Inverse reliability applications and performance-based design in timber engineering

Transcription

1 Inverse reliability applications and performance-based design in timber engineering Ricardo O. Foschi 1 and Hong Li 2 ABSTRACT Inverse reliability methods allow the direct determination of design parameters when target reliability levels are specified for each of the limit states considered in the design. The application of these methods is then at the core of performancebased design. This paper illustrates the application to two design problems: 1) the calculation of a floor span so that a target reliability is achieved in a vibration performance criterion; and 2) the mean transverse load is found for either a roof or a wall structural insulated panel, also in correspondence to a specified target reliability. The advantages and implications of these methods are discussed. INTRODUCTION The calculation of the reliability of a structural system for a certain limit state requires first the definition of that state in terms of a capacity C and a demand D. These quantities are, in general, functions of several random variables {X D }and {X C } associated, respectively, with D and C. In addition, C and D may be functions of design parameters {d C } or {d D }, which may be deterministic values controlling the design (for example, the depth of a beam, the number of connectors per unit length, etc.). The design parameters can also be the distribution parameters (mean, standard deviation) of some of the random variables {X}. The limit state is then described in terms of a performance function G, G = C X, d ) D( X, d ) [1] ( C C D D so that the probability of failure in the limit state corresponds to the probability of the event G<0. Correspondingly, the reliability of the design will be the probability of the event G>0. This probability may be obtained by simulation or by the application of approximate methods like FORM or SORM, and sophisticated software has been developed for this purpose. Among similar others, the computer program RELAN developed at the Civil Engineering department of the University of British Columbia implements the reliability calculation by FORM, SORM, standard or reduced variance simulation techniques (Foschi et al., 1999a). RELAN is a forward reliability evaluation tool. Given a target probability of failure or, correspondingly, a target reliability index β, a program such as RELAN could be used to assess the reliability of the design for a given set of design parameters. The process can then be repeated, varying the design parameters until the reliability achieved matches with the required target. This trial and error procedure is feasible, but rather inconvenient for the purposes of design. In an inverse problem, the target reliability is prescribed and the corresponding values of the design parameters are directly obtained, without the iterations implied by the forward procedure. The inverse problem is much less robust than the forward one, particularly for the case of several design parameters and several reliability constraints or performance criteria, since no solution may exist satisfying exactly the prescribed reliabilities for each limit state. Algorithms for the inverse problem have been recently proposed: for a detailed description of one such algorithm see Li and Foschi, (1998). This has been implemented in the software IRELAN (Foschi and Li, 1999b). Like RELAN, IRELAN requires the description of the performance functions G for each of the limit states of interest. In addition, IRELAN requires the specification of the desired target reliability levels for each of the limit states. If the number of design parameters exceeds the number of functions G, the problem is not well posed and has an infinity of solutions, unless it is complemented with an optimization, for example, for minimum cost. If the number of design parameters is less than or equal to the number of performance 1 Professor, Dept. of Civil Engineering, University of British Columbia, Vancouver, B.C. Canada V6T 1Z4 2 Research Engineer, Powertech Labs. Inc., Surrey, B.C.

2 functions (or reliability constraints) G, then the problem may have an unique solution for the design parameters or these may be adjusted to achieve the minimum deviation from the desired target reliabilities. The application of inverse reliability software allows for a straight-forward implementation of performance-based design. When complemented with the advantages of Windows-like software, programs like IRELAN can become standard engineering design tools, both for practicioners as well as for manufacturers of wood-based structural products. Engineers could directly design to the standard for performance or reliability prescribed by the performance-based Codes, and manufacturers could study and optimize their product to best meet the performance requirements of different applications and markets. The description of the limit states, through the performance functions G, could incorporate calls to other programs for the analysis and calculation of either the demand D or the capacity C. Thus, for the study of the reliability of wood floors, a finite strip floor analysis like FAP (Foschi, 1982) could be used for the structural model. Similarly, a finite element analysis of stressed-skin, structural insulated panels, could be used for the evaluation of roof or wall panelized construction. It is of course advantageous if the structural analyses are fast and efficient, since their linking to the reliability assessment software implies repeated calls to programs like FAP. As an alternative, an efficient approach can also be implemented: the structural analysis can be run first for a range of values of the design parameters and random variables, obtaining the corresponding response values of interest. With these, a response surface is constructed in terms of those variables and parameters and, finally, this response surface is called by the inverse reliability algorithm. While the development of the response surface may be relatively slow in some cases, its implementation in the inverse reliability procedure is very fast and permits very quick performance-based design. CASE STUDIES 1. Calculating a wood floor span for acceptable vibration performance Consider a wood floor with joists (beams) and nailed (or glued) subfloor plywood sheathing. The floor is simply supported on all four sides. For a given joist type (lumber or engineered I-Joists), for given sheathing and nailing characteristics, and for given joist spacing, the floor span L can be calculated to satisfy several design criteria. These usually include: 1) that the maximum bending stress in the joists does not exceed a tolerable level for the combination of a permanent and a uniformly distributed design live load, 2) that the static deflection under the same combination of loads does not exceed a fraction of the span (L/K), and 3) that the floor does not have vibration characteristics which are deemed unacceptable by the occupants. The limiting condition for floor spans is normally the vibration requirement, and static deflection limits like L/360 have been shown not to lead to floors with adequate vibration performance (in fact, floor spans recommended by manufacturers of engineered I-joists usually utilize criteria with at least L/480). The problem of floor vibration in the context of performance reliability has been studied by Foschi et al. (1995). In Canada, the National Building Code has accepted a criterion based on the static deflection of the floor under a concentrated load of 1 kn at midfloor, limiting that deflection to an absolute limit in mm which is a function of the span L (Onysko, 1986). The floor to which the criterion is applied is the average floor which could be built with the population of joists used. Using this limiting deflection D LIM (L) as the capacity C, and the demand D being the deflection produced by the 1kN load, calculated with the floor program FAP, the performance function G is written as G DLIM ( L) ( L, X ) = [2] where L is the floor span (the design parameter, in this case) and {X} is the vector of the random variables involved (including the moduli of elasticity of each of the joists, and the shear stiffness of the connections between the joists and the sheathing). Table 1 shows the calculated span L for two types of floors and two target reliability levels, expressed in terms of reliability indeces β. In the first floor, the joists are 38mm x 240mm dimension lumber (Spruce-Pine-Fir, No.2 grade), spaced at 400mm, with 15.9mm plywood sheathing (face grain perpendicular to the joists), nailed and glued to the lumber joists. The second floor has 240mm-deep engineered I-Joists, with 38mm x 89mm flanges and 12.5mm Oriented Strand Board web, also spaced at 400mm. Sheathing and connectors characteristics are the same as for the first floor. The criterion in the Canadian building code has been calibrated to results from occupant s responses to a survey, in such a way that an acceptable rating corresponds to the average floor construction. In the context of the reliability calculation, the acceptable floor should then correspond to a reliability level β = 0.0. The criterion has been calibrated in such a way

3 that acceptance means satisfactory performance for approximately slightly more than 50% of the population of occupants. Table 1 also shows the spans corresponding to a reliability level β = 1.645, for which, instead of just the average (or approximately 50% of the floors) being acceptable, 95% of the possible floors with the calculated span will pass the standard. The program FAP was linked with IRELAN to produce a direct, efficient calculation of the span L at the different desired reliabilities. The floors were analyzed with 12 joists, introducing then a total of 12 random variables associated with the corresponding moduli of elasticity. Since the program FAP is a complete and versatile structural analysis, its coupling with IRELAN also permits the calculation of the span L when the floor construction is altered, by changing the sheathing or the nailing characteristics, or modifying the statistics for the joists, or by introducing load-sharing bridging units between them. The approach can also be used to study the difference in spans, at the same target reliability, should the acceptance criterion be changed. An evaluation of different vibration criteria, using inverse reliability methods, has been presented in Foschi et al. (1999c), considering criteria based on floor frequency, rms of acceleration under an impact, or a criterion for human acceptance of vertical vibrations (Wiss and Parmelee, 1974). Table 1. Floor spans for vibration acceptance (National Building Code of Canada criterion) at specified reliability levels. Floor Type Lumber Joists (38mm x 240mm), glued, 400mm spacing, 15.9mm plywood sheathing. Engineered I-Joists (240mm), glued, 400mm spacing, 15.9mm plywood sheathing. Floor Span L (m) β = 0.0 β = This approach has already been extensively used by industry (Karalic, 2000) in producing performance-based span tables for floors using either conventional lumber or I-Joists, with or without load-sharing units between them. These spans tables were computed to provide one of two chosen target reliabilities for vibration serviceability performance, as done in Table Calculating the mean design transverse load for a Structural Insulated Panel A structural insulated panel is a stressed-skin construction utilizing a frame, two skins and rigid foam insulation material in the internal cavity. The assembly is glued and nailed, and panels can be manufactured to join with others in forming units for roofs or walls, Figure 1. The analysis of such a structure is complex and must be carried out with a finite element analysis, taking into account the nonlinear plate behavior of the thin skins (buckling or p- amplifications) and the nonlinear behavior of the rigid foam in compression, tension and shear. A computer program, PANEL, has been developed in the Civil Engineering department of the University of British Columbia for such an analysis. For wall units, receiving simultaneous vertical and transverse (wind) loads, the design parameter is the mean value of the transverse load that could be applied to a given panel configuration. Both vertical and transverse loads are considered random, with the mean value of the transverse load being calculated by IRELAN. For roof units, only transverse load is applied. Again, the design parameter is the mean value of such a load.

4 Vertical Load Racking Load Transverse Load Cover Frame Wall Rigid Insulation Shear Wall (racking) Roof Panel Wall (vertical and transverse loads) Figure 1. Structural Insulated Panel and Loading Applications Given the complexity of the structural analysis and the number of variables involved (skin and frame properties, shear stiffness of the nail/glue connections, properties of the rigid insulation), it is more efficient to develop a response surface for each of the desired outputs: the maximum overall panel deflection, the maximum skin deflection between frame members, the maximum compression strain in the insulation, the maximum bending stresses in the frame members and in each of the skins, the maximum load predicted from shear failure of the insulation. Working with such response surfaces, the design parameter (the mean of the transverse load) can be very efficiently calculated by IRELAN for each of the limit states, for the prescribed reliabilities, adopting the lowest calculated load. A sample calculation is now shown for a roof and for a wall panel. For the roof, the panel length L is 4.27m, with a cavity depth of 184mm, with lumber framing members 38mm x 184mm spaced at S = 1.22m, skins of 11.1mm Oriented Strand Board. A rigid insulation, with a density of 0.15 kn/m 3, fills the cavity. For the wall, the height of the panel is L = 2.44m, and the framing members are 38mm x 89mm spaced at S = 1.22m, with skins of 11.1mm Oriented Strand Board. The 89mm cavity is filled with the same rigid insulation as for the roof panel. A total of 14 random variables were considered for the roof panels, and 15 were involved in the wall. The transverse load was assumed to have a coefficient of variation of 0.20, and to be distributed according to an Extreme Type I or Gumbel distribution. For walls, the vertical load was applied, with a mean of 29.2 kn/m and a coefficient of variation of 0.05, and was also taken Gumbel-distributed. Other random variables included the modulus of elasticity of the framing members, the plate characteristics of the skins, and the range of material properties for the rigid insulation. The design criteria included: 1) the overall panel deflection, with a limit of L/360; 2) the skin deflection between frame members, with a limit S/600; 3) the maximum compression strain in the rigid insulation, with a limit of (to remain within the elastic range); 4) maximum stresses in the frame and 5) maximum stresses in the skins. The target reliability levels were β=1.645 for deflection limit states, and β= for strain or strength limit states. Since there is only one

5 design parameter, only one limit state could achieve the corresponding target reliability, but it was checked that the targets were exceeded in all other limit states. It was found that the dominant mode was always the overall deflection limit L/360, with a β= The results, including the mean transverse maximum load over a 30 year window, and the design 30 year-return load, are shown in Table 2. Table 2. Structural Insulated Panel, calculated design transverse loads at specified reliabilities. Panel Type Transverse load, mean max. over 30 years (kpa) Transverse load, 30 year return Design Load (kpa) Wall Roof CONCLUSIONS Inverse reliability methods permit the direct and efficient calculation of design parameters satisfying prescribed reliabilities or performance levels in different limit states. Thus, they offer a tool for performance-based design, when the performance level is specified as a target to be satisfied by the designer. However, these methods do not just offer a very valuable tool for the engineer. They can complement very effectively the development of engineered wood products, and the corresponding manufacturing process, by permitting to tailor the product or to evaluate manufacturing quality controls which will result in the required performance. Although it could be very well argued that, in the future, these methods, allowing direct performance-based design and the implementation of sophisticated structural analysis tools, could make superfluous today s restrictive code procedures, their main advantage may reside in applications to the performancetargeted manufacturing of valued-added wood products. In this context, the methods offer the manufacturer the possibility to assess and optimize the use of different resources, different manufacturing procedures and controls, vis-à-vis reliable performance demanded by the users in the marketplace. REFERENCES Foschi, R. O., Folz B., and Yao, F. 1999a. RELAN: Reliability Analysis, An User s Manual. Department of Civil Engineering, University of British Columbia, Vancouver, B.C. Foschi, R.O., and Li, H. 1999b. IRELAN: Software for Inverse Reliability Analysis, Department of Civil Engineering, University of British Columbia, Vancouver, B.C. Foschi, R.O., Yao, F., Li, H., and Karalic, M. 1999c. Inverse Reliability Methods in the Design of Wood Structures, Proceedings, ASCE Structures Congress, New Orleans, Louisiana, April Foschi, R.O Structural Analysis of Wood Floor Systems, ASCE Journal of the Structural Division, 108: Foschi, R.O., Neumann, G., Yao, F. and Folz, B Floor Vibration due to Occupants and Reliability-Based Design Guidelines. Canadian J. of Civil Engineering, 22: Karalic, M Assessment of Reliability Levels of Performance of Wood Floors Achieved with Different Design Approaches. Proceedings, World Timber Engineering Conference, Whistler, B.C. Canada. Li, H. and Foschi, R.O An Inverse Reliability Method and Its Applications. Structural Safety, 20: Onysko, D Serviceability Criteria for Residential Floors Based on a Field Study of Consumer Response, Forintek Canada Corp., Vancouver, B.C. Canada. Wiss, J.F. and Parmalee, R.A Human Perception of Transient Vibrations, ASCE Journal of the Structural Division, 100: