Design for earthquake-resistent reinforced concrete structural walls

Transcription

1 DOI /s EXPERIMENTAL SOLID MECHANICS Design for earthquake-resistent reinforced concrete structural walls N. St. Zygouris G. M. Kotsovos D. M. Cotsovos M. D. Kotsovos Received: 30 September 2013 / Accepted: 9 January 2014 Ó Springer Science+Business Media Dordrecht 2014 Abstract The work presented is concerned with the application of the compressive force path (CFP) method for the design of earthquake resistant reinforced concrete structural walls. It is based on a comparative study of the results obtained from tests on structural walls under cyclic loading mimicking seismic action. Of the walls tested, half have been designed in accordance with the CFP method and the remainder in accordance with the provisions of euro-codes 2 and 8. The results obtained show that both methods of design adopted lead to solutions which satisfy the requirements of current codes for structural performance in all cases investigated. Moreover, the solutions obtained from the application of the CFP method result in a significant reduction of the amount of stirrup reinforcement placed at the critical lengths of the walls vertical edges. In fact, such reinforcement is not specified by the latter method for the case of walls with a span-to-depth ratio smaller than 2.5; for the case of walls with a shear span-to-depth ratio larger than 2.5, not only is it placed over a length In Honour of Professor Emmanuel Gdoutos N. St. Zygouris G. M. Kotsovos Lithos Consulting Engineers, Athens, Greece D. M. Cotsovos Heriot Watt University, Edinburgh, Scotland, UK M. D. Kotsovos (&) National Technical University of Athens, Athens, Greece mkotsov@central.ntua.gr which is considerably smaller, but also its spacing is significantly larger, than the code specified values. Keywords Earthquake-resistant design Compressive force path method Reinforced concrete Short walls Seismic performance List of symbols b Wall width h Wall height l Wall length f c Uniaxial cylinder compressive strength f y Yield stress of steel bar f u Strength of steel bar M f Flexural capacity of wall M y Bending moment of wall corresponding at first yielding (either concrete of reinforcing steel) P exp Experimentally-established loadcarrying capacity P f Load-carrying capacity of wall corresponding to M f P y Load corresponding to M y V f Shear force corresponding to M f d Displacement d y,n Displacement at nominal yield d exp Displacement at failure l = d/d y,n Ductility ratio l exp = d exp /d y,n Ductility ratio at failure

2 1 Introduction Current code provisions for the design of earthquakeresistant reinforced concrete (RC) structural walls specify three types of reinforcement: (i) vertical reinforcement which safeguards flexural capacity corresponding to a load-carrying capacity at least equal to the design load; (ii) horizontal web reinforcement in an amount sufficient to prevent shear failure before flexural capacity is attained; and, (iii) stirrup reinforcement confining concrete along the two vertical edges of the wall so as to form concealed column (CC) elements capable of satisfying the code requirements for ductility [1, 2]. These design specifications, however, have a significant drawback: the dense spacing of the stirrups often results in reinforcement congestion within the CC elements and this may cause difficulties in concreting and, possibly, incomplete compaction of the concrete. On the other hand, experimental information on the causes of failure of RC structures has shown that the loss of load-carrying capacity suffered by linear RC elements in flexure is linked with triaxial stress conditions developing for purposes of deformation compatibility in the compressive zone irrespective of the presence of confining reinforcement [3, 4]. Moreover, in localized regions, these triaxial stress conditions are characterized by the presence of transverse tensile stresses that may lead to an abrupt loss of loadcarrying capacity, if the compressive zone is not properly reinforced; in fact, the amount of reinforcement required for this purpose has been found to be significantly smaller than the code specified amount for providing confinement to concrete when its calculation is made through the use of the Compressive Force Path (CFP) method [4]. In view of the above, although it is recognized that there may be a need for specifying transverse reinforcement within the compressive zone in order to increase the ductility of structural walls, it is argued that such reinforcement is not necessarily required for providing confinement to concrete at the walls edges, but in order to sustain the transverse tensile stresses that invariably develop in the compressive zone when the ultimate limit state is approached. A first attempt to clarify this matter has been based on a comparative study of the results obtained by finite-element analysis (FEA) on the response of RC walls subjected to various types of transverse loading simulating earthquake action [5]. All walls had similar geometric characteristics and flexural reinforcement, but differed in the amount and arrangement of the transverse reinforcement, which was designed either in compliance with the earthquake-resistant design provisions of current codes [1, 2] or in accordance with the CFP method [4]. The main finding of the work was that designing the walls in accordance with the CFP method does indeed lead to significant savings in both web and particularly edge transverse reinforcement without compromising the code performance requirements, with wall behaviour being essentially independent of the method used to design the transverse reinforcement. However, it is recognized that the results obtained by analysis may not always be realistic in fact, such results may even, sometimes, be treated with suspicion. As a result, although the FEA package used for the above work is considered reliable, as it had already been found to yield predictions of structural wall behaviour which correlated closely with experimental findings [6], the aim of the present work is to verify the validity of the analytical findings by experiment. Two types of walls are investigated: Slender walls, i.e. walls with a shear span-to-depth ratio a v /d [ 2.5 and short walls, i.e. walls with a v /d \ 2.5. As for the case of the walls investigated numerically, for each type of wall, the geometric characteristics and flexural reinforcement are the same, while the walls differ in the amount and arrangement of the transverse reinforcement; in half of the walls transverse reinforcement is designed in compliance with the earthquake-resistant design provisions of the European codes [1, 2] and in the other half in accordance with the CFP method. The work is based on the comparative study of the results obtained on the wall behaviour under static transverse load increasing to failure either monotonically or in a cyclic manner. The presentation and discussion of the results is preceded by a concise description of the CFP method, as the concepts which underlie it are in sharp contrast with current code tenets. 2 Causes of brittle failure The concepts which underlie the CFP theory are fully described elsewhere [4]; they are, therefore, only concisely discussed in what follows.

3 The CFP theory provides the basis for the implementation of the limit-state philosophy into the practical design of RC structures through (a) the identification of the regions that form the path along which the compressive forces developing within a structural element or structure under load are transmitted to the supports, and (b) the strengthening of these regions so as to impart to the element or structure the desired load-carrying capacity with sufficient ductility. The use of the name compressive force path is intended to highlight these two key features of the theory. Fig. 1 Schematic representation of the physical state of a simply supported RC beam at its ultimate limit state 2.1 Simply supported beam The modelling of a simply-supported beam is seminal for the application of the CFP theory in design, since it represents the portions extending between points of zero bending moment (i.e. points of contra-flexure, hinges, or simple supports) of any structure comprising linear elements. Figure 1 shows the model considered by this theory as the most suitable for providing a simplified, yet realistic, description of the physical state of a simply-supported beam-like element at its ultimate limit state under the action of a transverse point load. Failure is considered to occur due to the development of transverse tensile stresses within the region of the path of the compressive force, with the location of these stresses depending on the value of the shear span-to-depth ratio (a v /d). The manner in which a v /d affects the beam load-carrying capacity (the latter being expressed in a non-dimensional form as the ratio of the bending moment at failure, M u, to the flexural capacity, M f ) is indicated in Fig. 2 [4]. In fact, the trends exhibited by the variation of M u /M f with a v /d correspond to four distinct types of structural element behaviour. Of these, types II and III are characterized by brittle, non-flexural modes of failure, whereas for types I and IV the structural element may be designed to exhibit ductile behaviour without the provision of transverse reinforcement in excess of a nominal amount [4]; hence the latter types of behaviour need no further discussion in what follows. The brittle modes of failure associated with type II behaviour (encompassing, approximately, the range of a v /d between 2.5 and 5) is caused by tensile stresses developing either in the region of change of the CFP direction (location 1 within shear span a v1, in Fig. 1, Fig. 2 Trends of behaviour exhibited by the relationship between load-carrying capacity (M u /M f ) and shear span-todepth ratio (a v /d) and corresponding modes of failure assuming a v1 [ 2.5d) or in the region of the cross section at the left-hand side of the point load, where the maximum bending moment combines with the shear force (location 2 within shear span a v1, in Fig. 2). According to the CFP theory, the transverse stress resultant at location 1 is numerically equal to the acting shear force, and, by invoking St Venant s principle, its effect spreads to a distance equal to the cross-section depth d, on either side of location 1, where the CFP changes direction. Moreover, it has been proposed [7] that, for N = 0, the value of the

4 tensile force that can be sustained at this location can be realistically obtained by T II;1 ¼ 0:5bdf t ; ð1þ where b and d are the width and depth of the crosssection and f t the tensile strength of concrete. On the other hand, transverse tensile stresses within the compressive zone of the cross section where the maximum bending moment combines with the shear force (location 2 in Fig. 1) may develop due to the loss of bond between the longitudinal reinforcement and the surrounding concrete in the manner indicated in Fig. 3. The figure indicates a portion of the structural element between two cross-sections defined by consecutive cracks, together with the internal forces which develop at these cross-sections before and after the loss of bond s necessary to develop due to the increase in tensile force DF S. Setting the flexural moment M = Fz and observing that the shear force equals V = dm/dx = (df s /dx)z? F s (dz/dx), it can be seen that the two products on the right hand side of the equation correspond to beam (bond) and arch (no bond) action, respectively. From the figure, it can be seen that the loss of bond may lead to an extension of the right-hand side crack and, hence, a reduction of the compressive zone depth (x), which is essential for the rotational equilibrium of this portion as indicated by the relation F c ðx l x r Þ=2 ¼ Vx ð l =2Þ: ð2þ The reduction of the compressive zone depth increases the intensity of the compressive stress field, as compared to its value at the left-hand side of the portion, thus leading to dilation of the volume of concrete, which causes the development of transverse tensile stresses (r t in Fig. 3) in the adjacent regions. By considering these transverse tensile stresses and the ensuing triaxial stress conditions, it has been possible to express the shear force (V II,2 ) that can be sustained at locations 2 just before horizontal splitting of the compressive zone as follows [7]: V II;2 ¼ F c ½1 1= ð1 þ 5f t =f c ÞŠ ð3þ In contrast with type II behaviour, the brittle failure characterising type III behaviour (encompassing, approximately, the range of a v /d between 1 and 2.5) is a flexural mode of failure caused by the loss of loadcarrying capacity of the compressive zone (the depth of which decreases considerably due to the deep penetration of the inclined crack that forms within the shear span) before yielding of the tension reinforcement (see Fig. 3). For this type of behaviour, the loadcarrying capacity of an RC beam with a given value of a v /d ranging between 1 and 2.5 (see shear span a v2 in Fig. 2) may be obtained by linear interpolation of the values of M u /M f corresponding to a v /d = 1, for which M u = M f (see Fig. 2) and a v /d = 2.5, for which M u = Va v, where V = min (V II,1,V II,2 ) with V II,1 = - T II,1 [4, 7]. 2.2 Structural wall models The physical model of the simply-supported beam shown in Fig. 1 can also be used to model a structural wall as illustrated in Figs. 4 and 5. Figure 4 shows that the left-hand side (characterised by type II behaviour) of the simply-supported beam is equivalent to a Fig. 3 Redistribution of internal actions in the compressive zone due to loss of bond between concrete and flexural reinforcement

5 slender cantilever subjected to a transverse point load near its free end, since the boundary conditions at the fixed end of the cantilever are similar to the conditions at the beam s cross section through the load point. Similarly, Fig. 5 shows that the right-hand side (characterised by type III behaviour) of the simplysupported beam is equivalent to a short cantilever. Since a structural concrete wall under horizontal loading is essentially a cantilever beam, it can also be designed by adopting the CFP methodology. 3 Experimental programme 3.1 Specimens The overall dimensions of the structural walls investigated are shown in Fig. 6 which also provides an indication of the testing arrangement. All specimens are monolithically connected to two rigid prismatic elements at both their bottom and top end faces. They are fixed to the laboratory strong floor through the bottom prismatic element so as to simulate fixed-end conditions, whereas the load is applied through the top prismatic element. Both prisms are over-sized and over-reinforced so to essentially behave as rigid bodies. The total number of walls tested is eight: four slender and four short walls. The design details of the slender (SL) walls designed in accordance with the CFP method are shown in Fig. 7 [8], whereas those of the walls designed in accordance with the EC8 provisions for medium ductility class (DCM) are shown in Fig. 8. From the figures it can be seen that, for both methods of design, the walls have the same amount and arrangement of vertical and horizontal web reinforcement. The vertical reinforcement comprises eight pairs of 12 mm diameter steel bars at a centre-to-centre spacing of 100 mm with the distance of the bars centre from the closest wall face being equal to 25 mm, whereas the horizontal reinforcement comprises pairs of 8 mm diameter bars at a centre-tocentre spacing of 200 mm. In contrast with the web reinforcement, the amount and arrangement of the edge reinforcement depends on the design method adopted. For the walls designed in compliance with the EC8 provisions, the edge horizontal reinforcement extends throughout the specimen height and consists of 8 mm diameter stirrups at a centre-to centre spacing Fig. 4 Use of physical model of a simply supported beam of type II behaviour for modelling a cantilever Fig. 5 Use of physical model of a simply supported beam of type III behaviour for modelling a cantilever of 30 mm, whereas the stirrups of the walls designed in accordance with the CFP method have a 6 mm diameter and a centre-to-centre spacing of 40 mm and extend to a distance of about 500 mm from the wall lower end. The same concrete was used for all specimens; however, as the specimens were cast at

6 Fig. 6 Experimental arrangement and dimension of specimens different batches, the concrete strength (uniaxial cylinder compressive strength, f c ) exhibited variations as indicated in Table 1, which also includes the yield stress (f y ) and strength (f u ) of the steel reinforcement used. The design details of the short (SH) walls designed in accordance with the CFP method are shown in Fig. 9, whereas those of the walls designed in accordance with the code provisions for DCM are shown in Fig. 10. Unlike the slender walls, the short walls were cast through the use of ready-mix concrete in a single batch which, as indicated in Table 1, had an f c = 43 MPa for all specimens at the time of testing. The properties of the reinforcement used are provided in Table 2. As for the case of the slender walls, all short walls have the same amount and arrangement of vertical and horizontal web reinforcement. The former comprises eleven pairs of 12 mm diameter steel bars at

7 Fig. 7 Design details of slender walls designed in accordance with the CFP method a centre-to-centre spacing of 100 mm, with the bars centre line lying at a distance of 15 mm from the closest wall face. However, the left-hand side specimen in Fig. 10 has three additional 10 mm diameter bars within the CC elements, since these bars form part of the confining reinforcement cage. (It should be noted that the presence of these vertical bars is allowed in the calculation of the wall s flexural capacity.) The web horizontal comprises 8 mm diameter stirrups with a centre-to-centre spacing of 130 mm. As regards the edge horizontal reinforcement, this is only specified by the code and consists of 8 mm diameter stirrups at a 65 mm centre-to-centre spacing and extends throughout the wall height. 3.2 Loading regimes The walls are subjected to two types of static loading: monotonic (M), increasing to failure (considered to occur when the sustained load becomes smaller than 85 % the peak load value) and cyclic (C) applied in the form of imposed horizontal displacements varying between extreme predefined values initially set to ±10 mm and increasing in equal steps thereafter until

8 Fig. 8 Design details of slender walls designed in accordance with EC2, EC8 failure (as defined above). Three load cycles are carried out for each of the above predefined values with a displacement rate of 0.25 mm/s. 3.3 Design The walls are designed so that their load-carrying capacity is reached when their base cross-section attains its flexural capacity, the latter condition being referred to henceforth as plastic-hinge formation. Using the cross-sectional and material characteristics of the walls and the rectangular stress block recommended in [9], the flexural capacity M f of the elements is calculated from first principles allowing for the contribution of all vertical reinforcement and setting all material safety factors equal to 1. Using M f, the wall load-carrying capacity P f (and, hence, the corresponding shear force V f = P f ) is easily calculated from static equilibrium. The values of M f and V f = P f for each of the specimens tested are given in Table 2 together with the experimentally-established values of the load-carrying capacity, P exp. The table also includes the values of bending moment M y and load P y which correspond at yielding of the walls; the latter values are used for assessing (as described later) the ductility factors of the specimens tested.

9 Table 1 Concrete and steel properties used for constructing the specimens investigated Specimens f c (MPa) D6 (MPa) D8 (MPa) D12 (MPa) f y f u f y f u f y f u SL CFP M SL DCM M SL CFP C SL DCM C SH CFP M SH DCM M SH CFP C SH DCM C As discussed earlier, the horizontal reinforcement of the walls is designed either in compliance with EC2 and the earthquake-resistant design clauses of EC8 or in accordance with the CFP method. From Figs. 8 and 10, it is interesting to note the densely spaced stirrups confining the CC elements within the critical regions (extending throughout the wall height) specified by the Codes. Such spacing, resulting from expression 5.20 of EC8 (subclause (4)), is considered to safeguard ductile wall behaviour. As already discussed, in contrast with the code reasoning behind the calculation of the stirrups within the CC elements, the CFP method specifies a significantly smaller amount of such reinforcement only for the Fig. 9 Design details of short walls designed in accordance with the CFP method

10 Fig. 10 Design details of short walls designed in accordance with EC2, EC8 case of the slender walls exhibiting type II behaviour [4], a v /d = 1800/349 & 3.75, where d is the distance of the resultant of the forces developing in the tension reinforcement on account of bending from the extreme compressive fibre. On the other hand, the short walls exhibit type III behaviour, since a v /d = 1350/ 636 & 2.12, and for this type of behaviour the CFP method specifies only web reinforcement for the reasons discussed in Sect. 2. The horizontal web reinforcement designed in compliance with the code requirements (see clauses 6.2 and 9.6 in EC2) is considered to improve the wall s shear capacity so as to prevent shear failure of the walls before their flexural capacity is exhausted. As discussed in Sect. 2.1, this reasoning is also in conflict with that underlying the design of the horizontal reinforcement within the wall web in accordance with the CFP method. 4 Results of tests The structural walls investigated are designated by using a three part name, with the first part indicating the type of specimen (SL or SH), the second the method of design (CFP or DCM) and the third the type of loading (M or C). The main results of the work are given in Figs. 11, 12, 13, 14, 15, 16, 17, 18 and Table 2. Figures 11 and 12 show the curves describing the relationship between the applied load and the horizontal displacement of the load point for the cases of the slender and short walls under monotonic loading, whereas their counterparts tested under cyclic loading are shown in Figs. 14 and 15, respectively. All curves are shown in a normalised form by dividing the values of load with the calculated value of the load-carrying capacity P f and the values of displacements with the calculated value of the displacement at nominal yield. (As will be seen later the latter ratio is usually referred to a ductility ratio l.) Figures 16 and 17 show the variation of the energy dissipated during each load cycle with increasing values of l, for the type of cyclic loading adopted for the tests. The dissipated energy during each cycle (corresponding to a particular l) is provided in a form normalized with respect to a nominal value of the elastic energy expressed as E y = P y d y, where P y and d y are, as

11 Table 2 Calculated values of bending moment M y and corresponding force P y at yield, flexural capacity M f and corresponding loadcarrying capacity P f and experimentally-established values of load-carrying capacity P exp and ductility l exp M y (knm) P y (kn) M f (knm) P f (kn) P exp (kn) l exp SL CFP M SL DCM M SL CFP C (?)/154(-) 3 SL DCM C (?)/150(-) 3 SH CFP M SH DCM M SH CFP C SH DCM C discussed later, the values of the load and displacement at yield of the walls. The modes of failure exhibited by the walls tested are depicted in Figs. 13 and 18, and, finally, the calculated values of bending moment M y and corresponding force P y at yield, flexural capacity M f and corresponding load-carrying capacity P f, and the experimentally established values of load-carrying capacity P exp are given in Table 2, which also includes the values of l at which the specimens failed (l exp ), with failure being considered to occur as defined by EC8, i.e., when load-carrying capacity reduces below 0.85 the value of the peak load sustained by the walls. P/P y SL-CFP-M SL-DCM-M elastoplastic μ = δ/δ y Fig. 11 Load-deflection curves for slender walls tested under monotonic loading Discussion of results 5.1 Monotonic loading As indicated in Fig. 11, in spite of the differences in the reinforcement arrangement, both types of design resulted in similar slender wall behaviour under monotonic loading, with wall SL CFP M being characterised by a slightly larger load-carrying capacity and stiffness. Similarly, Fig. 12 shows that, for the case of short walls, the presence of the code specified stirrup reinforcement within the CC elements has an insignificant, if any, effect on structural behaviour. From Fig. 13, it is also interesting to note that all walls exhibited a similar mode of failure, in that the loss of load-carrying capacity was preceded by failure of the compressive zone at the wall base. Such behaviour clearly demonstrates that, under this type of loading, P/P y SH-DCM-M SH-CFP-M Elastoplastic μ = δ/δ y Fig. 12 Load-deflection curves for short walls tested under monotonic loading any amount of reinforcement lager than that specified by the CFP method is essentially ineffective. Figures 11 and 12 also show the location of the nominal yield point used for assessing the specimen ductility ratio. The location of this point was determined as follows:

12 Fig. 13 Failure modes of specimens tested under monotonic loading P/P y SL-CFP-C SL-DCM-C μ = δ/δ y P/P y SH-DCM-C -1 SH-CFP-C μ = δ/δ y Fig. 14 Load-deflection curves for slender walls tested under cyclic loading Fig. 15 Load-deflection curves for short walls tested under cyclic loading (a) The section bending moment at first yield, M y (assessed by assuming that yielding occurs when either the concrete strain at the extreme compressive fibre attains a value of or the tension steel bars closest to the extreme tensile fibre yields), and the section flexural capacity, M f, are first calculated.

13 E/E y (b) (c) SL-CFP-C SL-DCM-C μ Fig. 16 Variation of energy dissipated by the slender walls under cyclic loading with the ductility ratio E/Ey SH-CFP-C SH-DCM-C µ Fig. 17 Variation of energy dissipated by the short walls under cyclic loading with the ductility ratio Using the values of M y and M f derived in (a), the corresponding values of the transverse load at yield, P y = M y /a v, and at flexural capacity, P f = M f /a v, are obtained from the equilibrium equations, with a v (=1,800 or 1,350 mm for the slender and short walls, respectively) being the distance of the point of application of the applied load from the wall base (see Fig. 6). In Figs. 11 and 12, lines are drawn through the points of the load displacement curves at P = 0 and P = P y. These lines are extended to the load level P f, which is considered to define the nominal yield point, and to the corresponding value of the displacement d yn, which is used to calculate the ductility ratio l exp = d exp /d yn in Table 2, with d u being the displacement at failure as defined earlier. It is evident that all monotonically loaded specimens exhibited ductile behaviour. In fact, Table 2 indicates that the average value of the ductility ratio (l) of the specimens is larger than Cyclic loading Figures 14 and 15 indicate that, as for the case of the monotonic loading, both types of design led to similar types of behaviour. All walls exhibited similar ductility, with those designed in accordance with the CFP method being characterised by a slightly larger loadcarrying capacity in spite of the significantly less amount of stirrup reinforcement within the CC elements. Figures 16 and 17 show the variation of the energy dissipated during a load cycle (corresponding to a particular l) with successive load cycles, with the dissipated energy, as discussed earlier, being expressed in a form normalised with respect to a nominal value of the elastic energy expressed as E y = P y d y. The figures show a continuous increase of the dissipated energy which reaches its peak value when failure occurs, with the energy dissipated being significantly larger for the case of the slender elements. As regards the effect of the method adopted for designing the specimens on the dissipated energy, it appears that the energy dissipated by the slender walls exhibits a smooth increase up to load cycles corresponding to the normalised displacement (ductility ratio) causing failure, with the walls designed in accordance with the CFP method dissipating more energy with successive load cycles, except for the last when there is an abrupt large increase of the energy dissipated by the wall designed in accordance with the code specifications (see Fig. 16). On the other hand, form Fig. 17, it appears that the method of design has an insignificant, if any, effect on the dissipated for the case of the short walls. A comparison of the strength, deformation and failure characteristics of the walls tested under cyclic loading (see Figs. 14 and 15) with their counterparts established from tests under monotonic loading (see Figs. 11 and 12) indicates a small reduction in loadcarrying capacity of the order of 9 %, with the loss in ductility reaching a value of over 20 % for the case of short walls. On the other hand, as shown in Fig. 18, the failure mode (crushing of the compressive zone at base of the walls) was found to be independent of the loading regime imposed. It is interesting to note, however, that this mode of failure marks the start of an abrupt loss of load-carrying capacity in all cases investigated.

14 Fig. 18 Failure modes of specimens tested under cyclic loading 6 Conclusions Designing in accordance with the CFP method leads to significant savings in horizontal reinforcement without compromising the code performance requirements. More specifically, for the case of short walls the web reinforcement is found to be sufficient for safeguarding structural performance satisfying the code requirements. For the case of slender walls, the stirrup reinforcement specified by the CFP method within the concealed-column elements is significantly lesser than that specified by current codes; moreover, such reinforcement is placed within a portion of the concealed-column elements extending to just over one-third of the wall height, as compared with the full element height recommended by the codes. In contrast with the case of the stirrups, the amount of horizontal web reinforcement specified by the CFP method was similar to the code specified amount for the structural walls investigated. References 1. Eurocode 2 (EC2) (2004) Design of concrete structures. Part 1-1: general rules and rules of building. British Standards Institution, London 2. Eurocode 8 (EC8) (2004) Design of structures for earthquake resistance. Part 1: general rules, seismic actions and rules for buildings. British Standards Institution, London 3. Kotsovos MD, Pavlovic MN (1995) Structural concrete: finite-element analysis for limit-state design. Thomas Telford, London 4. Kotsovos MD (2014) Compressive force-path method: unified ultimate limit-state design of concrete structures. Springer, Hardcover 5. Cotsovos DM, Kotsovos MD (2007) Seismic design of structural concrete walls: an attempt to reduce reinforcement congestion. Mag Concr Res 59(9):

15 6. Cotsovos DM, Pavlovic MN (2005) Numerical investigation of RC walls subjected to cyclic loading. Comput Concr 2(3): Kotsovos GM, Kotsovos MD (2008) Criteria for structural failure of RC beams without transverse reinforcement. Struct Eng 86(23/24): Kotsovos GM, Cotsovos DM, Kotsovos MD, Kounadis AN (2011) Seismic behaviour of RC walls: an attempt to reduce reinforcement congestion. Mag Concr Res 63(4): Kotsovos GM (2011) Assessment of the flexural capacity of RC beam/column elements allowing for 3D effects. Eng Struct 33(10):