A generic fiber model algorithm for the analysis of arbitrary 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

Introduction Arbitrary cross section under bending and axial load National Technical University of Athens 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

National Technical University of Athens 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)

Generation of failure surface National Technical University of Athens 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)

Cross Section Arbitrary Cross Section National Technical University of Athens Cross Section Arbitrary Cross Section consisted of polygons and circles arcs are approximated by polygon chains to a specified accuracy

Cross Section Materials for graphical objects National Technical University of Athens Cross Section Materials for graphical objects “Foreground” and “Background” materials for each graphical object Positive “Foreground” material stresses Negative “Background” material stresses

Cross Section For a hollow circular steel section: Outer Circle National Technical University of Athens Cross Section Outer Circle Inner Circle Foreground (+) steel (+) none Background (-) none (-) steel For a hollow circular steel section:

Materials Custom material data: National Technical University of Athens 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

Calculations Rotation Direction of neutral axis is assumed National Technical University of Athens 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

Calculations Trapezoidal decomposition of polygons National Technical University of Athens Calculations Trapezoidal decomposition of polygons Using “Plane Sweep” algorithm Basic set of trapezoids calculated only once Circles are treated separately

Calculations Strains Circular sections Strains: National Technical University of Athens Calculations Strains Strains: Map transition points of stress – strain diagrams of materials: Extended set of trapezoids Circular sections

Calculations Stress resultants: Trapezoids National Technical University of Athens Calculations Stress resultants: Trapezoids

Calculations Stress resultants: Circular Sections National Technical University of Athens Calculations Stress resultants: Circular Sections

Calculations Moment – Curvature diagram Pick N, Initial Curvature step National Technical University of Athens 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

National Technical University of Athens Example 1 EC2 design charts

Example 1 EC2 design charts Equal reinforcement, top and bottom National Technical University of Athens Example 1 EC2 design charts Rectangular cross section Equal reinforcement, top and bottom Steel grade S500 d1/h = 0.10

Analysis with MyBiAxial National Technical University of Athens Example 2 Arbitrary cross section Cross section Analysis with MyBiAxial

Example 2 Arbitrary cross section Interaction curve for N = -4120kN National Technical University of Athens Example 2 Arbitrary cross section Interaction curve for N = -4120kN Complete failure surface

3D view of proposed connection National Technical University of Athens Example 3 Bolted connection 3D view of proposed connection

Stress solids in CAD software National Technical University of Athens Example 3 Bolted connection Example 3 in MyBiAxial Stress solids in CAD software

Example 4 Moment capacity of rigid footing National Technical University of Athens Example 4 Moment capacity of rigid footing

Example 4 Moment capacity of rigid footing National Technical University of Athens Example 4 Moment capacity of rigid footing