Ultimate Strength Analysis of Arbitrary Cross Sections under Biaxial Bending and Axial Load by Fiber Model and Curvilinear Polygons Aristotelis Charalampakis and Vlasis Koumousis National Technical University of Athens Institute of Structural Analysis and Aseismic Research

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 Moment – curvature diagram, interaction curves and failure surfaces can be used in non-linear analyses

Cross Section Arbitrary Cross Section National Technical University of Athens Cross Section Arbitrary Cross Section defined by curvilinear polygons (i.e. polygons with edges that may be straight lines or arcs) the polygons can be nested to any depth

Materials Custom material data Additional data: max or min strain, etc National Technical University of Athens Materials Custom material data Stress - strain diagram composed of any number and any combination of consecutive linear, parabolic or cubic segments Additional data: max or min strain, etc

National Technical University of Athens Materials Materials “Foreground” and “Background” materials for each curvilinear polygon. Positive “Foreground” / Negative “Background” material stresses

Calculations Deformed plane Angle θ Curvature k Strain ε0 National Technical University of Athens Calculations Deformed plane Angle θ Curvature k Strain ε0

Calculations Neutral axis is parallel to horizontal axis Y: National Technical University of Athens Calculations Direction of neutral axis (angle θ) is imposed by the algorithm By rotating the cross section by this angle θ, strains (and stresses) vary only in vertical axis Z: Neutral axis is parallel to horizontal axis Y:

Calculations Trapezoidal decomposition of curvilinear polygons National Technical University of Athens Calculations Trapezoidal decomposition of curvilinear polygons Basic set of curvilinear trapezoids calculated only once per angle θ

Calculations Basic integrals National Technical University of Athens Calculations Basic integrals Basic integrals calculated only once per curvilinear trapezoid

Calculations Stress resultants for a curvilinear trapezoid National Technical University of Athens Calculations Stress resultants for a curvilinear trapezoid Cubic stress – strain segment (αi material constants). Strain distribution in current deformed plane (θ fixed). Basic integrals Ij(m,n) have already been calculated. Independent of k, ε0. Summation over the trapezoids produces the overall stress resultants.

Moment – Curvature diagram (1) National Technical University of Athens Moment – Curvature diagram (1) Initialization Pick neutral axis direction (angle θ) Pick initial curvature step Rotate cross section Decompose into curvilinear trapezoids Calculate basic integrals Ij(m,n)

Moment – Curvature diagram (2) National Technical University of Athens Moment – Curvature diagram (2) Calculate initial ε0 for axial equilibrium with no curvature

Moment – Curvature diagram (3) National Technical University of Athens Moment – Curvature diagram (3) Neutral Axis Δk In a loop, apply a small increase in curvature...

Moment – Curvature diagram (4) National Technical University of Athens Moment – Curvature diagram (4) Neutral Axis … and find new ε0 using Van Wijngaarden – Dekker – Brent method

Moment – Curvature diagram (5) National Technical University of Athens Moment – Curvature diagram (5) Decrease curvature step when necessary (close to failure) Continue up to failure to produce the full moment – curvature diagram

National Technical University of Athens Interaction curve The interaction curve is produced by repeating the previous procedure for different directions of neutral axis (angle θ)

Calculation of Deformed Plane National Technical University of Athens Calculation of Deformed Plane Calculation of deformed configuration of a cross section under given external loads N, MYc, MZc Trial-and-error procedure All paths of the analyses stem from (M0Yc, M0Zc ) (Bending moments for axial equilibrium with no curvature)

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 Calculations for : ω = 0.00, 0.50, 1.00, 1.50, 2.00 and various axial load levels

Example 2 Analysis with MyBiAxial 2.0 Arbitrary cross section National Technical University of Athens Example 2 Arbitrary cross section Analysis with MyBiAxial 2.0

Example 2 Interaction curve for N = -4120kN Complete failure surface National Technical University of Athens Example 2 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 2.0 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

Conclusions Custom material data National Technical University of Athens Conclusions Generic algorithm for the analysis of arbitrary cross sections under biaxial bending and axial load Features: Custom material data Curved graphical objects with analytical expressions instead of approximations with simple polygons or dimensionless fibers for the reinforcement bars Construction of full bending moment – curvature diagram Fast, very stable algorithm Can be used for a variety of purposes as demonstrated in the examples