is normally regarded as a cantilever encastre at the foundation, and the building could be excited in the lateral, torsional and longitudinal modes. Effects of Vibration
These have to be considered in relation to the effect of the stresses induced on the fatigue life of the structure, and the effect of vibration on people in the building.
90
W. FRISCHMANN AND S. PRABHU
The human body is very susceptible to vibration, and the amplitude which can be tolerated is considerably less than would be permitted purely from stress considerations. A number of research workers have published limit values for intensity of vibration, depending on amplitude and frequency.!" The accepted amplitude of transverse lateral vibration is considerably less than that anticipated from static wind loading on the building, and this, therefore, must be the criterion for the design. Much of our understanding of the effects of wind on buildings has come from scaled models tested under controlled conditions in wind tunnels, reproducing such effects as pressures on cladding, air flow studies, aerodynamic stability and minimum structural damping. Effects of Earthquake Earthquake forces result from the vibratory motion of the grounds on which the structure is supported. The ground vibrates both vertically and horizontally, but the vertical component has a negligible effect on the structure due to its inherent vertical strength. The effectof earthquake forces on design of structures at the present time are empirical, bas-ed on the performance of structures in earthquake zones. In some countries building .regulations require that, in addition to the normal wind forces, the structure be designed to withstand a minimum total lateral seismic force which is assumed to act non-concurrently in the direction of each of the main axes of the building, equivalent to a given percentage of the weight of the building. The percentage generally varies from I to 13 %, depending on the location in relation to the earthquake zones. In addition, walls and partitions are anchored to the rest of the building to resist a force of 20 % of their weight and members such as parapets, ornamentation and cladding are usually required to be anchored for 100% of their weight. DEFLECTION
The maximum deflection due to lateral loads must be limited so that it does not cause discomfort to the occupants especially near the top of the building (as discussed previously) and prevents damage to the remaining structural elements and finishes, e.g. curtain walling, partitions, etc. The calculation of deflection should also take into account the effect of the rotation of the foundations due to compressibility of the soil. The following formula may be used for evaluating the maximum deflection at the top of the building of the example shown in Fig. 2 : wh4 h2 n Ii = maximum deflection = 8E I - 2E l 2 .2 aMi2n - a), d d n a=l
DESIGN AND CONSTRUCTION PROBLEMS
91
where w is the uniformly distributed horizontal loading, h is the total height of the shear wall, n is the number of storeys, I is the moment of inertia of each shear wall, M; is the moment a1 the ath storey due to the shear force Xa, and Ed is the dynamic modulus of elasticity (see Ref. 3). In the above formula the shear deformations have been ignored. The allowable deflection figure that seems generally in use in the United States for braced steel frames is due to H. V. Spurr's' and varies between h/500 and h/750, where h is the height to the building. The report on The Testing of Structures's' states that there is very little basis for any proposals and suggests a limit of lateral movement of 1/1200. In Hong Kong, where typhoons are frequent, a great deal of experience is available for building over 20 storeys. (61 It appears that limiting the height, depth ratio to 10 for a single shear wall and to 15 for a core (shear box), has proved satisfactory. This can also be expressed as a function of the height of the building as follows: ~
wh4 8E dl
wh2 h2 . -2 4Edl
= -- = -
M=L I y
where
Mh 2 f h2 = -- = -.-, 4Edl
Y 4Ed
Assuming concrete 4000 psi, works cube strength at 28 days,
Ed = 4 j(permissible)
Then
\
,
u
=
=
X
106 lb/sq, in.
1470 x 1-25Ib/sq. in.,
h2 1470 x 1-25 h/(2 x 10) 4 x 4 X 106
h = 434Thus the allowable deflection for single shear walls is approximately h/430. For cores (or shear boxes) it can be shown that the corresponding deflection is approximately h/300. There are no specific bye-law requirements for maximum permissible deflections and if conservative assumptions are made this could significantly increase the structural cost. In our opinion an approximate value of h/350 to h/400 appears to be reasonable, if the deflection calculations are based on bending (shear deformations if significant) and the effect of rotation of foundation, and provided it has been checked against the tolerable amplitude acceptable to the occupants.
92
W. FRISCHMANN AND S. PRABHU
METHODS OF CONSTRUCTION
The overall speed of construction for a building, and hence the cost, largely depends on the time taken for the construction of the shear or core walls. These should therefore be made as simple and uniform as possible so that a slipform or a mandrel type internal form of shuttering (to be described later) may be used. These methods have been found time and labour saving in practice. With in situ techniques the wall thickness should preferably be such as would carry all the loads with nominal reinforcement so that complicated arrangements of horizontal links to prevent lateral buckling are not required. In practice, however, it is very rarely that this ideal is achieved and the engineers are forced to accept high percentages of reinforcement, in many cases as high as 8 %, which is the maximum permissible according to the Code of Practice. The use of prefabricated welded reinforcement ladders has been found of value in such cases. The elements of the shear walls are normally found to carry the highest work load of the structure, and the construction cycle is almost entirely controlled by the speed at which the core can be built. Too high a concentration of workload within a small area will delay the cycle, so that in general a singlemassivecore should be avoided in favour of two smaller core elements. Accepting the premise that the cores are to be of in situ concrete construction, the formwork and fixingof reinforcement will control the cycle. Simplification of the core shape will greatly assist in raising output, and the same attentions should be given to this point where normal formwork is to be used as is invariably given where slipform is to be used. Any shape with a number of small closed cells should be avoided where possible, the fourth wall of ducts, for example, being subsequently constructed in brickwork. Where it is necessary to design lift shafts with four load-bearing walls, construction time may be saved by the use of mandrel type internal forms. Reinforcement should be limited to single storey lifts to avoid erection and dismantling of scaffolding during fixing. Time may also be saved by the use of prefabricated reinforcement panels for core and column construction. Cages should be restricted to 7 or 8 ft width for transport or handling purposes, and adequate bracing rods must be added to provide rigidity. The panels should be jig-assembled to give accurate bar spacing and neat lapping of vertical rods. The following construction methods can be used with advantage on tall buildings. Sliding Formwork
The core can be erected first by the use of continuously sliding shutters operated by hydraulic (oil) jacks. The shutters rise at the rate of about I ft
DESIGN AND CONSTRUCTION PROBLEMS
93
per hour. The reinforcement is fixed first in advance of the shutter. Concrete is deposited in the shutter from a hopper which is lifted by the main crane. Erection of the complete core takes only a few weeks. When the concrete has hardened an auxiliary crane may be erected on the top of the core by means of the main crane. The following points should be considered carefully. (i) The quantity of reinforcement in some cases may be so high that it may not be possible to keep the formwork moving. (ii) Difficulties may arise in supporting the climbing crane on recently poured concrete unless the crane is very light. (iii) The initial erection and subsequent dismantling of the sliding shutter may be time-consuming. (iv) Remedial work tends to be higher than for construction with normal formwork. It is very difficult to ensure that inserts do not become displaced. (v) Finishing trades may be introduced later than with normal construction.
Mandrel Form of Construction A constant internal shaft size is designed for the full height of the building, the variation in thickness of the walls taking place on the outside. The shafts are rectangular in plan with splays along the' corners. The splays in addition to their structural significance facilitate the shrinking of the forms for striking. The mandrel forms are of fully framed box units with an accurately made steel framework lined with high quality plywood reproducing the internal configuration of the shafts. The mandrel formers are constructed to the full storey height plus a projection above the general floor level (see Fig. 4). The mandrel formers can be climbed by the tower crane in one piece without the necessity of dismantling and can be used for the full height of the building with something like 25-30 re-uses, requiring only cleaning down and re-oiling after each successive floor cycle. PRECAST CONSTRUCTION
The success of the precast systems will very much depend on the design of the joint between panels forming the shear wall. The joints must be designed: (a) To provide adequate stability during erection, possibly in conjunction with temporary propping. (b) To reduce crane hook times to a minimum. (c) To enable the jointed members to accept stresses at an early date. (d) To enable the dimensional inaccuracies and tolerances of the units in manufacture and erection to be absorbed.
94
w.
FIG.
FRISCHMANN AND S. PRABHU
4. Mandrel former in position before casting the floor.
Where the dead loads are sufficient to counterbalance the stresses induced in the walls due to wind, etc., simple bedding joints may be used. If, however, the dead loads are insufficient, the precast units must be connected with full tension strength joints such as welding projecting reinforcement bars after erection, prestressing plated joints, grouted joints, coupler joints or some form of sleeved joint illustrated in some detail by W. W. Frischmann elsewhere.(7) When stability depends on the co-action between panels, the vertical joints must be designed to transfer the shear forces. In addition, since the vertical joints are invariably filled with concrete of high water-eement ratio, the joint material shrinks, which results in very high deflections under wind loading as, at the commencement of taking up the load, the panels tend to deflect as independent units. To ensure satisfactory co-action the joints are often corrugated, toothed or reinforced with stirrups to ensure mechanical connection.
DESIGN AND CONSTRUCTION PROBLEMS
95
FOUNDATIONS
With the knowledge and experience of soil mechanics large buildings can now be economically founded on relatively poor soils. Sites can, therefore, be better utilized than was possible in the past when foundation difficulties were often a major restriction. To determine the characteristics of the soil and predict its possible behaviour under the final loading, samples of soil are tested in the laboratory. From its mechanical properties it is then possible to estimate the probable distribution of stress in the soil, the settlement, and the sub-structure most suited to support the building. In proportioning foundations the allowable bearing pressure must be chosen to provide an adequate factor of safety against shear failure, and to ensure that both the overall and the relative settlements are within tolerable limits. On non-cohesive soils, e.g. sand and gravels, since the water can move freely the settlements are largely completed by the end of the construction. On cohesive soils, e.g. clays, the spaces between the particles are very small and the movement of water under pressure can only take place slowly. Theoretically consolidation can take place almost indefinitely, although perceptibly for some years only. It is this long, continued settlement which causes so much trouble with structures and finishes. A good deal of experience has been gained in big cities like London in founding tall blocks, in forming deep basements andin stabilizing neighbouring buildings against possible lateral and vertical movements. The London area is underlain by a great depth of clay which increases in strength and decreases in compressibility with increased depth. The choice of foundation for tall blocks therefore lies mainly between a raft, a buoyant foundation, a piled foundation or a combination of these. The principle of the buoyant foundation which has been applied to towers in London consists of excavating sufficient earth for the sub-structure to balance, as far as possible, the weight of the buildings and thus reduce the average settlement to a minimum. The practical difficulties of forming excavations deep enough to make a full buoyant foundation are expensive and normally, therefore, a compromise has to be accepted. This consists of either using a raft at, say, the subbasement level, extending well beyond the outline in plan of the tower block, or a raft placed at a shallower depth used in conjunction with piles which transfer the loads into the stiffer and less compressible soil. Large diameter augered piles transferring loads, partly in friction, partly in end bearing, have been found to be very economic. The bases of these piles can be enlarged by underreaming, if required. The function of a raft is to spread the load over a large area in order to reduce the bearing pressure to the allowable limit. Normally it is made stiff
96
W . FRISCHMANN AND S. PRABHU
enough either by itself or in combination with the sub-structure, so as to distribute the load and to reduce the differential settlements in the structure to acceptable values. MODEL ANALYSIS
The behaviour of shear wall structures for different loading conditions can be studied by using advanced theoretical analysis supplemented by model testing. This would help to determine the most suitable structural form and also to predict the behaviour of the structure at working and failure load. Thus the engineer can assess the true factor of safety. Much of the tedious work associated with theoretical analysis is simplified by the use of electronic digital and analogue computers. Less time is spent on the actual computingwork-thus leaving more time for the proper detailing of the structure to simplify site- operations. Testing can be carried out on models varying from Ijl5th to Ij50th scale, constructed from perspex, cement mortar, steel, celluloid, aluminium, etc., depending on the type of material to be used in the final structure and the accuracy of the results required. Examples where model testing has been used are the Pirelli Building in Milan and the Co-operative Insurance building in Manchester. The savings resulting from carrying out such tests more than offset the cost of model testing, since significant reductions in structural costs can be made by using a more appropriate factor of safety.
SECONDAR Y STRESSES
Iflarge massesof concrete are cast in members such as rafts, beams, columns, etc., care should be taken that the heat of hydration is not so high as to overheat the concrete. This can be achieved by careful positioning of construction joints, using low heat cements or cooling the water used in the mix. One of the major secondary stresses that arises in tall blocks is caused by the temperature differential between the exposed outside and the heated or cooled inside parts of the structure. This is specially so in air-conditioned buildings. The relative expansion with respect to the internal columns or cores due to temperature gradient can be considerable for very tall structures. The differential displacement is maximum at the top and decreases towards the bottom. If the displacement is beyond the allowable limits, cracking of finishes in the upper floors may occur. This displacement further induces moments and shears in the floor slab, and in addition a considerable increase in axial load and moments in the columns. In addition, the effect of differential stressing of various vertical members resulting in unequal strains and warpage of the floor slab should be investigated.
DESIGN AND CONSTRUCTION PROBLEMS
97
In particular, this can be of great significance when highly stressed perimeter columns are used in conjunction with shear walls which may not be so highly stressed under normal vertical loads, due to reserve required for wind loading. Reference 8 states how this problem was overcome in the design of the World Trade Centre which will be one of the world's tallest buildings. In a building where shear walls or cores are designed to resist the lateral forces the effect of their horizontal deflection on the remaining structure must be carefully investigated. REFERENCES
w. W., PRABHU, S. S., and TOPPLER, J. F., Multi-storey frames and interconnected shear walls subjected to lateral loads, Concrete and Constructional Engineering, June and July 1963. STEFFENS, R. J., The assessment of vibration intensity and its application to the study of building vibrations. National building studies. Special Report No. 19, D.S.tR., Building Research Station. H.M.S.O. 1952. B.S. Code of Practice C.P.115: 1959. The Structural Use of Prestressed Concrete in Buildings. SPURR, H. V., Wind Bracing' in High Towers, McGraw-Hili, 1939. Report of a committee on The Testing of Structures, The Institution of Structural Engineers, September 1964. PHILCOX, K. T., Some recent developments in the design of high buildings in Hong Kong, Structural Engineer, October 1962. Vol. 40, No. 10, pp. 303-323. FRISCHMANN, W. W., A Review of Structural Form-Structural Design of Tall BUildings, Prestressed Concrete Development Group, University College of South Wales and Monmouthshire. December 1964. Engineering News Record, April 1964.
1. FRIScHMANN,
2. 3. 4. 5. 6. 7.
8.
DISCUSSION J. E. CROFTS : Mr. Crofts (partner Nachshen, Crofts and Leggatt, Consulting Engineers) referring to design and construction problems, said that when tension occurred at the bottom of shear walls, the actual load factor was not obvious. Tension arose from subtraction of wind load stresses from dead load stresses and was generally the small differences of two large numbers. A modest increase in assumed wind speed or a small change in the distribution of wind load between shear walls would cause a large change in tension. This large tension increase would not generally be covered by the difference between working and ultimate stresses in the tension members. Special accuracy in defining dead and wind loads on the shear wall is justified in such a case so as to define the tension requirements with economy. Tall buildings founded in rafts or provided with a grillage type of structure at first floor levelare frequently sufficiently stiff to cope with local variations in ground bearing capacity which arise either from the natural ground condition or the method of construction. Conversely, flexible buildings founded on separate foundations with widely spaced shear walls rely to a greater extent on foundation assumptions being realized. The adequacy of the site investigation data and of construction supervision should be considered in the light of the proposed superstructure design. The specialization in superstructure design and foundation engineering which is common in research, tends to prevent this approach. Uniform wind load is not in all cases the critical wind condition when the main shear member is central, partial wind load resulting in twisting causing heavier loading on the off-centre shear walls. There seems no justification for relying on wind loading on a building being uniform. It is suggested that CP3 loading should be modified to allow for this. Quite moderate winds are a source of delay when cranes are being used for erection. This is a point which requires greater attention than is normally given in the initial design stage. Precast concrete walls are frequently used without reinforcement to form elements of shear walls, the justification being that shrinkage cracks do not appear in practice. This is an attractive concept and an analytical consideration of the practical limits of nonreinforced structural members might lead to their greater use and consequent savings.
P. R. BARNARD : I should like to ask a question concerning the part of the paper dealing with deflection. Could the authors please give the derivation of the formula for maximum deflection of a system of coupled shear walls given on page 90 of their paper. Not having their reference to hand, I assume that the formula applies to a system of two similar walls connected by a row of beams down the middle. I wonder if the same type of formula would be derived for walls of different stiffness and for the case where a system contains more than one row of beams. To be able to calculate the maximum deflection of each of a series of parallel but dissimilar wall systems in a building under unit horizontal loads will give an indication of the relative stiffnesses. If the floor system is stiff enough to act as a rigid diaphragm, then the stiffness of a wall system will be a measure of the amount of load carried by it. This partition of the total wind force to the various wall systems becomes particularly important in the "shear wall-slab" type of apartment structure over 20 or 25 storeys in height. W. W. FRISCHMANN :
The authors would like to thank both Dr. Barnard and Mr. Crofts for their contributions. 98
99
DISCUSSION
The formula for maximum deflection for a system of coupled shear walls may be derived as follows: wh' The deflection of the free cantilever = 8 . Ed . I The effect of BMs due to Xa's is to reduce the deflection by the following:
Mn.
o LJ] M2
~ .h ,
x
n-I
n '
a;.
h
h
h. '-----'
lYP-iW I In~~rarion
Qj@raM.
8.M. wilh unit loaa
at Top-
FlG. A
Hence deflection due to Xa's. I
ah
= - l L - . Ma /I
£ d " =1611
=£
I II
[11 _ a --h 11
~ ah Ma
I £., - 6 - (11 - a)h a =t n
I = £ I d
h2
L -2II • a . Ma(2/1 /I
a ~1
{(II ~, a)h + h} + "] + 4..:.-_-=--_-'-
+ 2h(2n -
2
a)
+ IIh]
.
aJ
Hence total deflection, wh.
= 8£d I
I - 2£ I d
n 2
La. Ma(211 - a)
11 a =1
It is true that this derivation is for a system of two similar walls connected by a row of beams but, as is evident from the Equivalent Column Method, this could give an approximate solution to a number of walls with different openings. The sketch above shows the combined effect of the Equivalent Column (which is the summation of individual wall stiffnesses) and the restraints applied at each floor level. The stiffness of various parallel but dissimilar wall systems can be found out from their deflections under unit load and the wind loading can be distributed accordingly. The authors agree with most of the points raised by Mr. Crofts. The problem of exact wind pressure is difficult and one can only go by the recommendations made by the Code of Practice or on the advice of the National Physical Laboratory. The effect of nonuniform loading may be considered only approximately and it is normally left to the d iscretion of the Designer. Some recommendations in the Code of Pract ice may, however, be very useful and would , moreover, standardize the design data over the whole country.