CONFERINŢA INTERNAŢIONALĂ DEDUCON 70

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1 CONFERINŢA INTERNAŢIONALĂ DEDUCON 70 DEZVOLTARE DURABILĂ ÎN CONSTRUCŢII Iaşi, 11 noiembrie 011 Conerinţă dedicată aniversării a 70 ani de învăţământ superior în construcţii la Iaşi A13 FREQUENC ANALSIS FOR CENTRAL CORE TALL BUILDINGS B BIANCA PARV * MONICA NICOREAC AND MIRCEA PETRINA Technical University o Cluj-Napoca, Faculty o Civil Engineering Abstract. The determination o natural requency in case o a tall structure is very important, or estimating the dynamic behavior o the structure. Most structural analysis sot-ware calculates the natural requency, but in the irst stage o structural design an approximate method based on simple mathematical relations is necessary or a quick determination o natural requency. Also, it is important to know the main structural characteristics that determine the natural requency, to know which characteristics create a more rigid structure. The result obtained using an approximate method is compared to those obtained using an exact method based on FEM. Thus, it can veriy and compare the results obtained by two the calculation methods proposed. The article contains two examples o central cores, to analyze the dierences obtained by the methods o calculations. Key words: requency analysis, central core, approximate method, FEM 1. Introduction The ocus o this article is to analyze the natural requency o a tall structure using two methods o calculation. An approximate method, based on mathematical relations and also an exact method based on FEM. According to the design code ASCE 7-05, the criteria or deining rigid structures compared with lexible: the natural requency or rigid structure is higher than 1Hz. By deault, or lexible structures, the natural requency is lower than 1Hz. The relations or determining the natural requency, proposed by design codes, are classiy according to structural system (rames, shear walls) and structural material (concrete, steel). A simple approximate method or determining the natural requency o the structure is obtained using the ratio: 150/H (t) or 46 / H (m) * Corresponding author: bianca.parv@mecon.utcluj.ro A-19

2 Bianca Parv, Monica Nicoreac and Mircea Petrina The mathematical relation takes into account only the most import dynamic characteristic o the structure, the building s height. The building s mass and bending stiness are expressed by the constant 46. This relation determines only the lateral requency o the structure, without taking into consideration the pure torsion requency or the coupled requency. For a global analysis the whole structure will be reduced to an equivalent cantilever ixed at the base. Thus, or determining the natural requency in case o a constant cantilever, the American design code ASCE 7-05 proposes an approximate method or natural requency: (1) 0,56 H m Where: bending stiness o the equivalent column m mass/unit height. Natural requency analysis Tall structures are considered as thin-walled bars. The dynamic analysis o thin-walled bars was studded by several scientists: Garland in 1940 using the Rayleigh-Ritz method o calculation, in case o a cantilever girder has obtained an approximate method assuming that the element is ininitely rigid in one o the main directions. Gere and Lin in 1958 has created a dierential equation system but obtained only the solutions or hinged girder-column. Gere has published in 1954 the dierential equations or pure torsion vibration but only or simply supported beam. Since no exact solutions have been achieved or determining the pure torsion vibration, the solutions obtained or coupled vibrations are only approximate [Kollar 1979; Rosman ; Vertes 1985; Goschy 1981]. The building s vibration is deined by partial dierential equations developed by Gere and Lin in 1958, assuming a uniormly distributed mass. Ater, simpliying the equation system and reducing the unctions that depends on time (or natural requency the unctions that depend on time are not necessary), so the equation system becomes [K. Zalka]: u'''' ρaωu 0 v'''' ρaω v 0 ω ϕ'''' GJϕ'' ϕaωϕi Where: ω ω, ω - circular requency, ϕ p ϕ 0 Starting with the dierential equation system presented above, the circular requencies are obtained in case o an equivalent column ixed at the A-130

3 DEDUCON 70 DEZVOLTARE DURABILĂ ÎN CONSTRUCŢII base [Timoshenko si oung, 1955]. With the circular requency determined can obtain the equation or natural requency in both directions: () 0,56 H ρa 0,56 H ρa The coeicient 0.56 represents the actor or lateral requency in case o equivalent cantilever column ixed at the base, or the irst mod o vibration. For the second mod the coeicient is and or the third mod o vibration the coeicient is [K. Zalka]. The main structural characteristic or lateral requency determination: building s height (the most important characteristic), lateral stiness and structural weight. K. Zalka proposes or determining the natural requency in both directions a reduction actor r that takes into account the act that the building mass is a concentrated load at each loor level, not a uniormly distributed load throughout the building s height as considered in the classical ormulation o a cantilever. The reduction actor: r sqrt (n/(n+.06)); where: n-number o loor levels The relations or determining the natural requency by taking into consideration the reduction actor r : (3) 0,56r H ρa 0,56r H ρa By solving the above system o equations and by taking into account the reduction actor, the relation or determining the pure torsion vibration becomes: (4) ηr i H ϕ p ω ρa Where: η- represents the parameter requency and is determined as a unction o torsion parameter k. The torsion parameter k takes into account the warping and St. Venant stiness. The main structural characteristics or determining the pure torsion requency: building s height, warping stiness ω, mass density ρ, radius o gyration ip, requency parameter η. The scientist Jeary, in 1981, said that tall structures are very sensitive to eccentricities. A 10% eccentricity can induce torsion vibration in the structure [Zhang, u si Knok, 1993]. A-131

4 Bianca Parv, Monica Nicoreac and Mircea Petrina The increase o pure torsion requency can be achieved by increasing the warping stiness ω or by reducing the radius o gyration value i p. 3. Coupled requency analysis In case o symmetrical structures, the centroid and the shear center o the structure coincide. Thus, the natural requency does not couple. The building s natural requency is considered to be the one having the lowest value (natural requency in both directions or pure torsion requency). I the structure is not symmetrical, the natural requency does couple (lateral requency pure torsion requency). An approximate method o calculation or determining the coupled critical requency, which is based on Foppl-Papkovich theory: (5) + + Where:,, φ represents the uncoupled requency in main directions and pure torsion requency This method is very simple but the solutions obtained are not economical, can provide errors. A more exact method, can achieved by solving the determinant proposed by S. Timoshenko. ω ω 0 ω y c (6) 0 ω ω ω x c 0 ω y ω x i ( ω ω ) c Where: ω, ω, ωϕ - lateral and pure torsion circular requency Solving the above determinant and knowing that ω / π is obtained a 3 degree equation or determining the natural requency. This method o calculation is also an approximate method but the results obtained are much closer to reality. 4. Numerical example c p ϕ ϕ 4.1. Symmetrical - central core The natural requency analysis is or a tall structure with 0 levels and a total height H70m (r 0,95). The concrete used is C0/5, the modulus o elasticity E3,0*10 7 kn/m and the transversal modulus G1,9*10 7 kn/m. A-13

5 DEDUCON 70 DEZVOLTARE DURABILĂ ÎN CONSTRUCŢII The central core is double symmetrical and the geometrical and stiness characteristics are given in table 1. Fig.1- Central core Table 1 Geometrical and stiness characteristics o the central core I I I ω [m 6 ] J γ [kn/m 3 ] i p [m] 38,831,674 39,97 0,0789 3,07,58 The torsion parameter k: kh*sqrt(gj/ ω ).039 Resulting, the requency parameters or the irst 3 modes o vibrations: η 1 η η 3 0,8637 3,944 10,31 This central core is symmetrical so the lateral requency and the pure torsion requency do not couple. Following the above relations will determine the lateral and pure torsion requency or the irst 3 modes o vibrations. The results obtained using the approximate methods will compare with the results obtained using an exact method FEM. Table Natural requency or the irst mod o vibration φ Approximate method 0,3 1,4 0,75 Approximate method (r ) 0,31 1,18 0,7 FEM 0,3 1,6 0,58 Dierence % 0% - 3% % - 6% 19% - % A-133

6 Bianca Parv, Monica Nicoreac and Mircea Petrina Table 3 Natural requency or the second mod o vibration φ Approximate method,03 7,75 3,43 Approximate method (r ) 1,94 7,37 3,6 FEM 1,97 6,75,63 Dierence % % - 3% 9% - 1% 0% - 3% Table 4 Natural requency or the third mod o vibration φ Approximate method 5,69 1,70 8,87 Approximate method (r ) 5,41 0,65 8,44 FEM 4,64 15,3 6,65 Dierence % 14% - 18% 6% - 9% 1% - 5% a b Fig. - First modes o vibration or symmetrical central core: a- irst mod: lateral vibration direction (x0.3hz) pure torsion vibration (0.58Hz) lateral vibration direction(y1.6hz) b- second mod: lateral vibration direction (x1.97hz) pure torsion vibration (.63Hz) lateral vibration direction (y6.75hz) A-134

7 DEDUCON 70 DEZVOLTARE DURABILĂ ÎN CONSTRUCŢII Fig. 3- Third mod: lateral vibration direction (x4.64hz) lateral vibration direction (y15.3hz) pure torsion vibration (6.65Hz) The dierence between the results obtained using the approximate method and the exact method are relatively small. Especially, the results obtained or the irst natural requency having the lowest values are similar. 4.. Mono-symmetrical central core The natural requency analysis is or a tall structure with 0 levels and a total height H70m (r 0,95). The concrete used is C0/5, the modulus o elasticity E3,0*10 7 kn/m and the transversal modulus G1,9*10 7 kn/m. The central core is mono-symmetric and the geometrical and stiness characteristics are given in table1. Fig. 4- Central core Table 5 Geometrical and stiness characteristics I I I ω [m 6 ] J γ [kn/m 3 ] i p [m] 6,079 38,831 64,0 0, ,53 A-135

8 Bianca Parv, Monica Nicoreac and Mircea Petrina Resulting, the torsion parameter k1.61. Table 6 Frequency parameters or the irst 3 modes o vibration η 1 η η 3 0,77 3,79 10,07 In case o mono-symmetry, the lateral vibration in plan o symmetry is independently developed, without coupling to pure torsion vibration. The lateral vibration perpendicular to the plane o symmetry, couples with the pure torsion vibration. For mono-symmetrical structures, the centroid and the shear center are situated somewhere on the axe o symmetry. The coupled requency is determined according to x, or -representing the axe o symmetry, and according to y, or -the axe o symmetry. (7) coupled ε (8) Where ε coupled parameter, determined according to: τ and r. τ y i C p r Initially, determine the lateral requency and the pure torsion requency uncoupled or the irst 3 modes o vibrations. Ater that, using the relations or mono-symmetric structure will determine the coupled requency o the structure. First mod: τ and r 0.56, resulting ε Table7 Natural requency or the irst mod o vibration φ Approximate method Approximate method (r ) FEM ,5 Approx. method F coupled Second mod: τ and r 0.15, resulting ε0.138 ϕ A-136

9 DEDUCON 70 DEZVOLTARE DURABILĂ ÎN CONSTRUCŢII Table8 Natural requency or the second mod o vibration φ Approximate method Approximate method (r ) FEM.87.5 Approx. method F coupled Third mod: τ and r 0.70, resulting ε0.486 Table9 Natural requency or the third mod φ Approximate method Approximate method (r ) FEM 5.33 Approx. method F coupled The lateral requency on y direction does not couple. Thus, the results obtained using the approximate method o calculation are compared with the results obtained using FEM. The dierence between the results is very small. On x direction, perpendicular to the axe o symmetry, lateral requency couples with pure torsion requency according to couple parameter. a b Fig. 5- Main mode o vibration or a mono-symmetric central core: a- irst mod: lateral vibration (0.51Hz); b- coupled vibration (0.5Hz) A-137

10 Bianca Parv, Monica Nicoreac and Mircea Petrina a b Fig. 6 - Main mode o vibration or a mono-symmetric central core: a- second mod: lateral vibration (.87Hz); coupled vibration (.5Hz); b- third mod: coupled (5.33Hz) 5. Conclusions Analyzing the results obtained by the two methods o calculation, the approximate and the exact method, can observe that the obtained results are similar or the irst 3 modes o vibration analyzed. The dierence between the results is reduced, both or uncoupled requency and or coupled requency. Thus, the approximate method can be used or checking and comparing the results obtained by FEM but also to determine the natural requency in the irst stage o structural design. Aknowledgements. This paper was supported by the project "Doctoral studies in engineering sciences or developing the knowledge based society-sidoc contract no. POSDRU/88/1.5/S/60078, project co-unded rom European Social Fund through Sectorial Operational Program Human Resources REFERENCES 1. Timoshenko S. P., Gere J.M., Theory o Elastic Stability. nd Ed., McGraw- Hill, New ork, Smith B.S., Coull A., Tall Building Structures Analysis and Design, A Wiley-Interscience Publication, 1991 A-138

11 DEDUCON 70 DEZVOLTARE DURABILĂ ÎN CONSTRUCŢII 3. Zalka K.A., Global Structural Analysis o Buildings, Taylor & Francis e- Library Publication, Taranath B.S., Reinorced Concrete Design o Tall Building, CRC Press, Taylor & Francis Group, 010 A-139