Composite hollow truss with multiple vierendeel panels

Treliça tubular mista com múltiplos painéis vierendeel

CIVIL ENGINEERING ENGENHARIA CIVIL

Composite hollow truss with multiple vierendeel panels

Treliça tubular mista com múltiplos painéis vierendeel

Augusto Ottoni Bueno da Silva^I; Newton de Oliveira Pinto Júnior^II; João Alberto Venegas Requena^III

^IDoctor in Civil Engineering, Enginner of the Municipality of Campinas. [e-mail]

^IIDoctor in Civil Engineering, Professor of the College of Civil Engineering and Arquitecture at the State University of Campinas - Unicamp. [e-mail]

^IIIDoctor in Civil Engineering, Professor of the College of Civil Engineering and Arquitecture at the State University of Campinas - Unicamp. [e-mail]

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.

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.

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 & 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 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).

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 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 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 = 205000 MPa and n_s = 0.3, and, E_c = 23800 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.

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/m², leading to a distributed load of 9.33 kN/m, and a maximum bending moment in the middle span of 262.41 kN.m. The composite truss had been subjected to an occupancy load of 5 kN/m², leading to a distributed load of 30.22kN/m, and a maximum bending moment at a mid-span of 849.94 kN.m. The loads were factored in accordance with the standard NBR 8800 (ABNT, 2008).

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 N_Rd are, respectively, the design normal force and design resistance 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.

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 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.

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 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, 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.

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, according to Equation 4, where N_Sd and N_Rd 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, M_v, 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 two portions of reduced moments on the bottom chord, M_Rd,bc,red. M_v is then compared to the applied moment V_{Sd x} l_v, (Inequality 5) where V_Sd is the design external shear force acting upon the panel, and l_v the length of the Vierendeel panel. If the bending resistance, M_v, exceeds the applied moment the project is satisfactory. If not, a heavier section must be chosen or bend stiffeners shall be introduced.

5. Maximum vertical displacements

Immediate maximum vertical displacement, δ_max, of a isostatic bi-supported composite truss subjected to a uniformly distributed load p can be calculated through Equation 6, determined by the basic theory 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.

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-by-step 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.

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.

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, 80-90 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 272.7 MPa.

The reduced moments of the top and bottom chords value, respectively, 17.64 kN.m and 25.50 kN.m. The moment due to composite action between the slab and the upper chord is 18.36 kN.m. The shear force acting on the panel is 60.44 kN. Thus, the group resistance was checked by replacing the values in Inequality 5:

M_v= 18.36 + 2x17.64 + 2x25.50 = 104.64 > 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.

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 respect to the Vierendeel bending moments (Table 4). When building 9 panels, the structure supports even less than the structure with 7 panels, and the bar BC7 enters flow before BC3. The group resistance is very close to the maximum Vierendeel bending moment. With 13 panels, the structure supports a smaller load than a structure with 9 panels and once again, the member of the bottom chord located at the mid-span, BC7, enters into flow before BC1. In addition, the composite truss is not able to withstand the local bending moments acting on the Vierendeel panel.

The behavior of the composite truss with 5 panels showed a state of transition or design equilibrium, which led to simultaneous yielding of the upper and lower faces of the bottom chord.

9. Vertical displacements in function of quantity of vierendeel panels in the composite truss

The vertical displacement limit allowed for a floor beam, according to NBR 8800 (ABNT, 2008) is span/350 therefore, 42.9 mm. The truss, with none or with one Vierendeel panel, taking into account the effects of creep estimated at 15% of initial displacement, can be constructed with or without cambering (Table 5). The ratio of the immediate values found via elastic modeling and through procedure proposed by NBR 8800 (ABNT, 2008) is r_d,st = 1.016 (19.4mm / 19.1 mm) for the non-composite truss and r_d,ct = 1.186 (14.0mm / 11.8 mm) for the composite truss (Table 6), or also, the presence of a panel or none, makes for a correct prediction of the elastic maximum vertical displacement of the non-composite and composite trusses should be made by multiplying the value found by means ABNT for 1.016 and 1.186, respectively (Figure 7). Similar values were presented for the case of trusses with 3, 5, 7, 9 and 13 panels.

10. Conclusions

The proposed method for the analysis of a composite truss bi-supported with a 15 m span, consisting of Vierendeel panels in the middle third position, showed that the ultimate limit state occurs by combination of the tension and bending moment, simultaneously at the underside of the BC7 member and upper face of BC5, provided the horizontal shear in the connectors and the resistances of the slab, steel members and joints are maintained within safe limits. When increasing the space where the panels are arranged, the structure is no longer capable of supporting the maximum bending moment required in the project, showing in some cases vertical displacements above the limit specified in the NBR 8800 standard and insufficient group resistance. Thus, the study found the proportion span/3 - span/3 - span/3 is the ideal relationship between the stretches trusses - Vierendeel - trusses, because the composite structure with 5 panels showed a transition state, or project balance, where there occurs a simultaneous yield of the upper and lower faces of the bottom chord. The values of elastic maximum vertical displacements provided by equating for composite trusses were not suitable for use in a composite truss with multiple Vierendeel panels. The modeling via Ftool and Ansys softwares demonstrated that as the number of Vierendeel panels increases, the structure becomes less rigid, and the analytical values move away from that obtained by modeling, supposed to be more realistic.

11. References

Artigo recebido em 17 de dezembro de 2012.

Aprovado em 07 de agosto de 2013.

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Publication Dates

• Publication in this collection
  08 Jan 2014
• Date of issue
  Dec 2013

History

• Received
  17 Dec 2012
• Accepted
  07 Aug 2013