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DYNAMIC STABILITY OF SINGLE LAYER RETICULATED DOME UNDER STEP LOAD Ce Wang 1 and Shizhao Shen 2 1Department of Civil Engineering, Tsinghua University, Beijing, 100084 2Harbin University of Civil Engineering and Architecture, 150008 ABSTRACT The present paper is concerned with dynamic stability of single layer reticulated domes. The updated Lagrangian formulation is employed to develop three dimensional beam elements nonlinear analysis which includes joints large displacements, large rotations and nonlinear material constitutive relation. Dynamic stability of latticed domes under step load are studied through various parameters such as span, rise-span ratio, elastic or elastic-plastic constitutive relation, including damping and without damping. The influence factors of material non-linearity, damping, initial geometry imperfection and initial static load for structure dynamic stability is analyzed. The simplified dynamic critical load calculating method is also suggested. KEYWORDS: Dynamic Stability, Dynamic Stability Critical Load, Nonlinear Analysis, Reticulated Dome, Step Load, Updated Lagrangian Formulation INTRODUCTION Single layer reticulated dome is imperfection sensitive structure which may lose its stability under strong earthquake action and strong wind load. There are several methods that are adopted by numerous investigators to solve latticed dome static stability but few concerned with dynamic stability. Dynamic stability means structural stability under dynamic disturbance which is a research field closely related to stability theory and vibration theory. In the paper members of reticulated dome assumed as three dimensional beam element, the non-linearity of latticed domes include geometric non-linearity caused by joint large displacement, large rotation and nonlinear material constitutive relation. Nonlinear dynamic finite element method is the basis of latticed dome dynamic stability analysis. According to the continuum mechanics principle, the updated 201 202 C. Wang and S. Shen Lagrangian formulation is employed to develop three dimensional beam element geometry nonlinear analysis which include joints large displacements and large rotations (Wang(1997)). The joint large rotation is modified because it doesn't accord with law of exchange so that Euler angle which describes rigid body motion around fixed point is used to simulate large joint rotation. In material nonlinear analysis the Mises yield criterion and Prandtl-Reuss flow rule are adopted to describe elastic-plastic constitutive relation. The Newmark integration combined with Newton-Raphon equilibrium iteration are used to solve structural nonlinear vibration equation that can improve the calculation precision and numerical stability. The first task in structure dynamic stability analysis is to determine structure dynamic stability critical load which is very time consuming while structural geometry and material non-linearity considered in each numerical integration procedure. Trial calculations have to be employed in order to exactly judge dynamic critical load which makes the work more difficulty. The critical criterion which have theory basis and convenient in practical application is very important. The equation of motion approach is a famous method adopted by Budiansyk (1967). Structure vibration equations are numerically solved for various of the load parameters, thus obtaining the system responses. The load parameter at which there exists a large change in the response is called critical. Budiansyk criterion failed in some cases when structure dynamic response isn't sensitive to load changes. According to the concept of Liapunov stability, a motion is said to be stable if all of other neighbor motions stay close to it at all time; otherwise it said to be unstable. If structure tangent stiffness matrix is negative definite then structure transient response exponentially diverge. During Newmark integration procedure structure tangent stiffness matrix is triangle decomposed if there are negative values found in the diagonal elements the tangent stiffness matrix is negative definite. In the present paper the dual criterion is used to determine the critical load (Wang(1993)): if structure tangent stiffness matrix remains negative definite during several time steps and structure transient responses diverge then the load is called dynamic stability critical load. Step load is the simplest dynamic load with constant amplitude at all time. Structure stability under step load represents its resistance to dynamic disturbance, it's also the basis for studying structure dynamic stability under strong earthquake. The are many factors which influence structure dynamic stability such as geometric parameters, material constitutive relation, damping, initial imperfection, initial static load, etc. The various influence factors including material non- linearity, damping, initial geometry imperfection are studied through a numerical example. By using parameter analysis method, structural dynamic stability critical load of various spans and rise-span ratio are calculated when material constitutive relation being elastic, elastic-plastic, including damping and without damping. Finally, a simplified dynamic critical load calculating method is suggested. DYNAMIC STABILITY INFLUENCE FACTORS Trial calculation is only valid method for structure dynamic stability critical load analysis. Increasing load step by step then calculating structure nonlinear dynamic response, structure vibration amplitude increased with the load. When structure vibration time history curve bifurcate and diverge, at the same time structure stiffness matrix is negative definite structure vibrate from stable state to unstable state, the load is called dynamic stability critical load. A 90 members dome is analyzed by various parameters which may influence structure dynamic Dynamic Stability of Single Layer Reticulated Dome under Step Load 203 stability including material nonlinear, mass quantity, damping, half span load, impulsive load and initial geometry imperfection. The K6 type reticulated dome of 10 meters span which is modeled as space frame with 1:10 rise to span ratio, shown in Fig. 1. The supports of the dome are assumed to be pinned and restrained against translational motion. Vertical uniformly distributed pressure load is applied at the nodes symmetrically. The members are steel tube ~60mm • 3.5mm. Fig. 1 Geometry of 90-member shallow dome For static case, structure elastic stability load is 36.9 KN/m 2 solved by spherical constant arc-length method. Assuming material is elastic perfect plastic with yield stresses 2.35e5 KN/m 2, the static stability load reduced to 16.1 KN/m 2, the critical load reduced about 56% when material non-linearity is included. Because elastic stability critical load of reticulated dome is high, the steel tubes became plastic before the structure reached its elastic critical load. Material non-linearity must be considered in reticulated dome stability analysis. For dynamic case, step load distributed as static case is applied at the dome and uniformly distributed mass 500kg/m 2 lumped to the nodes. Structure fundamental period Tf=0.21s, time step At=0.005s, damping matrix is neglected. Several levels of load are calculated, Fig. 2 shows node 3 displacement history at elastic stage. Structure dynamic stability is sensitive to small load perturbation when applied load is 14.70 KN/m 2 structure tangent stiffness matrix remains positive definite structure vibrate stable, when the load reached 14.75 KN/m 2 structure tangent stiffness matrix became negative definite at 0.39 seconds structure vibrate curve bifurcate, structures vibrate unstable dynamic responses increase very fast. Assuming elastic perfect plastic material with yield stress 2.35e5 kN/m 2 structure dual nonlinear dynamic response is calculated again. When step load is 9.0KN/m 2 structure vibrate stable, until the load reached 9.1KN/m 2 structure lost stability (Fig. 3). If only material nonlinear is considered and geometric non-linearity neglected structure dynamic critical load is 11 KN/m 2 compare with only geometric non-linearity considered the critical load 14.75KN/m 2. According to the numerical example, material nonlinear influence is large then geometric non-linearity. Structure dynamic stability load is smaller than that of static stability load no matter material is elastic or elastic perfect plastic. Damping influence is important in structure dynamic analysis which can largely reduces vibration peak value and maintaining structure dynamic stability. The Rayleigh damping is used with damping ratio ~=0.05. Elastic dynamic stability critical load increase from 14.75KN/m 2 to 23KN/m 2, elastic-plastic dynamic stability load increase from 9.1 KN/m 2 to 15.5 KN/m 2, the increase ratio is 56% and 70%, respectively. Node 3 elastic and elastic plastic time history with damping is shown in Fig. 4. 204 C. Wang and S. Shen Fig. 2 Node 3 elastic displacement history Fig. 3 Node 3 elastic-plastic displacement history Fig. 4 Node 3 displacement history with damping Fig. 5 Node 3 elastic-plastic response Fig. 6 Response with different load distribution Fig. 7 Node 3 response under impulsive load Assuming two uniform distribution mass M=300kg/m 2, M=100kg/m 2, the other parameter is the same as previous analysis. The dynamic stability critical load is 9.5 KN/m 2 compare with 9.1KN/m 2 of M=500kg/m 2 (Fig. 5). It can be seen that the quantity of mass have less influence in structure dynamic stability. Load distribution is also important in structure dynamic analysis assuming only half span applied uniform load with mass of M=300kg/m 2, elastic-plastic critical load is 11.8 KN/m 2 large then full span load distribution critical load of 9.1KN/m 2, but the total load is half of full span load distribution. Increasing left half span load 50% and reduce right half span load 50% the total load remain the same, the critical load is 7.6KN/m 2 which reduce 16% compare with full span load distribution (Fig. 6). Loading time also should be considered, infinite loading time is step load, very short loading Dynamic Stability of Single Layer Reticulated Dome under Step Load 205 time is impulsive load. There are two loading case calculated with time duration t o = 0.05 s and t o = 0.025s, respectively. Structure elastic-plastic critical loads are 9.7 KN/m 2 and 14.4KN/m 2 while step loading case is 9.1KN/m 2. The critical load decreased when loading time increased, under impulsive loading structure lost stability during free vibration state (Fig. 7). Reticulated dome is imperfection sensitive structure with lower load bearing capacity than perfect structure. The imperfection distribution and values are impossible to predict, here structure static buckling modes are used to simulate initial geometry imperfection. Buckling mode is the tendency of structure displacement at critical status, if imperfection mode is the same as buckling mode it will cause the worst influence to structure vibration. The first tenth static linear buckling modes are calculated from which choosing the detrimental imperfection mode. Assuming the maximum imperfection value is 5cm for each buckling mode then calculating structure static linear buckling load, the lowest buckling load with corresponding mode is chosen as imperfection mode. Without damping the elastic dynamic critical load and elastic-plastic critical load are 10KN/m 2 and 6KN/m 2, the reduction ratio is 32% and 34% compare with perfect dome. Dynamic buckling mode is different from perfect structure, the perfect dome buckling mode is symmetric large area collapse on whole structure, but the imperfect structure buckling mode is part collapse near the maximum imperfection point (Fig. 8). Fig. 8 Maximum imperfection point time history and collapse mode DYNAMIC STABILITY PARAMETER ANALYSIS In parameter analysis there are total 12 latticed domes fixed at the edge with spans 30m, 40m, 50m, 60m, each span have three rise to span ratio 1/10, 1/8, 1/6, respectively. The tube material assumed as perfect elastic-plastic with yield stress 235KN/m 2. From previous analysis mass quantity have few influences on stability so that the mass distribution is chosen constant 200kg/m 2 lumped at each joint. Rayleigh damping ratio is ~=0.05, time step At = 0.02. The members for each span are tube 90 nos. q~140 • 5, 156 nos. q~159 • 5,240 nos. ~180 • 8, 342 nos. 00194 • 10, respectively. Structure static stability load is also calculated using load incremental method. There are total 48 cases with different spans, elastic or elastic-plastic constitutive relation, including or without damping, the results are shown in table 1. 206 Span Rise to (m) Span 1/10 3O 1/8 1/6 1/10 40 1/8 1/6 1/10 50 1/8 1/6 1/10 60 1/8 1/6 C. Wang and S. Shen TABLE 1 STATIC AND DYNAMIC CRITICAL LOAD (KN/M 2) Static Static Elastic Dynamic Elastic Plastic damp no damp 7.91 6.09 7.4 5.0 9.88 7.56 8.4 6.0 11.85 9.50 9.8 7.3 7.56 4.86 7.3 4.6 9.80 6.16 8.4 5.6 11.52 7.84 9.7 6.8 9.50 6.65 9.5 7.0 14.70 8.42 13.1 9.3 19.53 10.50 15.5 11.0 9.20 6.68 9.1 8.0 13.95 8.82 13.6 10.8 22.28 11.38 18.6 13.2 Plastic Dynamic damp no damp 5.4 3.4 6.0 4.2 6.5 5.0 4.6 2.8 6.0 3.4 6.8 4.0 5.8 4.2 7.5 5.0 10.0 6.0 5.8 5.0 7.3 5.2 10.5 6.2 It can be seen from table 1 that structure dynamic stability critical load is less than static critical load no matter material is elastic or elastic-plastic, elastic critical load is less than elastic-plastic critical load. If damping is included then the critical load increase 40%~ 50% when material is elastic or elastic-plastic. The critical load increase with the rise to span ratio at the same span. Structural dynamic stability is closely related to static stability, dynamic to static ratio is defined as dynamic stability load divided by static stability load. Without damping elastic dynamic to static ratio is 0.56 "-~ 0.87, elastic-plastic dynamic to static ratio is 0.28 ~ 0.54. With damping elastic dynamic to static ratio is 0.79 ~ 1.0, elastic-plastic dynamic to static ratio is 0.47 "~ 0.68. In order to assess the reasonable range of dynamic to static ratio the follow simplify is proposed: without damping elastic dynamic to static ratio is 0.6, elastic-plastic dynamic to static ratio is 0.3; with damping elastic dynamic to static ratio is 0.8 and elastic-plastic dynamic to static ratio is 0.5. The influences of initial geometry imperfection change with different imperfection mode and values. The imperfect 40m span dome with rise to span ratio 1/10, 1/8, 1/6 are analyzed, the imperfection mode is chosen from the first tenth static buckling mode as previous analysis with maximum imperfection values 4cm, 8cm, 12cm, respectively. The influence of material non- linearity and damping are considered, the results shown in table 2. Structure static and dynamic stability loads decrease a lot with increasing imperfection value. With imperfection value 4cm 12cm, including damping, elastic dynamic critical load decrease about 35%'~ 62% , elastic- plastic dynamic critical load decrease about 41%~- 59% compare with perfect dome. Without damping, elastic dynamic critical load decrease about 21%~ 38%, elastic-plastic dynamic critical load decrease about 13%~- 36%. Including damping the dynamic stability load is 0.4 0.6 times of perfect structure, 0.6 ~- 0.8 times without damping. With damping elastic imperfect structure dynamic to static ratio decrease from 0.8 to 0.4, elastic-plastic dynamic to static ratio decrease from 0.6 to 0.4. Without damping elastic imperfect structure dynamic to static ratio decrease from 0.6 to 0.25, elastic-plastic dynamic to static ratio decrease from 0.4 to 0.25. Dynamic Stability of Single Layer Reticulated Dome under Step Load TABLE 2 IMPERFECTION STATIC AND DYNAMIC CRITICAL LOAD (KN/M 2) 207 Rise to Imperfec Static Span :tion (cm) Elastic 0.0 7.56 1/10 4.0 4.62 8.0 3.10 12.0 2.88 0.0 9.80 1/8 4.0 7.94 8.0 5.11 12.0 3.60 0.0 11.52 1/6 4.0 11.70 8.0 9.60 12.0 6.48 Static Elastic Dynamic Plastic damp no damp 4.86 7.3 4.6 3.50 4.5 4.0 2.50 3.1 3.0 1.90 2.8 2.9 6.16 8.4 5.6 4.92 7.7 5.2 3.70 5.0 4.5 2.80 3.6 3.5 7.84 9.7 6.8 6.72 9.1 6.3 5.52 8.7 6.0 4.32 6.3 5.4 Plastic Dynamic damp no damp 4.6 2.8 3.0 2.7 2.5 2.1 1.9 1.8 6.0 3.4 4.0 3.2 3.5 3.0 2.8 2.5 6.8 4.0 6.0 3.9 5.0 3.8 4.0 3.5 In practical engineering application there are initial static loads before suddenly applied dynamic load. The 40m span dome with rise to span ratio 1/6 applied initial static load P0 = 2.0KN/m2, P0 = 5-0KN/m2, P0 = 7.5KN/m2, other parameters is the same as previous analysis. First calculate structure static nonlinear response under P0, second calculate structure dynamic response after suddenly applied step load. The critical load is the sum of initial static load and dynamic load. When static load P0 increase the newly applied step load PD decrease, but the total load P approach to structure static stability load Ps = 7.84KN/m2 (Fig. 9). Fig. 9 Apex time history with initial static load SIMPLIFY CALCULATION METHOD Solving dynamic stability critical load is time consuming because it is dual nonlinear dynamic analysis using many times trail calculation so that the simplify calculation method is needed for practical engineering application. If structure dynamic critical load can be assumed among the reasonable value then it can save much time during nonlinear FEM analysis. Referring to the quasi-shell method in static stability analysis the simplified calculating method for structure dynamic critical load is suggested as follows: 208 C. Wang and S. Shen Pz~ = K~ x Kz x Ps (1) Where K l is dynamic to static ratio, without damping elastic K 1 =0.6, elastic-plastic K 1 =0.3, with damping elastic K 1 =0.8, elastic-plastic K 1 =0.5. K 2 is modified factor of imperfection, with damping K 2 =0.4 ~ 0.6, without damping K 2 =0.6 ~- 0.8. Ps is static linear buckling load, Ps: 0.8 • ~~: (2) Where R is radius of dome, E is modulus of elasticity, t is average equivalent member 2A - thickness t =-~-, A is average member area, l is average member length, 6 is average - 1 equivalent bending thickness 6 - (12~f3/) ~ , ] is average member moments of inertia. CONCLUDING Under step load structure dynamic stability critical load is less than static critical load no matter material is elastic or elastic-plastic, elastic critical load is less than elastic-plastic critical load. The influence of material non-linearity is large than geometric non-linearity, when elastic-plastic is considered dynamic critical loads reduce 40%'~ 50%. Including damping structure dynamic critical loads increase 40%~ 50% compare with no damping case. Initial geometry imperfection largely decrease structure dynamic stability load, the decrease ratio vary with different imperfection mode and increasing with imperfection values. With damping imperfect structure dynamic stability critical load is only 0.4 ~ 0.6 times of perfect dome and 0.6 ~ 0.8 times without damping. Initial static load also should be considered in structure dynamic stability analysis. The proposed simplify method can be used in assessing structure dynamic stability load in practical engineering application. REFERENCES Bathe K. J. and Bolourchi S. (1979). Large Displacement Analysis of Three Dimension Beam Structures. International Journal for Numerical Methods in Engineering Vol. 14, 961-986. Budiansky, B. (1967). Dynamic Buckling of Elastic Structures: Criteria and Estimates. Dynamic Stability of Structures, Pergamon, New York. Simtses G. J. (1990). Dynamic Stability of Suddenly Loaded Structures. Springer-Verlag New York Inc. Wang Ce and Shen Shizhao. (1993). Nonlinear Dynamic Response and Collapse Analysis of Spatial Truss Structures. Symposium on nonlinear analysis and design for shell and spatial structures. Tokyo. Wang Ce, Shen Shizhao and Chen Yunbo. (1996). Dynamic Stability of Reticulated Dome. Proceedings of International Conference on Advances in Steel Structures. ICASS'96, Hong Kong. Pergamon, Oxford, UK, Vol II, 1065-1070. Wang Ce. (1997). Dynamic Stability of Single Layer Reticulated Dome. PhD Dissertation. Harbin University of Civil Engineering Civil Engineering and Architecture. EXPERIMENTAL STUDY ON FULL-SIZED MODELS OF ARCHED CORRUGATED METAL ROOF Liu Xiliangl, Zhang Yongl, Zhang Fuhai 2 1. Department of Civil Engineering, Tianjin University, Tianjin, 300072, China 2. Beijing Milky Way Metal Roof Forming Technology Institute, Beijing, 100021, China ABSTRACT Nine full-sized models of arched corrugated metal roof were tested to failure under static loads, and four of these specimens were modeled using two different kinds of finite elements separately. Through the description of the experimental processes and the analysis of the experimental results, the load bearing performance and the failure model of this kind of structure could be seen clearly. Based on theoretical and experimental results, some valuable conclusions were summarized and some recommendations for further studies were proposed. KEYWORDS Arched Corrugated Metal Roof, full-sized model test, load bearing performance, thin-walled structure, arch, shell, finite element method 1 INTROUDUCTION Arched corrugated metal roof is an alternative to stressed skin diaphragm structures. It is composed of a series of arched trough plates which are made of color-coated galvanized steel sheets (thickness ranges from 0.6mm to 1.5mm) and coldly formed by special cold roll forming machine. The steel sheets are firstly rolled into straight trough plates, and to obtain the desired curvature of an arched roof, straight trough plates are cold-rolled again. With their lower sidings rolled out 209 . 1. Department of Civil Engineering, Tianjin University, Tianjin, 300072, China 2. Beijing Milky Way Metal Roof Forming Technology Institute, Beijing, 100021, China ABSTRACT Nine full-sized. such as span, rise-span ratio, elastic or elastic-plastic constitutive relation, including damping and without damping. The influence factors of material non-linearity, damping, initial geometry. material constitutive relation, damping, initial imperfection, initial static load, etc. The various influence factors including material non- linearity, damping, initial geometry imperfection are