Unbraced Frame as Simple Series System Ing. Leo Václavek, CSc., VŠB - Technical University of Ostrava, Czech Republic Session 5

2 W+EQ 2 5 F2‘F2‘ F1‘F1‘F3‘F3‘ F4‘F4‘ q5q5 q6q6 q7q7 Unbraced frame with leaning columns If fixed column 1 fails, collaps of all structure occure, structure behaves as series system. Failure of elements 2,3,…,7 need not necessary cause failure of the whole system.

 +a F4F4 F3F3 F1F1 F2F2  +a 3  +a 1  +a 2 W+EQ Input values, random variables given by pertinent histograms: 21 mutually uncorrelated vertical loadings, wind, earthquake (correlated with vertical loadings), 4 equivalent geometrical imperfections, cross section characteristics of fixed column, yield stress. Output values: Horizontal displacement  of the upper end of fixed (cantileverd) column, stress at the critical cross section of fixed column. Scheme of deformed load structure

If the limit of resistance R (Limit Value - LV) is defined by the onset of yielding in the critical cross section, analytical transformation model based on the second order theory can be made and the load effect Q (corresponding stresss) expressed. 2 W+EQ 2 5 F2‘F2‘ F1‘F1‘F3‘F3‘ F4‘F4‘ q5q5 q6q6 q7q7 When FEM transformation model must be used, much more powerfull computers and computational techniques are needed to realise, using Monte Carlo simulation, such large number of the simulation steps. The safety function, SF = R - Q can be analyzed by M-Star™ program, see SBRA method. When analytical model is used, million of simulation steps take several minutes on current PC computer.

Concluding remark Monte Carlo based analysis of reliability function allowes for calculating probability of failure in transparent way

(a) (b) (c) Combination of APT and MLE approach Following histograms and LDC could be used for modelling of the different time duration of the extraordinary loading. When (a), (b) and (c), is used, then 7.6%, 69.8% and 97.5% respectively of the total service life the structure is exposed to extraordinary loading. Whe zero value is eliminated, most frequent values are around ±0.75. LDC and histograms (a) Earth51 N eq /N total 7.6%, (b) Earth60 N eq /N total = 69.8%, (c) Earth65 N eq /N total =97.5%

(a) (c) (b) Load Duration Curves and histograms (a) dead load - histogram Dead1 (b) random long lasting load - histogram Long2 (c) random short lasting load - histogram Short1

Load Duration Curve and histogram Wind1 Steel A36yield stress, histogram A36-m