Aristotelis Charalampakis and Vlasis Koumousis National Technical University of Athens

1. A generic fiber model algorithm for the analysis ofarbitrary cross sections under biaxial bending and axial load Aristotelis Charalampakis and Vlasis Koumousis National Technical University of Athens Institute of Structural Analysis and Aseismic Research

2. NationalTechnicalUniversity ofAthens Introduction Arbitrary cross section under bending and axial load • Failure:Corresponds to top values of moment – curvature diagram • Conventional failure:Smaller values dictated by Codes

3. NationalTechnicalUniversity ofAthens Problem definition Task: Analysis of arbitrary cross sections under biaxial bending and axial load Using a “fiber model” based on the Bernoulli – Euler assumption: • Simple calculation of strains (plane sections remain plane)• Used in design Codes• Close agreement with experimental results for monotonic / proportional loading• Failure surface data can be used in plastic analysis (i.e. damage index)

4. NationalTechnicalUniversity ofAthens Generation of failure surface Three different techniques: • interaction curves for given bending moments ratio (blue) • interaction curves for given axial load N (magenta) • 3D interaction curves for assumed neutral axis direction (red)

5. NationalTechnicalUniversity ofAthens Cross Section Arbitrary Cross Section • consisted of polygons and circles • arcs are approximated by polygon chains to a specified accuracy

6. NationalTechnicalUniversity ofAthens Cross Section Materials forgraphical objects • “Foreground” and “Background” materials for each graphical object • Positive “Foreground” material stresses • Negative “Background” material stresses

7. NationalTechnicalUniversity ofAthens Cross Section For a hollow circular steel section:

8. NationalTechnicalUniversity ofAthens Materials Custom material data: • Stress - strain diagram composed of any number and any combination of consecutive parabolic or linear segments • Additional data: max or min strain, etc

9. NationalTechnicalUniversity ofAthens Calculations Rotation • Direction of neutral axis is assumed • Rotation: neutral axis is parallel to horizontal axis Y • Strains (and stresses) vary only in vertical axis Z

10. NationalTechnicalUniversity ofAthens Calculations Trapezoidal decomposition of polygons • Using “Plane Sweep” algorithm • Basic set of trapezoids calculated only once • Circles are treated separately

11. NationalTechnicalUniversity ofAthens Calculations Strains • Strains: • Map transition points of stress – strain diagrams of materials: • Extended set of trapezoids • Circular sections

12. NationalTechnicalUniversity ofAthens Calculations Stress resultants: Trapezoids

13. NationalTechnicalUniversity ofAthens Calculations Stress resultants: Circular Sections

14. NationalTechnicalUniversity ofAthens Calculations Moment – Curvature diagram • Pick N, Initial Curvature step • For zero curvature, find • In a loop:Add curvature step, find new material failure (max – min strain) check: custom restrictions (“Point C”) axial equilibrium if necessary decrease curvature step and retry

15. NationalTechnicalUniversity ofAthens Example 1 EC2 design charts

16. NationalTechnicalUniversity ofAthens Example 1 EC2 design charts • Rectangular cross section • Equal reinforcement, top and bottom • Steel grade S500 • d1/h = 0.10

17. NationalTechnicalUniversity ofAthens Example 2 Arbitrary cross section Analysis with MyBiAxial Cross section

18. NationalTechnicalUniversity ofAthens Example 2 Arbitrary cross section Interaction curve for N = -4120kN Complete failure surface

19. NationalTechnicalUniversity ofAthens Example 3 Bolted connection 3D view of proposed connection

20. NationalTechnicalUniversity ofAthens Example 3 Bolted connection Example 3 in MyBiAxial Stress solids in CAD software

21. NationalTechnicalUniversity ofAthens Example 4 Moment capacity of rigid footing

22. NationalTechnicalUniversity ofAthens Example 4 Moment capacity of rigid footing