Determination of fastener stiffness and application in the structural design with semi-rigid connections

Transcription

1 Determination of fastener stiffness and application in the structural design with semi-rigid connections Claudia Lucia de Oliveira Santana Researcher, Institute for Technological Researches of State of Sao Paulo (IPT), Sao Paulo, Brazil Nilson Tadeu Mascia Associate Professor, State University of Campinas (UNICAMP), Campinas, Brazil Summary Stiffness of fasteners, as well as its strength, is a fundamental parameter in the design of structures with semirigid connnections. In this paper, an analysis of the elastic response of wood under embedment and its quantitative influence in structures with semirigid connections is presented. We mae initially a brief review of the study of the response of wood under embedment. We present the derivation of stiffness of fasteners (slip modulus) from the beam on elastic foundation model and some experimental aspects and results for hardwood species, and underline some criteria aiming normalization for the embedment properties of wood. Finally we apply these values and criteria on the analysis of structures with semirigid connections. We conclude that provided that wood embedment properties are determined with good exactness, the linear and multiple linear models for the behavior of fasteners lead to reliable results in structural design.. Introduction The stiffness of fasteners is understood as the relationship between load and the relative displacement or slip and is also called slip modulus, although the real behavior of connections is non-linear and elastoplastic. Besides, the behavior of connectors, or more precisely, the behavior of the setup formed by connectors and wood in which they are embedded, reflects the behavior of wood, and presents time-dependent behavior. It may be supposed that the behavior of wood may be approximated by a multi-linear behavior. In computational analysis, the non-linear behavior may be regarded by means of a stepwise increasing slip modulus. It is well nown that the deformation of connections affects the deformation of the whole structure. In moment resisting connections, the stiffness of connectors influence in the sharing of internal efforts, especially bending moment, among connectors, and affects the stiffness of the moment resisting connection (rotational stiffness). In the structure as a whole, it affects the distribution (diagram) of bending moments. Eurocode 5 (CEN, 995) presents rules for the calculation of slip modulus of connectors, based on experimental results. Theoretical methods based on beam on elastic foundation give accurate explanation about the behavior of fastener. But it depends on the way in which the wood properties are determined. By means of a simple analysis of the expression for slip modulus, one may conclude that a deviation in the value of wood property affects signicantly the final value. Before treating the slip modulus, we have to treat the wood properties. The wood properties are basically the embedment strength and the embedment stiffness, that is, the foundation modulus of wood considered as a foundation or more precisely, a Wrinler type foundation. Wrinler type foundation is supposed to have no shear stiffness, that is, any area of the foundation may be compressibly deformed without shear interaction with the surrounding area (Wilinson, 97). In this paper, we show the aspects related to the determination of strength and stiffness and its relationship with the design of the structure. We show experimental results of a simulation of the behavior of a moment resisting connection and an application of the design and calculation of a structure with semi-rigid connections.

2 . Embedment Strength and Stiffness and Slip Modulus The embedment strength and stiffness properties of wood are conventional properties, which describe the performance of wood, in general terms, from the displacement measured under a certain load. The strength is usually defined as the stress which leads to a conventional maximum displacement, with the stress defined as the load divided by the projected area of contact (d t), where d is the connector diameter and t is the wooden piece thicness. In the scope of design of connectors, conventional displacements determine the limit states of wood under embedment. If the increment of displacement is compared to an increment of load per unit length of connector, the resulting parameter is independent of the connector length. For connection with linear response, the stiffness expressed in this way is usually named foundation modulus. The foundation modulus is an important property of wood in the scope of study of connections. In this wor, it will be denoted generally by. Wilinson (97) founded experimentally a correlation of the foundation modulus with diameter of the connector used in the embedment test, and proposed a property, the wood bearing elastic capacity, attributed exclusively to wood and given by: c o = [] d The discussion above applies to linear relationship between load and vertical displacement, what in fact may be assumed up to a certain load level. However, wood has a nonlinear behavior, which, together with the non-linear behavior of connectors steel, is responsible for the nonlinear behavior of connection. Kuenzi (955) developed his theory of connections woring as a beam on elastic foundation. It is based on Wrinler type foundation. The reference system is shown in Figure. F x x x x y y undeformed nail or bolt y deformed nail or bolt y Fig Reference system for connector deformation modeling F According to the reference system shown at Figure, the relative slip is given by: = v t ) + (0) [] ( v where v is the vertical displacement of bolt in member (system x -y ), t is the thicness of member (maximum value for x ), and v is the vertical displacement of bolt in member (system x -y ). In the linear range, the slip modulus is the constant relating F and, that is: F = K [3]

3 By developing the solution for the differential equation of the beam on elastic foundation, by applying boundary conditions, and the compatibility equation, Kuenzi arrived that the slip modulus may be given by: ( J J ) K ( L + L ) [4] K + K where: J J K K L L sinh sinh sinh sinh + sin + sin 3 sinh λt cosh λt sinh λt 3 sinh λt cosh λt sinh λt sinh λt cosh λt sinh λt sinh λt cosh λt sinh λt with λ and λ defined by: = + sin cos + sin cos cos cos [5.a] [5.b] [5.c] [5.d] [5.e] 4 λ [5.g] 4EI 4 λ = [5.h] 4EI Here, EI is the bending stiffness of the nail and and are the foundation modulus of the wood in the pieces. Foschi (974) extended the beam on elastic foundation model to an elastoplastic model and considered a non-linear behavior for wood. He developed a finite element analysis of the bolt. By implementing his model and by taing the initial linear response of materials (wood and bolt steel), it is possible to arrive to the same solution obtained with Kuenzi s expressions, since the finite element approximation for beam element coincides with the exact theoretical solution. It is very important to notice that in each member, the wood fibers may be oriented in a different direction in the plane perpendicular to the plane x -y or x -y (see Fig ). According to Eurocode 5 (CEN, 995), the slip modulus in the serviceability limit state for bolted connections is given by: K ser,5 d ρ m = [6] 3 where ρ m is the density in g/m 3, and d is the bolt diameter in mm. In the ultimate limit state, the slip modulus is taen as K u =/3 K ser. [5.f]

4 Experimentally, the slip modulus may be obtained from connection tests. 3. The connection element for framed structures The definition of a connection element is very favorable for computational programming based on matrix analysis. Each connection element has two nodes (ends), each of them able to be connected to one node (end) of a beam element. At undeformed shape, both nodes of connection element have the same position in plane. At deformed shape, due to relative displacements of connectors, each of the nodes displaces to a different position. These positions are described by a set of six (three for each node) displacements joined in a vector u. Figure shows the scheme of a real joint and the scheme of the general form of the lin element, with their coordinates, in the deformed shape, and the relative displacement and force in a generic connector. Undeformed 3 connection element 3 Deformed connection element A^ =A^ u j u i 4 4 uj u i v j v i i j F V i M i θ i N i θ j V j M j N j v j V i M i v i Ni θ i θj F V j Nj Fig Connection element with external displacements and internal force and relative displacement in a generic connector. In accounting for the connection deformation in the matrix analysis, the nodal displacements and loads may be considered as external and the connectors slip and loads are considered as internal. The general relationship F=f( ) is required, where f is a scalar function from to F, nonlinear in generic sense. This relationship is supposed to have different parameters according to the direction of load in relation to the fiber direction. For practical reasons, this generic relationship may be determined in two perpendicular global reference directions, say x (horizontal) and y (vertical) in the plane of the structure (see Fig ). From these relationships, the relationship F () =f( () ) for each connector in the direction of the load and relative displacement (which are supposed to be the same) may be derived. Be u i, i ranging from to 6, the nodal displacements of the joint, and a i the corresponding nodal loads. Be () the slip of a generic connector of the joint, and F () the corresponding load. The equation of the Principle of the Virtual Wor(PVW) may be written as: 6 i= n a δ u = F δ cos( α φ ) [7] i i = M j

5 where n is the number of connectors, δ means virtual, α () is the angle between the connector load F () and the reference direction and φ () is the angle between the virtual slip δ () and the reference direction. From geometric relationships only, the slips may be written as a function of the nodal displacements, such that: x y and: = Q. u = Q u [8.a] x y xi yi i = Q. u = Q u [8.b] i = ( x ) + ( y ) [9] where: Q = { 0 r sin β 0 r sin β } [0.a] x and: Q = {0 r cos β 0 r cos β } [0.b] y If linear behavior is adopted for connectors, then the ratio F () / () in is constant and we may find a linear equation relating a and u, given by: a=f(u) [] where f is a vectorial function of u whose components are given by: f j n yj y = K [( Q Q + Q Q ). u] [] = xj x where j is a free index varying from to 6. This last equation describes the mechanical behavior of connection. The diagonal stiffnesses K x, K y and K z are especially important to the design of rotational connections. They command the distribution of internal efforts (bending moment, shear force and normal force) among connectors, as shown by Santana and Mascia (00) and Racher(995). On the other hand, they have important application in the comparison between connections, especially the last one, named rotational stiffness, even for non-linear rotational connections, as they are considered as initial stiffnesses. The rotational stiffness is given by: z n K = K r [3] = 4. Experimental results for embedment properties and slip modulus Some results of embedment tests carried out by the authors (Santana, 00) according to Brazilian Code NBR 790:997 (ABNT, 997) are shown in Table. The nail diameter was 6.4 mm, the thicness of the wood member was t=d, and the wood species was Cupiuba (Goupia glabra), green wood, with modulus of elasticity in compression parallel to fibers 5 0MPa. Table Results of embedment tests (3 specimens each direction). Parallel to fibers Perpendicular to fibers Embedment stiffness K e (N/mm) Foundation modulus =K e /t (N/mm ) We applied these values of foundation modulus in the Kuenzi s expression for slip modulus. For nail steel we used Young Modulus 0 000MPa. The theoretical values were compared with slip modulus were compared with the results obtained from connection tests. Results are 6060N/mm obtained from test ( specimens) and 4970N/mm obtained from Kuenzi s model.

6 Results differs in about 0%, however, very few results were avaiable. The theoretical values obtained with the theory of beam on elastic foundation depend on the accuracy of the determination of the foundation modulus. More researches and normalization on this area are needed. 5. Application In this section we present an analysis of framed structures with semi-rigid connections with linear behavior. We considered a frame with span of 6.5m, height of 3.5m and ultimate load of 4N/m, as shown in Figure 3. In the same figure, the schematic bending moment diagram is shown. We considered E=9 78.5MPa, A=0.m and I= m 4 for both members, with width of 00mm. We considered columns with double rectangular section and beam with single rectangular section. q M C M E Connection element y M B B C D MD E F M F H x R =; R =; R =0 x y z L Fig 3 Geometrical definitions for structure. A M A =0 M G =0 G After solving the structure through conventional analysis, we obtained the results shown at Table. Table Results obtained from conventional analysis. Section N(N) V(N) M(Nm) B C D We designed the connections by calculating the forces due to internal efforts (M, N and V) in each connector and comparing the resulting force in each connector with its resistance calculated according NBR790:997(ABNT, 997). For the design, we used a spread sheet implement by the authors. The method was detailed by Santana and Mascia(00). Two possible solutions are shown in Table 3. Table 3 Geometrical parameters of the distributions of bolts. Connection Diameter of bolt (mm) Diameter of outer circular row (mm) Total number of bolts (double shear) Rotational Stiffness (Nm) Ultimate bending moment (Nm) C C With the connections determined, we proceeded to the semi-rigid connection analysis. By using the method proposed by Foschi (974), through a computational program implement by the first author

7 with the second author as advisor (Santana, 00), we found the parameters describing the linear behavior of the bolts. For each of the bolts used in the connections, d=0mm and d=0mm we obtained the parameters shown in Table 4 (x-direction and y-direction are defined in Fig 3). The parameters in both directions are the same because angle between members is 90 degrees. They are valid only for the connection with that particular geometrical configuration (members width and angle between members), and of course, to wood specie considered. Values from Eurocode 5 are shown in the same table. We considered the embedding parameters of Eucalipytus grandis estimated from embedding tests results found in Valle (999), shown in Table 5. Data of steel were Young Modulus 0 000MPa and Design Yield Strength 545MPa. Table 4 Parameters of the connectors used in the connections. Slip Modulus (N/mm) Theoretical Simulation Slip Modulus (N/mm) Eurocode 5 d=0mm Table 5 Estimated embedment parameters of wood. Embedment Stiffeness (N/mm ) Tests d=0mm x-direction y-direction x-direction y-direction Parallel to Fibers Perpendicular to Fibers With the bolt parameters shown in Table 4, and with a computational program implemented for linear matrix analysis, we obtained the results for the frame with semi-rigid connections and linear behavior. Results are shown in Table 6. Table 6 Internal efforts and displacements obtained from Linear Analysis. Cross Sec. Connection N(N) V(N) M(N.cm) A B C D 6. Discussion Rigid C(d=0mm) C(d=0mm) Rigid C(d=0mm) C(d=0mm) Rigid C(d=0mm) C(d=0mm) Rigid C(d=0mm) C(d=0mm) The comparison of the rigid connections with semi-rigid calculations shows that the semi-rigid connections lead to a redistribution of bending moment diagram. Different connections cause different redistributions. The parameters of connections important to the design and solution of framed structures are the slip modulus and strength of individual connectors for the design of the rotational connections and the parameters for embedment behavior of wood (foundation modulus) for calculation of bolt linear parameters.

8 The slip modulus is not promptly available in most codes for parallel and perpendicular directions and for mixed directions in two connected members. The embedment stiffness of wood is not available as well. The slip modulus of each connector in the direction of slip is needed for the design of rotational connection (distribution of forces among connectors) - for that, more important than the values, is the relative difference between values in parallel and perpendicular directions in relation to wood fibers) - and for the calculation of rotational stiffness and solution of internal efforts. According to the usual afety model, the actions are majored and the properties of materials are minored. However, the stiffness of connections has a different relationship with framed structures. The lower the stiffness, the lower will be the moments transferred by the connections and the moments in the supports. The moments in the middle spans will be larger. 7. Conclusions In this wor, we show the interrelations among the aspects related to the connections properties and the structural design. The main aspects are: the properties of wood as a foundation, the individual connector elastic parameter (slip modulus), rotational stiffness and the connection as an element of the frame structure. Structures with semi-rigid connections may be solved with a matritial analysis, by the way, widely available inside finite element analysis commercial programs. Linear analysis may be reasonably used for the design of semi-rigid connections, according to a linear or multilinear behavior of connections. Anyway, the effect of connection deformation is significant, especially in moment distribution. The aspects that lac normalization are the embedment strength and stiffness, slip modulus for connections with members oriented in general directions in relation to direction of wood fibers; the rotational connection behavior; criteria for solving the structure (design of connections). Effects as second order effects and shear effects were not considered in the comparative analysis between rigid and semi-rigid connections. 8. Acnowledgements The authors than to FAPESP to financial support for this research (grant # 00/ ) and to LaMEM / EESC / USP for the infrastructure and human resources for the experimental program. 9. References [] EUROPEAN COMMITTEE FOR STANDARDIZATION, Bruxelas. EUROCODE 5; design of timber structures - part -: general rules and rules for buildings. Bruxelas, p. [] WILKINSON, T. L. Theoretical Lateral Resistance of Nailed Joints. Journal of Structural Division. v. 97, n. ST-5, p May 97. [3] KUENZI, E. W. Theoretical design of a nailed or bolted joint under lateral load. Madison: Forest Products Laboratory, Forest Service, United State Department of Agriculture, p. (Rep. D95). [4] FOSCHI, R. Load-Slip Characteristics of Nails. Wood Science. v. 7, n., p , 974. [5] SANTANA, C. L. O., MASCIA, N. T. Automated design of rotational connections in wooden structures. In: IBERO LATIN AMERICAN CONGRESS ON COMPUTATIONAL METHODS FOR ENGINEERING. Giulianova, Italy, 00. Proceedings. L Aquila, Italy, 00. [6] RACHER, P. Moment Resisting Connections. In: BLASS, H. J. et al. Timber Engineering -Step. Almere: Centrum Hout, 995. Lecture C/6. [7] SANTANA, C. L. O. Analysis of framed structures with semi-rigid connections. Campinas: State University of Campinas, p. (PhD Thesis). [8] ABNT (997). Rio de Janeiro. NBR790; projeto de estruturas de madeira (design of timber structures) p. [9] VALLE, A. Rigidez de ligações com parafusos em estruturas de madeira laminada colada (Stiffness of bolted connections in structures in GLULAM timber). Sao Paulo: USP, p. (PhD Thesis)