Efficient three-dimensional seismic analysis of a high-rise building structure with shear walls

Efficient three-dimensional seismic analysis of a high-rise building structure with shear walls In many cases, high-rise building structures are designed as a framed structure with shear walls that can effectively resist horizontal forces. Many of the high-rise apartment buildings recently constructed in the Asian region employ the box system that consists only of reinforced concrete walls and slabs as the structural system. In most of these structures, a shear wall may have one or more openings for functional reasons. It is necessary to use a refined finite element model for an accurate analysis of a shear wall with openings.

Engineering Structures 27 (2005) 963–976 www.elsevier.com/locate/engstruct Efficient three-dimensional seismic analysis of a high-rise building structure with shear walls Hyun-Su Kima, Dong-Guen Leea,∗, Chee Kyeong Kimb a Department of Architectural Engineering, Sungkyunkwan University, Chun-chun-dong, Jang-an-gu, Suwon, 440-746, Republic of Korea b Department of Architecture, Sun moon University, Kalsan-ri, Tangjeong-myeon, Asan-si, Chungnam, 336-708, Republic of Korea Received November 2003; received in revised form December 2004; accepted 17 February 2005 Available online April 2005 Abstract In many cases, high-rise building structures are designed as a framed structure with shear walls that can effectively resist horizontal forces Many of the high-rise apartment buildings recently constructed in the Asian region employ the box system that consists only of reinforced concrete walls and slabs as the structural system In most of these structures, a shear wall may have one or more openings for functional reasons It is necessary to use a refined finite element model for an accurate analysis of a shear wall with openings But it would take a significant amount of computational time and memory if the entire building structure were subdivided into a finer mesh Thus an efficient method that can be used for the analysis of a high-rise building structure with shear walls regardless of the number, size and location of openings in the wall is proposed in this study The proposed method uses super elements, substructures and fictitious beams Static and dynamic analyses of example structures with various types of opening were performed to verify the efficiency and accuracy of the proposed method It was confirmed that the proposed method can provide results with outstanding accuracy requiring significantly reduced computational time and memory © 2005 Elsevier Ltd All rights reserved Keywords: Shear wall with openings; Super elements; Substructuring technique; Matrix condensation; Stiff fictitious beam Introduction It is common to design high-rise building structures in a framed structure with shear walls to resist horizontal loads such as wind or seismic loads This structural system may have many openings in the shear walls to accommodate the entrances to elevators or staircases etc., as shown in Fig In the analysis of this kind of building structure, commercial software such as ETABS [1] and MIDAS/ADS [2] is generally used In general, plane stress elements and beam elements are used to model the shear walls and frames respectively in the analysis of this kind of building structures Drilling degrees of freedom are required in the plane stress elements for the connection of shear ∗ Corresponding author Tel.: +82 31 290 7554; fax: +82 31 290 7570 E-mail address: dglee@skku.ac.kr (D.-G Lee) 0141-0296/$ - see front matter © 2005 Elsevier Ltd All rights reserved doi:10.1016/j.engstruct.2005.02.006 walls and frames Otherwise, beams cannot be rigidly connected to shear walls, resulting in the underestimation of the lateral stiffness of a building structure For this reason, the use of plane stress elements with drilling degrees of freedom was proposed by Allman [3], and Bergan and Fellipa [4] The concept has been further elaborated by many other researchers to obtain improved elements [5–8] Choi et al added non-conforming modes to the translational and rotational degrees of freedom (DOFs) to obtain an improved element [9] Kwan et al developed a finite element with rotational DOFs defined as vertical fiber rotations, which is compatible with beam elements [10,11] Lee provided a 12 DOFs plane stress element having two translational DOFs and one rotational DOF per node based on the 16 DOFs plane stress element proposed by Barber [12,13] The displacement shape functions along the boundary of the Lee element are identical to those of a typical beam element and the Lee element can accurately represent the shear stress 964 H.-S Kim et al / Engineering Structures 27 (2005) 963–976 (a) Floor plan (b) A-A section Fig Typical frame structure with shear walls (a) Typical plan of apartment (b) Window type opening (c) Door type opening Fig Shear wall with openings distribution in an element Therefore, a Lee element can be used appropriately for the modeling of the shear wall in the building structures Recently, many high-rise apartment buildings have been constructed in the Asian region using the box system, which consists only of reinforced concrete walls and slabs Shear walls in a box system structure may have openings to accommodate windows, doors and duct spaces, as shown in Fig 2(a), and window and door type openings in shear walls are shown in Fig 2(b) and (c) The number, location and size of these openings would affect the behavior of a structure as well as stresses in the shear wall Therefore, it is necessary to use a refined finite element model for an accurate analysis of a shear wall with openings But it would be inefficient to subdivide the entire apartment building structure into a finer mesh with a large number of elements because of the tremendous amount of analysis time and computer memory costs Therefore, many researches on the efficient analysis of a shear wall with openings have been performed [14–17] Ali and Atwall have presented a simplified method for the dynamic analysis of plates with openings based on Rayleigh’s principle of equilibrium of potential and kinetic energies in a vibrating system [14] Tham and Cheung have also presented an approximate analytical method for a laterally loaded shear wall system with openings [15] Each opening is taken into account by incorporating a negative stiffness matrix into the overall stiffness matrix through the super element concept Choi and Bang have developed a rectangular plate element with rectangular openings [17] The stiffness matrix of the element was formed by numerical integration in which the region for the opening in the element was excluded But the efficiency and accuracy of these analysis methods largely depended significantly on the location, size and number of openings Approximate modeling methods for a shear wall with openings are frequently adopted to avoid the troublesome preparation of refined models and significant amount of computational time in practical engineering When the size of an opening is significantly smaller than that of the shear wall, the opening is usually ignored, as shown in Fig 3(a) In the case of a door type opening, the lintel may be modeled by an equivalent stiffness beam, as shown in Fig 3(b) If the opening is quite large, the surrounding part of the shear wall would be modeled using beam elements, as shown in Fig 3(c) and (d) However, this type of models may lead to inaccurate analysis results, especially in dynamic analyses [18] An efficient method for an analysis of a shear wall with openings was proposed by Lee et al using stiff fictitious beams to enforce the compatibility at the boundary of super elements [18,19] Fig 4(a) shows the deformed shape of a shear wall with window type openings due to lateral loads obtained using a refined finite element model The model using super elements derived without stiff fictitious beams could not satisfy the compatibility condition at the interfaces, as shown in Fig 4(b) As could be observed in Fig 4(c), stiff fictitious beams used in a super element could result in the deformed shape of the structure very close to that of the refined mesh model A similar result could be obtained, as shown in Fig 5, for a shear wall with door type openings This method is very efficient for a two-dimensional analysis of a shear wall with openings Therefore, similar results can be expected in a threedimensional analysis of high-rise building structures if a three-dimensional super element developed in a similar manner were used An efficient method for a three-dimensional analysis of a high-rise building structure with shear walls is proposed in this study Three-dimensional super elements for shear walls and floor slabs were developed and a substructure was formed by assembling the super elements to reduce the time required for the modeling and analysis The proposed method turned out to be very useful for an efficient and accurate analysis of high-rise building structures based on the analysis of example structures Use of a fictitious stiff beam The use of a fictitious stiff beam is one of the most important techniques used in the proposed analytical H.-S Kim et al / Engineering Structures 27 (2005) 963–976 965 Fig Approximate modeling methods for shear wall with openings (a) Refined mesh (a) Fine mesh model (b) Super element w/o fictitious beams (c) Super element w/ fictitious beams Fig Deformed shape of a shear wall structure with window type openings (b) Super element w/o fictitious beam (c) Super element w/ fictitious beam Fig Deformed shape of box system structure (a) Refined mesh (b) Super element w/o fictitious beam (c) Super element w/ fictitious beam Fig Deformed shape of a shear wall structure with door type openings method Therefore, the procedure of the use of a fictitious beam is theoretically explained in this section Three types of modeling methods are used to verify the efficiency of the proposed method, as shown in Fig Fig 6(a) represents the refined mesh model that is assumed to be the most accurate Each shear wall in a story can be modeled with a single element, as shown in Fig 6(b), for more efficient analysis The proposed model in this study is illustrated in Fig 6(c) The equilibrium equation for the refined mesh model can be rearranged as shown in Eq (1) by separating the active DOFs for the corners of shear walls from the inactive DOFs for the boundary and inner area of shear walls and floor slab as follows: SD = = Sii Sia Sai Saa Di Da ) ) S(W S(S) S(S) S(W ii ia ii ia + ) (S) ) S(S) S(W S(W aa Saa Ai Di = Da Aa (1) where subscripts a and i are assigned to the DOFs for the active and inactive nodes respectively, the matrix S(S) is the 966 H.-S Kim et al / Engineering Structures 27 (2005) 963–976 stiffness matrix for a floor slab, and S(W ) is the stiffness matrix for a shear wall A Gaussian elimination process can be employed to condense Eq (1) into the equation consisting of only active DOFs at corner nodes of the slab and wall GSD = GA (2) where the matrix G makes the stiffness matrix S into an upper triangular matrix If the equation is represented by separating the active and inactive DOFs, then Gii O Gai I Sii Sia Sai Saa Di Gii O = Da Gai I Ai Aa = Gii Sia Gii Sii O Gai Sia + Saa = Gii Ai Gai Ai + Aa Di Da Di Da (4) T Sia = −Sii Gai (5) The second row of Eq (4) can be expanded as follows: (Gai Sia + Saa )Da = Gai Ai + Aa (6) Substitution of Eq (5) into Eq (6) leads to the following result: T + Saa )Da = Gai Ai + Aa (−Gai Sii Gai (7) This equation can be represented by using the slab stiffness matrix (S(S)) and the shear wall stiffness matrix (S(W )) as follows: (S) (W ) T T (W ) − Gai Sii Gai + S(S) (−Gai Sii Gai aa + Saa )Da = Gai Ai + Aa (8) On the other hand, modeling a shear wall using a single element and joining a shear wall to a slab only at corner nodes leads to the following equilibrium equation: (S) (S) O O Sii Sia (S) (S) + O S(W A) Sai Saa aa Di Ai = Da Aa (9) A) where the matrix S(W is the stiffness matrix for shear walls aa that is modeled by a single element It is different from ) S(W aa , which is the stiffness matrix for active DOFs of shear walls modeled with a refined mesh In order to make the equilibrium equation consist of only active DOFs at common nodes of the slab and wall, a Gaussian elimination process can be employed as follows: Hii O Hai I O O S(S) S(S) ii ia + A) (S) O S(W Sai S(S) aa aa Hii O = Hai I Ai Aa Hii S(S) Hii S(S) ii ia (S) (W A) O Hai Sia + S(S) aa + Saa Di Da Hii Ai Hai Ai + Aa = (S) (3) If Eq (3) is developed, the stiffness matrix is transformed to an upper triangular matrix and Sia can be represented as follows: Gii Sia Gii Sii Gai Sii + Sai Gai Sia + Saa where the matrix HHaiii OI makes the stiffness matrix into an upper triangular matrix Eq (10) can be represented as an upper triangular stiffness matrix by the Gaussian elimination process and Sia can be given as Eq (12): (11) (S) T Sia = −Sii Hai (12) After expansion of second row of Eq (11), substitution of Eq (10) into that expanded equation leads to Eq (13) (S) T (W A) (−Hai Sii Hai + S(S) aa + Saa )Da = Hai Ai + Aa (13) It can be easily noticed that the stiffness in Eq (13) is different from that of the equilibrium equation constituted by the refined mesh model (Eq (8)) To remove this difference, a fictitious beam is employed in this study From the proposed method using a fictitious stiff beam, the equilibrium equation can be represented as follows: O O S(S) S(S) S(B) S(B) ii ia ii ia + A) (S) (B) (S) + O S(W Sai Saa Sai S(B) aa aa Di Da Ai Aa = (14) where S(B) denotes the stiffness matrix of the fictitious beam A Gaussian elimination process was used to make the equilibrium equation consist of only active DOFs at common nodes of the slab and wall as follows: Jii O Jai I B) B) O O S(S S(S ii ia (S B) (S B) + O S(W A) Sai Saa aa O O O S(G) aa − Di J O = ii Jai I Da Ai Aa (15) where S(S B) = S(S) + S(B) and S(G) aa represents the stiffness matrix of the beam that is to be subtracted From the Gaussian elimination process, Eq (15) can be (S B) transformed into an upper triangular matrix and Sia can be represented as Eq (17) B) B) Jii S(S Jii S(S ii ia (S B) B) (W A) O Jai Sia + S(S − S(G) aa + Saa aa Di Da Jii Ai Jai Ai + Aa (16) B) B) T S(S = −S(S Jai ia ii (17) = Substitution of Eq (17) into the second row of expanded Eq (16) gives: Di Da (10) (S B) T Jai (−Jai Sii B) (W A) + S(S − S(G) aa + Saa aa )Da = Jai Ai + Aa (18) H.-S Kim et al / Engineering Structures 27 (2005) 963–976 967 From the equation S(S B) = S(S) + S(B), Eq (18) can be further expanded as follows: (B) T T (S) (B) (W A) (−Jai S(S) − S(G) aa ) ii Jai − Jai Sii Jai + Saa + Saa + Saa × Da = Jai Ai + Aa (19) If the stiffness (S(B) aa ) of the fictitious beam is the same ) as the stiffness (S(W aa ) of the refined shear wall, the following relationships can be noticed from comparison of the equation of the refined mesh model (Eq (8)) and that of the proposed model (Eq (19)) In conclusion, it can be expected that the proposed method can approximately represent the behavior of the refined mesh model G≈J (S) T (S) T Gai Sii Gai ≈ Jai Sii Jai (W ) T (B) T ≈ Jai Sii Jai Gai Sii Gai A) (W ) (B) ≈ S(G) S(W aa aa → Saa ≈ Saa (a) Refined mesh (20) (21) (22) A) + S(W aa − S(G) aa (23) Generally, the in-plane stiffness of a shear wall or floor slab is significantly large compared with the out-ofplane stiffness Therefore, a fictitious beam can employ sufficiently large stiffness for the compatibility condition as long as it may not cause numerical errors in the matrix condensation procedure As stated previously, it would be more efficient to model each shear wall in a story with one element to minimize the number of nodal points used, which is shown in Fig 6(b) In this case, however, the compatibility condition will not be satisfied at the interface of the slabs and the shear walls, because most of the nodes at the boundary of the slabs are not shared with those in the shear walls The lateral stiffness of this model becomes smaller than that of the refined model The stress distributions in the floor slab for these two models are significantly different from each other, as shown in Fig 7(a) and (b) The number of elements used in the proposed model shown in Fig 6(c) is identical to the model in Fig 6(b), but much less than that of the refined mesh model in Fig 6(a) The deformed shape and stress distribution of the model with fictitious beams are, however, similar to those of the refined mesh model in Figs 6(a) and 7(a), which are considered to be the most accurate Modeling of a shear wall structure with openings 3.1 Finite element for modeling of shear walls and floor slabs The plane stress element used by Lee et al for the development of 2D super elements for the analysis of a shear wall structure with openings was the Lee element [12] with 12 DOFs, as shown in Fig 8(a) Because the edge of the Lee element deforms in a cubic curve just like the beam element, the in-plane deformation of the edge of a slab or shear wall including fictitious beams will be nearly consistent with that of the neighboring shear wall or slab (b) Super element w/o fictitious beam (c) Super element w/ fictitious beam Fig Von-Mises stress distribution in slab The finite element to be used in this study should be able to represent the out-of-plane deformation as well as the inplane deformation of walls and slabs for a three-dimensional analysis of building structures with shear walls For a threedimensional analysis of a high-rise building structure with shear walls, a shell element with DOFs per node shown in Fig 8(c) was introduced by combining the Lee element and a plate bending element For this purpose, the MZC element [22] with a rectangular shape as shown in Fig 8(b) was selected because of the convenience in the combination of stiffness matrices 3.2 Modeling of a shear wall structure using super elements The efficiency in the modeling and analysis of a building structure can be significantly improved by using super elements A super element derived from the assemblage of 968 H.-S Kim et al / Engineering Structures 27 (2005) 963–976 (a) 12 DOFs plane stress element (Lee element) (b) 12 DOFs plate bending element (MZC element) (a) Refined model (b) Separate blocks (c) 24 DOFs shell element Fig Finite element for shear walls and floor slabs several finite elements for a shear wall or floor slab in the structure has much fewer DOFs compared to the original assemblage of finite elements Therefore, the computational time and memory can be significantly reduced And the modeling of the building structure would be more efficient since a super element can be used repeatedly in many places Fig 9(a) illustrates a refined mesh model of a shear wall structure This refined mesh model can be separated into several blocks of finite elements having the same configuration in each story, as shown in Fig 9(b) Super elements for shear walls and floor slabs can be generated, as shown in Fig 9(c), if all of the DOFs for the inactive nodes are eliminated by using the matrix condensation technique to have only active nodes in the super element The active nodes indicated by solid circles in Fig 9(c) and (d) are used to connect the shear walls and floor slabs Then, the entire structure is assembled by joining the active nodes of super elements, as shown in Fig 9(d) The equation of motion for a block of finite elements can be rearranged as shown in Eq (24) The subscripts a and i are assigned to the DOFs for the active and inactive nodes respectively Mii Mia Mai Maa ăi D Sii Sia ă a + Sai Saa D Di Ai = Da Aa (24) Eliminating the DOFs by the matrix condensation procedure [23], the equation of motion for the super element can (c) Generate super elements (d) Assemble super elements Fig Modeling procedure using super elements be obtained as follows: ă a + Saa Da = A∗a M∗aa D (25) M∗aa T T = Maa + Tia Mia + Mai Tia + Tia Mii Tia , where −1 −1 ∗ ∗ Aa = Aa − Sai Sii Ai , Saa = Saa − Sai Sii Sia and Tia = ∗ ∗ −S−1 ii Sia The matrix Maa is the mass matrix, Saa is the ∗ stiffness matrix, Aa is the reduced action vector and Da is the vector of nodal degrees of freedom for a super element with only active nodes If this super element is used in the numerical model, the compatibility condition will not be satisfied at interfaces of super elements because the nodes only at the corners of the super elements are shared by adjacent super elements Therefore, the lateral stiffness of the entire structure may be underestimated in comparison to that of the refined model Thus, it is necessary to enforce H.-S Kim et al / Engineering Structures 27 (2005) 963–976 969 the compatibility without using additional nodes along the interface of super elements for an accurate and efficient analysis 3.3 Super elements for shear walls and floor slabs Stiff fictitious beams introduced by Lee et al [18–21] were used to enforce the compatibility at the interface of super elements in this study The use of fictitious beams in the development of a super element for the floor slab shown in Fig 9(b) is illustrated in Fig 10 Fictitious beams are added to the interface of the floor slab and five shear walls, as shown in Fig 10(a) Because the analysis is expanded from two dimensions [18] into three dimensions, the fictitious beams used in this procedure are threedimensional elements Then, all of the DOFs except those for the active nodes located at the ends of each fictitious beam are eliminated as shown in Fig 10(b) using the matrix condensation technique The surplus stiffness introduced by the fictitious beams should be eliminated by subtracting the stiffness of fictitious beams from the stiffness matrix of the super element, as shown in Fig 10(c) It should be noticed that the fictitious beams in Fig 10(a) are subdivided into many elements to share nodes with the refined mesh of the floor slab, while the fictitious beam in Fig 10(c) has nodes only at both ends Finally, a super element with the effect of fictitious beams can be generated, as shown in Fig 10(d) Figs 11–15 illustrate the use of fictitious beams in the development of super elements for shear walls A, B, C, D and E shown in Fig 9(b) The location of fictitious beams added to the refined model for a shear wall depends on the location of the shear walls, and the selection of nodes to be maintained in the super element depends on the type and location of the openings in the shear wall In a 2D analysis of a shear wall structure, the compatibility condition is to be satisfied on the boundary between the shear walls in the adjacent stories However, the compatibility condition on the boundary between the neighboring shear walls in a floor or between floor slabs and shear walls in addition to the boundary between the shear walls in the adjacent stories should be satisfied in a 3D analysis A fictitious beam is added to each side of the shear wall A as shown in Fig 11 to enforce the compatibility between this shear wall and the shear wall B and D The compatibility condition between this shear wall and the slab in this floor or the floor above can be approximately satisfied by the fictitious beam added at the top or bottom of this wall The short fictitious beam added in between two openings is to enforce the compatibility with the shear wall C The fictitious beams on both sides of the shear wall B are to enforce the compatibility between this shear wall and the shear wall A and E The compatibility at the boundary between this wall panel and the floor slab is enforced by two short fictitious beams at the bottom of the wall, and the same fictitious beams are added at the top, as shown in Fig 12 Since the opening is located at the left edge of the shear (a) Add fictitious beams (b) Condense matrices (c) Subtract fictitious beams (d) Super element Fig 10 Use of fictitious beams for floor slab C, as shown in Fig 13, a short fictitious beam is added on the left side of the wall for a similar reason of using a short fictitious beam at the bottom of the wall The short fictitious beam used for the shear wall A in Fig 11 and this fictitious beam will enforce the compatibility at the boundary between the shear walls A and C The fictitious beams on the perimeter of the shear walls D and E, as shown in Figs 14 and 15, are to enforce compatibility at the boundary with shear walls or floor slabs 970 H.-S Kim et al / Engineering Structures 27 (2005) 963–976 (a) Add fictitious beams (c) Subtract fictitious beams (b) Condense matrices (a) Add fictitious beams (b) Condense matrices (c) Subtract fictitious beams (d) Super element (d) Super element Fig 11 Use of fictitious beams for shear wall A in Fig 9(b) Fig 14 Use of fictitious beams for shear wall D in Fig 9(b) (a) Add fictitious beams (b) Condense matrices (c) Subtract fictitious beams (a) Add fictitious beams (b) Condense matrices (c) Subtract fictitious beams (d) Super element (d) Super element Fig 12 Use of fictitious beams for shear wall B in Fig 9(b) Fig 15 Use of fictitious beams for shear wall E in Fig 9(b) 3.4 Use of coarse mesh super elements (a) Add fictitious beams (c) Subtract fictitious beams (b) Condense matrices (d) Super element Fig 13 Use of fictitious beams for shear wall C in Fig 9(b) connected to this wall panel The compatibility condition at the boundary between shear walls C and E is approximately satisfied by the fictitious beam located inside the shear wall E In general, building structures have various arrangements of shear walls and columns in plan And the size, type and location of openings in shear walls and floor slabs may vary depending on their use Therefore, the finite element mesh for each block of a structure such as a floor slab or wall panel is modeled to account of the location of openings, shear walls and columns The nodes on the boundary of neighboring blocks should be shared in each block, as shown in Fig 16(a), to satisfy the compatibility condition Thus, it is necessary to use a finer mesh finite element model to consider various openings and locations of structural members for an accurate analysis of building structures However, when super elements with a limited number of nodes are used, coarse mesh models for shear walls and floor slabs can be used, as shown in Fig 16(b), because the compatibility at the boundary of the super elements is enforced by the fictitious beams Therefore, the location of nodes except the nodes shared with neighboring super H.-S Kim et al / Engineering Structures 27 (2005) 963–976 (a) Model A 971 (b) Model B (a) Fine mesh model (c) Model C (d) Model D (b) Coarse mesh model Fig 16 Mesh type of proposed analysis method elements and the mesh size are not restricted Thus, it would be very efficient to use super elements in modeling as well as in the analysis of a building structure Static analysis of the 5-story example structure shown in Fig was performed to verify the accuracy of the proposed method using five types of models, as shown in Fig 17 Model A is a fine mesh model which is assumed to provide the most accurate results Models B and C replace the link beam above the opening by an equivalent stiff beam, as shown in Fig 17(b) and (c) The rigid diaphragm assumption is applied to each floor in model C and the flexural stiffness of the floor is ignored Model D employs the super element proposed in this study generated from a fine mesh model while model E is derived from a coarse mesh model, as shown in Fig 17(d) and (e) The lateral displacements of each model subjected to a lateral load of 10 000 kg at roof level in the transverse direction are compared in Fig 18 In the case of models B and C, the lateral displacements were significantly larger than those of model A This overestimation in displacements was introduced by the overestimation of the shear deformation in the upper part of the shear wall at both sides of the opening because the lintel is modeled by an equivalent beam element Since the flexural stiffness of the floor slab was ignored in model C, the lateral displacements were even larger than those of model B Model D could provide lateral displacements very close to those of model A, indicating that the compatibility is well enforced at the (e) Model E Fig 17 Name of analytical models Fig 18 Lateral displacement of example structure boundary of super elements by the effect of fictitious beams Since the lateral stiffness of a coarse mesh model is usually 972 H.-S Kim et al / Engineering Structures 27 (2005) 963–976 overestimated compared to that of a fine mesh model, model E resulted in slightly smaller displacements compared to those of model D Three-dimensional modeling of a building structure using substructures (a) Refined mesh model of shear walls Most of the high-rise buildings may have the same plan repeatedly in many floors Thus, it may be very efficient to apply the substructuring technique in the preparation of the numerical model In this section, the procedure in modeling a building structure using substructures is presented for the case of a high-rise apartment building Shear walls in a story are modeled as a substructure by assembling super elements, and a floor slab is modeled by combining super elements for the floor slab of each residential unit and staircase (b) Blocks for shear walls 4.1 Modeling of shear walls using substructures The modeling procedure for shear walls in a story using a substructure is illustrated in Fig 19 The refined finite element model for the shear walls in a typical floor shown in Fig 19(a) is to be modeled as a substructure As illustrated in Fig 19(b), the refined mesh model is separated into many blocks for the generation of super elements The separated blocks for shear walls can be classified into several types according to their configuration If several shear walls are of the same type, they can be modeled by the same super element Then, the super elements derived from corresponding blocks, as shown in Fig 19(c), are assembled into a substructure for shear walls in a typical floor, as shown in Fig 19(d) (c) Generation of super elements (d) Generation of substructure 4.2 Modeling of floor slabs by using substructures The procedure to model floor slabs in a floor into a substructure is illustrated in Fig 20 The refined finite element model for floor slabs in a floor is shown in Fig 20(a) The floor slab in a floor can be separated into three blocks for the residential units and staircase, as illustrated in Fig 20(b), to develop super elements Super elements are derived for corresponding residential units and staircase respectively, as shown in Fig 20(c) Since super element SE-A’ is the mirror image of super element SE-A, the stiffness and mass matrices for this super element can be obtained easily by rearranging the DOFs and changing the algebraic sign of terms correspondingly The number of super elements to be used in modeling the floors in a building structure will be limited, because the type of residential units in a high-rise apartment building is usually limited to one or two A substructure for the floor slab in a floor can be formed by assembling the super elements, as illustrated in Fig 20(d) The nodes in the substructure are selected for the connection of the slab and shear walls Fig 19 Modeling process of shear walls by using a substructure 4.3 Three-dimensional modeling of building structures using substructures The entire structure can be modeled by assembling the substructures representing the floor slabs and the shear walls, respectively Fig 21 illustrates the modeling procedure for a typical story by combining the floor slab substructures with the shear wall substructures This substructure can be used repeatedly for all of the stories with the same floor plan in a building structure If the rigid diaphragm assumption is applied, the number of in-plane DOFs in a floor can be reduced to three, and out-of-plane DOFs can be eliminated by the matrix condensation procedure again Therefore, building structures, for which the slab and the shear wall are subdivided into plate elements, can be modeled as a stick having DOFs per story Therefore, the computational time and memory for the analysis can be significantly reduced in comparison with the refined mesh model when H.-S Kim et al / Engineering Structures 27 (2005) 963–976 973 Analysis of example structures (a) Refined mesh model of floor slab (b) Division of floor slab (c) Generation of super element (d) Generation of substructure Fig 20 Modeling process of floor slab by using a substructure Analyses of two example structures were performed to verify the efficiency and accuracy of the proposed numerical method A framed structure with a shear wall core and a box system structure were used as example structures in the analyses Equivalent lateral forces were applied for the static analysis and the ground acceleration record of the El Centro (1940, NS) was used as input ground motion for the dynamic analysis 5.1 A framed structure with a shear wall core Recently, many high-rise buildings have been constructed using a frame with shear wall cores Static and dynamic analyses of a 10-story building structure with door type openings in the shear wall core, as shown in Fig 22, were performed Equivalent static, eigenvalue and time history analyses were performed and the results are shown in Fig 23 Models A, C and D were prepared in the same manner as explained in Fig 17 Model C is frequently used by many practical engineers, and the method proposed in this study was used in model D The lateral displacements of model D are similar to those of model A, as could be observed in Fig 23(a), while model C significantly overestimated the lateral displacements for the same reason as explained in Section 3.4 for the similar overestimation in Fig 18 The natural periods of model C were longer than those of the other models as expected based on the lateral displacements, as shown in Fig 23(b) The roof displacement time histories of models A and D are very close, while model C resulted in somewhat different displacement from the others, as shown in Fig 23(c), because of the difference in the natural periods 5.2 A shear wall structure Fig 21 Modeling process of typical story by using substructures the proposed method is used in the analysis, because the model can represent the behavior of a refined mesh model using only a limited number of DOFs Furthermore, the time and effort required for the preparation of a numerical model can be significantly saved if the building has an identical plan in many floors This kind of stick model is employed by conventional analysis software such as ETABS or MIDAS/ADS However, this conventional stick model does not include the flexural stiffness of a floor slab or the effects of openings in shear walls The second example structure is a 20-story reinforced concrete shear wall building with window type and door type openings, as shown in Fig 24 The example structure has two residential units arranged symmetrically with a staircase in between The thickness of shear walls and floor slabs is 20 cm and 15 cm, respectively Analyses of the example structure were performed, and the results shown in Fig 25 were obtained The lateral displacements of the proposed model D turned out to be almost identical to those of model A while model C significantly overestimated the lateral displacements, as shown in Fig 25(a), because of the underestimation of the lateral stiffness The natural periods of vibration are overestimated by model C, as shown in Fig 25(b) Therefore, the displacement time history from model C is somewhat deviated from the others, as shown in Fig 25(c) Model D could provide a roof displacement time history 974 H.-S Kim et al / Engineering Structures 27 (2005) 963–976 Table Comparison of DOFs and computational time required for analysis Models Number of DOFs Model A Model C Model D 56 640 60 60 Computational time (s) Assembly Static M&K analysis Eigenvalue analysis Time history analysis Total 78 10 156 16 623 97 97 387 12 13 18 071 125 272 983 6 (a) Typical floor (a) Lateral displacements (b) Natural periods (b) Floor plan (c) Displacement time histories Fig 23 Seismic analysis results of the example structure from the models A, C and D (c) A-A section Fig 22 Example structure almost identical to that of model A, as could be expected from the accuracy in the periods of vibration The computational time and the number of DOFs of each numerical model used for the analysis of the example structure are compared in Table Models C and D used only 60 DOFs because they applied the rigid diaphragm assumption to reduce the DOFs in a floor to 3, while model A, which is a refined finite element model, used more than 900 times the number of DOFs compared to the others Models A and C required 78 and 10 s to obtain stiffness and mass matrices while model D required 156 s because of the additional computation required to derive super elements and substructures The total computational time for the model A was 18 071 s including static and dynamic analyses while model C required only 125 s, demonstrating the reason why this model is commonly used by practicing engineers However, static and dynamic responses obtained using model C were significantly different from those of H.-S Kim et al / Engineering Structures 27 (2005) 963–976 975 (a) Typical floor (b) Floor plan (a) Lateral displacements (b) Natural periods (c) Roof displacement time history Fig 25 Seismic analysis results of the example structure from the models A, C and D (c) 3D view of example structure Fig 24 Example structure model A Model D could perform the analysis in 272 s because the computational time required for the procedure except for the formulation of mass and stiffness matrices is almost the same as that of model C, because of the same number of DOFs used in the analysis The accuracy in the static and dynamic analysis results of model D was at a similar level to that of model A, while the computational time required by model D is about 1.5% of that for model A In the case of larger building structures such as 30- or 40-story buildings with or residential units in a floor, the efficiency of the proposed model will be more significant because the same super elements can be used for the additional residential units and the same substructures can be used for the additional stories Therefore, the proposed method can be an efficient means for the analysis of a highrise building structure with shear walls A personal computer with Pentium 500 MHz processor and 512 MB RAM was employed in this study Conclusions An efficient three-dimensional model for the analysis of building structures with shear walls was proposed in this study using super elements and substructures The super elements were derived by introducing fictitious beams to satisfy the compatibility condition at the interfaces of super elements The accuracy and the efficiency of the proposed method were investigated by performing analyses of example structures Based on this study, the main features of the proposed method considered are summarized below: The refined finite element model of a high-rise building structure with shear walls is expected to cost a significant amount of computational time and memory while it would provide the most accurate results Thus the refined mesh model may not be feasible for practical engineering purpose The model using equivalent beams for the lintel above the openings and ignoring the flexural stiffness of the floor slab may lead to analysis results with somewhat deteriorated accuracy while computational time is significantly reduced Thus, it is undesirable to use this model for the analysis of an important or complicated building structure Therefore, it is desirable 976 H.-S Kim et al / Engineering Structures 27 (2005) 963–976 for the engineers in practice to be aware of the limitation in the accuracy of the results obtained by this model The proposed method could provide static and dynamic analysis results with an accuracy comparable to that of a refined mesh model with the cost of slightly increased computational time compared to the model using equivalent beams for the lintel Therefore, the proposed method can be an efficient means for the analysis of a high-rise building structure with shear walls The super elements are connected only through the active nodes and fictitious beams are used to enforce the compatibility at the boundary of super elements because the inactive nodes at the boundary are eliminated in the proposed method Thus, the location of inactive nodes in the finite element mesh to be used for a super element is not required to coincide with the counterpart in a neighboring super element Therefore, a super element can be developed easily, accounting for the location of active nodes independently Acknowledgements The Brain Korea 21 Project supported this work, and this work was partially supported by the Korea Science and Engineering Foundation (KOSEF) through the Korea Earthquake Engineering Research Center (KEERC) at the Seoul National University (SNU) References [1] Wilson EL, Habibullah A ETABS-three dimensional analysis of building systems users manual Berkeley (CA): Computers and Structures Inc.; 1995 [2] Lee HW, Park IG MIDAS/ADS-shear wall type Apartment Design System MIDAS Information Technology Co., Ltd; 2002 (http://webmaster@midasit.com) [3] Allman DJ A compatible triangular element including vertex 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Building and Environment 1999;34:109–27 [18] Kim HS, Lee DG Analysis of shear wall with openings using super elements Engineering Structures 2003;25(8):981–91 [19] Lee DG, Kim HS Analysis of shear wall with openings using super elements In: Proceeding of EASEC-8, 2001 Paper No 1378 [20] Lee DG, Kim HS, Chun MH Efficient seismic analysis of highrise building structures with the effects of floor slabs Engineering Structures 2002;24(5):613–23 [21] Lee DG, Kim HS The effect of the floor slabs on the seismic response of multi-story building structures In: Proceeding of APSEC2000 2000 p 453–61 [22] Zienkiewicz OC, Cheung YK The finite element method for analysis of elastic isotropic and orthotropic slabs Proceedings of the Institution of Civil Engineers 1964;28:471–88 [23] Weaver Jr W, Johnston PR Structural dynamics by finite elements Prentice Hall; 1987 p 282–90  ... obtained as follows: ă a + S∗aa Da = A a M∗aa D (25) M∗aa T T = Maa + Tia Mia + Mai Tia + Tia Mii Tia , where −1 −1 ∗ ∗ Aa = Aa − Sai Sii Ai , Saa = Saa − Sai Sii Sia and Tia = ∗ ∗ −S−1 ii Sia... and slabs for a three-dimensional analysis of building structures with shear walls For a threedimensional analysis of a high-rise building structure with shear walls, a shell element with DOFs... boundary and inner area of shear walls and floor slab as follows: SD = = Sii Sia Sai Saa Di Da ) ) S(W S(S) S(S) S(W ii ia ii ia + ) (S) ) S(S) S(W S(W aa Saa Ai Di = Da Aa (1) where subscripts a