Euro-SiBRAM’2002 Prague, June 24 to 26, 2002, Czech Republic
Session 5
Reliability analysis of a steel frame (pilot study)
Abstrakt
The following study demonstrates on concrete example the potential of SBRA method in reliability assessment of statically indeterminate steel frame. Input quantities, such as loadings, geometrical and material properties, are considered as random variables.
Key Words: SBRA method, random variable, bounded histogram, statically indeterminate steel frame.
The goal of this study is application of SBRA (Simulation-Based Reliability Assessment) method (see [1], [2], [3]) in reliabiity analysis of statically indeterminate steel frame (see [5]).
Fig. 1 Scheme of statically indeterminate planar steel frame
In Fig. 1 is shown scheme of statically indeterminate planar steel frame. Frame is exposed to random variable forces F1, F2, F3, F4, F5 and uniformly distributed load f1. Frame is attached to the foundations in point h by fixed hinge and in points g and j is fixed. The goal of the study is: {a} analysis of internal forces in frame and {b} reliability assessment of member 3 in cross-section j from the viewpoint of safety (carrying capacity). Reliability assessment has be performed for target probability Pd = 0,00007, corresponding to the usual level of reliability (see.[6]).
In the analysis is considered transformation model based on 1st order theory and static response of the structure to the loading. As the reference level is considered steel yield stress.
Individual random variable loads are calculated as the product of corresponding maximum load value and random variable represented by bounded histogram (see. Tab 1 and Fig. 2).
Tab. 1 Maximum values and corresponding random variables of loads
Fig. 2 Used bounded histograms of random variable loads
Individual members of the frame are made of rolled shapes IPE, as is evident from Tab. 2. Cross-section characteristics of individual members are considered as random variables (see Tab. 3, Tab. 4, Tab. 5 and Fig. 3). Cross-section heights hi, areas Ai and moments of inertia Ii of individual members are calculated according to these equations:
(m) (1)
(m2) (2)
(m4) (3)
Lengths of individual members are considered as deterministic values (see. Tab. 6).
Tab. 2 Member profiles
Tab. 3 Cross-section heights
Fig. 3 Used bounded histogram representing random variable geometrical properties
Tab. 4 Cross-section areas
Tab. 6 Member lengths
The calculation of internal forces in the frame was performed using simplified deformation method (see [4]). Resulting scatters of normal forces, shear forces and bending moments calculated using SBRA method and expressed by „2D anthills“ are shown in Fig. 4a, 4b and 4c. (Note: negative values of normal and shear forces are drawn on bottom side of individual members (denoted by dashed line – see Fig. 1), bending moments are drawn on tensioned side of individual members).
Fig. 4a Scatter of normal forces in the frame
Fig. 4b Scatter of shear forces in the frame
Fig. 4c Scatter of bending moments in the frame
Load effects combination in arbitrary point of the structure (in case of shear forces neglecting) can be expressed by normal stress according to the equation:
,
(Pa) (4)
where N is normal force, M is bending moment, A is cross-section area, I is moment of inertia and z is distance from considered point to neutral exis.
As the reference function, regarding to safety (carrying capacity) of the whole structure, is applied steel yield stress. Exceeding of this value in arbitrary point of the structure is assumed as the structure failure. The reference function is given by the equation:
,
(Pa) (5)
where fy,var is random variable yield stress represented by the histogram „St235.dis“ (see Fig. 5). Coefficient 0,9 expreses deviation of steel the yield stress value measured in laboratory from the real value.
Fig. 5 Used bounded histogram of random variable steel yield stress
Reliability assessment from the viewpoint of the safety (carrying capacity) was performed using reliability function analysis:
(Pa) (6)
where fy is steel yield stress (reference function) calculated according to (5) and sx is normal stress calculated from equation (4). Reliabiliy criterion, applied in reliability assesment according to the SBRA method, has form:
,
(-) (7)
where Pf is calculated probability of failure and Pd is target probability of failure given by code.
The goal of the study was reliability analysis of member 3 in cross-section j (see Fig. 1). From the resulting scatters of bending moments in the frame (see Fig. 4) is clearly, that in the considered cross-section j is necessary to assess the magnitude of the normall stress in lateral fibres of the left and right flange with consideration of positive and negative bending moments occurence (consequence of two-directional wind acting – see histogram “Wind1.dis” in Fig. 2). Resulting values of reliability functions RF± , calculated using SBRA method, are shown in Fig. 6 and Fig. 7 (left flange) and in Fig. 8 and Fig. 9 (right flange). From these figures is clearly, that for target probability of failure Pd = 0,00007 is in all cases met reliability criterion (7).
Fig. 6 Reliability function RF– (MPa) – left flance
Fig. 7 Reliability function RF+ (MPa) – left flance
Fig. 8 Reliabiltiy function RF– (MPa) – right flange
Fig. 9 Reliabiltiy function RF+ (MPa) – right flance
The study demonstrates the potential of the SBRA method in reliabilty analysis of statically indeterminate steel frames. All input variables, such as loadings, material and geometrical imperfections, can be considered as random variables. For more information regarding of reliability assessment of statically indeterminate frames see [7] and [8].
The support of the Grant Agency of the Czech Republic (Projects No. 103/01/1410 and 105/01/0783) is gratefully acknowledged.
[1] Marek, P., Guštar, M., Anagnos, T.: Simulation-Based Reliability Assessment for Structural Engineers. CRC Press Inc., Boca Raton, Florida, 1995.
[2] Marek, P., Guštar, M. a Bathon, L.: Tragwerksbemessung. Von deterministischen zu probabilistischen Verfahren. Academia, Praha 1998.
[3] Marek, P., Brozzetti, J., Guštar, M.: Probabilistic Assessment of Structures using Monte Carlo Simulation. Background, Exercises and Software. Institut of Theoretical and Applied Mechanics, Academy of Sciences of the Czech Republic, Praha, 2001.
[4] Kadlčák, J. Kytýr, J.: Statika stavebních konstrukcí III. VUT Brno, 1992, ISBN 80-214-0438-8.
[5] Pustka, D., Disertation work: Využití spolehlivostní metody SBRA při navrhování ocelových, betonových a ocelobetonových konstrukcí. VŠB – Technical University of Ostrava, Ostrava, September 2002.
[6] ČSN 73 1401: Navrhování ocelových konstrukcí. Český normalizační institut, Praha, 1998.
[7] Pustka, D., Marek, P.: Moving variable loads in reliability analysis of a statically indeterminate steel frame by probabilistic method SBRA. Sborník příspěvků: International conference Reliability and diagnostics of transport structures and means. University of Pardubice, Pardubice, 2002.
[8] Pustka, D., Marek, P.: Reliability analysis of statically indeterminate reinforced concrete frame by probabilistic method SBRA. Sborník příspěvků: International conference Reliability and diagnostics of transport structures and means. University of Pardubice, Pardubice, 2002.