Combined approximations method for reanalysis of natural
Transkrypt
Combined approximations method for reanalysis of natural
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 Porto, Portugal, 30 June - 2 July 2014 A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.) ISSN: 2311-9020; ISBN: 978-972-752-165-4 Combined approximations method for reanalysis of natural frequencies of reinforced buildings in mining areas 1,2 Krystyna Kuzniar1, Maciej Zajac2 Institute of Technology, Pedagogical University of Cracow, Podchorazych 2, 30-084 Krakow, Poland email: [email protected], [email protected] ABSTRACT: In the paper combined approximations hybrid method (CA) is proposed as an effective reanalysis approach for computation of natural frequencies of actual reinforced 12 storey prefabricated buildings located in the most seismically active mining region in Poland – Legnica-Glogow Copperfield (LGC). The reanalysis method is intended to analyse modified structures using some information about the structures before modification and without solving the complete set of equations in case of the new (modified) object. This idea leads to the computational effort reduction. Changes of building stiffness and mass resulted from the inner or exterior structural reinforcements were taken into account. Buildings were modelled using finite element method and the special code of CA algorithm was drawn up in Matlab software. CA procedure was applied in case of the same number as well as in case of increasing number of degrees of freedom (DOF) in structure after modification. Computed values of natural frequencies of building vibrations were compared with experimental ones obtained for actual structures. The results show that in case of the small as well as the large modifications of building structure, the eigenvalue analysis using CA method is much faster with no significant decrease of the accuracy. KEY WORDS: Eigenvalue reanalysis; Combined approximations method; Paraseismic excitations; Prefabricated buildings. 1 INTRODUCTION Precast reinforced concrete apartment and public utility buildings constructed in 20th century represent a significant part of all buildings, especially in Central and Easter Europe. Using precast reinforced concrete panels technology made the building production process much easier, faster and cheaper. However some disadvantages of this technology solution are observed nowadays. The limited building design flexibility as well as the age of the structures seem to be the biggest problem for current users. Taking into account for instance new environmental regulations, safety requirement and expectations of today’s lodgers, the modernization of precast concrete buildings become necessary. Mainly, modernization deals with heat insulation, additional storey, new door openings or widening the existing ones. For public utility buildings main works deal with adaptation to the needs of the disabled. A lot of prefabricated buildings are located in mining areas and they are subjected to paraseismic excitations induced by mining tremors. Although these tremors are strictly connected with the human activity and can usually be observed only in the mining regions, they differ considerably from other paraseismic vibrations. They are not subject to human control and they are random events with respect to the time, place and magnitude likewise earthquakes. Most of the structures have not been designed to carry this kind of load. So modernization and reinforcement of the buildings becomes necessary to assure safety in use of them [6, 17]. Such modernization increases the stiffness of building. For instance, various changes in structures of prefabricated buildings are carried out in the most seismically active mining region in Poland – Legnica-Glogow Copperfield (LGC) where mine-induced rockbursts excite tremors comparable to low intensity earthquakes, characterized by energies run up to 1E10 J, surface horizontal vibrations reaching even 0.2 acceleration of gravity (g) and vertical components reaching 0.3g. Every significant change in structure causes the changes in its dynamic properties, among them – in natural frequencies of vibrations which are used in response spectrum method [8] in case of kinematic excitations. Therefore new computations are necessary in every case of significant modifications of building structure, especially for buildings subjected to paraseismic or seismic excitations. Computation of natural frequencies of vibrations can be carried out using various methods. In case of simple models of structures it is possible to use analytic formulas. The eigenproblem for more complicated ones needs to be solved using numerical methods. Nowadays finite element method (FEM) [23, 24] is one of the most popular. In spite of fast progress in hardware and software, computation of the natural vibration frequencies of the modified actual buildings using finite element method usually needs a very long computational time because of a huge number of degrees of freedom of the models, the large scale problem and various modification variants discussed. That is why application of reanalysis techniques [4, 9, 10, 12] for computation of natural frequencies of modified structures is proposed in this paper. Reanalysis methods are intended to analyse modified structures using some information about the structure before modification and without solving the complete set of equations in case of the new (modified) object. This idea leads to the computational effort reduction. In the paper combined approximations (CA) hybrid method was applied and verified as an eigenvalue reanalysis approach in case of modifications of actual typical apartment high (12 3861 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 storey) buildings located in LGC mining region in Poland. Changes of building stiffness and mass resulted from the inner or exterior structural reinforcements were taken into account. Buildings were modelled using finite element method and the special code of CA algorithm was drawn up in Matlab [25] software. CA procedure was applied in case of the same number as well as in case of increasing number of degrees of freedom (DOF) in structure after modification. Computed values of natural frequencies of building vibrations were compared with experimental ones obtained for actual structures. 2 COMBINED APPROXIMATIONS HYBRID METHOD The main aim of combined approximations (CA) method [10, 13] application is to quicken computing by reduction of the number of eigenproblem equations. In the CA algorithm, basis vectors (global approximations), containing information about the initial structure and modifications, are computed by the terms of the binomial series (local approximation). Equations (1) and (2) describe the eigenproblems for structure before and after modification, respectively, KΦ i = λi MΦ i , i = 1, K , p, (1) K M Φ Mi = λMi M M Φ Mi , i = 1,K, p, (2) where: K, KM - stiffness matrices; M, MM - mass matrices; Φi, ΦMi - mode shapes; λi, λMi - eigenvalues before and after structure modification, respectively. Substitution matrices KM and MM in equation (2) for the relations (3) K M = K + ΔK M M = M + ΔM , (3) (where ΔK, ΔM - changes in stiffness and mass matrices corresponding to the geometrical changes of the structure) allows to write the eigenproblem for the modified structure using equation (4) with information about the initial model parameters and changes caused by modifications. ( K + Δ K )Φ M i = λ M i ( M + Δ M )Φ M i . (4) Computation of the basis vectors matrix rB according to equation (5) is the next step of CA approach [14]. rB = [r1 , r2 , K, rs ], (5) where: r1,…,rs – the basis vectors, s – the number of basis vectors; s is much smaller than the number of degrees of freedom. The first basis vector r1 is given by formula (6) [14]: r1 = K −1 M M Φ i . (6) Each next vector is calculated using the previous one and matrix B: rk = −Brk-1 , k = 2,3, K , s, 3862 (7) B = K −1 Δ K . (8) The following step is the evaluation of reduced stiffness KR and mass MR matrices: K R = rBT K M rB M R = rBT M M rB . (9) The first (lowest) eigenvalue λR1 of reduced eigenproblem described by Equation (10), represents the value of natural vibration frequency corresponding to the adequate value which is obtained from full modified eigenproblem using Equation (2) (y1 - vector of coefficients): K R y1 = λR1M R y1. (10) In case of modification where the number of DOF is increased [2, 18, 19, 22], the modification of CA algorithm is necessary because of changing of the number of equations. Sizes of the stiffness matrix and mass matrix are increased accordingly in such case of structure modification. Having a different number of DOF of the models of the initial and modified structure, it is impossible to define ΔK, ΔM matrices which represent real changes between initial and modified structure and are necessary for basis vectors computing, cf. Equations (3)-(8). The new pseudo initial model with added degrees of freedom needs to be created, because the sizes of eigenproblem before and after structure modification have to be equal. Pseudo initial model consists of Kf (Eq. 11) and Mf (Eq. 12) matrices that are created using stiffness and mass matrices of initial model (K, M) and new sub-matrices: KMn, KMp, KMnp, MMn, MMp, MMnp that represent the parts of KM, MM corresponding to the new DOF. αK Mp ⎤ , αK Mnp ⎥⎦ (11) α M Mp ⎤ ⎡ M Mf =⎢ ⎥. ⎣αM Mn αM Mnp ⎦ (12) ⎡ K Kf =⎢ ⎣αK Mn In case of the modification where the number of DOF is increased it is necessary to choose the proper value of α parameter that is a scalar multiplier to be selected from the range (0, 1] [11, 15]. Now, equations (3) can be written as follow: K M = K f + ΔK f M M = M f + ΔM f , (13) where ΔKf, ΔMf represent the changes between pseudo initial and modified model. Additionally it is important to define new pseudo initial eigenvector Φfi for pseudo initial model [3], which is applied for calculation of the first basis vector according to Equation (5). Vector Φfi consists of two parts: the elements of initial Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 model eigenvector Φi and the vector ΔΦi containing elements related to new degrees of freedom. Further part of CA algorithm remains unchanged. 3 ANALYSED BUILDINGS In the paper three actual 12 storey precast concrete apartment buildings constructed according to one of Polish prefabricated systems – system WWP [6, 7] were analysed. The first one (building “A”) is typical for WWP system. The second one (building “B”) was reinforced because of mine-induced kinematic excitations with the inner and exterior structural reinforcements [5, 28]. In the building “C” (the third one) only the inner structural reinforcements were introduced as the modifications. All the analysed buildings are located in the seismically active mining region in Poland – Legnica-Glogow Copperfield (LGC). Transverse-longitudinal load-bearing concrete wall system has been applied in WWP prefabricated system. The thickness of the reinforced concrete walls is 14cm and the storey height is 2.7m [6, 7]. Precast gable and curtain walls consist of three layers: the thermal insulation layer and the inner and outer reinforced concrete layers. The buildings are founded directly on the ground using concrete strip foundations and there are solid reinforced concrete basements (wall thickness - 30cm) in the analysed buildings [5, 28]. Figure 2. Floor plan of typical segment of WWP system [6]. a) x Detail “1” y Detail “2” a) b) b) Figure 1. Analysed structures: a) building “A” - typical for system WWP; b) building “B” with inner and exterior structural reinforcements. c) Figure 3. Reinforcement of building “B”: a) Floor plan with reinforcements marked with bold line, where: (1) and (7) – new monolithic concrete spans; (3) and (5) – new monolithic load-bearing longitudinal walls; (2), (4), (6) – reinforcements of load-bearing transverse walls; b) Detail “1”; c) Detail “2”. 3863 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 Each of the three buildings is divided into single and double stairway segments by expansion gaps. The segments are connected to each other using the lightweight “wood concrete” called arbolit [26] in each floor plan. Examples of the analysed buildings are presented in Figure 1. Figure 2 shows the floor plan of typical segment of such buildings. The range of strengthening in case of the building "B" is presented in Figure 3. New monolithic concrete spans, built at both gable walls are labelled as (1) and (7) in Figure 3a. The height of extra spans is 7 and 10 storeys respectively [5, 28]. New monolithic load-bearing longitudinal walls described as (3) and (5) in Figure 3a (thickness of 20cm) were also constructed. Additionally, some of the existing transverse load-bearing walls (described as 2, 4, 6 in Figure 3a) as well as the basement walls were reinforced by wall thickening [5, 28]. FEM models of the building “B” parts with reinforcements are shown in Figure 3b and Figure 3c. 4 NUMERICAL MODELS OF BUILDINGS Numerical computations of natural frequencies of vibrations of analysed buildings were carried out in known FEM system – Ansys [27]. Building elements such as continuous concrete strip foundations, monolithic walls of basement, precast concrete load-bearing walls, curtain walls, stairways, lift shafts, air-shafts, flat roof were taking into account and modelled using 4-node elastic shell elements with 6 degrees of freedom. a) Figure 5. Model of foundation and basement of reinforced building “B”. The connections of load-bearing prefabricated walls with the floors were modelled as fixed and 40% of changing load was attached at each floor level. The number of degrees of freedom of models of buildings before modification (model “A”) and after strengthening of the load-bearing walls and reinforcing by the extra building spans (model “B”) is 612732 and 672138 respectively. Finite element models of analysed structures are presented in Figure 4. In case of elements of structural monolithic concrete members (foundation, basement, new walls and wall strengthening) one value of Young modulus E equal 23.1GPa was used. Figure 5 shows the model of foundation and basement of reinforced building “B”. The influence of panel joints on the wall stiffness was taken into consideration in the models. Young modulus (E) reduction of structural elements was proposed for this purpose. Several attempts to select the proper value of the reduced Young modulus were carried out and E = 20 GPa was applied in the computations. In the analysed models non-structural elements such as for instance division walls, stairways, lift shafts and air-shafts were also taken into consideration. From the results of investigations presented in [16] it is visible that the influence of non-structural elements on dynamic characteristics of buildings can be significant. The division walls in the analysed buildings are made of cavity brick, thickness – 6.5cm. They were modelled as shells fixed connected with load-bearing panels. Shell models of the division walls (dark color) in one-stairway segment of building are presented in Figure 6. The applied material parameters of structural elements are shown in Table 1. Soil flexibility was taken into consideration using translation and rotation spring elements according to Savinov soil model [20]. b) Figure 4. Finite element models of analysed structures: a) building “A”; b) building “B”. 3864 Figure 6. Shell models of division walls (dark color) in onestairway segment of building. Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 Table 3. Comparison of computed values of the natural vibration frequencies of the model of typical building “A” and the model of modified building “C”. Table 1. Material parameters for FEM elements. FEM element Foundation Basement Prefabricated elements Reinforced elements Cavity brick wall Arbolit [26] 3 E [GPa] 23.1 23.1 20 ν 0.2 0.2 0.2 ρ [kg/m ] 2500 2500 2500 23.1 0.72 0.9 0.2 0.25 0.25 2500 1400 700 To verify finite element models material and structure assumptions, the computed results were compared with experimental ones obtained in [21] for actual structures. Values of natural frequencies of horizontal vibrations of analysed buildings are presented in Table 2. It is visible that the computational accuracy is sufficient in engineering practice. Table 2. Comparison of computed and experimental values of natural frequencies of building horizontal vibrations before and after modification. Building “A” “B” Natural frequencies of horizontal building vibrations [Hz] computed experimental [21] f1x 1.59 1.60 – 1.63 f1y 1.73 1.71 – 1.76 f1x 1.79 1.64 – 1.71 f1y 2.47 2.17 – 2.28 The another type of modifications (introduced to typical WWP building – building “A”) does not change the number of DOF of the FEM model. As was said, in the building “C” only the inner structural reinforcements were introduced. The modifications relate to strengthening by thickening of the existed monolithic basement and precast load-bearing concrete walls (thickening of shell elements in the model of the building “C”). Thus the number of DOF of the model of building “A” and the number of DOF of the model of building “C” are the same and equal 612732. Floor plan of the model “C” with reinforcements marked with bold line is shown in Figure 7. Thickened gable walls are labelled as (1) and (3) and the thickened transverse loadbearing wall is described as (2). x y Natural frequencies of horizontal building vibrations [Hz] Building “A” Building “C” f1x 1.59 1.62 f1y 1.73 1.74 Comparison of computed values of the natural horizontal vibration frequencies of the model of typical building “A” and the model of modified building “C” is presented in Table 3. It is visible that the inner reinforcements (small modifications of the structure) applied in building “C” increase the values of natural frequencies of horizontal building vibrations, especially in transverse (“x”) direction. 5 REANALYSIS RESULTS The accuracy of combined approximations hybrid method algorithm (CA) as the reanalysis approach was verified in evaluation of the first natural frequency of horizontal vibrations of modified buildings subjected to mine-induced tremors. Two cases of eigenvalue reanalysis corresponding to two types of building modifications were analysed. The first one concerns strengthening of existing walls without changing the number of DOF of FEM model (building ”C”) in relation to the number of DOF of model of the initial structure (building “A”). In the second one the inner and exterior structural reinforcements (building “B”) are taken into account – increasing the number of DOF of FEM model in relation to the number of DOF of model of the initial structure. To illustrate accuracy of the results obtained using CA method fractions r1 and relative errors Errf1 were computed: r1 = f1CA f1FEM , Errf1 = f1FEM − f1CA (14) ⋅ 100% , f1FEM (15) where: f1CA, f1FEM – the first natural frequency of building vibrations computed using CA method and FEM, respectively. As the example, relative errors Errf1 of the first natural frequency of horizontal vibrations in transverse direction of building “B” obtained using CA method, in dependence of the various number of basis vectors are presented in Table 4. As one can see, using CA method with only 10 basis vectors ensures relative errors not greater than 8% by the reduction of number of eigenproblem equations from almost 700000 to 10. Comparison of computed and experimental values of the first natural vibration frequency of the modified building “B” is presented in Table 5. Table 4. Relative errors Errf1 of the first natural frequency of transverse vibrations of the modified building “B” obtained using CA method. Figure 7. Floor plan of the model “C” with reinforcements marked with bold line; where: (1) and (3) – the thickened gable walls, (2) – the thickened transverse load-bearing wall. No. of basis vectors 2 5 10 20 Errf1 [%] 55 27 7.8 6.7 3865 Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014 Table 8. Values of fraction r1 in case of model of modified building “B” in the function of number of basis vectors and value of α parameter. Table 5. Comparison of computed and experimental values of the first natural vibration frequency of the building “B”. Experimental [21] f1 [Hz] FEM model 1.64-1.71 CA (20 basis vectors) α =10-6 1.91 1.79 α -3 It is visible that a little worse accuracy of the computed frequency was achieved using CA method instead of FEM. But the result (Errf1 less than 7%) is acceptable in engineering practice. In Table 6 fraction r1 and Errf1 for model “C” is presented. As one can see, using CA method with only ten basis vectors ensures relative errors not greater than 3% (reduction of number of equations from almost 613000 to 10). Table 6. Fraction r1 and Errf1 of the first natural vibration frequency computed in case of the model “C”. No. of basis vectors r1 2 1.13 5 1.04 10 1.03 20 1.01 Errf1 [%] 12.5 3.9 2.5 0.8 Additionally, the number of numerical operations which are necessary to obtain natural frequencies of building vibrations using CA was estimated and compared with the number of numerical operations in very popular subspace iteration method [1]. Values of fraction NSIM / NCA [13] (NSIM, NCA – number of numerical operations for subspace iteration method and combined approximations method in case of single eigenvalue computation) depending on number of basis vectors in case of the model of building “B” are presented in Table 7. It is visible that using CA method with 20 basis vectors reduces the number of necessary numerical operations more then 20 times in proportion to application of subspace iteration method. In case of modification that increase the number of degrees of freedom, it is very important to choose proper value of α parameter that assure convergence of CA method. In case of models of actual buildings with significant number of DOF and large modifications, choosing α = 0.001 that is suggested in some papers [4, 12] does not guarantee to obtain satisfying results. Therefore the tests to find a relationship between the value of α parameter and “size” of model and its modifications were carried out. In Table 8 values of fraction r1 obtained in case of building “B” in the function of number of basis vectors and value of α parameter are presented. It is visible that the choice of α equal or smaller than 10-6 provides the results accurate enough. Table 7. Values of fraction NSIM / NCA in the function of number of basis vectors in case of the model of modified building “B”. No. of basis vectors 2 3 4 5 10 20 NSIM / NCA 216 144 108 86 43 21 3866 10 10-6 10-8 6 1 185 7.02 7.02 r1 = f1CA / f1FEM No. of basis vectors 3 5 10 180 174 116 1.49 1.27 1.08 1.41 1.31 1.09 20 82 1.07 1.07 CONCLUSIONS In the paper combined approximations hybrid method (CA) was proposed as a tool for reduction of computational effort in case of reanalysis of natural frequencies of reinforced wallbearing prefabricated buildings subjected to mine-induced tremors. Large and small modifications of high apartment buildings were analysed. It is clear from the obtained results that application of CA method leads to significant decreasing of number of necessary algebraic operations what makes the numerical analysis much faster. CA method is efficient reanalysis procedure for computing of natural vibration frequencies of structures with an accuracy sufficient in engineering practice. 7 ACKNOWLEDGMENTS Numerical analysis in the paper has been supported by ACK CYFRONET AGH software. 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