Advances in Steel Structures - part 25 pptx

220 L. Yuxin and L. Zhitao {X}, {X} ,-1 q- [D],_, {a},_l (14) in which {o~} ,-1 is unknown. According to the criteria of energy, we can select {a}~_ 1 -~b[D] T = i-1 {f} (15) Here, it is clearly that ~b is a size parameter that controls the size of displacement increments in the gradient direction. Substituting Eqn. 15 to Eqn. 13 yields {d}t = ~b[D]~_ 1 [D][, {f} (16) Repeating the procedure from Eqn. 13 to Eqn. 16, we shall find out the nodal coordinates in GSES step by step. Finally, the final space coordinates of nodes can be expressed as follows oo {x}= {X}o + ~ ' {d}, (17) i=l The convergence of Eqn. 17 depends on the extemal wind force {f}. This is the same behavior for any unstable structure (Liu and Motro,1995). STRUCTURAL ANALYSIS AND VECTORIAL SUBSPACES Eqn.6 which describe the kinematic and static relations in a reticulated system connect two vectorial spaces, namely the node space R, and the member space Rb. External forces and displacements are related to the first one, internal forces and elongation to the second one. If we call r the rank of the equilibrium matrix [A], a Gaussian elimination procedure on the augmented matrix [A:I], [/] being a diagonal unit matrix, leads to the transformation below (Pellegrino and Calladine,1986) Gaussian transformation ~- ~- [A:I] >[A:I] (18) which can be put in the form [A'I]= Im (19) Applying the transformation to static relationship Eqn.6, we get Quasi- Tensegric Systems and Its Applications 221 The external force space {f} can be split in two subspaces: one is r-dimensioned and is a fitted external force subspace (forces can be equilibrated), the second one, of m-dimension with m n-r corresponds to the forces which activate the mechanisms. Similar derivation (with [J] being a diagonal unit matrix) can be achieved on the compatibility equation 2 and lead to The deformation space is composed of two subspaces, a "fitted" and a "non-fitted" subspace. Only the former is compatible with the displacements. If we consider Eqns.19, 20 under an energy view, each row of [It] and [lm], can be assimilated to displacements and each row of [Jr], [Js] to internal forces. In Eqn.20, when components of {d} belong to the mechanism subspace, corresponding values of {e} are equal to zero and corresponding rows of [Br], which are related to the external forces are orthogonal to the displacements. The m mechanism vectors are included in [lm] from Eqn.19 and the corresponding displacements can be computed by {am } = [im IT {a} (21) with {a} being composed of arbitrary constants combining elementary mechanisms.[D]=[lm] r is known as the displacement mode matrix. Similar analysis leads to the computation of any self-stress vector by {t, } : [J~ ]r {/3} (22) with {/3} being composed of arbitrary constants combining elementary self-stress states.[D]=[J,] r is known as the self-stress mode matrix. We note that for a structure that verifies compatibility condition, the self-stress subspace is orthogonal to the elastic deformation one. If the structure is in equilibrium, the mechanism displacement space is orthogonal to the external force space. Governing Equations for GSES Considering the compatibility equation, the matrix [B] is a bxn matrix, and generally this is not a square and even in this case its rank is not equal to n (=b); we can't use traditional procedures to solve it. We must introduce the Moore Penrose inverse and more precisely the { 1 }-inverse [B] With this, the general solution of the compatibility equation can be put in the form {d} = [B]- {e} + ([I] - [B]-[B]){y} (23) Where {7"} is a n-dimensional vector. The second term in Eqn.23 belongs to the mechanism displacement space and Eqn.23 is equivalent to ([I]- [B]-[B]){y} = [D] {a} (24) 222 L. Yuxin and L. Zhitao When {e}={0}, i.e. for a rigid displacement of the structure, the space coordinates can be found either by general inverse method or by an iterative altemate method (Liu and Motro,1995), [D] is called as a displacement model matrix Similar derivations give access to the intemal forces as {t} = [A]- {f} + [S] {fl} (25) Matrix [S] is called as a self-stress model matrix, {fl} is an unknown vector. Elastic Stable Equilibrium State In order to reach the ESES with a known GSES, it is necessary to determinate the values of vectors { a} and {fl} in Eqns.24, 25. Assuming that {t}0 is the initial value of internal forces in GSES, we can obtain the elastic deformation vector {e} by combining Eqn.25 and 7 which describes the constitutive law {e} = [F]([A]- {f} - {t}0 ) + [F] [S] {/3} (26) From the previous discussion, for an incompatible structure, the deformation space is orthogonal to the self-stress space. This condition gives the needed equation for computing {fl} [S] v {e} = [S] r [F]([A]- {f} - {t}0 ) + [S] r [F][S]{fl} = {0} (27) Since the product [S] T [F][S] is not singular and the first term of Eqn.27 is known, we can find out {fl} ,and the relevant internal force {t} (Eqn.25) and {e} with the constitutive law (Eqn.7). The second term can be equated by considering the orthogonality between the mechanism displacement and the non- fitted external forces, when structure is in equilibrium. Matrix [D] is a base of non-fitted external force space, so we get with Eqn.23 and 24 [D] ~ {d} = [D] r [B]- {e} + [D] T [D] {a} = {0} (28) where [D] r [D] is not singular, so we can find out { a} and simultaneously {d}. Then the position vector {x} is calculated with the GSES coordinates as first value. A new matrix [B] is derived and the process is repeated until the increments become close to zero. Hence the ESES is found. The convergence depends on the properties of the structure. If the rank is equal to the number of members there is no self-stress state and the elastic deformation {e} is compatible with {d}: the process is always convergent. When r<b, {e} doesn't fit with {d}. In this case, the external forces must be divided into small increments, so that the fitting between {e } and {d} can be nearly satisfied. NUMERICAL EXAMPLES Unstable Cable Quasi-Tensegric Systems and Its Applications 223 The first example is an unstable cable that has been researched by F. Baron and M.S. Vendatesan (1971). The sectional area of the cable is 1.465 cm 2 and the stiffness is EA=12119.51kN. The initial ESES is reached under two symmetric loads 17.8kN and an added 13.3kN local load applied to node 2. Comparison with Baron's results is done for the GSES and the ESES. Coordinates of nodes are listed in Table 1 and internal forces in Table 2. Baron's results are derived from a Newton-Raphson method. TABLE 1 SPACE COORDINATES CALCULATED Direction Node 1 Error(%) Node 2 Error(%) 31.1475 61.5544 xl (m) F. Baron x2 (m) 13.9172 16.4714 xl(m) 31.1611 0.04 61.5217 -0.05 GSES x2(m) 13.7810 -0.98 16.3073 -1.00 xl(m) 31.1540 0.02 61.5609 0.01 ESES x2(m) 13.8517 -0.47 18.4969 0.15 TABLE 2 STRESES IN BARS (MPa) Member No. Baron 1971 GSES Error(%) ESES Error(%) 1 367.908 371.255 0.91 365.276 -0.71 2 337.479 340.707 0.96 335.032 -0.73 3 383.088 386.694 0.94 381.266 -0.48 Our results are close to those of Baron. It can be noticed that the difference between GSES and ESES is very small and could be neglected: a non-extension hypothesis could be acceptable. Tensegrity System A tensegric system (see Figure 1) is analyzed, in which an initial state is shown in Figure l a. The calculating results by the method proposed in this paper are Shown in Figure lb and c. The initial nodal 6 i 2 X2 0.5m 0.5m (a) Initial state 6 J ~1 x2 8,7 5,6 (b) Final State: vertical view (c) Final State: elevation X 2 Figure 1" Tensegric element 224 L. Yuxin and L. Zhitao coordinates are listed in Table 3. The nodal 5,6,7,8 are fixed, the nodal 1,2,3,4 are free. The final free nodal coordinates are listed in Table 4. In dynamic relaxation method (Liu, 1998), the coefficient damping takes 500kg-m/s, the time incremental is 0.2s, nodal mass is 5000kg at each node. After 73695 times iteration get the final form. For incremental iterative method, the coefficient ~b =0.001, iterate 14166 times get the final results. Finally we reach the results which show in Table 4 and Table 5, respectively. The two methods are effective for calculating tensegric system. TABLE 3 THE INITIAL COORDINATES OF THE TENSEGRIC SYSTEM Nodal No. 1 2 3 4 xl(m) -0.35355 -0.35355 -0.35355 0.35355 x2(m) 0.35355 0.35355 -0.35355 -0.35355 x3 (m) 0.97832 0.97832 0.97832 0.97832 5 0.5 0.5 0 6 -0.5 0.5 0 7 8 -0.5 0.5 -0.5 -0.5 0 0 TABLE 4 THE FINAL FREE COORDINATES OF THE TENSEGRIC SYSTEM Methods Nodal No. x,(m) x2(m) x3(m) Dynamic Relaxation (Liu, 1998) 1 2 3 4 0.0000 -0.50000 0.00000 0.50000 0.5000 0.00000-0.50000 0.00000 0.86603 0.86603 0.86603 0.86603 Incremental iterative 1 2 3 4 -0.00001 -0.50023 -0.00001 0.50021 0.50022 0.00000 -0.50022 -0.00000 0.86701 0.86701 0.86701 0.86701 TABLE 5 THE INITIAL INTERNAL FORCES OF THE TENSEGRIC SYSTEM Methods Dynamic Relaxation Incremental iterative Members 12,23,34,41 15,26,37,48 18,25,36,47 12,23,34,41 15,26,37,48 18,25,36,47 Stresses(kN) 0.50000 0.70711 -1.00000 0.49952 0.70673 -1.00000 THE CONSTRUCTION OF QUASI-TENSEGRIC SYSTEM Cable net is a good structural system that has ability to span large space, but the stiffness is a big problem. Tensegric system is a maximum economic form of structure, but its construction is very complicate. This paper tries to combine the advantages of the two systems to form quasi-tensegric systems. According to the area to be covered, we at first design a cable-net, for example as shown in Figure 2a. Of cause, in practical engineering the plan of structure may not be a rectangular. Then using the tensegric element of Figure lb,c, we connect the tensegrity to cable-net one element by one element, as shown in Figure 2b. Finally we can form a double-layer quasi-tensegric spatial frame. In construction, the key problems are the application of prestressed force technology and the connecting Quasi-Tensegric Systems and Its Applications 225 form at nodes. These will be researched further in the future. And the whole structural analysis after integrating quasi-tensegric system is also important. O Cable Tensegric element Tensegric element i I Tensegric element Tensegric element Cable Tensegric element Cable (a) Cable-net Cable Cable Cable ' Cable (b) Tensegric elements build on cable-net Figure 2: Quasi-tensegric system CONCLUSION The results of our numerical examples assess the validity of the iterative alternate method we developed. As far as unstable system are concerned this method could be useful and specifically in the field of tensegric systems. Quasi-tensegric system is a new system that has the advantage of cable-net and tensegrity. It will be applied widely large span structure in practice. ACKNOWLEDGMENT This work was supported by National Natural Science Foundation of China (Project number 59508010 ). We express heartfelt thanks. REFERENCES Liu Y. (1998). Analysis of unstable systems and of tensegrity by dynamic relaxation. Chinese Journal of Spatial structures, 4:3, 26-30 Liu Y. and Motro R.(1995). Shape analysis and internal force in unstable structures. Journal of Southeast University, No.IA, 262-267 Liu Y., Lu Z., Han X., Jing J. (1995). Analysis for unstable cable-nets under static wind loads. Journal of Southeast University, No.IA, 531-535 Pellegrino S. and Calladine C.R.(1986). Matrix analysis of statically and kinematically indeterminate frameworks, lnt. J. Solids Structures, 22:4, 409-428 Baron F. and Vendtesan M.S. (1971). Nonlinear analysis of cable and truss structures. Journal of the structure Division, ASCE, 97:2, 679-710 This Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left Blank Connections This Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left Blank THE DESIGN OF PINS IN STEEL STRUCTURES R.Q. Bridge School of Civic Engineering and Environment, University of Western Sydney, Nepean, Kingswood, NSW 2747, Australia ABSTRACT There has been a recent upsurge in the use of pins, particularly in architectural steel structures with visible tension and compression members. However, the rules for the design of pins vary quite considerably from code to code and this has caused some confusion amongst consulting structural engineers operating internationally. A comprehensive testing program has examined the influence of such parameters such as pin diameter, material properties of the pin, thickness of the loading plates, material properties of the loading plates and the distance of the pin to the edge of the loading plates. Modifications to current design procedures are proposed that take into account the different possible modes of failure. KEYWORDS Bearing, design, failure, pins, shear, steel structures, strength, tests INTRODUCTION As they have no head and are not threaded, pins cannot carry any axial forces and can only carry shear forces transverse across the pin. Despite this limitation, they are often used in structural applications by designers and architects for steel structures with visible tension and compression members, particularly in applications such as canopies, sporting stadiums, convention centres and bridges. In these cases, the pins are essentially subjected to static conditions and rotations are generally small. The design procedures for pins can be found in most structural steel codes, standards and specifications. However, there is some disparity in the design values for three major of the major design conditions: shear of the pin; bearing on the pin; and bearing on the plies (plates) that load the pin. For instance, the Australian Standard AS4100-1998 has an apparently high design value for the strength of a ply (plate) in bearing and yet a low value for the strength of a pin in shear whereas Eurocode 3-1992 has a low value for plate bearing strength but a high value for pin shear strength. To explore this disparity, the behaviour of pins under load has been examined experimentally to determine the effects of the material and geometrical properties of both the pin and the loading plate on the 229 . determinate the values of vectors { a} and {fl} in Eqns.24, 25. Assuming that {t}0 is the initial value of internal forces in GSES, we can obtain the elastic deformation vector {e} by combining. 3 4 xl(m) -0 .35355 -0 .35355 -0 .35355 0.35355 x2(m) 0.35355 0.35355 -0 .35355 -0 .35355 x3 (m) 0.97832 0.97832 0.97832 0.97832 5 0.5 0.5 0 6 -0 .5 0.5 0 7 8 -0 .5 0.5 -0 .5 -0 .5 0 0 TABLE. BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left Blank THE DESIGN OF PINS IN STEEL STRUCTURES R.Q. Bridge School of Civic Engineering and Environment,