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Augusto Ottoni Bueno da Silva et al. Engenharia Civil Civil Engineering Composite hollow truss with multiple vierendeel panels Treliça tubular mista com múltiplos painéis vierendeel Augusto Ottoni Bueno

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Augusto Ottoni Bueno da Silva et al. Engenharia Civil Civil Engineering Composite hollow truss with multiple vierendeel panels Treliça tubular mista com múltiplos painéis vierendeel Augusto Ottoni Bueno da Silva Doctor in Civil Engineering, Enginner of the Municipality of Campinas. Newton de Oliveira Pinto Júnior Doctor in Civil Engineering, Professor of the College of Civil Engineering and Arquitecture at the State University of Campinas - Unicamp. João Alberto Venegas Requena Doctor in Civil Engineering, Professor of the College of Civil Engineering and Arquitecture at the State University of Campinas - Unicamp. Resumo O objetivo desse trabalho foi avaliar, através de cálculo analítico e modelagens elástica bidimensional e plástica tridimensional, a capacidade resistente e o modo de ruptura de treliças tubulares mistas biapoiadas com 15 metros de vão, variando-se o número de painéis Vierendeel centrais. O estudo apontou a proporção vão/3 - vão/3 - vão/3 como a ideal para a relação entre trechos treliçado - Vierendeel - treliçado, pois, ao se aumentar a proporção do trecho central, ocupado pelos painéis Vierendeel, os novos sistemas perdem muita rigidez, passando a não suportar mais a carga estipulada em projeto. Além disso, podem passar a apresentar deslocamentos verticais excessivos e resistência às forças cortantes externas atuantes sobre os painéis insuficiente. Palavras-chave: Treliça mista, painel Vierendeel, estrutura tubular. Abstract The aim of this study was to evaluate through analytical calculation, two-dimensional elastic modeling, and three-dimensional plastic modeling, the bearing capacity and failure modes of composite hollow trusses bi-supported with a 15 meter span, varying the number of central Vierendeel panels. The study found the proportion span/3 - span/3 - span/3, as the ideal relationship for the truss - Vierendeel - truss lengths, because by increasing the proportion of the length occupied by the central Vierendeel panels, the new system loses stiffness and no longer supports the load stipulated in the project. Furthermore, they can start presenting excessive vertical displacements and insufficient resistance to external shear forces acting on the panels. Keywords: Composite truss, Vierendeel panel, hollow structure. REM: R. Esc. Minas, Ouro Preto, 66(4), , out. dez Composite hollow truss with multiple vierendeel panels 1. Introduction The composite beams of steel and concrete began to be used with stud bolt type connectors in the 1950s. They were composed of a section I filled web which worked in conjunction with a concrete slab, sometimes or not with a steel deck. However, with the need to gain larger spans, with the height limitations often imposed by zoning regulations and, as a high-headroom is normally required to allow for the passage of pipes and ducts of large diameter on full web beams, new systems of composite beams have emerged: among them, the composite beams with variable inertia, the composite beams with openings in the web (Clawson & 2. Structural characteristics The structure consisted of a reinforced concrete slab over a steel deck, supported by a plane vertical metal structure of parallel chords made up of welded hollow sections and multiple Vierendeel panels in the central part of the span (Figure 1). The metallic structure was designed with circular hollow section (CHS) bottom chord with a diameter of d=168.3 mm and thickness of t = 7.1 mm; square hollow section (SHS) top chord with a width b = 95 mm and thickness Darwin, 1982), the composite cellular beams, the stub girders, the composite steel joists (Tide & Galambos, 1970) and, finally, the composite trusses (Chien & Ritchie, 1984). The composite trusses, a more efficient alternative to overcome large spans, are generally used in commercial and industrial buildings, and rail and road bridges. In many cases, in order to enable the passage of ducts, with complications in the frames with the presence of diagonals, a central Vierendeel panel is built, but in some situations, if this single panel is insufficient, then new panels need to be created to meet the intended use for the structure. In this t=6.4 mm; and CHS brace members with a diameter of d=73mm and thickness of t = 6.4 mm. The connection between the parts was made by stud bolts. For a better understanding of the work, the names of the members and frames are shown in Figure 2. The steel deck MF-50 with a nominal thickness of 1.25 mm, as found in catalog Metform (Metform, 2010), was used together with connectors of 19 mm diameter. The total thickness used for the case, the objective of the study was to determine, through analytical calculation, two-dimensional elasticity modeling and three-dimensional plastic modeling, the bearing capacity and failure modes of a truss with a 15 meter span, and the entire central third consisting of Vierendeel panels. Then, keeping the span of 15 meters and the sections determined in the design, a parameterization of the results was made for structures having 3, 7, 9 and 13 panels. This structure, here called composite Vierendeel-truss, was designed with hollow members to present perspectives of an efficient structural solution, combining constructive strength and speed (Samarra et al., 2012). slab was 110 mm and the nominal resistance to concrete and hollow steel bars were, respectively, f ck = 25 MPa and f yk = 300 MPa. The modulus of elasticity and Poisson s ratio for the steel and concrete were, respectively, E s = MPa and n s = 0.3, and, E c = MPa and n c = 0.2. The supports for the Vierendeel-truss were designed to simulate a bi-articulated structure, on a support device or roller to allow turning without suffering the influence of supporting links. front view side view reinforced concrete slab + steel deck 2500 mm 5 x 1000 mm 5 x 1000 mm 5 x 1000 mm 1000 mm Figure 1 Dimensions of the composite Vierendeel-truss. TC1 TC2 TC3 TC4 TC5 TC6 TC7 TC8 members D1 V1 D2 V2 D3 D4 D5 V3 V4 V5 V6 V7 BC1 BC2 BC3 BC4 BC5 BC6 BC7 frames 1L 2L 3L 4L 5L 6L 7L 8 Figure 2 Names of the members and frames. 3. Resistance of the members, joints and stud bolts, and ultimate limit state The Canadian Standards Association (CSA) through standard CAN/CSA- S16-01 (CSA, 2001) and the Brazilian Association of Technical Standards (ABNT) by NBR 8800 (ABNT, 2008) indicate that the design of the top chord shall be done by loading the non-composite truss, and that the design of the bottom chord, verticals, diagonals and shear connectors must be done by loading the composite truss. The isolated steel truss was subjected to a construction load of 0.5 kn/m 2, leading to a distributed load of 9.33 kn/m, and a maximum bending moment in the middle span of kn.m. The composite truss had been subjected to an occupancy load of 5 kn/m 2, leading to a distributed load of 30.22kN/m, and a maximum bending moment at a mid-span of kn.m. The loads were factored in accordance with the standard NBR 8800 (ABNT, 2008). 432 REM: R. Esc. Minas, Ouro Preto, 66(4), , out. dez. 2013 The design process firstly checked the resistance of the steel members. The safety condition to be met by welded hollow steel members subjected to the combined efforts of axial force and bending moments, loaded so that there is no torsion, is provided by Equations 1 and 2, according to NBR 8800 (ABNT, 2008), where N Sd and are, respectively, the design normal force and design resistance ( ) Augusto Ottoni Bueno da Silva et al. to normal force, M x,sd and M y,sd are, respectively, the design bending moments about x and y axis, and, M x,rd and M y,rd are, respectively, the design resistances for bending moments about x and y axis. If N Sd 0.2 : N Sd + 8 M x,sd + M y,sd M x,rd M y,rd If N Sd 0.2 : N Sd + M x,sd + M y,sd 1.0 (2) 2. ( M x,rd M y,rd ) Then, the project kept the longitudinal shear in the connectors, and, the resistances of the slab and welded joints within safe limits, thus avoiding the appearance of undesirable ultimate limit states, which would lead to the composite structure breaking sharply. It was guaranteed the complete interaction between slab and upper chord leading the ultimate state achieve with the yielding of the lower chord, as desired for the case of composite trusses, but at this 4. Group resistance The introduction of multiple Vierendeel panels gives rise to considerable bending moments located in the bottom chord of the composite truss as the shear force is transferred through the panels. Although, the bottom chord doesn t resist alone. It resists in conjunction with the performance of the top chord and with a The flexural strength in the chords may be reduced due to the effect of shear force and shall be reduced due to the presence of the axial force. The interaction between bending and axial force is complex. However, Lawson and Hicks (Lawson & Hicks, 2011) evaluated in the composite phase that when the section is compact, the reduced moments, M Rd,red, can be determined from the plastic resistant bending moment, M Rd,pl, composite action between the slab and the top chord. This is called the group resistance, described in the works of Lawson (Lawson, 1987) and Lawson and Hicks (Lawson & Hicks, 2011). The bending resistance due to the composite action between the slab and the top chord, M tc,s, is given by Equation 3, M tc,s n. Q t t + h f Rd + x ( tc 2 ) [ ( )] M Rd,red = M Rd,pl 1 - N Sd 2 according to Equation 4, where N Sd and are respectively, the design normal force and design resistance to normal force on the chords. For the case of full slab-truss connection, the total resistance of Vierendeel local bending moments,, is then calculated by summing up the moment of resistance due to the composite action, M tc,s, the two portions of reduced moments on the top chord, M Rd,tc,red, and time, due to the presence of Vierendeel panels, by the combination of tension and bending moments. The calculation of the required number of connectors was carried out according to NBR 8800 (ABNT, 2008), and the installation of 48 stud bolts with a 19 mm diameter and 305 mm spacing was determined. To avoid stress concentration at the ends of the steel members, overlapping welded connections were chosen, and the performance was verified as provided by the European Committee for Standardization in Eurocode 3 - Part 1-8 (ECS, 2005), respecting all the conditions prescribed for the validity of geometric relations between hollow members. As the strength of the welds are greater than the resistance of the sections, and, assuming that the welds would be well executed at the time of construction of the steel structure, the hypothesis that they would not create a limit state was considered. where n is the number of stud bolts above the Vierendeel panel, Q Rd is the design resistance to shear force in one connector, t t is the total thickness of the concrete slab, h f the rib height in the steel deck, and x tc the distance from the upper face of the top chord to the center of gravity of the top chord. (3) (4) two portions of reduced moments on the bottom chord, M Rd,bc,red. is then compared to the applied moment x l v, (Inequality 5) where is the design external shear force acting upon the panel, and l v the length of the Vierendeel panel. If the bending resistance,, exceeds the applied moment the project is satisfactory. If not, a heavier section must be chosen or bend stiffeners shall be introduced. = M tc,s + 2 M Rd,tc,red + 2M Rd,bc,red. I v (5) d max = 5. p. L E s. I e,ct (6) REM: R. Esc. Minas, Ouro Preto, 66(4), , out. dez Composite hollow truss with multiple vierendeel panels 5. Maximum vertical displacements Immediate maximum vertical displacement, d max, of a isostatic bisupported composite truss subjected to a uniformly distributed load p can be calculated through Equation 6, deter m i ned by t he basic t he - ory of strength of materials, where L is the span, E s the modulus of elasticity of steel, and I e,ct the effective moment of inertia of the composite truss, which takes into account the effect of shear deformations. The prescribed values for I e,ct by the Canadian rule CAN/CSA-S16-01 (CSA, 2001) and the Brazilian standard NBR 8800 (ABNT, 2008) are shown in Equations 7 and 8, respectively. The moment of inertia of a composite truss, I ct, is calculated by reducing the area of the concrete slab to an equivalent steel area. I st is the moment of inertia of the non-composite steel truss. To consider the effects of creep the Canadian rule suggests a 15% increase in the value found for the initial vertical displacement of the composite truss. I e,ct = 0,85.I ct + 0,1275.I st (7) I e,ct = I ct - 0,15.I st (8) 6. The analytical calculation and the elastic and plastic modeling Based on the requirements described in items 3, 4 and 5, an analytical calculation was performed leading to the determination of the structural characteristics already presented in item 2. The step-bystep process can be clearly checked in the thesis of Silva (Silva, 2013). The sections described in item 2 were then used as input data for elastic and plastic modeling. The two-dimensional elastic modeling was performed through the software Ftool (Martha, 2008) building up the geometry of the structure and defining to each member the material, the cross-sectional area (Figure 3) and moment of inertia. The connecting element between the upper chord and the slab was considered to be concrete, with moment of inertia calculated by a rectangular element with a width equal to the influence width, b e, of the slab and height equal to the average width of the rib. Modeling within the finite element method was achieved with the use of the Ansys software version 10.0 (ANSYS INC., 2005) and the element type shell181 for all parts of the composite structure. Shell181 is suitable for analyzing thin to moderately-thick shell structures and well-suited for large strain nonlinear applications. It is a 4-node element with six degrees of freedom at each node: translations and rotations about x, y and z axes. The accuracy in modeling is governed by the first order shear deformation theory, usually referred to as Mindlin-Reissner theory of plates, which involves a constant through-the-thickness transverse shear distribution. For the element domain, reduced integration scheme was choosen (Keyopt(3) = 0), in other words, the number of points in Gauss-Legendre integration was reduced, and the same number of integration points of the stiffness matrix was adopted for the portions of shear and bending. The non-linearity of the materials was considered by building the stress x strain diagrams for steel and concrete. The diagram for steel used in the hollow members was incorporated by modeling Ansys bilinear, with material type bilinear isotropic hardening (BISO). The design diagram of the concrete (Figure 4A) was applied through the multilinear type material with isotropic hardening (MISO), and the points for the chart to plot stress x strain were determined by taking the base parabola-rectangle diagram prescribed in NBR 6118 (ABNT, 2003). For the geometry construction of the composite truss, 533 areas were utilized and 51,427 items were created during the process of generating the mesh. The types of elements used were preferably the smart sized quadrangle used in the bottom chord, diagonals and verticals, as shown in Figure 4B. However, due to analysis of geometric nonlinearity, these elements can provide maximum corner angles not allowed. In these cases, it was chosen the free mesh, obtaining triangular elements to some areas of the model, like the top chord and the supports. For the concrete slab and ribs it was decided for tri-mapped, 3 or 4 sided mesh. Box Circle Rectangle Top chord Bottom chord and web members Slab and ribs Figure 3 Element types for two-dimensional elastic modeling. 434 REM: R. Esc. Minas, Ouro Preto, 66(4), , out. dez A B Figure 4 Modeling with shell181. (A) Concrete stress x strain curve. (B) Mesh in a joint between diagonal, vertical and bottom chord. Augusto Ottoni Bueno da Silva et al. 7. Behavior of a composite truss with five central vierendeel panels The comparative internal forces obtained via analytical calculation and elastic modeling are shown in Table 1, and the ratio between them demonstrates that the values obtained via analytical approach were close to the more precise process developed via elastic modeling. According to Table 2, with a loaded truss, it was found that the limit state of the composite structure occurs simultaneously on the upper face of BC5 member (frames 6L and 6R) and on the underside of BC7 (frame 8) defined by the safety condition presented in Equation 1. Table 1 Bending moments and axial forces obtained via analytical and elastic modeling calculation, according to ultimate limite state of the composite beam. Table 2 Safety condition in the bottom chord of the composite beam. Member (frame) Analytical calculation Elastic modeling Ratio BC7 M Sd = 12 kn.m M Sd = kn.m (frame 8) N Sd = kn N Sd = kn BC5 M Sd = kn.m M Sd = kn.m (frames 6L and 6R) N Sd = kn N Sd = kn ratio obtained by dividing analytical by elastic modeling calculation. Member (frame) Design Resistance Internal forces Safety Condition BC7 M N (frame 8) t,rd = kn Sd = kn.m M Rd = kn.m N Sd = kn 0.93 1.0 BC5 TC d = mm M Sd = kn.m (frames 6L and 6R) t = 6.4 mm N Sd = kn 0.93 1.0 obtained via elastic analisys. For the Ansys nonlinear analysis, 3 sub-steps of loading were used (80%, 90% and 100% of the ultimate load). Figure 5A shows the von Mises stresses in the deformed composite truss. When applying full imposed load throughout the span, mm of the top face of the bar BC5 comes into flow, presenting stresses between 270 and 310 MPa (Figure 6A). At the same time, as illustrated in Figure 6B, a long stretch of the underside of the bottom chord (bars BC7 and 50% of BC6, sides left and right) also presents stresses of this magnitude demonstrating, as expected from the design process, that the two sections would come into simultaneous yield under stresses in the order of MPa. The reduced moments of the top and bottom chords value, respectively, kn.m and kn.m. The moment due to composite action between the slab and the upper chord is kn.m. The shear force acting on the panel is kn. Thus, the group resistance was checked by replacing the values in Inequality 5: = x x25.50 = 60.44kN.m The values of the immediate maximum vertical displacements calculated according to the theories proposed by CSA and ABNT were, respectively, 12.4 mm and 11.8 mm, and the values determined via elastic and plastic modeling (Figure 5B) were, respectively, 17.3 mm and 19.9 mm. A B Figure 5 Overviews of the composite beam subjected to: (A) Ultimate load. (B) Service load. A B Figure 6 von Mises stresses in the composite beam subjected to ultimate load: (A) In the upper face of the bar BC5. (B) In the underside of the bar BC7. REM: R. Esc. Minas, Ouro Preto, 66(4), , out. dez Composite hollow truss with multiple vierendeel panels 8. Behavior of the composite truss when varying the number of vierendeel panels Assuming that the proposed analytical calculation was satisfactory, or rather, leading to the correct interpretation of the structural behavior of a composite Vierendeel-truss composed of 5 central Vierendeel panels, the method was repeated for similar trusses composed of 3, 7, 9 and 13 panels. The study applied to a truss with 3 Vierendeel panels made to bear a greater load (Table 3), but with the disadvantage of having less available panels for duct passage. The limit state occurs with the yield of the upper face of BC6 and the composite truss is safe with respect to the Vierendeel bending moments (Table 4). When constructing 7 panels, the structure no longer supports the load stipulated in design (1.24 1.0), and when the truss is fully loaded, the bar BC7 flows before it enters BC4. The composite beam is safe with resp

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