1. 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

2. Introduction Arbitrary cross section under bending and axial load
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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. 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)

4. Generation of failure surface
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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. Cross Section Arbitrary Cross Section
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Cross Section
Arbitrary Cross Section
consisted of polygons and circles
arcs are approximated by polygon chains to a specified accuracy

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

7. Cross Section For a hollow circular steel section: Outer Circle
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Cross Section
Outer Circle
Inner Circle
Foreground
(+) steel
(+) none
Background
(-) none
(-) steel
For a hollow circular steel section:

8. Materials Custom material data:
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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. Calculations Rotation Direction of neutral axis is assumed
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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. Calculations Trapezoidal decomposition of polygons
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Calculations
Trapezoidal decomposition of polygons
Using “Plane Sweep” algorithm
Basic set of trapezoids calculated only once
Circles are treated separately

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

12. Calculations Stress resultants: Trapezoids
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Calculations
Stress resultants: Trapezoids

13. Calculations Stress resultants: Circular Sections
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Calculations
Stress resultants: Circular Sections

14. Calculations Moment – Curvature diagram Pick N, Initial Curvature step
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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. National Technical University of Athens
Example 1
EC2 design charts

16. 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

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

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

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

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

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

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