CONFERINŢA INTERNAŢIONALĂ DEDUCON 70

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1 CONFERINŢA INTERNAŢIONALĂ DEDUCON 70 DEZVOLTARE DURABILĂ ÎN CONSTRUCŢII Iaşi, 11 noiembrie 011 Conferinţă dedicată aniversării a 70 ani de învăţământ superior în construcţii la Iaşi A14 STABILITY ANALYSIS OF TALL BUILDINGS SUPPORTED BY REINFORCED CONCRETE CENTRAL CORE BY BIANCA PARV * MONICA NICOREAC and BOGDAN PETRINA Technical University of Cluj-Napoca, Faculty of Civil Engineering Abstract.. The aim of this article is a structural stability analysis for a reinforced concrete tall building. The stability analysis will be studied using an approximate method based on equivalent column theory. Will take into consideration the boundary conditions of a cantilever fixed at one end. For determining the critical load will analyze the predominant buildings characteristics that influents the stability analysis: building s height, bending stiffness, torsion stiffness and radius of gyration. Mean while will study the stability analysis using structural analysis programs based on FEM. Thus, will realize a comparative study between the results obtained using the approximate method and the exact method. The article contains also three numerical examples of tall structures. Key words: stability analysis, equivalent column, tall structures, central core 1. Introduction In the last decades it can be observed a tendency of higher and more slender buildings. Tall buildings and structural efficiency improvements lead to reduced reserves of stiffness and therefore to reduced reserves of stability. The more slender is the structure the more vulnerable to the appearance of the bending instability becomes. Thus, the stability analysis for tall buildings represents an important stage for structural design. For flexible structures collapse may occur due to stability loss. The structure tends to deform under the action of loads. If the loads value is continuously increasing, the structure goes from initial undistorted position, to the deformed position. If the structure presents small deformations under bending, can say that the structure is in stable equilibrium. The load corresponding to this deformation is the critical load (Pcr). By exceeding critical * Corresponding author: bianca.parv@mecon.utcluj.ro A-140

2 DEDUCON 70 DEZVOLTARE DURABILĂ ÎN CONSTRUCŢII load, it results stability loss (the structural system failure by stability loss or structural failure by buckling) [Mircea Teodorescu].. General assumptions for stability analysis The aim of the article is to achieve a buckling analysis under its own weight. The whole structure will be reduced to a single cantilever column, fixed at the base. The building s weight is uniformly distributed throughout the building s height. The cantilever fixed at the base tends to buckling under its own weight as presented in figure 1. In case of stability analysis is assumed that: the structure is perfectly straight, without any initial deformation initial eccentricities does not occur when vertical loads are applied the structure is considered as a whole, ignoring the possibility of local buckling shear deformation, characteristic for small and medium structures, does not influence the building s stability the simultaneous action of lateral and vertical loads is ignored [B. Taranath] For a structure these assumptions can not be totally fulfilled. For example, the tolerance allowed regarding building verticality, inevitable eccentric distribution of mass throughout the building s height causes an initial bending, increasing the P-Δ effect. The structure must resist lateral loads (wind and earthquake), resulting simultaneous action of vertical and horizontal loads. This consequence reduces the critical load value previously determined for a particular case of inelastic stability [B. Taranath]. For stability analysis it is necessary to analyze the entire structure as a single element, but also to analyze each structural element individually. Structural analysis for each element in case of tall building is the same as for the case of small or medium buildings. Thus, in this article will consider only the stability analysis for the entire structure as a single cantilever element, fixed at the base. 3. Computation models for stability analysis For stability analysis are used differential equations for the equivalent column. The solution for differential equation to determine the deformation curve was first described by Leonard Euler ( ), who tried to perform buckling analysis for a bar under its own weight, but without reaching a precise solution. This domain was going to be studied and developed by researchers such as: Stephen P. Timoshenko ( ), Z. Vlasov ( ). Researchers that studied the stability analysis applied to tall buildings: Brian Smith, Bungale S. Taranath, Karoly A. Zalka. A-141

3 Bianca Parv, Monica Nicoreac and Bogdan Petrina Tall structures are considered as thin-walled bars. S. Timoshenko presented the solution for bending and torsion buckling for thin-walled bars under the action of concentrated loads at the ends of the bars. To solve this case S. Timoshenko reached a 3 degree equation obtained by solving the determinant presented bellow. Karoly A. Zalka managed to present an approximate solution to solve the 3 degree equation. The accuracy of approximate method has not been fully investigated. Thus, the aim of this article is to achieve the stability analysis for several tall buildings and to compare the results obtained by approximate method of calculation with the results obtained by FEM using structural analysis program ANSYS 1.1. The critical load having the lowest value represents the critical load of the structure. The behavior of the building is influenced predominantly by the first modes of vibration; the following modes of vibration influence the building s behavior in a small proportion, except for relatively flexible buildings [3]. In case of tall structures, the resistance elements will be transformed into an equivalent column. The stability analysis will be study for the equivalent column starting by the 4 order homogeneous differential equations; S. Timoshenko: (4) ' EI u + [N(z)(u' y ϕ')] 0 y c = (4) ' x v + [N(z)(v' + x cϕ')] = (4) ω ϕ (GJϕ' N(z)i pϕ')' + EI 0 ' EI [N(z)(x cv' y cu')] = 0 In case of double symmetry the equations presented above will simplify since x = y 0. c c = For analyzing the buckling of a bar under the action of uniformly distributed axial loads, the differential equation of the deformed has variable coefficients. Thus, the solution for differential equation is obtained by applying the infinite series or by applying an approximate method based on energy method. Fig.1- Structural deformation [1] A-14

4 DEDUCON 70 DEZVOLTARE DURABILĂ ÎN CONSTRUCŢII Fig.- Vertical loads distribution Differential equation of the deformed curve (fig.1) [1]: d y EI dx = l x q( η y)dξ The right side of the equation represents the bending moment in mn section, under uniformly distributed load q. The critical load value is obtained starting from the differential equation of deformed curve, by expressing the Bessel differential equation [1] EI (1) Pcr = H For determining critical load using energy method will start by using πx the deformed curve equation: y = δ(1 cos ) and by using the equilibrium l condition ΔU=ΔT. Where: ΔU- energy bending deformation ΔT- total mechanical work Critical load value thus obtained, 7.89EI () Pcr = H Comparing the results obtained by energy method to the results obtained integrating the differential equation, it can be notice the de difference between the two methods is very small, thus, both approximate methods can be used for determining the critical load for lateral buckling. Karoly A. Zalka, based on the relations demonstrated by S. Timoshenko for determining the critical load in case of a cantilever under axial uniformly distributed load introduces the reduction factor r s. For the buckling analysis A-143

5 Bianca Parv, Monica Nicoreac and Bogdan Petrina demonstrated by S. Timoshenko, axial load is considered to be uniformly distributed over the building s height, but in reality the axial load is a concentrated load at each floor level (fig.). After the equation derivations, it does not take into account the load from the last level. Thus, for a distribution of axial load closer to reality, K. Zalka introduces the reduction factor r s : r s =n/(n+1.588), where n-number of levels rsEI (3) Pcr = H Critical load in case of pure torsional buckling determined using the relation [3]: αrsei (4) Pcr, ϕ = i H For thin-walled bars (as tall buildings are considered), when warping stiffness is zero, the relation for determining the critical load becomes [3]: GJ (5) P cr, ϕ = i In this case the critical load does not depend on building s height. The parameter for determining critical load as a function of ks: k GJ k s = where: k- torsional parameter: k = H r EI s ω GJ-bending stiffness of the structure EI ω - warping stiffness of the structure i p - radius of gyration The radius of gyration represents an import part in determining the critical load for pure torsion. As the building plan dimensions are larger and the distance between centroid and shear center of the building is higher, the value of radius of gyration is higher and the critical load for pure torsion is lower. To determine the combined critical load can apply an approximate method developed by Foppl-Papkovich. The advantage of this method is represented by the fast method of calculation. In certain cases can lead to considerably uneconomical structural solution, as the error of the formula can be as great as 67%[3]. (6) 1 Pcr, combinat = p 1 Pcr, x p + ω 1 Pcr, y + 1 Pcr, ϕ A-144

6 DEDUCON 70 DEZVOLTARE DURABILĂ ÎN CONSTRUCŢII An exact method for determining the critical load is by solving the determinant proposed by Timoshenko: (7) P P 0 Py cr,y c 0 P P Px cr,x c Py Px i (P P p c c cr, ϕ ) =0 The determinant provides a 3 degree equation, with 3 roots, and the lowest value represents the critical load of the structure. For determining the critical load, has developed a computer program in Matlab to solve the 3 degree equation. The equation is solved using the source code developed by Professor Nam Sun Wang, providing the 3 roots in a very short time. If the structure is mono-symmetric, the 3 degree equation can be simplified and the critical load taking into consideration the interaction effect of the 3 modes is [3]: (8) Pcr,combinat = εpcr, Y (X symmetry axis) (9) Where the value of parameter ε is obtained according to: x c τ X = and i p P r = P If the shear center and the geometrical center of the structure coincide, the critical load does not couple and the global critical load of the structure is equal to the lowest value of the 3 critical loads calculated: P cr,x, P cr,y, P cr,φ. Thus, it is not necessary to calculate the above determinant [3]. The maximum error made by neglecting coupling is 100%; when the two basic critical loads P cr,y, and P cr,φ are equal and eccentricity is maximum ( τ = 1. X 0 ), coupling reduces the combined critical load to half of the basic critical load[3]. cr, ϕ 4. Numerical example cr,x 4.1. Circular core with constant section In this case is study the stability analysis of a tall structure having 50 levels with a total height of H=18.50m. The lateral stiffness of the structure is provided by the reinforced concrete central core having a constant thickness over the building s height t h =1,5m. The elasticity modulus E=3*10 7 kn/m and the moment of inertia is I=7833m 4. In this case, the shear center and the A-145

7 Bianca Parv, Monica Nicoreac and Bogdan Petrina geometrical central of the structure coincide, thus, the critical load is equal to the lowest value of the three critical loads.. Fig.3- Buckling analysis of the circular tub ANSYS 1.1 The mathematical relation for determining the critical load(1): 7 Pcr = 5,5*10 kn For the axial load distribution as concentrated loads on each floor levels will take into consideration the reduction factor r s =0,969 (for a 50 leveles building) (3) 7 Pcr = 5.35*10 kn. The dead loads and live loads are 1kN/mp. The structural weight is estimated as 11.03*10 5 kn. Comparing the critical load to the structural weight of the structure can notice that there is a high degree of safety regarding the buckling analysis under its own weight. If the buckling analysis takes into consideration the cracked concrete which reduces the moment of inertia than Pcr=3,31*10 7 kn. Even so there is a high degree of safety; the global safety factor: 3,31*10 7 /11,03*10 5 = The researchers I. MacLeod and K. Zalka in 1996 introduced a global critical load ratio: (10) P ν = P Where: P total - total vertical load P cr - global critical load Theoretically if the global critical load ratio is lower than 1, than the structure is considered in stable equilibrium. According to Eurocode3, is proposed the limiting ratio to 0,1. ν 0,1 If this condition is satisfied, then the vertical loadbearing elements can be considered as braced and neglecting the second-order effects may result in a maximum 10% error [3]. total cr A-146

8 DEDUCON 70 DEZVOLTARE DURABILĂ ÎN CONSTRUCŢII The global critical load must be lower than 0,5 for any kind of adopted structure. If the ratio is not respected, must adopt a more accurate method of calculation for determining the critical load value or must take into consideration the second order effect. The buckling analysis using an exact method of calculation is achieved using the structural analysis program ANSYS 1.1. Fig.4 Buckling analysis using ANSYS 1.1 Table 1 Comparing the results (central core-constant section) Calculation method Pcr Global critical load ratio ν [kn] approximate 5,5* exact ANSYS 1.1 6,4* difference % 11% 4.. Circular core with variable section For the second case will analyze a tall structure having 50 levels with a total height of 18.5m. The reinforced concrete central core has a variable section over the building s height. In general, the vertical elements of a tall building will not have a constant section throughout the height of the building, for economical reasons. Thus, it will analyze a structure with variable section. The first 16 levels (l =58,40m) have the shear wall thickness t=1,65m and the moment of inertia I =837.4m 4 and for the last 4 levels (l 1 =14,10m) have the shear wall thickness t=1,10m and the moment of inertia I 1 =5476.3m 4. A-147

9 Bianca Parv, Monica Nicoreac and Bogdan Petrina Fig.5 - Variable section For determining the critical load S. Timoshenko starts by writing the differential equations for both deformed curves and takes into consideration the boundary conditions of the cantilever fixed at the base. The relation developed in case of a cantilever fixed at the base having a variable section: mei 7 Pcr = = 5,989*10 kn l Where: m- numerical factor and depends by ratio I 1 /I =0.664 and by a/l=0.3; resulting that m=8,07. Table Comparing the results (central core-variable section) Calculation method Pcr Global critical load ratio ν [kn] approximate 5,989* exact ANSYS 1.1 5,459* difference % 9% 4.3. Shear walls and perimeter frames In third case will study a 0 levels building with a total height of 80m. The structure consists of perimeter frames and shear walls which are disposed in the building s central. The reinforced concrete has the modulus of elasticity E=.5*10 7 kn/mp. The structure is proposed by Professor Brian Smith. A-148

10 DEDUCON 70 DEZVOLTARE DURABILĂ ÎN CONSTRUCŢII Fig.6 - Building s plan Fig.7 - Buckling analysis In case of a tall building consisted of shear walls or central cores and perimeter columns, the structure can not be considered anymore as a cantilever which has only bending deformation. In this case appears the possibility of deformation under shear forces. Thus, for buckling analysis in case of tall building having shear walls (central cores) and perimeter columns, must take into consideration the possibility of deformation under bending (shear walls and central cores characteristics) as well as the possibility of deformation under shear forces (perimeter frames characteristics associated by double curveting of the columns and girders). For buckling analysis in case of a shear walls frames structure, B. Smith developed an approximate method of calculation for determining the critical load. This method is based on the equivalent continuum method (gravity load and structural elements rigidity are constant over the building s height). A-149

11 Bianca Parv, Monica Nicoreac and Bogdan Petrina Mathematical relations for determining the critical loads are obtained by solving the equilibrium differential equations. critical load for lateral buckling: s x (EI) x P cr,x = =33.5*10 4 Kn H critical load for pure torsion buckling: s θ (EI ω ) P =50.9*104 kn = cr, θ H Where: (EI) x the two shear walls stiffness in X direction (EI ω ) the torsion stiffness of the shear walls The coefficients value s x and s θ is determined as a function of transversal and torsion parameters: (αh) X respectively (αh) θ. Total vertical load: P=80000 kn The shear central and centroid of the structure coincide, thus the critical load does not couple: global critical load being the critical load having the lowest value: P cr =33,5*10 4 kn. Table 3 Comparing the results (central shear walls-perimeter frames) Calculation method Pcr Global critical load ratio ν [kn] approximate 33,5* exact ANSYS ,9* differential % 1 % According to a widely accepted rule of thumb in the structural design of buildings, there is an absolute bottom line, as far as global safety is concerned: the value of the global safety factor must be at least four[3]. 33.5*10 4 /8*10 4 =4.18>4 or the global critical load ν max < Conclusions Analyzing the results obtained using the approximate method of calculation and the exact method by using the structural analysis program ANSYS 1.1, can notice that the results value are similar. Thus, it can be said that both methods of calculation have been applied correctly and can be used tall buildings stability analysis. In general, in case of tall structures, the gravity load represents a small proportion of the load necessary for buckling occurrence. The rate is limited by safety considerations to 0.5. The critical load must by at least 4 times higher than the total gravity load to ensure the structural safety to buckling. The 3 examples presented in this article respect the stability condition. Thus, it can proceed to next steps for designing high structures. A-150

12 DEDUCON 70 DEZVOLTARE DURABILĂ ÎN CONSTRUCŢII Aknowledgement. This paper was supported by the project "Doctoral studies in engineering sciences for developing the knowledge based society-sidoc contract no. POSDRU/88/1.5/S/60078, project co-funded from European Social Fund through Sectorial Operational Program Human Resources REFERENCES 1. Timoshenko S. P., Gere J.M., Theory of Elastic Stability. nd Ed., McGraw- Hill, New York, Smith B.S., Coull A., Tall Building Structures Analysis and Design, A Wiley-Interscience Publication, Zalka K.A., Global Structural Analysis of Buildings, Taylor & Francis e- Library Publication, Taranath B.S., Reinforced Concrete Design of Tall Building, CRC Press, Taylor & Francis Group, Teodorescu M.E., Stabilitatea Structurilor, Editura Conspress, Constantin I., Stabilitatea si dinamica Constructiilor, Iasi, 004 A-151