Advances in Steel Structures - part 24 pptx

210 L. Xiliang et al. many tiny cross corrugations, they become curved, forming arched trough plates. Because arch structure can translate the applied loads mainly into forces in the plane of its surface, so such arched trough plate can be employed in larger span buildings (more than 30m) not only as an accessory material to be used for simple coating, but also as load bearing skeleton. With the plate- skeleton-combined structural style and the highly mechanized construction procedure, arched corrugated metal roof possesses such advantages as strong spanning ability, light self-weight, fast and easy construction, good waterproofing quality and attractive appearance etc. The combination of these advantages certainly can result in cost saving. It is very suitable to be used in single layer buildings, and if hoisting condition permits, it can also be used in multistory buildings. According to the different sectional configurations of arched trough plates, this kind of structure can be classified into several types. In China there are mainly three kinds of sectional configurations, so there are three types of this kind of structure, which are respectively named MMR-118, MMR-178 and MMR-238. Figure l shows the outlines of their cross sections. Figure 1 Design specifications and recommendations for cold-formed steel structural members are now available in many countries, but none of the rules for the design and construction of arched corrugated metal roof have been published all over the world till now. Even though it is a typical kind of thin-walled steel structure, because of its peculiar characteristics, its performance under load differs in several significant respects from that of ordinary cold-formed structural members. As a result, design specifications for cold-formed steel structural members cannot possibly cover the design features of this kind of structure completely, so it needs an appropriate design specification. With no provision of certain design code, engineering accidents will be inevitable. In the winter of 1996, a heavy fall of snow in the northeast of China caused more than 30000 m 2 of this kind of roof to collapse. According to former research work, there are mainly two kinds of mechanic models for this kind of structure, namely arch and shell. However, for some reasons, none purely theoretical analysis on this kind of structure can make satisfactory results [1 ], so experimental studies are essential here. Nevertheless, just because of its special construction characteristics, it is almost impossible to carry out scale model test, full-sized model tests are indispensable to the research of this kind of structure. After the engineering accidents mentioned above, the authors had carried out nine groups of large- span model experiments on the spots of these accidents. Through these model tests the cause of Study on Full-Sized Models of Arched Corrugated Metal Roof 211 these accidents and the load bearing performance of this kind of structure could be understood. By comparing the theoretical results with the testing results, the great divergences between them could be seen clearly. Aiming at reducing these divergences, some recommendations for further studies are proposed. 2 OUTLINE OF EXPERIMENT 2.1 Model specimens All of these tests were on-the-spot tests. The models studied here were the very structures that survived from that heavy fall of snow. The steel plate used in these models had the yield strength of 280Mpa and Young's modulus of 2.00 • 105 MPa. The sectional configurations of these trough plates of these models were the same as that shown in fig.lc, namely trapezoid section. Five of these models spaned 33m and the others spaned 22m. For the convenience of the application of load, only one model was made up of six pieces of arched trough plates, the others were all made up of four pieces. The cross section is shown in fig.2. In order to search for an effective measure to raise the load bearing capacity of this kind of structure, three models were reinforced with tension chords. The reinforcing pattern is shown in fig.3. The geometrical size and load patterns of these models are described in tab.1. Because the width-to-span ratios of these models were very small, their lateral rigidity was quite low. To avoid lateral buckling and something unwanted scaffolds were placed under and by both sides of these models. The outlook of a model after being put in order is shown in fig.4 Figure2: The cross section of models Figure 3: The reinforcing pattern Figure 4: Testing ground 212 No. 1 2 3 4 5 6 7 8 9 L. Xiliang et al. TABLE 1 Arch span 33(m) 33(m) 33(m) 33(m) 33(m) 22(m) 22(m) 22(m) 22(m) Arch rise 6.6(m) 6.6(m) 6.6(m) 6.6(m) 6.6(m) 4.4(m) 4.4(m) Plate thick, 1.25(mm) 1.25(mm) 1.25(mm) 1.25(mm) 1.25(mm) 1.00(mm) 1.00(mm) Lateral width 2440(mm) 2440(mm) 3660(mm) 2440(mm) 2440(mm) 2440(mm) 2440(mm) Load pattern Full span Half span Half span Full span Half span Remarks Local distributed load Reinforced Reinforced Full span Half span Half span Triangular load distribution Half span Reinforced 4.4(m) 4.4(m) 1.00(mm) 1.00(mm) 2440(mm) 2440(mm) 2.2 Loading method As a kind of thin-walled structure, arched corrugated metal roof is very sensitive to concentrated load which may cause local buckling of the structure at a relatively low load lever. In actual engineering, large concentrated load should be avoided. To simulate the actual load-bearing pattern, distributed loads were applied by using sandbags. From tab.1 we can see that No.3 model bore local half-span distributed load, which means that only four out of the six trough plates bore half- span distributed load, while the two edge trough plates were free from any external direct loads. Tab.1 tells us that No.8 model bore triangularly distributed load. This loading pattern is to imitate non-uniformly distributed snow load. 2.3 Observation method Because this is a kind of flexible structure, its deformations are so large that any displacement measuring instruments with conventional precision can not cover its deformation scope, therefore levelling instruments were used to survey the vertical displacements, and theodolites were used to measure the rotary angles of those surveying points. Through the values of these rotary angles, we can figure out the horizontal displacements. 7V08 static electrical resistance strainometer was employed to observe the distribution of strains in the models. The surveying points of displacements and strains were arranged at such locations as two springs, L/8 section, L/4 section, L/2 section, 3L/4 section and 7L/8 section. Study on Full-Sized Models of Arched Corrugated Metal Roof 3 EXPERIMENTAL RESULTS 213 The ultimate load, maximum horizontal displacement (U) and its location, maximum vertical displacement (V) and its location of each model are listed in tab.2 No. 1 2 3 4 5 6 7 8 9 TABLE 2 Ultimate load U Location V Location 0.87kN/m 2 38cm L/8 43cm L/2 0.56kN/m 2 52cm 3 L/4 57 cm 3 L/4 0.27kN/m 2 53cm L/4 54cm L/4 0.92kN/m 2 36cm L/8 42cm L/2 1.02kN/m 2 19cm L/4 23cm L/4 1.06kN/m 2 18cm L/8 27cm L/2 0.54kN/m 2 32cm 3 L/4 41 cm 3 L/4 1.02kN/m 2 31 cm 3L/4 39cm 3 L/4 1 lcm 1.28kN/m 2 L/4 16cm L/4 Studying the data got from electric resistance strainometer, it's hard to find the laws of the stress distribution in these models' sections. Although the cross sections of the models and patterns of external load were symmetric, the stresses in one section didn't show symmetry. The direction of principal stress of a certain point changed form time to time with the load added. The tiny ripples in the trough plates and the out door wind load may account for this to a certain extend. Certainly the stresses measured couldn't reflect the laws of the distribution of the actual stresses, but as few of them exceeded the yield point stress of the material, so they could qualitatively tell us it isn't strength that determines this kind of structure's load bearing capacity. Though the width-to-thickness ratios of the trough plates in these models are very large, local buckling models which is common for thin-walled members didn't appear during these tests. This demonstrates clearly that the tiny ripples can strengthen the local stability of the plates. Both No.1 and No.6 models bore full-span uniformly distributed load, so their performances were similar. When the load level wasn't high, their deformations were symmetric, as shown in fig.5. But when the load was close to the ultimate load (shown in tab.2), a sudden change from symmetric deformation to non-symmetric deformation happened, which caused the internal forces around L/8 in this side to increase steeply. With a little more loads, the model lost its stability and buckled. The failure model of this kind of structure under half-span distributed load was shown in fig. 6. It's easy to understand that the stability bearing capacity of this kind of structure under half-span load is 214 L. Xiliang et al. much lower than that under full-span load, while the stresses and displacements were much bigger. The reason accounted for this was that the deviation between arch axis and pressure line in the half-span load model was much larger than that in the full-span load model, so bending moments were prominent here, which was very disadvantageous to any structures. According to the data provided by the local meteorological department, after that fall of snow the basic snow load of the zone where the accidents happened was about 0.521kN/m 2, and the gale also blow snow from windward side to leeward side during the snow-fall. So the uneven half-span snow load was close to ultimate half span load listed in tab.2. It's quite sure that half-span load pattern is the most dangerous load pattern for this kind of structure. Original shape ~ L / 8 4 L / 8-~L L / 8 ~L / 8 4 L / 8-~L L / 8 4 L / 8 4 L / 8 *J Original I, L. I B -'-4 L I B J L I B I L I B J' L I B-'-~ L I B I L I B J' L I B "-~ Figure5: Deformation Shape of Full Span Load Model Figure6: Deformation Shape of Half Span Though there were two pieces of trough plate free from direct external load, the ultimate load of No.3 model is not bigger than that of No.2 model. This model test indicates that as a kind of thin- walled member with open cross Load Model I~' L / 8 "#- L / 8 Jr L / 8"-"IL- L / 8 i L / 8~L L / 8 J,"-'L / 8 l L / 8 4 Figure7: Deformation Shape of Reinforced Half-Span Load Model section, the trough plate's torsional rigidity was very small and its capacity of resisting torsional load was poor. From this test we also can see that the coordination between two pieces of plates was bad, and the lateral widths of other models had little effects on their load bearing capacity. Fig.7 shows the deformation shape of the models reinforced with tension chords subjected to half span load. Tab.2 tells that the reinforcing pattern shown in fig.3 has little effect on the load bearing capacity of the structure under full span load, while under half span load the load bearing capacity of the same structUre can be doubled. From fig.7 we can see that two chords restrain the 3L/4 section, where the largest deformation will take place without these chords. The tension chords can make the distribution of the internal forces even more uniform. 4 COMPARISON OF EXPERIMENTAL AND THEORETICAL RESULTS Because of the symmetry of the configuration and the load distribution along the longitudinal direction of this kind of structure, it can be looked as a kind of arch structure and modeled with thin-walled beam elements. The material constants, such as bending rigidity, axial rigidity, etc, are Study on Full-Sized Models of Arched Corrugated Metal Roof calculated according to the geometric size of unit width of its cross section. 215 To reflect such structure characteristics as thin wall, tiny ripples, doubly curved space configuration, shell element is the most ideal one. The shell element used here is a kind of generalized conforming quadrilateral flat shell element [2]. A piece of arched trough plate is chosen as calculating model. Because the length-to-width ratio of the trough plate is very big, in order to avoid deformed grid dividing, the size of shell element should be very small. So the number of the shell elements in a piece of trough plate is great. This of course increases the amount of calculation, while on the other hand this also can raise the calculating precision. In general, the steel material used in this kind of structure is isotropic. But because of the ripples on the webs and flanges of the trough plate, the webs and flanges will respond to load orthotropically. To analyses the effect of the ripples an equivalent orthotropic fiat sheet is defined for the shell FEA model. The material constants of the equivalent flat sheet can be acquired according to the equivalent condition [3]. The above experiments had indicated that it is global stability, not material strength, that control the load bearing capacity of this kind of structure, so only geometric nonlinearity is considered in this paper. For the same reason, local buckling isn't considered. To avoid the problem of material nolinearity in theoretical analysis, yield criterion is adopted as the failure criterion. By the programs based on above mentioned theory, specimen 1, 2, 6 and 7 had been calculated. The ultimate loads of theoretical analysis and experiments and the errors of theoretical results compared with experimental results are listed in tab.3. TABLE 3 Experiment Arch model Error Model No. Shell model Error 1 0.87kN/m 2 2.17kN/m 2 149.4% 1.26kN/m 2 44.83% 2 0.56kN/m 2 1.06kN/m 2 89.29% 0.67kN/m 2 19.64% 6 1.02kN/m 2 5.76kN/m 2 464.7% 3.23kN/m 2 216.7% 1.89kN/m 2 0.54kN/m 2 250.0% 1.14kN/m 2 111.1% As a kind of thin-walled steel structure, it is very sensitive to defects. Because the models used in these experiments were the survivors of accidents, initial deformation and initial stress were inevitable. In addition, all the tests were carried out outdoor, wind load will bring harmful effect on the tests too. So from tab.3 we can see all the theoretical results are much higher than the corresponding experimental results. But compared with half-span loading models, the errors of full-span loading models are even larger, which indicates that this kind of structure under full-span load is more sensitive to defects than that under half-span load. It's obviously that the results calculated with shell FEA model are much closer to the experimental 216 L. Xiliang et al. results than that calculated with arch FEA model. This indicates that even though it's symmetric along longitudinal direction, the arched trough plate, the structure's components have the property of space load carrying because of its characteristics of thin wall and local ripple shape. The construction of ripples on the plates certainly can strengthen the stiffness along longitudinal direction, which makes the structure free from wavelike local buckling, but they weaken the stiffness along span direction which is very disadvantageous for this kind of bearing structure. Shell FEA model can reflect these factors to a certain extend. From the analysis above, it's not difficult to find out that purely theoretical analysis on this kind of structure has a distance from real application. Model test is indispensable here. But experimental study requires testing of full-sized models, which are very expensive and the result is only applicable for some special situations. So studying the relation between theoretical analysis and the experimental results and finding out the appropriate calculating constants from experiments so as to revise the calculation programs have great significance for the research of this kind of structure. The authors of this paper are now preparing several groups of member tests in order to observe the material constants of the arched corrugated trough plate. The material constants got from experiments will be used in FEA. 5 CONCLUSION Through the description of these full-sized model tests, the load bearing performance and the failure model of arched corrugated metal roof are clear now. After pointing out that local buckling and material strength are not the control factors to its load carrying capacity, two kinds of FEA models were established for the its theoretical analysis. Though the theoretical results didn't agree well with the test results, these deviations indicate that such structural characteristics of this kind of structure as thin wall and local ripple shape have great effect on its load bearing performance. To reduce the difference between theory analysis and experiment study, recommendations for further research are proposed. References 1. Zhang Yong, Liu Xiliang and Zhang Fuhai (1997) Experimental Study on Static Stability Bearing Capacity of Milky Way Arched Corrugated Metal Roof. J. of Building Structures, 18:6, 46-54 2. Zhang Fuhai, Zhang Yong and Liu Xiliang (1997) A Generalized Conforming Quadrilateral Flat Shell Element for Geometric Nonlinear Finite Element Analysis. J. of Building Structures 18:2, 66-71 3. Erdal Atrek, Arthur H.Nilson (1980) Nonlinear Analysis of Cold-Formed Steel Shear Diaphragms, J. of the Structural Division 3,693-710 QUASI-TENSEGRIC SYSTEMS AND ITS APPLICATIONS Liu Yuxin 1 and Lti Zhitao 2 1Nanjing Architectural & Civil Engineering Institute; Nanjing 210009, China 2Southeast University, Nanjing 210018, China ABSTRACT Tensegric system is an optimum structural form in which the behavior of high strength in steel cable can be utilized, but the reliability of this system is not very good because of the quasi-variable characteristics. Cable-nets are also an effective structure that could span large space. This paper proposes a new concept of spatial structure in which we combine tensegrity with cable-nets to form a quasi-tensegric system. So we can make use of the advantages of these two systems. A construction manner is developed. A quasi-tensegric system could be formed by the tensegric elements. This paper divides the equilibrium state of quasi-tensegric system into two states: one is geometrical stable equilibrium state, the other is elastic state equilibrium state. A method is developed to calculate the form and internal forces in the geometrical stable equilibrium state and the convergence is provided. The results of calculating show that the method proposed has a good convergence and a high precision. Comparing incremental iterative method with dynamic relaxation method, the two methods are effective and reliable in engineering design. KEYWORDS Quasi-tensegric system, tensegrity, cable-nets, geometrical stability, equilibrium state, prestressed force, incremental iterative method, dynamic relaxation method INTRODUCTION Among reticulated structures composed of struts and cables, which require formfinding processes a specific class can be defined as funicular system's class (Liu and Motro,1995). Their stable shape is directly related to a set of external actions. Two equilibrium states are defined. The first one which doesn't take into account the member deformations corresponds to geometrical stable equilibrium state (GSES), the second one is related to a computation of the equilibrium in the deformed shape under 217 218 L. Yuxin and L. Zhitao extemal actions and is named the elastic state equilibrium state (ESES). A method for computing the coordinates for the GSES was obtained by using the theory of generalized inverse matrix (Liu and Lu et a1,1995). In order to determine the ESES leading to the value of node coordinates and internal forces, an alternate method was put forward. Computed results are compared with those obtained with a Newton Raphson method. We shall introduce briefly these main results in this paper. After giving the method of unstable systems, we discuss calculating procedure of cable-nets and simple tensegric system. And finally give the construction rule of quasi-tensegric systems. INCREMENTAL ITERATIVE METHOD FOR UNSTABLE SYSTEM Kinematic Relationship Static and kinematic equations are established assuming classical hypothesis for reticulated structures with struts and cables. Assuming that free nodal displacements there are b members and n degreeS of freedom, the kinematic relationship can be expressed in matrix as follows {e} = ([B] + [AB]){d} (1) {e } is elastic deformation vector, [B] is the compatibility matrix and [AB] an increment of [B], {d} is the displacement vector in which boundary condition being included by deleting the corresponding values. When II {d} II is very smaller, the second term can be neglected, then [B] {d} = {e} (2) For an unstable structure, there is no elastic deformation until the geometric stable equilibrium state and Eqn.2 become [B] {d} = {0} (3) Static Equilibrium Relationship Static equilibrium equation can be derived from principle of virtual work. For a set of extemal actions {f} and a virtual displacement { rid}, corresponding elongation { de } and internal forces {t} must satisfy {f} r {d} = {t} r {e} (4) Substituting Eqn.2 into Eqn.4 yields ({f}r _{t}r [B]){fd} = {0} (5) It holds for arbitrary { 6at}, so that Quasi-Tensegric Systems and Its Applications [B] r {t}= {/} or [A]{t}={/} The constitutive law can be expressed in matrix form as follows {e} = [F]{t} 219 (6) (7) Where [F] is b-order diagonal matrix with F, = (L / EA) , i=l,b (8) Stability Criterion and Convergence When analyzing the form in geometric stable equilibrium state, we use the compatibility Eqn.3 instead of static equilibrium equation Eqn.7. At equilibrium the total potential YI of the structure takes a local minimum value. The necessary and sufficient conditions for equilibrium are oTI=0 (9) 521 I 0 (10) Where 5 is a variational symbol related to the displacement space. The equilibrium is arbitrary or stable according to the value of 6 2H. Condition expressed by Eqn.9 will be used in next section in order to choose a parameter leading to the stable equilibrium state. For the problem of GSES, we use linear incremental method to solve the system of linear homogenous equation 3. That is to say an iterative procedure will be used. As the incremental {d} is small, in each iterative step (say i-l), take the first order approximation, then {d},_ 1 ={x'}i_lt (11) Where t is a small parameter, {x'},_~ is the first order derivation of the displacement vector with regard to t. If {d},_~ have been found out, [B]i. 1 can be calculated[2]. So we have IN]i_ 1 {dIi {0) (12) Based on the generalized inverse theory of matrix algebra, thesolution of Eqn.12) is {d}, =([I]-[Bl+[B]{y},_l = [D],_I {a},_~ (13) where { y} t-1 as well as {a} ,_~ are n and (n-r) dimension algebra vector, respectively. In this paper we chose { 1 }-inverse, so { c~} i-1 is a (n-r) dimension vector. [D] ~-1 will be called as a displacement model matrix. Hence from i-1 step to i step, nodal position coordinate vector is . code, engineering accidents will be inevitable. In the winter of 1996, a heavy fall of snow in the northeast of China caused more than 30000 m 2 of this kind of roof to collapse. According to. for this kind of structure. Original shape ~ L / 8 4 L / 8-~ L L / 8 ~L / 8 4 L / 8-~ L L / 8 4 L / 8 4 L / 8 *J Original I, L. I B -& apos ;-4 L I B J L I B I L I B J' L I B-&apos ;-~ L I B. Structural Division 3,69 3-7 10 QUASI-TENSEGRIC SYSTEMS AND ITS APPLICATIONS Liu Yuxin 1 and Lti Zhitao 2 1Nanjing Architectural & Civil Engineering Institute; Nanjing 210009, China 2Southeast