Nonlinear analysis and reliability of metallic truss structures

doi:10.1007/s11709-017-0458-y

Frontiers of Structural and Civil Engineering

2018, Vol. 12

Issue (4): 577-593 https://doi.org/10.1007/s11709-017-0458-y

本期目录

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

全文: PDF(1163 KB) HTML

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.

Key words： modeling nonlinearity mechanical truss probability reliability response surface probability of failure

收稿日期: 2017-03-20 出版日期: 2018-11-20

Corresponding Author(s): Karim BENYAHI

引用本文:

. [J]. Frontiers of Structural and Civil Engineering, 2018, 12(4): 577-593.
Karim BENYAHI, Youcef BOUAFIA, Salma BARBOURA, Mohand Said KACHI. Nonlinear analysis and reliability of metallic truss structures. Front. Struct. Civ. Eng., 2018, 12(4): 577-593.

链接本文:

https://academic.hep.com.cn/fsce/CN/10.1007/s11709-017-0458-y
https://academic.hep.com.cn/fsce/CN/Y2018/V12/I4/577

Fig.1

Fig.2

Fig.3

Fig.4

Fig.5

Fig.6

Fig.7

Fig.8

Fig.9

Fig.10

Fig.11

Fig.12

vector X	random variables	distribution law	mean $μ X$	standard deviation $σ X$
X1 X2	P $δ$	exponential lognormal	45.0 0.00463588	32.4037035 0.00419236

Tab.1

Fig.13

Fig.14

Fig.15

reliability index	probability of failure	direction cosine ( $α 1$ , $α 2$ )	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

Tab.3

Fig.16

case	reliability index	probability of failure	direction cosine ( $α 1$ , $α 2$ )	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

Fig.17

vector X	random variables	distribution law	mean $μ X$	standard deviation $σ X$
X1 X2	$λ$ U/L	exponential lognormal	0.589735 0.0016639	0.34098745 0.00179006

Tab.5

Fig.18

Fig.19

Fig.20

reliability index	probability of failure	direction cosine ( $α 1$ , $α 2$ )	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

Tab.7

Fig.21

case	reliability index	probability of failure	direction cosine ( $α 1$ , $α 2$ )	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

E_a: Young modulus of steel,

ε e

?: Steel yielding strain,

σ e

: Steel yielding stress,

ε u

: Steel ultimate strain,

ε x

?: Strain in the gravity center of the total area caused by the normal force N,

{F_mn}: Contribution caused by the concrete and / or metal profile,

S_m: The metal profile’s cross-section.

E_m (y,z)?: The longitudinal elastic modulus at a current point of the metal profile cross-section.

Δ σ m (y, z)

: Normal stress in a current point of the metal profile,

[k_mn]: Section stiffness matrix,

{F_sn}: Vector of the sections normal forces,

e?: Element length increase,

L₀?: Element initial length,

L: Element length after deformation,

[B]: Geometric transformation matrix,

[K_L]: Element stiffness matrix in the local coordinate,

[R₀]: Geometric transformation matrix,

[K_X]: Element stiffness matrix in the absolute coordinates,

[K_N]: Bar element’s stiffness matrix in the intrinsic system coordinate.

[K_U]: Element Stiffness matrix in the intermediate local coordinate system,

{F_X}: The nodes load vector in the absolute system coordinate,

{S_X}: The nodes displacements vector in the absolute coordinate system,

[F_L]?: The nodes load in the local coordinate system.

[S_L]?: The nodes displacements vector in the local system coordinate.

[F_U]?: The nodes loads vector in the intermediate system coordinate.

[S_U]?: The nodes displacements vector in the intermediate system coordinate.

u_i, v_i, w_i_?: Components of the displacement vector in the local coordinate system_,

[S S] i − 1

: Sections flexibility matrix of the iteration (i-1),

ε s

: Strains balanced in the previous step,

{Δ F s} r

: Forces increase in the step r,

{Δ ε} 0

: Initial strains increase,

[K]_i: Structure stiffness matrix at the iteration (i),

{U_s}: Node displacement vector at the latest stable step,

{Δ P} r

: Applied load increase in the r step,

{P}: External structures applied loads,

{P^int}: Internal structures applied loads,

ϕ

?: The normal law distribution function reduced centered (mean 0 and standard deviation 1),

m_R?: Means strength,

m_S ?: Means loads,

σ R

: Standard deviations of the strength,

σ S

?: Standard deviations of the loads,

P*?: Point of the most probable failure,

α (k)

: Vector cosine directors.

1	Yaghmai S. Incremental analysis of large deformations in mechanics of solids with applications to axisymmetric sheus of revolution. Technical Reprot SESM 68–17, Univ. Califorma, Berkeley, 1968
2	Grelat A. Nonlinear analysis of reinforced concrete hyperstatics frames, Doctoral thesis Engineer. University Paris VI, 1978
3	Grelat A. Non-linear behavior and stability of reinforced concrete frames, Annals of I.T.B.T.P., N° 234, 1978, France
4	Sargin M. Stress-Strain relationship for concrete and the analysis of structural concrete sections. Solid Mechanics division, University of waterloo, Canada, 1971
5	Rabah O N. Numerical simulation of nonlinear behavior of frames Space, Doctoral Thesis. Central School of Paris, France, 1990
6	Robert F. Contribution to the geometric and material nonlinear analysis of space frames in civil engineering, application to structures, Doctoral Thesis. National Applied Sciences Institute, Lyon, France, 1999
7	Espion B. Contribution to the nonlinear analysis of plane frames. Application to reinforced concrete structures, Doctoral Thesis in Applied Science, vol. 1 and 2. Free University of Brussels, Belgium, 1986
8	Machacek J, Charvat M. Design of Shear Connection between Steel Truss and Concrete Slab. Structures and Techniques, 2013, 57: 722–729
9	Feng R, Chen Y, Gao S, Zhang W. Numerical investigation of concrete-filled multi-planar CHS Inverse-Triangular tubular truss. Thin-walled Structures, 2015, 94: 23–37 https://doi.org/10.1016/j.tws.2015.03.030
10	Adjrad A, Kachi M S, Bouafia Y, Iguetoulène F. Nonlinear modeling structures on 3D, in Proc. 4th Annu.icsaam 2011. Structural Analysis of Advanced Materials, Romania, pp. 1-9, 2011
11	Kachi M S. Modeling the behavior until rupture beams with external prestressing, Doctoral Thesis. University of Tizi-Ouzou, Algeria, 2006
12	Ravindra M K, Lind N C, Siu W. Illustration of reliability-based design. Journal of the Structural Division, 1974, 100(9): 1789–1811
13	Karamchandani A, Cornell C. Sensitivity estimation within first and second order reliability methods. Structural Safety, 1992, 11(2): 95–107 https://doi.org/10.1016/0167-4730(92)90002-5
14	Vu-Bac N, Lahmer T, Keitel H, Zhao J, Zhuang X, Rabczuk T. Stochastic predictions of bulk properties of amorphous polyethylene based on molecular dynamics simulations. Mechanics of Materials, 2014, 68: 70–84 https://doi.org/10.1016/j.mechmat.2013.07.021
15	Vu-Bac N, Lahmer T, Zhang Y, Zhuang X, Rabczuk T. Stochastic predictions of interfacial characteristic of polymeric nanocomposites (PNCs). Composites. Part B, Engineering, 2014, 59: 80–95 https://doi.org/10.1016/j.compositesb.2013.11.014
16	Vu-Bac N, Silani M, Lahmer T, Zhuang X, Rabczuk T. A unified framework for stochastic predictions of mechanical properties of polymeric nanocomposites. Computational Materials Science, 2015, 96: 520–535 https://doi.org/10.1016/j.commatsci.2014.04.066
17	Vu-Bac N, Lahmer T, Zhuang X, Nguyen-Thoi T, Rabczuk T. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100: 19–31 https://doi.org/10.1016/j.advengsoft.2016.06.005
18	Vu-Bac N, Rafiee R, Zhuang X, Lahmer T Uncertainty quantification for multiscale modeling of polymer nanocomposites with correlated parameters. Composites. Part B, Engineering, 2015, 68: 446–464 https://doi.org/10.1016/j.compositesb.2014.09.008
19	Crisfield M A. Non-linear Finite Element Analysis of Solids and Structures – Vol 1. John Wiley & Sons Ltd., Chichester, England, 1991
20	Crisfield M A. Non-linear Finite Element Analysis of Solids and Structures – Vol 2. John Wiley & Sons Ltd., Chichester, England, 1997
21	Carlos A. Felippa. Nonlinear finite element methods. University of Colorado Boulder, Colorado 80309-0429, USA., 2001
22	Liu G R, Quek S S. The Finite Element Method: A Practical Course, Butterworth-Heinemann. Elsevier Science Ltd., 2003
23	Fish J, Belytschko T. A First Course in Finite Elements. John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ. England, 2007
24	Greco M, Venturini W S. Stability analysis of three-dimensional trusses. Latin American Journal of Solids and Structures, 2006, 3: 325–344
25	Lemaire M.Probabilistic approach to dimensioning: Modeling uncertainty and approximation methods, Technical editions engineering, 2012
26	Deheeger F. Reliability-mechanical coupling: SMART- Statistical learning methodology in reliability, Doctoral Thesis. University BLAISE PASCAL- Clermont II, 2008
27	Pendola M, Mohamed A, Hornet P, Lemaire M. Structural reliability in context of uncertainty Statistics, XV mechanics French Congress, Nancy, 3-7, 2001
28	Köylüoǧlu H U, Nielsen S R K. New approximations for SORM integrals. Structural Safety, 1994, 13(4): 235–246 https://doi.org/10.1016/0167-4730(94)90031-0
29	Low B K. FORM, SORM, and spatial modeling in geotechnical engineering. Structural Safety, 2014, 49: 56–64 https://doi.org/10.1016/j.strusafe.2013.08.008
30	Lemaire M. Structural reliability: static reliability-mechanical coupling. Editions Hermes Lavoisier, Paris, 2005
31	Sun X, Chan S L. Design and second-order analysis of trusses composed of angle sections. International Journal of Structural Stability and Dynamics, 2002, 02(03): 315–334 https://doi.org/10.1142/S0219455402000610
32	Kim S E, Park M H, Choi S H. Practical advanced analysis and design of three-dimensional truss bridges. Journal of Constructional Steel Research, 2001, 57(8): 907–923 https://doi.org/10.1016/S0143-974X(01)00015-3

Viewed

Full text

Abstract

Cited

Shared

Discussed