- Open Access
Preliminary Structural Design of Wall-Frame Systems for Optimum Torsional Response
© The Author(s) 2016
- Received: 12 March 2016
- Accepted: 20 November 2016
- Published: 27 December 2016
Recent investigations have pointed out that current code provisions specifying that the stiffness of reinforced concrete elements is strength independent, and therefore can be estimated prior to any strength assignment, is incorrect. A strength allocation strategy, suitable for preliminary structural design of medium height wall-frame dual systems, is presented for allocating strength in such buildings and estimating the dependable rigidities. The design process may be implemented by either the approximate continuous approach or the stiffness matrix method. It is based on the concept of the inelastic equivalent single-degree-of-freedom system which, the last few years, has been used to implement the performance based seismic design. The aforesaid strategy may also be used to determine structural configurations of minimum rotation distortion. It is shown that when the location of the modal centre of rigidity, as described in author’s recent papers, is within a close distance from the mass axis the torsional response is mitigated. The methodology is illustrated in ten story building configurations, whose torsional response is examined under the ground motion of Kobe 1995, component KJM000.
- earthquake engineering
- inelastic structures
- strength dependent stiffness
- asymmetric buildings
- modal center of rigidity
Force-based methods for seismic design, as recommended by current building codes, have been questioned during the last two decades, in the sense that the selection of a more or less arbitrary force reduction (behavior) factor may not lead to a safer design, as potential damage is related more on the deformation capacity of the structure rather than on its strength. In more explicit terms, the concept that multistory buildings may be designed on the basis of a single force-reduction factor, depending on the structural type and not on the structural geometry (Priestley et al. 2007; Priestley 2000) is lacking in that the deformation capacity of the system under a horizontal loading is unknown and hence its vulnerability to seismic actions. As stated by Priestley (2000), two different buildings designed to the same code and with the same force-reduction or ductility factors may experience different levels of damage under a given earthquake. In other words, a proper design should be based on the principle that the seismic demand (the deformations induced by the seismic excitations) should be matched with the capacity of the structure to sustain such deformations. This approach, generally termed as ‘performance-based design’, is recommended the last few years to complement the current seismic design philosophy (Chopra and Goel 1999, 2000; Fajfar 2000; Heo and Kunnath 2013). Two major strategies are adopted by this approach: at first, based on experimental evidence, it became clear that because of the inelastic behavior of structural members and the associated ductility property, the magnitude of the induced inertia forces may be much lower than that predicted by an elastic analysis. The second refers to the realization that it is not the level of the design base shear the key point of a structural design, but the distribution of strength among the various members of a given structure, according to the capacity design concept, as it was developed in the seventies by Park and Paulay (1975).
For a well-detailed structure, an inelastic step by step time-history analysis is probably the most realistic procedure to evaluate its seismic response. With this methodology, the elastic or inelastic deformation state of response is taken into account at every step of the analysis, but, evidently, such a method is much more complex than a static procedure, costly and time consuming for structural applications. Besides, it requires an ensemble of representative ground motions to receive reliable information about the response of a given structure. In a few words, it is not practical for every day design use. As a compromise, the pushover analysis has been recommended to provide an estimate of the deformation capacity of structures exposed to seismic actions. In general terms, this procedure, which is the backbone of the ‘performance-based design’, consists an incremental static analysis under a monotonically increasing static loading until the top of the structure reaches the target displacement. This is usually bounded by predetermined limits of story drifts (critical for non-structural members) or strains capacities (critical for structural members), or when the base shear—top deflection diagram (pushover curve) drops by more (say) than 20% and the building model is considered unstable further on. This procedure requires that the structural model is well defined: at first a strength assignment of the various members should be implemented. Usually the lateral loading recommended by the code is used to determine yield moments at the locations of potential plastic hinges. The second requirement is that the moment-rotation relationships at these critical sections should also be well defined to allow for plastic deformation when the building is displaced beyond the elastic limits. Having defined the structural model, the inelastic static procedure, under increasing lateral loads, can easily be performed. A key point of this analysis is the shape of the distributed lateral loading, which, ideally, should represent the deflection profile of the system when it is stressed well into the post-elastic phase. Most of the proposals recommend this procedure for buildings which respond mainly in the translational mode and therefore the pushover analysis can be performed on the symmetrical counterpart of the real building. Three-dimensional pushover analyses performed on plan-asymmetric multi-story buildings have shown that seismic demands at or near the flexible edge are higher due to torsional effects (Moghadam and Tso 2000) and in another paper (Fajfar et al. 2005) it is suggested that the results obtained from the 3D pushover analysis should be combined with those of a linear dynamic analysis in order to assess the torsional amplifications. In fact, modern codes require special precautions for such cases (e.g. EC8 2004, clause 126.96.36.199.2.7).
The pushover curve, thus obtained is then expressed into an idealized bilinear force–displacement relationship (capacity curve) of an equivalent SDOF system (Chopra and Goel 1999, 2000; Fajfar 2000). Both branches of the later curve are drawn by engineering judgment and the slope of the initial branch specifies the ‘elastic’ period of the equivalent SDOF system, together with its yield displacement. The seismic displacement demand can then be determined from the acceleration design spectrum, when it is transformed in an acceleration–displacement (A–D) format (demand diagram).
This paper presents a simple procedure, suitable for preliminary structural design of low to medium height wall-frame dual systems. This type of structures has considerable merits in withstanding seismic actions and it is recommended by some modern codes (e.g. EAK 2000). The formation of the undesirable soft story mechanism is prevented and dual building systems combine the advantages of the two constituent sub-systems: wall and frame (Paulay and Priestley 1992; Garcia et al. 2010). The first objective of the paper is concerned with the assessment of the element flexural strengths and their dependable flexural rigidities, when the building is designed to form a beam-sway plastic mechanism into the inelastic phase. In particular, it is examined whether with this procedure the ‘elastic’ characteristics of the equivalent SDOF system (frequency and yield displacement or yield acceleration) may be accurately assessed from the first mode data of the elastic structure having the aforementioned rigidities. It is worth reminding here that the current forced-based design procedure of low or medium height structures is practically based on the first mode frequency and on a more or less arbitrary reduction factor. In most of the codes this frequency may be taken as that of the symmetrical counterpart structure and it is calculated on the grounds of flexural rigidities equal to a fraction of member’s gross sections.
In Sect. 2, expressions are provided for element flexural strengths and their dependable flexural rigidities, when the building is designed to be displaced as a beam-sway mechanism. In Sect. 3 the limits of inelastic displacements are investigated with respect to the code provisions and member plastic rotation capacities and predictions are made about the onset of yielding. In Sect. 4, it is shown how quick estimates of the ‘elastic’ characteristics of the equivalent SDOF system can be made by using the approximate continuum approach methodology. More accurate assessments of the ‘elastic’ characteristics of the equivalent SDOF system can also be made from the first mode data of the elastic discrete multistory system (with flexural rigidities as described in Sect. 2) when it is analyzed by the traditional stiffness method and in the numerical example presented at the end of this paper it is notable the closeness of these values with those provided by the capacity curve of the SDOF system.
The second objective of the paper refers to the torsional behavior of inelastic asymmetric structures. The design procedure described above refers to buildings responding in a more or less translational mode, and the pushover analysis demonstrates the displacement capacity of planar structures. It is generally accepted that eccentricity in buildings is the main cause of the rotational response during strong ground motions, and that in many cases this response may lead to partial or total collapse. In recent years a number of investigations have been carried out to demonstrate the seismic vulnerability of these buildings and qualitative papers have been published from time to time on this issue (e.g. Chandler et al.1996; Paulay 1998, 2001; Rutenberg 1998; De Stefano and Pintucchi 2008; De Stefano et al. 2015; Anagnostopoulos et al. 2015a, b; Bosco et al. 2015; Kyrkos and Anagnostopoulos 2011a, b, 2013). The recognition of the seismic vulnerability of such buildings has also raised the issue of mitigating the torsional effects during a strong ground motion. Most of the studies are based on systems with elements having the traditional strength independent stiffness, but a few of them involve systems with wall elements in which the stiffness is strength dependant (e.g.: Aziminejad et al. 2008; Aziminejad and Moghadam 2009). This issue has also been the subject of author’s recent research (Georgoussis 2008, 2009, 2010, 2012, 2014, 2015) in multistory systems with traditional strength independent element stiffnesses. It has been demonstrated that the seismic behavior of linear systems (composed by different types of bents: walls, frames, coupled wall assemblies, etc.) can be accurately assessed by analyzing two simpler systems: (i) the corresponding uncoupled multi-story structure which provides the first mode frequency and effective mass, M e * , and (ii) a torsionally coupled equivalent single story system, which has a mass equal to M e * , and is supported by elements with stiffnesses equal to the product of M e * with the squared frequencies of the corresponding real bents (element frequencies) of the assumed multi-story structure. In the case of uniform structures composed by very dissimilar bents, a higher accuracy of the aforementioned analysis can be attained with the use of the effective element frequencies, (Georgoussis 2014). The stiffness centre of the equivalent single story system constitutes the modal centre of rigidity (m-CR) and when this point lies on (or close to) the axis passing through the centers of floor masses, the rotational response sustained by an elastic asymmetric building system is minimum (Georgoussis 2009, 2010, 2012, 2014, 2015). In Sect. 5, the procedure of constructing a structural configuration of minimum torsional response is demonstrated by means of the formulation of the approximate continuous approach, using the strength dependent flexural rigidities of Sect. 2. This is a direct procedure, since the effective element frequencies of walls and frames are given by simple formulae and therefore the location of the stiffness center (m-CR) of the equivalent single story system is easily assessed. The same quantities can also be obtained by the familiar to designers stiffness method and this is demonstrated in the ten story model building examined in Sect. 6.
The third objective of the paper is to demonstrate that the elastic response of minimum torsion is preserved into the inelastic region when the element strength assignment is ‘compatible’ with static analyses under a lateral loading simulating the first mode of vibration. This has already been shown in asymmetric buildings with traditional, strength independent element rigidities (Georgoussis 2012, 2014, 2015) and can be explained as follows: when a medium or low height building structure, in the linear phase, is responding in a practically translational mode, the effective seismic forces developed are basically proportional to the first translational mode of vibration. Therefore, a strength assignment obtained from a planar static analysis under a set of lateral loads simulating the aforesaid mode of vibration, represents a system in which all potential plastic hinges at the critical sections are formed at about the same time. The almost concurrent yielding of these elements preserves the translational response, attained at the end of the elastic phase, to the post elastic one. This procedure of constructing a structural configuration of minimum rotational response is now investigated in asymmetric systems with elements having strength dependent stiffnesses and this is demonstrated in a ten story eccentric dual building under the ground motion of Kobe 1995, component KJM000.
The yield acceleration, A y , and the corresponding yield deformation, u y , are also shown in Fig. 1d, together with the slope of the initial elastic branch, ω e 2 , which represents the square value of the effective frequency.
2.1 Assigning Strength and Rigidity to the Frame Sub-system
Evidently, in exterior beam-column joints the column moment will be equal to half of that of Eq. (8) and therefore the stiffness of edge columns will be half of that given by Eq. (9a). Note here that induced axial compression loads in column sections affect their flexural strengths (typical bending moment-axial load (M–N) interaction diagrams indicate this relationship). When the axial load is below the ‘balance point’ of the mentioned diagrams, the column bending moment capacity assessed by Eq. (8) is underestimating the true column capacity, but this assessment does not really affect the yield drift of the frame as explained further below.
For edge columns, the base yield moments will be half of those assessed by the equation above.
2.2 Assigning Strength and Rigidity to the Wall Sub-system
Wall elements should be designed to resist an overturning moment equal to V dw H e . Again the designer has the choice to allocate different fractions of this bending moment to the various wall elements: from the classical method, in proportion to the traditionally defined (elastic) stiffness, to the methodology of having the same longitudinal steel ratio in all walls, as proposed by Paulay (1998).
Note that axial compression loads sustained by wall sections affect their flexural strengths and, when the mean compression stress reflects a low fraction of the concrete compression strength (as suggested by many building codes), the moment capacity may be higher. As a result the effective second moment of area determined by Eq. (12b) represents a conservative estimate of this property at the base of the wall. In practice however, the reinforcement content is gradually reduced at higher levels, resulting in lower bending capacities and therefore in reduced flexural rigidities. Therefore, for a preliminary structural design, it is considered satisfactory to assess the wall effective second moment of area by means of Eq. (12b).
4 The Fundamental Frequncy by the Continuous Approach and Estimates of the Reduction Factor and Ductility Demand
For uniform over the height buildings, responding in a translational mode, estimates of the first mode dynamic data (frequency, effective mass, yield displacement) can be made by means of the approximate continuous approach, where the structure is treated as a continuous medium. As follows, the evaluation of these quantities can be implemented by hand calculations without the need to perform any structural analysis. It is therefore useful for the preliminary stage of a practical application. Note that the following formulation is based on the grounds that the flexural rigidities (as calculated in Sect. 2) are ‘compatible’ with the concept of the beam-sway mechanism and further below, in the numerical example of Sect. 6, the results of this analysis are compared with the corresponding values derived from the capacity curve of the equivalent SDOF system. It is reminded here that the first objective of the paper is to examine whether the first mode dynamic characteristics of the structure, with the aforesaid rigidities, are close to the ‘elastic’ characteristics of the equivalent SDOF system (frequency and yield displacement or yield acceleration).
For the practical range of λ from 0.3 to 0.4, and d w /H from 0.10 to 0.18, the parameter αΗ varies from 1.1 to 1.9, when k w = 1.8, θ y = 1% and ε y = 0.002. For such values of αΗ the variation of the corresponding first mode effective mass, M e * , is very narrow [between 0.623 and 0.645 of the total mass (Georgoussis 2014)] and therefore it can be taken approximately equal to 0.635M tot .
By comparison of values shown in the Eqs. (24) and (25), it is evident that the ‘initial’ characteristics of the equivalent SDOF system (frequency, yield displacement and yield acceleration) are very close to the first mode data of the linear system, when the strength assignment (and the associated flexural rigidities) is ‘compatible’ with the beam-sway plastic mechanism. The results provided by the approximate continuous approach (Eq. 23), are less accurate, but quite satisfactory for the preliminary stage of a practical application. In all cases, as the period of the structure falls into the velocity sensitive range of commonly used design spectra, the reduction and ductility factors (R and μ) are equal. Note here that the period Te does not represent a Rayleigh quotient of the first mode period of the linear system, which is found by solving the eigenvalue problem (Chopra 2008).
The frequency and the reduction factor, which are the main parameters in the force-based design philosophy of low to medium height buildings, can be evaluated with reasonable accuracy by a simple methodology which (i) allocates strengths in wall-frame dual systems and, (ii) enables the determination of the dependable flexural rigidities in the various structural members. The method can be implemented by both the approximate continuum approach, which is very simple since it is based on a well known formulation and, also, by the stiffness matrix method using a commercial software of structural analysis. This methodology can easily be incorporated in the strategy of constructing structural configurations of minimum rotational response, which is the main requirement in the design of structures expected to sustain strong ground motions. The approach presented is based on simple principles and it is design oriented, useful in the preliminary stage of a practical application, where efficient, practical and economic solutions are sought by the practicing engineer.
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