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Frontiers of Structural and Civil Engineering

Front. Struct. Civ. Eng.    2018, Vol. 12 Issue (4) : 577-593     https://doi.org/10.1007/s11709-017-0458-y
RESEARCH ARTICLE |
Nonlinear analysis and reliability of metallic truss structures
Karim BENYAHI1(), Youcef BOUAFIA1, Salma BARBOURA2, Mohand Said KACHI1
1. LaMoMs Laboratory, University Mouloud Mammeri of Tizi-Ouzou, 15000 Tizi-Ouzou, Algeria
2. C.N.R.S. LSPM – UPR 3407 Laboratory, Paris 13 University, Paris, France
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Abstract

The present study goes into the search for the safety domain of civil engineering structures. The objective is to show how a reliability-evaluation brought by a mechanical sizing can be obtained. For that purpose, it is necessary to have a mechanical model and a reliability model representing correctly the behavior of this type of structure. ?It is a question on one hand, to propose a formulation for the nonlinear calculation (mechanical nonlinearity) of the spatial structures in trusses, and on the other hand, to propose or to adapt a formulation and a modeling of the reliability. The principle of Hasofer-Lind can be applied, in first approach, for the reliability index estimation, scenarios and the probability of failure. ?The made check concerned metallic in truss structures. Finally, some structures are calculated using the method adapted by Hasofer-Lind to validate the probability approach of the reliability analysis.

Keywords modeling      nonlinearity mechanical      truss      probability      reliability      response surface      probability of failure     
Corresponding Authors: Karim BENYAHI   
Online First Date: 20 March 2018    Issue Date: 20 November 2018
 Cite this article:   
Karim BENYAHI,Youcef BOUAFIA,Salma BARBOURA, et al. Nonlinear analysis and reliability of metallic truss structures[J]. Front. Struct. Civ. Eng., 2018, 12(4): 577-593.
 URL:  
http://journal.hep.com.cn/fsce/EN/10.1007/s11709-017-0458-y
http://journal.hep.com.cn/fsce/EN/Y2018/V12/I4/577
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Karim BENYAHI
Youcef BOUAFIA
Salma BARBOURA
Mohand Said KACHI
Fig.1  Equilibrium of a metallic section
Fig.2  Flowchart for the equilibrium state search in the metallic section
Fig.3  Bar element in the coordinate space
Fig.4  Research the state of the truss structure equilibrium flowchart
Fig.5  Flowchart Hasofer-Lind-Rackwitz-Fiessler (HL-RF)
Fig.6  Flow chart of a mechanical-reliability coupling response surface
Fig.7  Plane truss beam
Fig.8  Plane truss beam- load curve – arrow
Fig.9  Bridge at truss
Fig.10  Loading condition of the bridge at truss
Fig.11  Plane truss- load curve – arrow
Fig.12  Probability distributions of the random variable P; (b) Probability distributions of the random variable δ
vector X random variables distribution law mean μX standard deviation σX
X1
X2
P
δ
exponential
lognormal
45.0
0.00463588
32.4037035
0.00419236
Tab.1  Parameter of random variables laws to limit state
Fig.13  Limit state in physical space G (P, δ)
Fig.14  Limit state approached in the reduced space centered H^ (P, δ)
Fig.15  Absolute value of the direction cosines of the random variables (P,δ)
reliability index probability of failure direction cosine
( α 1,α2)
design point
(U1, U2)
design point
(X1, X2)
1.29306 0.09853 (-0.3047, 0.9524) (0.3940, -1.2315) (28.8314, 0.0018)
Tab.2  Results of the mechanical reliability analysis
variables distribution law
case 1 case 2 case 3 case 4 case 5 case 6 case 7
P
δ
normal
normal
lognormal
lognormal
exponential
exponential
exponential
lognormal
exponential
normal
lognormal
normal
lognormal
exponential
Tab.3  The different case laws of distributions of random variables
Fig.16  Limit state approached different cases in the reduced centered space
case reliability index probability of failure direction cosine
( α 1,α2)
design point
(U1, U2)
design point
(X1, X2)
case 1 0.20974 0.4169 (-0.1094, 0.9939) (0.0229, -0.2084) (51.755, 0.0045)
case 2 0.11476 0.4542 (-0.1157, 0.9932) (0.0132, -0.1139) (36.724, 0.0032)
case 3 0.09772 0.4611 (0.2409, -0.9705) (-0.0235, 0.0948) (11.84, 0.00086)
case 4 1.29306 0.09800 (-0.3047, 0.9524) (0.3940, -1.2315) (28.831, 0.0018)
case 5 0.70959 0.2390 (-0.2901, 0.9569) (0.2058, -0.6790) (69.763, 0.0074)
case 6 0.82593 0.2044 (-0.3148, 0.9491) (0.2600, -0.7839) (71.391, 0.0079)
case 7 0.4054 0.3426 (04997, -0.8661) (-0.2026, 0.3511) (33.498, 0.0022)
Tab.4  Results of the mechanical reliability analysis for the various cases treated
Fig.17  (a)Probability distributions of the random variable λ; (b)Probability distributions of the random variable U/L
vector X random variables distribution law mean μX standard deviation σX
X1
X2
λ
U/L
exponential
lognormal
0.589735
0.0016639
0.34098745
0.00179006
Tab.5  Parameter of random variables laws to limit state
Fig.18  Limit state in physical space G ( λ,U/L )
Fig.19  Limit state approached in the reduced space centered H^ ( λ,U/L )
Fig.20  Absolute value of the cosine directors of random variables ( λ,U/L )
reliability index probability of failure direction cosine
( α 1,α2)
design point
(U1, U2)
design point
(X1, X2)
1.0300 0.1515 (-0.2794, 0.9601) (0.2878, -0.9889) (0.3622, 0.000639)
Tab.6  Results of the mechanical reliability analysis
variables distribution law
case 1 case 2 case 3 case 4 case 5 case 6 case 7
λ
U/L
normal
normal
lognormal
lognormal
exponential
exponential
exponential
lognormal
exponential
normal
lognormal
normal
lognormal
exponential
Tab.7  The different case laws of distributions of random variables
Fig.21  Limit state approached different cases in the reduced centered space
case reliability index probability of failure direction cosine
( α 1,α2)
design point
(U1, U2)
design point
(X1, X2)
case 1 0.3385 0.3674 (-0.4174, 0.9086) (0.1413, -0.3076) (0.6942, 0.0014)
case 2 0.2506 0.4010 (-0.5398, 0.8417) (0.1353, -0.2109) (0.6406, 0.0012)
case 3 0.2360 0.4067 (0.2460, -0.9692) (-0.0580, 0.2287) (0.226, 0.00033)
case 4 1.0300 0.1515 (-0.2794, 0.9601) (0.2878, -0.9889) (0.362, 0.00063)
case 5 0.7535 0.2255 (-0.2558, 0.9667) (0.1928, -0.7283) (0.6811, 0.0013)
case 6 0.4710 0.3188 (-0.2491, 0.9684) (0.1173, -0.4561) (0.7426, 0.0014)
case 7 0.4364 0.3312 (0.4279, -0.9037) (-0.1867, 0.3944) (0.476, 0.00077)
Tab.8  Results of the mechanical reliability analysis for the various cases treated
Ea: Young modulus of steel,
εe?: Steel yielding strain,
σe: Steel yielding stress,
εu: Steel ultimate strain,
εx?: Strain in the gravity center of the total area caused by the normal force N,
{Fmn}: Contribution caused by the concrete and / or metal profile,
Sm: The metal profile’s cross-section.
Em (y,z)?: The longitudinal elastic modulus at a current point of the metal profile cross-section.
Δσm(y ,z): Normal stress in a current point of the metal profile,
[kmn]: Section stiffness matrix,
{Fsn}: Vector of the sections normal forces,
e?: Element length increase,
L0?: Element initial length,
L: Element length after deformation,
[B]: Geometric transformation matrix,
[KL]: Element stiffness matrix in the local coordinate,
[R0]: Geometric transformation matrix,
[KX]: Element stiffness matrix in the absolute coordinates,
[KN]: Bar element’s stiffness matrix in the intrinsic system coordinate.
[KU]: Element Stiffness matrix in the intermediate local coordinate system,
{FX}: The nodes load vector in the absolute system coordinate,
{SX}: The nodes displacements vector in the absolute coordinate system,
[FL]?: The nodes load in the local coordinate system.
[SL]?: The nodes displacements vector in the local system coordinate.
[FU]?: The nodes loads vector in the intermediate system coordinate.
[SU]?: The nodes displacements vector in the intermediate system coordinate.
ui, vi, wi?: Components of the displacement vector in the local coordinate system,
[ SS]i1: Sections flexibility matrix of the iteration (i-1),
εs: Strains balanced in the previous step,
{ΔFs} r: Forces increase in the step r,
{Δε} 0: Initial strains increase,
[K]i: Structure stiffness matrix at the iteration (i),
{Us}: Node displacement vector at the latest stable step,
{ΔP}r: Applied load increase in the r step,
{P}: External structures applied loads,
{Pint}: Internal structures applied loads,
ϕ?: The normal law distribution function reduced centered (mean 0 and standard deviation 1),
mR?: Means strength,
mS ?: Means loads,
σR : Standard deviations of the strength,
σS?: Standard deviations of the loads,
P*?: Point of the most probable failure,
α (k): Vector cosine directors.
  
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