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10 cumbersome to navigate in a regular ASCII text editor. To plastic penetration for corrugated metal and crack depth enhance the review of the output report, a topical browser for reinforced concrete. A local plot of the pipe is also pro- is included as part of CANDE-2007. This browser provides vided (see Figure 7). an organized, bookmarked table of contents for the output Results Generator. This is an interactive text report gen- file that provides quick access to any table in the report (see erator that features user-defined report selection to create Figure 5). dynamic output reports. Options are available for tabular- CANDE Log File. The log file is a short file that is displayed izing soil responses and pipe group responses as a function on the monitor screen during execution. It contains the of load step. master input selections along with a history list of each load step analyzed and a trace of iterations required to solve each load step. If the solution is unsuccessful, the log file New Analysis Capabilities also provides error messages and, when possible, guidance to correct the error. In addition to code restructuring, improvements were Mesh Plot. The mesh plot is an interactive plotting tool for made in every subroutine of the CANDE Engine, some creating and viewing the finite element mesh topology significant new analytical capabilities were added to CANDE- (Levels 2 and 3) including element numbering, nodal con- 2007 that did not exist in the previous versions of CANDE. nectivity, material zones, construction increments, and A companion document, CANDE-2007 Solution Methods boundary conditions. Likewise, the tool is used to create and Formulations, describes in detail the complete derivation and plot solution output such as deformed shapes and of these new capabilities developed in the course of this color contours of soil stresses and strains. For an overview research effort. Provided is a brief synopsis of the new analy- of the Mesh Plot options, see Figure 6. sis capabilities. Graphs. This is an interactive plotting tool for creating and viewing the structural response of beam-element groups, Multiple Pipe-Type Capability (i.e., pipe types). Structural responses are plotted contigu- ously over the pipe shape for any load step or sets of load A new architecture called "multiple pipe-type capability" steps. Structural responses include moments, thrusts, and is embedded in CANDE-2007 that removes the old restric- shears as well as responses specific to the pipe type such as tion of limiting the analysis to just one continuous pipe-type Figure 5. CANDE output viewer.

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11 Figure 6. CANDE mesh plot options. structure. The new strategy allows for multiple element through the soil stiffness. Alternatively, element groups may groups rather than just a single group. The connection be arbitrarily joined together at common nodes to model cell- between element groups is completely arbitrary and can be like structures or composite structures such as a corrugated defined by the user. For example, two groups may be assigned metal arch roof placed on a reinforced concrete base. The new independent node numbers (no nodes in common) so that architecture provides virtually unlimited modeling capabili- they become independent structures that only interact ties to define any configuration within a 2-D framework. The

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12 Bending moment(lb-in/in) Figure 7. CANDE Graphs window. implementation of the multiple pipe-type capability is oper- observed by noting the peak load at which the system fails due able in both the analysis and design mode. to large or unbounded displacements. In order to predict the buckling capacity at the end of each load increment, a lin- earized buckling prediction methodology is also incorporated Large Deformations and Buckling into the program based on determining the scalar multiple of The large deformation development is based on the so- the geometric stiffness that renders the determinant of the called "Updated-Lagrange" methodology, which includes the combined system stiffness matrix to be zero. standard small strain-displacement term plus a nonlinear term related to rotation. This strain-displacement expression Pipe-Type Design Criteria accurately represents the beam element's internal strain field and LRFD Methodology due to moderate stretching and large rotations. The word "up- dated" means that the nodal coordinates of all elements are The design criteria for all pipe types have been extensively updated to their displaced position after each load step instead revised in accordance with the AASHTO LRFD Specifications of the original undeformed position being used. Incorporat- (6). The design criteria are equally applicable to either work- ing the nonlinear component of the strain-displacement rela- ing stress or LRFD design methodologies wherein working tionship into a beam-column element results in a new matrix stress employs service loads and actual resistance, and LRFD called the geometric stiffness matrix, proportional to current employs factored loads and factored resistance. Listed below thrust level, and a new load vector called the rotational-stretch are the design criteria for the three major categories of pipe vector. This method of analysis produces a nonlinear load- materials with emphasis on the new design criteria developed deformation curve wherein the load causing instability may be under this research effort.

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13 Corrugated metal. Corrugated steel and corrugated alu- Almost all the typical profiles that are currently manufactured minum have the same design criteria, differing only by the can be represented by a subset of these subelements includ- numerical value of design criterion limits such as the yield ing ribbed profiles, T-ribbed profiles, unlined corrugations, stress. The strength design criteria include thrust stress lined corrugations, and box-like profiles. limits for material yielding, global buckling, and seam The key feature of the Profile wall-type option is that it in- strength. In addition, a new strength criterion is incorpo- cludes the nonlinear phenomenon of local buckling in accor- rated to warn against full plastic penetration from thrust dance with the local buckling equations specified in Section 12 and bending. Other new additions include revised tables of the AASHTO LRFD Specifications. That is, depending on for available steel and aluminum corrugation sizes and the level of the compressive strain, some or all of the subele- updated flexibility factors for handling considerations. ments may experience local buckling, which is simulated by Reinforced concrete. The strength-related design criteria removing a central portion of the subelement's area in pro- for reinforced concrete include yielding of the reinforcing portion to the degree of local buckling. Solutions are iterated steel, crushing of concrete, diagonal shear failure (three until convergence occurs. forms), and radial tension failure of concrete due to ten- sion steel. The theoretical development of the last criterion, Unification of Nonlinear Iteration Schemes which is not well addressed in the AASHTO LRFD Speci- fications, was developed in the course of this research Previous versions of CANDE employed individual itera- effort. Similarly, the Heger-McGrath prediction for crack tion counters and strategies for each nonlinear algorithm width, a service-related design criterion, is also a product (i.e., pipe material type, soil models, interface models, and of this research effort. now large-deformation models). CANDE-2007 unifies the Thermoplastic pipe. Thermoplastic pipe materials in- iteration schemes of all nonlinear models to be easily con- clude high-density polyethylene, polyvinylchloride, and trolled by the user by one command. Now, the user may select polypropylene. All thermoplastic pipe materials have the one iteration limit that governs all nonlinear models (soil, same general design criteria, differing only in pipe material pipes, interfaces, and large deformations). Also, the user now data and duration of loading. Strength-related design cri- has the option to continue or stop the analysis if the iteration teria include thrust stress limits for material failure and limit is exceeded. In either case, diagnostics of the nonlinear global buckling, and maximum outer fiber strain limits models are printed showing which model(s) did not converge from thrust and bending. Performance-related criteria along with a measure of the convergence error. include allowable displacements and tensile strain. Bandwidth Minimization A bandwidth minimizer has been installed in CANDE- Plastic Profile Wall 2007 that reduces the maximum bandwidth of the system The previous version of CANDE was limited to smooth equations characterizing the finite element mesh topology. walls for all plastic pipes. As part of this research effort, the The motivation for reducing the maximum bandwidth is to wall type is extended to include a variety of profile wall sec- reduce computer storage space for the system of equations tions defined by the length and thickness of subelements that and to increase the speed of solving large sets of equations. form a repeating profile shape along the length of the pipe. Bandwidth minimization is achieved by strategically renum- The user-specified subelements include two web elements bering the node numbers that were originally input by the and four horizontal subelements as shown in Figure 8. user. Node renumbering is accomplished by swapping node numbers, one pair at a time, starting with the node number with the highest bandwidth and exchanging it with another Period node number that optimally reduces the maximum band- width at the first location while generating a bandwidth at the Link (4) Crest (3) Link (4) second location that is less than the original maximum band- width. The algorithm (contained in subroutine BMIZER) Web Web was developed during this research effort and is unique to the Height CANDE-2007 program. In addition to the above new features, many additional Valley (1) Liner (2) Valley (1) modifications have been made to improve the program and Figure 8. Profile wall with user-defined ease the burden of user input. For example, a simplified subelements. method of defining interface angles is now operational in the