1. 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

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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

3. Cross Section Arbitrary Cross Section
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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

4. Materials Custom material data Additional data: max or min strain, etc
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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

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Materials
Materials
“Foreground” and “Background” materials for each curvilinear polygon. Positive “Foreground” / Negative “Background” material stresses

6. Calculations Deformed plane Angle θ Curvature k Strain ε0
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Calculations
Deformed plane
Angle θ
Curvature k
Strain ε0

7. Calculations Neutral axis is parallel to horizontal axis Y:
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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:

8. Calculations Trapezoidal decomposition of curvilinear polygons
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Calculations
Trapezoidal decomposition of curvilinear polygons
Basic set of curvilinear trapezoids calculated only once per angle θ

9. Calculations Basic integrals
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Calculations
Basic integrals
Basic integrals calculated only once per curvilinear trapezoid

10. Calculations Stress resultants for a curvilinear trapezoid
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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.

11. Moment – Curvature diagram (1)
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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)

12. Moment – Curvature diagram (2)
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Moment – Curvature diagram (2)
Calculate initial ε0 for axial equilibrium with no curvature

13. Moment – Curvature diagram (3)
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Moment – Curvature diagram (3)
Neutral Axis Δk
In a loop, apply a small increase in curvature...

14. Moment – Curvature diagram (4)
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Moment – Curvature diagram (4)
Neutral Axis
… and find new ε0 using Van Wijngaarden – Dekker – Brent method

15. Moment – Curvature diagram (5)
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Moment – Curvature diagram (5)
Decrease curvature step when necessary (close to failure)
Continue up to failure to produce the full moment – curvature diagram

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

17. Calculation of Deformed Plane
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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)

18. Example 1 EC2 design charts Equal reinforcement, top and bottom
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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, and various axial load levels

19. Example 2 Analysis with MyBiAxial 2.0 Arbitrary cross section
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Example 2
Arbitrary cross section
Analysis with MyBiAxial 2.0

20. Example 2 Interaction curve for N = -4120kN Complete failure surface
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Example 2
Interaction curve for N = -4120kN
Complete failure surface

21. 3D view of proposed connection
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Example 3
Bolted connection
3D view of proposed connection

22. Stress solids in CAD software
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Example 3
Bolted connection
Example 3 in MyBiAxial 2.0
Stress solids in CAD software

23. Example 4 Moment capacity of rigid footing
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Example 4
Moment capacity of rigid footing

24. Example 4 Moment capacity of rigid footing
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Example 4
Moment capacity of rigid footing

25. Conclusions Custom material data
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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