A MATLAB Code for the Design of Built-up Box Columns Subjected to Axial-Bending

A built up box column supports a 16.5 m-tall signboard (Fig. 1) designed in accordance with AISC 360-16 specifications. An all-in-one MATLAB code for wind load analysis and design is developed and used to investigate the box section as shown in Fig. 2.

Fig. 1 Signboard Structure

Fig. 2 Box column section

Based on AISC 360, this column has a slender element in compression, and a non-compact flange due to Mny. The demand/capacity ratio (DCR) can be computed from the minimum value of

Where Mnx, and Mny are computed from the minimum value of

The notations LTB, FLB and WLB stand for

LTB = Lateral-Torsional Buckling (Mn vs unbraced Length)FLB = Flange local buckling (Mn vs b/t, flange)

WLB = Web local buckling (Mn vs b/t, web)

Computing Mn considering LTB

For a box section, LTB usually does not govern the bending strength. However, a generic AISC LRFD formula for critical flexural stress (Fcr) is given to relate Mn and Lb, i.e.

zone 1 : constant Mp

zone 2 : Linear interpolation from Mp --> 0.7My

zone 3 : Euler's zone

I

function Mn_LTB = boxLTBStrength_AISC_style(Mp, E, S, Lb, r_lateral, J, h0, Cb)
% Approximate AISC-style lateral-torsional buckling strength
% for rectangular box / HSS section.
%
% Inputs:
%   Mp        = plastic moment, kN-m
%   E         = elastic modulus, MPa = N/mm2
%   S         = elastic section modulus, mm3
%   Lb        = unbraced length, mm
%   r_lateral = radius of gyration about lateral-buckling axis, mm
%   J         = torsional constant, mm4
%   h0        = distance between flange centroids, mm
%   Cb        = LTB modification factor
%
% Output:
%   Mn_LTB    = nominal LTB moment strength, kN-m
%
.
.
.
.

    % --------------------------------------------------------------------
    % Three-zone LTB strength
    % --------------------------------------------------------------------
    if Lb <= Lp

        % Zone 1: fully plastic LTB strength
        Mn_LTB = Mp;

    elseif Lb <= Lr

        % Zone 2: inelastic interpolation
        ratio = (Lb - Lp)/(Lr - Lp);

        Mn_LTB = Cb*(Mp - (Mp - My07)*ratio);

        % AISC-style cap
        Mn_LTB = min(Mn_LTB, Mp);

    else

        % Zone 3: elastic LTB branch
        Fcr_LTB = Fcr_fun(Lb, Cb);

        Mn_LTB = Fcr_LTB*S/1e6;   % kN-m

        % AISC-style cap
        Mn_LTB = min(Mn_LTB, Mp);

    end

    % ---------------------------------------------------------------------
    % Safety cleanup
    % ---------------------------------------------------------------------
    Mn_LTB = max(Mn_LTB, 0);

end

Computing Mn considering FLB and WLB

The bending strength due to non-non compact and slender section, i.e., about x axis, is determined from

Calculation Procedure

1. Define wind load and design input

2. Compute box section properties

3. Classify plate slenderness

4. Compute effective area for axial compression

5. Compute axial compression strength

6. Compute local flexural buckling strengths:

FLB = flange local buckling

WLB = web local buckling

7. Compute lateral-torsional buckling strength:

LTB = 3-zone AISC-style check

8. Select governing flexural strength:

Mn_used = min(Mn_FLB, Mn_WLB, Mn_LTB)

9. Apply LRFD resistance factors

10. Check beam-column interaction

11. Print report

12. Plot:

- X-axis local buckling strength

- Y-axis local buckling strength

- LTB Mn vs Lb curve

- beam-column interaction diagram

The calculation flowchart can be summarized as follows

Wind load integration over height for the base shear (Fw) and moment (Mw)

Fw = ∫ Ce(z) Cp q b dz , Mw = ∫ Ce(z) Cp q z b dz

↓

Required strengths Pr, Mrx, Mry

↓

Box section properties Ag, I, S, Z

↓

Slenderness and effective area Q

↓

Column buckling strength Pc

↓

Flexural strength candidates:

flange local buckling

web local buckling

lateral-torsional buckling

↓

Mnx = minimum controlling strength

Mny = minimum controlling strength

↓

Beam-column interaction DCR

↓

Print detailed TXT report & plots

A sample graphical plot illustrating the calculation of Mnx, Mny, LTB, and the column interaction chart is shown in Fig. 3. The plots clearly show the design point for the nominal bending moment considering LTB, FLB, and WLB, as well as the safe region for combined axial-bending action.

Fig. 3 Graphical Plots

Conclusion

In this project, we developed a MATLAB code for explicit calculation of box beam secti9n subjects to axial and bending. The program performs checking for demand/capacity of the box beam. It's not difficult to extend the code to other built up sections considering non compact ot slender elements.

Reference:

JC McCormac, SD Csernak, Structural Steel Design, 5th ed., 2017 AISC 360-16