DYNAMIC ANALYSIS OF TALL BUILDING UNDER PULSATION WIND EXCITATION

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1 ASIAN JOURNAL OF CIVIL ENGINEERING (BUILDING AND HOUSING) VOL. 7, NO. 1 (2006) PAGES DYNAMIC ANALYSIS OF TALL BUILDING UNDER PULSATION WIND EXCITATION S.M. Auta and A.M. Maslennikov St. Petersburg State University of Architecture & Civil Engineering, Department of Structural Analysis, 2 nd Krasnoarmeiskaja Street, 4, St. Petersburg, Russian Federation ABSTRACT This paper presents analysis of tall building under wind load excitation taking account of pulsation. The path of the analysis using MatLAB, Lira and Russian national code SNiP are presented. Use of wind velocity-time graph is employed where the change in wind pressure (pulsation) with time is assumed to be proportional to a similar change in wind velocity. Results are compared and show a conformation to dynamic behaviour of the buildings under such excitations as pulsation, thus making the analysis adequate for assessing the dynamic behaviour of a structure. Keyword: dynamic analysis, numerical method, wind load, pulsation, drift, acceleration 1. INTRODUCTION Wind effects on structures continue to pose danger that continues to attract the attention of scientist round the globe. This is because of its trend of effect that changes with time, a dynamic problem, and inadequate information on the response of structures to wind action. In a most recent report by the US National Institute of Standards and Technology makes 30 recommendations to improve the safety of tall buildings on the collapse of the World Trade Centre. Among the 8 core structural recommendations is: Increased structural integrity-the standards for estimating the load effects of potential hazards (such as progressive collapse, wind) and that the design of structural systems to mitigate the effect of those hazards should be improved to enhance structural integrity [1]. This suggests the importance and necessity of continues research on dynamic behaviour of structures. In this study three buildings are considered: 10-storey [2]; 15-storey [2]; and 20-storey [3]. 10 and 15-storey consist of uniform cross-sectional area in struts and girders from reinforced concrete. The masses M1=M2=M3=..M14=51ton, except M10 and M15 which are 60 ton each for 10 and 15-storey respectively. The 20-storey consists of structural steel frame from the class 26КУ6 with masses M1=M2=.M19=20ton and M20=30ton. All -address of the corresponding author: smahuta@yahoo.com

2 96 S.M. Auta and A.M. Maslennikov the analytic frames have constant stiffness along the heights and all columns are fixed to the foundation and are all moments resisting frames. The cross sectional analytic frame models are shown in figures 1. M20 M15 M14 M13 M12 M19 M18 M17 M16 M15 M10 M9 M8 M7 3.3x15 = 49.5m M11 M10 M9 M8 M7 2.78x20 = 59.4m M14 M13 M12 M11 M10 M9 3.3x10 = 33.0m M6 M5 M4 M3 M6 M5 M4 M3 M8 M7 M6 M5 M4 M2 M1 M2 M1 M3 M2 M1 6.0m 6.0m 6.0m 6.0m 5.4m 5.4m Figure storey, 15-storey and 20-stoey cross-sectional analytic frames 2. THEORY OF METHODS The computations of wind loads on tall buildings from Russian national code [4], is taken as the sum of statically obtained average wind load and pulsation wind load. While the former is computed from equation (1), the later is acceptable to be computed from equation (2) if the stiffness, mass and width of windward surface along height of the building are constant: w s =w 0 kc (1) w p =1.4(z/h)ξw ph (2) w 0 =0.61v 0 2 (3) w ph =w s ζυ (4) Where w 0 is the nominal wind pressure (for Area II =30 kgf/m 2 ); k height factor; c aerodynamic coefficient +0.8 for windward side and -0.6 leeward side; ξ dynamic coefficient; w ph nominal wind pressure due to pulsation at height h to be calculated from equation (4); v 0 wind velocity; υ coefficient of space correlation due to wind pulsation; ζ

3 DYNAMIC ANALYSIS OF TALL BUILDING UNDER PULSATION WIND coefficient of wind pulsation. Lira is a software for structural analysis. Wind load values obtained from SNiP are also processed and analysed here for the sake of comparison. Numerical method of analysis is also employed in this study. Differential equations of a moving mass can be integrated using computer analysis [5, 6]. In dynamic analysis these equations at any given time t are written in matrixes which take the form of equation (5): MΫ t + CỶ t + KY t = R t, (5) where M, C, K matrix of masses, damping and stiffness; R t vector representing external excitations; Y t, Ỷ t, Ϋ t vectors of displacements, velocities and accelerations of a system with definite number of degree of freedom. Numerical integration of these equations is done through an iterative process. Table 1. Summary of wind load for 10-storey frame Average wind load, Pulsation wind load, Total wind load, F in kn w si, kn/m 2 F si =A(w si ), kn A=14.85m 2 w pi, kn/m 2 F pi =A(w pi ), kn A=14.85m 2 Windward, F wi = F si + F pi Leeward, F li = 0.75(F si + F pi ) w s1 = 0.18 F s1 = w p1 = F p1 = F w1 = F l1 = w s2 = 0.24 F s2 = w p2 = F p2 = F w2 = F l2 = w s3 = 0.24 F s3 = w p3 = F p3 = F w3 = F l3 = w s4 = 0.30 F s4 = w p4 = F p4 = F w4 = F l4 = w s5 = 0.30 F s5 = w p5 = F p5 = F w5 = F l5 = w s6 = 0.30 F s6 = w p6 = F p6 = F w6 = F l6 = w s7 = 0.36 F s7 = w p7 = F p7 = F w7 = F l7 = w s8 = 0.36 F s8 = w p8 = F p8 = F w8 = F l8 = w s9 = 0.36 F s9 = w p9 = F p9 = F w9 = F l9 = w s10 = 0.36 F s10 = w p10 = F p10 = F w10 = F l10 = 3.665

4 98 S.M. Auta and A.M. Maslennikov Table 2. Summary of wind load for 15-storey frame Average wind load, Pulsation wind load, Total wind load, F in kn w si, kn/m 2 F si =A(w si ),kn A=14.85m 2 w pi, kn/m 2 F pi =A(w pi ), kn A=14.85m 2 Windward, F wi = F si + F pi Leeward, F li = 0.75(F si + F pi ) w s1 = 0.18 F s1 = w p1 = F p1 = F w1 =3.435 F l1 =2.576 w s2 = 0.24 F s2 = w p2 = F p2 = F w2 =4.197 F l2 =3.148 w s3 = 0.24 F s3 = w p3 = F p3 = F w3 =4.959 F l3 =3.719 w s4 = 0.30 F s4 = w p4 = F p4 = F w4 =5.720 F l4 =4.290 w s5 = 0.30 F s5 = w p5 = F p5 = F w5 =6.036 F l5 =4.527 w s6 = 0.30 F s6 = w p6 = F p6 = F w6 =6.799 F l6 =5.099 w s7 = 0.36 F s7 = w p7 = F p7 = F w7 =7.560 F l7 =5.670 w s8 = 0.36 F s8 = w p8 = F p8 = F w8 =7.876 F l8 =5.907 w s9 = 0.36 F s9 = w p9 = F p9 = F w9 =8.192 F l9 =6.144 w s10 =0.36 F s10 = w p10 = F p10 = F w10 =8.509 F l10 =6.382 w s11 =0.36 F s11 =5.346 w p11 = F p11 = F w11 =8.825 F l11 =6.619 w s12 =0.36 F s12 = w p12 = F p12 = F w12 = F l12 =6.856 w s13 =0.41 F s13 = w p13 = F p13 = F w13 = F l13 =7.372 w s14 =0.41 F s14 = w p14 = F p14 = F w14 = F l14 =7.888 w s15 =0.41 F s15 = w p15 = F p15 = F w15 =5.417 F l15 = WIND LOAD CALCULATIONS FROM SNIP [3] AND LIRA The analysis of the frames in figure 1 for wind load using SNiP [4] is calculated from the appropriate formulae (1, 2, 3 and 4). The buildings are assumed to be sited in type A open country with no obstruction (Area II =30 kgf/m 2 ). The wind load calculations is summarised in tables 1, 2 and 3. These values were then processed in Lira to calculate the maximum

5 DYNAMIC ANALYSIS OF TALL BUILDING UNDER PULSATION WIND moment in the middle columns and as well the drift at the top of the structure. Lira was also used to calculate displacements of the nodes from unit force which were then used in subsequent dynamic analysis with MatLAB. It is to be noted that the self weight of the frame was not considered while calculating for the bending moments of columns. Table 3. Summary of wind load for 20-storey frame Average wind load, Pulsation wind load, Total wind load, F in kn w si, kn/m 2 F si =A(w si ),kn A=10.692m 2 w pi, kn/m 2 F pi =A(w pi ), kn A=10.692m 2 Windward, F wi = F si + F pi Leeward, F li = 0.75(F si + F pi ) w s1 = 0.18 F s1 = w p1 = F p1 = F w1 =2.414 F l1 =1.811 w s2 = 0.24 F s2 = w p2 = F p2 = F w2 =2.903 F l2 =2.177 w s3 = 0.24 F s3 = w p3 = F p3 = F w3 =3.393 F l3 =2.545 w s4 = 0.30 F s4 = w p4 = F p4 = F w4 =3.883 F l4 =2.912 w s5 = 0.30 F s5 = w p5 = F p5 = F w5 =4.051 F l5 =3.038 w s6 = 0.30 F s6 = w p6 = F p6 = F w6 =4.540 F l6 =3.405 w s7 = 0.36 F s7 = w p7 = F p7 = F w7 =5.030 F l7 =3.773 w s8 = 0.36 F s8 = w p8 = F p8 = F w8 =5.198 F l8 =3.899 w s9 = 0.36 F s9 = w p9 = F p9 = F w9 =5.367 F l9 =4.025 w s10 =0.36 F s10 = w p10 = F p10 = F w10 =5.536 F l10 =4.152 w s11 =0.36 F s11 =3.849 w p11 = F p11 = F w11 =5.705 F l11 =4.279 w s12 =0.36 F s12 = w p12 = F p12 = F w12 = F l12 =4.405 w s13 =0.41 F s13 = w p13 = F p13 = F w13 = F l13 =4.732 w s14 =0.41 F s14 = w p14 = F p14 = F w14 = F l14 =5.060 w s15 =0.41 F s15 = w p15 = F p15 = F w15 =6.914 F l15 =5.186 w s16 =0.41 F s16 = w p16 = F p16 = F w16 =7.083 F l16 =5.312 w s17 =0.41 F s17 = w p17 = F p17 = F w17 = F l17 =5.439 w s18 =0.41 F s18 = w p18 = F p18 = F w18 = F l18 =5.565 w s19 =0.41 F s19 = w p19 = F p19 = F w19 = F l19 = w s20 =0.41 F s20 = w p20 = F p20 =1.687 F w20 = F l20 = 2.909

6 100 S.M. Auta and A.M. Maslennikov 4. DYNAMIC ANALYSIS ON MATLAB Structural analysis becomes a dynamic analysis when load becomes a function of time. The change in wind pressure (pulsation) with time is assumed to be proportional to a similar change in wind speed. With this assumption it is sufficient to use a real wind velocity-time graph taken from [7]. From this graph, the wind pulsation load is deduced proportionately to nodal wind load from tables (1, 2, and 3) for each floor of the entire buildings. These values are by numerical method arranged in matrixes according to equation (5), which is then processed in MatLAB in order to produce dynamic characteristics of the buildings. Input data and computation analysis are shown in appendix I for 10-storey frame. The same process as shown in appendix I for 10-storey was followed for 15-storey and 20-storey frames respectively. 10 Load F, kn/s Time t, sec. Figure 2. Wind load variation with time at the 9 th floor of the 10-storey frame Load F, kn/s Time t, sec.

7 DYNAMIC ANALYSIS OF TALL BUILDING UNDER PULSATION WIND Figure 3. Wind load variation with time at the 14 th floor of the 15-storey frame An example of this operation is on the 10-storey frame. The maximum amplitude is in the 9th floor, 3.983kN which is added to the statistical wind load 5.35kN to produce a total of 9.33kN. The illustration is shown in figure 2. For the period of 40seconds the peak is located in the 6th second. All other floors are equally computed for in relation to the 9th floor. The same procedure is followed for the 15-storey and 20-storey frames with their maximum amplitude appearing at 14th and 19th floor respectively as illustrated in figure 3 and 4. 8 Load F, kn/s Time t, sec. Figure 4. Wind load variation with time at the 19 th floor of the 20-storey frame 5. RESULTS AND DISCUSSION The result of the analysis is presented in table 4. Results from the dynamic analysis on MatLAB and equivalent calculations from SNiP and Lira are closely similar with slight deflections. The following deductions can be made: 1. Maximum drift δ max generally remain within the limit H/500 [4] except in the case of 10-storey where δ max obtained from MatLAB was beyond limit by +29%, thus indicating an effect of wind pulsation; 2. Maximum acceleration in all cases remains within limit of 0.1m/s 2 [8], however this was obtainable only from dynamic analysis on MatLAB; 3. Maximum bending moment M max obtained from MatLAB was always more than that obtained from LIRA and SNiP by 26%, 33% and 22% for 10, 15 and 20-storey frame respectively. This is also indicating an effect of wind pulsation; 4. The summary of wind effect due to pulsation can be presented graphically in figure 5, where w s is statically defined average wind pressure; w p is the pulsation wind pressure. In this illustration it is shown that at any given time the value of pulsation wind pressure can be more than its equivalent calculated value given by SNiP. This can only be detected by dynamic analysis on MatLAB.

8 Wind pressure w, kn/m^2 102 S.M. Auta and A.M. Maslennikov Table 4. Summary of result 10-storey analytical frame 15-storey analytical frame 20-storey analytical frame Mode frequency, f 1, Hz f 2, Hz f 3, Hz SNiP[3] Lira MatLAB Drift, δ max, mm Drift limit, δ l, mm [4] 66 M max, kn.m Acceleration, a max, m/s Accelaration limit, a l, m/s 2 [8] Mode frequency, f 1, Hz f 2, Hz f 3, Hz Drift, δ max, mm Drift limit, δ l, mm [4] 99 M max, kn.m Acceleration, a max, m/s Accelaration limit, a l, m/s 2 [8] Mode frequency, f 1, Hz f 2, Hz f 3, Hz Drift, δ max, mm Drift limit, δ l, mm [4] 119 M max, kn.m Acceleration, a max, m/s Accelaration limit, a l, m/s 2 [8] W P W S W + 0.5W S P Time t, sec Figure 5. Summary of the wind effect due to pulsation.

9 DYNAMIC ANALYSIS OF TALL BUILDING UNDER PULSATION WIND CONCLUSIONS Dynamic analysis of tall building under pulsation wind load excitation is presented. Results show reliably, dynamic behaviour of the building since they are generally similar. The traditional way of limiting the lateral drift at the top of the building to a fraction of the height under the statically equivalent design wind load prescribed in SNiP is not sufficient to spare occupants discomfort. This is because comfort of occupants on the top floor is closely related to building acceleration which is a dynamic problem due to wind pulsation as analysed in this paper, therefore dynamic analysis on MatLAB will reasonably serve in designing and studying the behaviour of a multi-storey building. Acknowledgement: The St. Petersburg State University of Architecture & Civil Engineering, St. Petersburg Russian Federation and the Federal University of Technology, Minna Nigeria, is hereby greatly acknowledged for the grant of scholarship and fellowship respectively for the research. The success of this research is also attributed to the supervising professor, Prof. Maslennikov A. M. REFERENCES 1. New Steel Construction, UK, October, No. 9, 13(2005) Hachiyan, E. Ye., Seismic action on tall buildings and structures, Aistan Publishing House, Iravan, 1973 (Russian edition). 3. Henn V., Hart F. et al., Atlas of steel structures. Multistorey buildings, Translation from German edition, Moscow, 1977 (Russian edition). 4. SNiP * Loadings and influences, Official publication, Moscow, 2004 (Russian edition). 5. Maslennikov, A. M., The structural response to any non-periodical dynamic loading, Publishing House of Leningrad University, Leningrad, 1991 (Russian edition). 6. Maslennikov, A. M., Basics of dynamics and stability of skeletal systems, Text book for students of structural engineering specialities (Industrial & Civil Constructions) SPBSUACE, ACB Publishing House, St.Petersburg, 2000 (Russian edition). 7. Hand book on dynamics of structures, Building Press (Barshtein M. F.), Moscow, 1972 (Russian edition). 8. Residential and public buildings. Short hand book for engineers designers, Building Press (Dihovichny U. A.), Moscow, 1991 (Russian edition).

10 104 S.M. Auta and A.M. Maslennikov Appendix I % Numerical method of analysis of dynamic equations on MatLAB % Input data dt=1;a=0.5;b=0.25;a0=1/(b*dt^2);a1=a/(b*dt);a2=1/(b*dt);a3=(a-2*b)*dt/(2*b); a4=(a-b)/b;a5=(0.5-b)/b; bs=[-0.667;-0.758;-0.819;-0.861;-0.870;-0.875;-0.876;-0.877;-0.877;-0.877]; sf=89.97; f=1e-3*[ ]; M1=[ ];m=diag(M1); be1=[ ];be=diag(be1); % Input data wind load at joints of 10 storey frame p0=1.75*[ ; ; ; ; ; ; ; ; ; ]; k=inv(f); c=m*be;r0=1.75*[3.12;3.56;4.01;4.46;4.46;4.90;5.35;5.35; ]; yn=inv(m)*r0;k1=a0*m+a1*c;k2=a2*m+a4*c;k3=a5*m+a3*c;k01=k+k1;g1=inv(k01);w1=g1* r0; v1=a1*w1;y1=a0*w1; % Iterative process for j=1:40 p=p0(:,j);r1=p+k1*w1+k2*v1+k3*y1;w3=g1*r1;v3=a1*(w3-w1)-a4*v1-a3*y1; y3=a0*(w3-w1)-a2*v1-a5*y1; w1=w3;v1=v3;y1=y3;z=(w1(10))';y=(y1(10))';w=z ;t=1*j;s=-m*y1;my=bs'*s+sf; t=1.0*j; sm=my';yz=[t sm];%yz=[t w]; %yz=[t y]; disp(yz); end.

FRAME ANALYSIS. Dr. Izni Syahrizal bin Ibrahim. Faculty of Civil Engineering Universiti Teknologi Malaysia

FRAME ANALYSIS. Dr. Izni Syahrizal bin Ibrahim. Faculty of Civil Engineering Universiti Teknologi Malaysia FRAME ANALYSIS Dr. Izni Syahrizal bin Ibrahim Faculty of Civil Engineering Universiti Teknologi Malaysia Email: iznisyahrizal@utm.my Introduction 3D Frame: Beam, Column & Slab 2D Frame Analysis Building