XTRACT: A Tool for Axial Force - Ultimate Curvature Interactions. C.B. Chadwell 1 and R.A. Imbsen 2

Transcription

1 XTRACT: A Tool for Axial Force - Ultimate Curvature Interactions C.B. Chadwell 1 and R.A. Imbsen 2 1 Assistant Professor, Department of Civil and Environmental Engineering, California Polytechnic State University, San Luis Obispo, CA 93407: chadwell@calpoly.edu. 2 President, Imbsen & Associates, Inc., 9912 Business Park Drive, Suite 130, Sacramento, CA 95827: raimbsen@imbsen.com. Abstract XTRACT (Chadwell and Imbsen, 2002) started as an academic and research tool at the University of California at Berkeley as a program titled UCFyber. At its infancy, UCFyber was the first interactive Microsoft Windows based program that performed moment curvature analysis for reinforce concrete cross sections. The program had the capability of performing realistic analysis of cross sections incorporating the effects of increased strength and ductility of confined concrete as well as nonlinear steel behavior within a graphical environment. While XTRACT has become an invaluable instrument for concrete research within earthquake engineering, it has also evolved to become a production tool for analysis and design of concrete systems within design offices around the World. XTRACT is an important tool for earthquake engineering analysis when a realistic assessment of moment and curvature capacities of a cross section is required. XTRACT has been used on many high profile projects for both buildings and bridges ranging from analysis of the existing columns for the Salt Lake City, City Hall seismic upgrade project to analysis of the suspension bridge columns on the new Caquenez Straits bridge in the San Francisco Bay Area. In addition, XTRACT is currently being used in the design of the temporary bypass structure for the new San Francisco-Oakland Bay Bridge as well as in numerous state transportation agencies across the country. This paper discusses some of the basic analytical features within XTRACT and introduces some innovative uses of the program for seismic assessment of reinforced concrete columns. During seismic excitation of concrete moment resisting frames axial forces vary due to overturning demands. This, in turn, affects the ultimate curvature capacity and consequentially, the seismic displacement capacity of the concrete columns. By generating a plot of axial force verses ultimate curvature, curvature demands can be checked directly for a concrete column within a seismic force resisting frame. This paper outlines the methodology behind the creation of this type of diagram and includes example diagrams for both unconfined and confined rectangular concrete cross sections. 1

2 Moment Curvature Analysis Methodology in XTRACT Moment curvature analysis is often used in earthquake engineering as a necessary step toward assessment of the displacement capacity of reinforced concrete components. A moment curvature analysis establishes the ductile capacity of a cross section by plotting the curvatures against corresponding moments. Material Models. Analysis begins with the specification of nonlinear material models as stress-strain diagrams. For reinforced concrete, three typical material models must be defined: steel, unconfined concrete, and confined concrete. The stress-strain behavior of steel depends on the material type and strength but generally can be described as a three part relation: linear, constant zero slope (the yield plateau), and a strain hardening branch. Numerous models for confined and unconfined concrete have been proposed by many researchers (Kent and Park, 1971; Vallenas et al, 1979; and Sheikh and Uzumeri, 1980). Confined concrete mathematical models incorporate effects of increased compressive strain capacity in addition to an increased compressive strength as a function of passive confinement from transverse reinforcing steel. One commonly used model in moment curvature analysis is the Mander Model proposed by Mander et al (1988). The Mander model accounts for the effects of confinement by variation of the input parameters such that the same mathematical formulation can be used to describe unconfined, lightly confined, and heavily confined concrete behavior. Discretization. With the material models defined, the cross section must be cut into a series of layers, if moments about one axis are considered, or fibers if moments about two axes are considered. Tighter mesh sizes will give more accurate results within the confines of the material models but at the cost of increased computation time. When a loose (or course) mesh is used, unconservative or incorrect results may ensue. Each fiber (or layer) within the discretized cross section is associated with a tag identifying it with a specific material type as defined by the material model. Analytical Methods. With the materials defined and mapped to the fibers (layers) of the cross section, an applied axial load is specified along with the analytical method for finding the moment curvature coordinate pairs. There are two general methods used in moment curvature analysis: displacement control and force control. Displacement Control. In displacement control, the curvatures (φ x, φ y ) are imposed (about the x and/or y axes) and corresponding moments are found. By imposing a curvature, and knowing the centroid of each individual fiber, the strain can be found for every fiber in the cross section with the assumption that plane sections remain plane. The strain defined relative to the strain at the (0,0) coordinate (ε a ). It is often convenient to set the (0,0) coordinate at the centroid of the cross section as shown in Figure 1 to avoid eccentricity induced by an applied axial force. Given the strain in each fiber, and knowing the stress-strain relations from the material models, the corresponding stress in the fiber can be found. The fiber stress multiplied by the fiber area gives the force in the fiber (F i ). From equilibrium considerations, the overall axial force (P) and corre- 2

3 y ε i ( x i, y i ) = ε a φ y x i + φ x y i Figure 1. Strain Distribution in the Discretized Cross Section. sponding moments about the X and Y axes (M x and M y ) can be found as: P = F i M = x y i F M i = x F y i i i Because the total axial force found from summing the individual forces within each fiber is not necessarily equal to the applied axial load on the cross section, an iteration between the applied axial load and resisting axial load (P) is done by changing the strain at the centroid. Once the applied axial force and resisting axial force matches to within a defined tolerance, the curvature is incremented and the process is repeated. Termination of the incrementing curvatures occurs when a desired limit state is reached within the material. The ultimate limit state is defined for reinforced concrete cross sections as the curvature at which either fracture of the longitudinal reinforcement or crushing of the confined concrete occurs as defined by the respective material models. Force Control. In force control, the forces (or moments) are incremented until a material limit state is reached. In force control, there is iteration within each analysis step using the tangent cross section stiffness in conjunction with a Newton-Raphson type iteration. XTRACT provides two options for force control, iteration at a minimum unbalanced displacement normal and iteration at a constant arc length (Clarke and Hancock, 1990). In a force control solution strategy, the inverse problem is solved. With this type of solution, problems with convergence can occur when there are severe discontinuities defined within the material models. Once the series of moment curvature coordinate pairs have been calculated for a target axial load, using either displacement or force control, a plot of these points is referred to as a moment curvature diagram. Each moment curvature analysis is performed for a target axial force. The results reveal the ultimate curvature that corresponds to the specific axial force as the last point in the analysis. A series of moment curvature analyses is typically performed to consider a range of axial force-ultimate curvature pairs. i x i i th Fiber in the Discretized Cross Section Geometric Centroid 3

4 Axial Force-Moment Interaction Diagrams In addition to moment curvature analyses, XTRACT can perform axial force-moment interaction diagrams. Using the axial force-moment interaction type analysis, an axial force-ultimate curvature surface can be found directly. In XTRACT there is an option for finding axial force-moment interaction surfaces at target strains. By allowing the input of target strains, XTRACT provides the user with the capability to generate interaction surfaces for various limit states: i.e. first material yield, unconfined concrete spalling, etc. The target strains are defined as the limits of analysis such that for each coordinate pair of axial force and moment, at least one of the limit strains will be reached but none will be exceeded. Typical Axial Force-Moment Interaction Surface. An axial force-moment interaction surface for a typical rectangular reinforced concrete cross section is given in Figure 2. For this analysis the target concrete strain was and the target steel strain Figure 2. Typical PM Interaction Diagram. was specified as the strain at the onset of strain hardening (.008). The change of the surface from being controlled by the concrete in compression to being controlled by the steel in tension is the so called balanced point. Analysis Methodology. Internally, XTRACT loops through all fibers and reinforcing steel bars identifying critical fibers (and/or bars) for each material. The critical fibers (or bars) are defined as fibers (or bars) that are the greatest distance away from the centroid in the four quadrants relative to the geometric centroid - (+x, +y), (-x, +y), (-x, - y) and (+x, -y). Finding the bounds between pure tension and pure compression, the top and bottom points of the interaction surface and the range of centroidal strains are identified. The range of strains is divided by the number of user defined points on the interaction surface. For each centroid strain, curvatures are found associated for each of the critical fibers. Using the minimum curvature, a corresponding moment is calcu- 4

5 lated. The axial force associated with the particular strain diagram along with the corresponding moment becomes the axial force-moment coordinate pair where a limiting strain has been reached. Figure 3 depicts a schematic of varying centroidal strains with Varying Imposed Strain Profiles each Resulting in Corresponding Moments and Curvature ε a = Varying Strain at the Centroid Concrete Cross Section ε = Tensile Strain Capacity in the Steel - Pure Tension Strain Profile Strain Profile for the Balanced Point ε = 0 ε = Compression Strain Capacity of the Concrete - Pure Compression Strain Profile Figure 3. Consecutive Strain Diagrams for varying Centroidal Strain. corresponding strain profiles when considering the axial force-moment interaction surface for bending about the y-axis of a symmetric cross section. The strain profile where both the tension and compression limit is reached simultaneously is, by definition, the balanced point of the axial force-moment interaction surface. The angle of each strain profile is the curvature specific to the strain profile, moment, and axial force. The limiting strains need not be specified solely for two materials (steel and concrete) as is typically done when generating an axial force-moment interaction surface; but rather, any number of materials, at any location, with specified limiting strains can be defined. This will result in a surface defined by multiple governing material limit states. Axial Force-Ultimate Curvature Diagram. Using the approach of specifying the limiting strains, an axial force-ultimate curvature diagram can be generated. By specifying ultimate strain values for each material model within the cross section, XTRACT generates an axial force-moment surface where the limiting strain values are reached but not exceeded. This surface will have a slight variation from the nominal axial forcemoment interaction surface typical of reinforced concrete. However, the output data will contain moments and curvatures that correspond to this ultimate limit state as defined by the limiting strains. By creating the axial force-ultimate curvature diagram using an interaction surface, there is no iteration which is required when performing a series of moment curvature 5

6 analyses to construct this diagram. Furthermore, it is of interest to note that because the strain diagram is being varied, the solution strategy is essentially displacement controlled without iteration. Example To demonstrate, two rectangular sections are created with the same cross section and longitudinal reinforcement layout. The sections are 76.2cm (30in) by 76.2cm (30in) with 12 - φ25mm longitudinal reinforcing bars of A615 Grade 60 steel. One of the cross sections is transversely reinforced with φ13mm reinforcing bars at 10.2cm (4in) on center (Figure 4). The other section has no transverse reinforcement. Figure 4. Rectangular Example Section. Confined Concrete Core Unconfined Concrete Cover The unconfined confined concrete model assumes a 27.6MPa (4,000psi) 28-day concrete strength used for both cross sections. The confined concrete model, used for the confined section, and the longitudinal steel reinforcing model, used for both sections, are shown in Figure 5. The confined concrete model has been reduced to incorporate Figure 5. Confined Concrete and Longitudinal Reinforcing Steel Models. 6

7 the effects of concrete arching action (Paulay and Priestley, 1992). Two analyses are performed: one with material yield strains set as target strains and the other with ultimate strains set as target strains. Results. Figure 6 shows the axial force-yield curvature and the axial force-ultimate Yield Surface Axial Force (kn) Ultimate Surface Curvature (1/m) Figure 6. Axial Force verses Curvature Diagram for the Confined Concrete Section. curvature diagrams found from analysis with the confined cross section. Figure 7 Yield Surface Axial Force (kn) Ultimate Surface Curvature (1/m) Figure 7. Axial Force verses Curvature Diagram for the Unconfined Concrete Section. shows results from analysis with the unconfined concrete section. For the confined concrete cross section, the maximum curvature capacity occurs at low levels of axial compression; for the unconfined concrete section, the maximum ultimate curvature capacity occurs when the section is in tension. Figure 8 shows that both ultimate surfaces exhibit the same rapid drop in ultimate curvature with increasing axial load, however, with the unconfined cross section, this occurs at a lower value of axial load. This result is not unexpected. 7

8 20000 Axial Force (kn) Ultimate Unconfined Surface Ultimate Confined Surface Figure 8. Comparison of Ultimate Axial Force-Curvature Surfaces for Unconfined and Confined Concrete. Curvature (1/m) The alternative to calculation of the axial force-ultimate curvature relation, as described above, is to perform a sequence of moment curvature analyses with differing axial forces. To demonstrate this, four moment curvature analyses are performed with four different axial forces applied to the confined concrete cross section shown in Figure 4: 500kN, 2000kN, 4000kN, and 6000kN. Figure 9 shows the four moment curvature Curvature (1/m) Ultimate Surface Moment (kn-m) Moment Curvature Analyses with 500kN, 2000kN, 4000kN, and 6000kN. Figure 9. Moment Curvature Diagrams for Various Axial Loads Plotted with Moment and Curvature Coordinate Pairs from the Axial Force-Ultimate Curvature Interaction Surface. analysis results plotted with moment curvature data output from the axial force-ultimate curvature surfaces. It is clear from Figure 9 that the axial force-ultimate curvature surface calculated with an axial force-moment interaction analysis, constructed with ultimate strains, does result in the desired failure envelope. 8

9 Conclusions As earthquake engineering design is turning to a displacement based design methodology, realistic assessment of the nonlinear behavior of systems is necessary. A highly important limit state for reinforced concrete components subject to inelastic deformation demands is the ultimate curvature. It is the ultimate curvature, in part, that determines the ultimate rotations as well as the ultimate displacement capacity. To determine this value for different axial loads, a series of moment curvature analyses are typically performed and the last point calculated within a moment curvature relation is taken as the ultimate curvature corresponding that particular axial load. Using the axial force-interaction analysis in XTRACT with target strains set to ultimate values, an axial force-ultimate curvature diagram can be constructed. This diagram shows the variation in axial load with corresponding curvatures but the analysis is performed without iteration. Rapid construction of this curve can be executed in XTRACT for determination of ultimate curvature capacities; which in turn, can be compared to curvature demands resulting from seismic excitation. References Chadwell, C.B., Imbsen & Associates, (2002), "XTRACT - Cross Section Analysis Software for Structural and Earthquake Engineering". xtract.htm Clarke and Hancock (1990), "A Study of Incremental-Iterative Strategies for Non-Linear Analysis", International Journal of Numerical Methods in Engineering, Vol. 29, Kent, D.C., and Park, R. (1971), "Flexural Members with Confined Concrete," Proceeding ASCE, Vol. 97, No. ST7, July 1971: Mander, J.B., Priestley, M.J.N., and Park, R. (1988), "Observed Stress-Strain Behavior of Confined Concrete," Journal of Structural Engineering, ASCE, Vol. 114, No. 8, Aug. 1988: Paulay, T., and Priestley, M.J.N. (1992), Seismic Design of Reinforced Concrete and Masonry Buildings, John Wiley & Sons, New York. Sheikh, S.A., and Uzumeri, S.M. (1980), "Strength and Ductility of Confined Concrete Columns, " Proceedings ASCE, Vol. 106, No. ST5, May 1980: Vallenas, J.M., Bertero, V.V., and Popov, E.P. (1979), Hysteretic Behavior of Reinforced Concrete Structural Walls, Report UCB/EERC-79/20, Earthquake Engineering Research Center, University of California, Berkeley, Aug