Innovative Buckling Design Rules for Structural Hollow Sections FINAL WORKSHOP, Oslo, 6th June 2019 HOLLOSSTAB is an EU funded programme under RFCS, the Research Fund for Coal and Steel, under grant agreement 709892

PROGRAMME 1. Welcome and introduction 2. Trends for hollow sections 3. Physical and numerical test campaign 4. Development of GSRM/CSM Design 5. Software: calculation core 6. Examples 7. Closing statement

Software: calculation core The calculation core consists of two executables: the calculation of the critical bifurcation load, based on Generalized Beam Theory (GBT); the calculation of the plastic load parameter. The executables were written in C++. i/o with the main program through txt files. The fundamentals of each executable are detailed next. The validation studies carried out are also briefly presented.

Critical bifurcation load The program is based on a semi-analytical Generalized Beam Theory formulation and the linear stability analysis concept. This ensures very fast computation times. Fundamental assumption: the local buckling half-wavelength is small enough not to be affected by the longitudinal variation of stresses. The semi-analytical approach does not require a longitudinal discretization and thus eigenvalue NDOFs = cross-section NDOFs.

Critical bifurcation load Example of cross-section geometry and discretization Eigenvalue problem for single half-wave buckling (C, D, B, X1, X2 are GBT modal matrices)

Input file for RHS/SHS Line 1 external width of web [L] external width of flange [L] thickness [L] number of walls per web [positive integer] number of walls per flange [positive integer] corner midline radius [L] number of walls per corner (0 for straight-edge model) [non-negative integer]   Line 2 Area [L^2] Inertia about axis parallel to flange [L^4] Inertia about axis parallel to web [L^4] Line 3 Young's modulus [F/L^2] Poisson's ratio [-] Line 4 N (compression positive) [F] My [FL] Mz [FL] Line 5 Iterative procedure? (1-yes,0-no) [1 or 0] Half-wavelength tolerance (must be non-null but is only used if Iterative procedure = 1) [L] Line 6 max number of iterations (if Iterative Procedure = 1) or max number of steps (if Iterative procedure = 0) [positive integer] Line 7 Export critical buckling mode (1-yes, 0-no) [1 or 0]

Critical bifurcation load For the iterative procedure, the Golden Section Search Method is used to calculate cr,min with respect to the half-wavelength, with an initial search interval between 0.001B and 1.5H. Otherwise, the signature curve is determined for equally spaced half- wavelengths up to 1.5H, with user-input no. of steps. The executable was validated against CUFSM for the complete set of RHS and SHS cross-sections in EN10219-2 and considering several combinations of N + My + Mz. Discretization: 5 elements per wall & rounded corner half-wavelength tol. = 1% of lower external dimension max. no. iterations = 100.

Critical bifurcation load Two sets of analyses were performed: all 305 cross-sections, subjected to load cases A through G a subset of 4 cross-sections for multiple pairs (Ψw; Ψf), defining a mesh with intervals of 0.1 along both directions, in which each ratio ranges from -2 to 1

Critical bifurcation load Statistical summary (over 8500 cases)

Critical bifurcation load The GSS failed in 4 cases involving major axis bending and very unlikely cross-sections. In one case the local minimum can be determined using the signature curve procedure; in the remaining 3 no local minima exist. Disclaimer.txt warns about this fact.

The minimum is sought in ]0; c*max(A,B)], in s steps. Input file for EHS/CHS Line 1 semi-major axis (mid surface dimension - A) [L] semi-minor axis (mid surface dimension - B) [L] thickness [L] number of walls/nodes [positive integer]   Line 2 Area [L^2] Inertia about axis y [L^4] Inertia about axis z [L^4] Line 3 Young's modulus [F/L^2] Poisson's ratio [-] Line 4 N (compression positive) [F] My [FL] Mz [FL] Line 5 Steps s [positive integer] Parameter c [L] Line 6 Export critical buckling mode (1-yes, 0-no) [1 or 0] The minimum is sought in ]0; c*max(A,B)], in s steps.

Critical bifurcation load Validation (standard cross-sections):

Critical bifurcation load Case 1: Case 2: Case 3:

Critical bifurcation load Non-standard cross-sections:

Plastic load parameter The procedure is based on that proposed by Plakolb (2015) and is validated using CUFSM. The cross-section is discretized into rectangles or “fibers”. For RHS the walls are subdivided along y and z. For EHS the discretization is in the radial direction. a) b)

Plastic load parameter An iterative procedure with load control is implemented. If an upper bound is found the bisection method is used. The initial increment is defined through a first yield criterion. In each increment Newton- Raphson iterations are carried out to achieve equilibrium.

Input file for RHS/SHS Line 1 external width of web [L] external width of flange [L] thickness [L] corner midline radius [L] Line 2 number of fibers in webs along flange direction [positive integer] number of fibers in webs along web direction [positive integer]; Line 3 number of fibers in flanges along flange direction [positive integer] number of fibers in flanges along web direction [positive integer]; Line 4 number of fibers in corners along flange direction [positive integer] number of fibers in corners along web direction [positive integer]; Line 5 Young's modulus [F/L^2] Yield stress [F/L^2] Line 6 N (positive) [F] My (positive) [FL] Mz (positive) [FL] Line 7 maximum number of load steps [positive integer] alpha (load-stepping relative tolerance) [double] maximum number of iterations per step [positive integer] beta (residual relative tolerance in Newton-Rhapson iterative procedure) [double] Line 8 export cross-section stresses (1-yes, 0-no) [1 or 0]

Input file for EHS/CHS Line 1 semi-axis parallel to Y (mid surface dimension - A) [L] semi-axis parallel to Z (mid surface dimension - B) [L] thickness (t) [L] number of walls (nw) [positive integer] number of fibers (nfib) [positive integer] Line 2 Young's modulus [F/L^2] Yield stress [F/L^2] Line 6 N (positive) [F] My (positive) [FL] Mz (positive) [FL] Line 7 maximum number of load steps [positive integer] alpha (load-stepping relative tolerance) [double] maximum number of iterations per step [positive integer] beta (residual relative tolerance in Newton-Rhapson iterative procedure) [double] Line 8 export cross-section stresses (1-yes, 0-no) [1 or 0]

Plastic load parameter Warning messages are written if: the max. no. load steps is reached without satisfying convergence tolerance α no upper bound is found If the percentage of average absolute stress with respect to yield stress is lower than 90% (revision of discretization, maximum number of load steps and/or maximum number of iterations per step are required)

Plastic load parameter RHS/SHS Validation for:

Plastic load parameter EHS/CHS Validation for standard cross-sections:

Plastic load parameter EHS/CHS Validation for non- standard cross-sections:

Graphical User Interface General

Graphical User Interface General

Graphical User Interface General

Graphical User Interface General

Graphical User Interface General

Graphical User Interface Input – definition of the cross-section

Graphical User Interface Input – definition of the cross-section

Graphical User Interface Input – definition of design parameters

Graphical User Interface Input – definition of loads

Graphical User Interface Input – definition of loads

Graphical User Interface Output – calculation

Graphical User Interface Output – graphical representation of the results

Graphical User Interface Output – graphical representation of the results 235 -195,8

Graphical User Interface Output – calculation sheet

Graphical User Interface Output – calculation sheet

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