Nonlinear analysis and reliability of metallic truss structures

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. ƒIt 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. ƒThe 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

Introduction

The hypothesis of the behavior of linear structures has a restricted domain of validity. Indeed, the modeling of the material’s real behavior made calls has a non linear relationship (stress-strains).

The basics of material nonlinear analysis were the first to be established because of their simplicity. Two different approaches have been proposed: the initial deformation method and the tangent modulus (now known as the incremental method). After a series of specific applications, the equations of a general formulation of geometric nonlinear analysis were established by S. Yagmai [¹] in 1968 for the updated Lagrangian description.

A. Grelat [²,³] was among the first to take into account in the simulation these two types of nonlinearity. The behavior of the compressed concrete was modeled by Sargin law [⁴] and that of the tensed zones was simulated using an original form of behavior law where the concrete is represented by a triangular fictitious stress diagram from the neutral fiber. His work was complemented by O. Naït-Rabah [⁵] who extended numerical simulations to three-dimensional structures. In order to improve the simulations of large displacements and large rotations, F. Robert [⁶] proposed to introduce a corotational description of motion for the treatment of geometric nonlinearity.

Other authors have complemented existing codes by developing specific finite elements such as, for example, B. Espion [⁷] which proposed a finite planar beam element with 9 degrees of freedom.

A numerical elastic 2D and elastic-plastic 3D analysis up to shear connection collapse in the framework of the longitudinal shear in shear connection of both floor and bridge composite steel and concrete trusses is presented by Josef Machacek et al. [⁸]. He studied the influence of the main parameters and presented recommendations for practical design.

Ran Feng et al. [⁹]. proposed to use the finite element method for the numerical analysis of concrete-filled multi-planar CHS Inverse-Triangular tubular truss, which he modeled according to a Pratt truss configuration with symmetric geometry. The material and geometric non-linearity’s of the concrete are taken into account, a parametric study is performed in evaluating the effects of main influential factors on the strength and behavior of concrete-filled multi-planar CHS Inverse-Triangular tubular truss.

Truss structures are commonly used in construction because they can lighten the weight of a building while ensuring greater stability. In the first part of this article, we propose a formulation for the nonlinear material calculation of truss structures (bars elements) in static behavior under an increasing loading until fracture. Our formulation will be implemented in a calculation program in FORTRAN language processing the beams elements, which was developed in references [¹⁰,¹¹].

The safety of a mechanical system is ensured by a safety coefficient: the ratio between a variable loading and a variable strength determined by a deterministic approach. For a complex structure these efforts are poorly known, and its strength is uncertain, so there is always a risk of the structure fracture. To do this, the probabilistic approach allows evaluating the risk by methods of analysis of reliability of the mechanical systems developed during the last years.

In structure reliability, the FORM and SORM methods are an approximation methods allowing to determine a particular structures design point and therefore to estimate the failure probability. These methods are therefore intrinsically linked to this famous design point it permit to define the distance between the design point and the failure surface point in the original space. This distance is called Reliability: Reliability or Safety Index.

The so-called safety index method was also used by Ravindra et al. [¹²]. to design reinforced concrete beams and structural steel members.

Karamchandani and Cornell [¹³] developed a method that approximates the parameter sensitivity with respect to distribution parameters that can take second order effect into account, based on SORM and the finite difference method.

N. Vu-Bac et al. [¹⁴]. performed a sensitivity analysis (SA) based on their MD results to quantify the influence of uncertain input parameters on the predicted yield stress and elastic modulus. The sensitivity analysis (SA) is based on response surface (RS) models (polynomial regression and moving least squares). They used partial derivatives (local SA) and variance-based methods (global SA) in this study an coefficient of determination (COD) is computed for allowing an estimation of the quality of the approximation.

N. Vu-Bac et al. [¹⁵]. used simulations of molecular dynamics (MD) or studied the effect of single-walled carbon nanotube (SWCNT) radius, the temperature and the pulling velocity on interfacial shear stress (ISS). For computational efficiency, the sensitivity analysis (SA) is based on surrogate models (polynomial regression, moving least squares (MLS) and hybrid of quadratic polynomial and MLS regressions).

N. Vu-Bac et al. [¹⁶]. proposed a stochastic framework based on sensitivity analysis (SA) methods to quantify the key-input parameters, which used to evaluate the Young modulus of the polymer (epoxy) clay nanocomposite (PCNs). They compared the computation of sensitivity indices and to the simulation time between the kriging regression (KR) model and the quadratic regression (QDR) model.

N. Vu-Bac et al. [¹⁷]. carried out a sensitivity analysis consisting in quantifying the influence of uncertain input parameters on uncertain model outputs. The results are based on a probability density function (PDF) provided for the input parameters.

It has been demonstrated that the spline regression model is more robust than polynomial regression model. It is necessary to take penalized spline regression models using global separate penalties or separate global penalties into account in order to approximate the observed data.

N. Vu-Bac et al. [¹⁸]. used a hierarchical multiscale model brindging four (nano, micro, meso and macro) scales to study the effect of uncertain model inputs on the macroscopic Young’s modulus and the Poisson’s ratio. They used sensitivity analysis (SA) methods to estimate the effect of the uncertain correlated (dependent) inputs on the Young's modulus and the Poisson's ratio for the multiscale model in the context of a global SA. Estimates for correlated parameters are performed for both first-order and total sensitivity indices.

It is proposed in this article in the second part, first a classical technique for the calculation of the reliability (method Hasofer Lind level 2) which uses the probability and the statistic and makes it possible to check the reliability of the metallic structures and ensuring dialogue between mechanics and reliability model with indirect coupling method. The reliability model and its coupling with the mechanical model are writing in FORTRAN language.

Finally, in this paper we propose a non linear computational calculation method for the truss structures. We have modeled the reliability problem by the Hasofer-Lind principle for the reliability index estimation the failure probability. Once the mechanical model and the reliability model have been available, they have to be coupled by a response surface, and then the probability laws of the random variables retained to approximate the statistical law (real law).

Methodology of nonlinear analysis of truss structures

Introduction

The elements of a truss work only in tension or compression, thus they are modeled by finite elements such bars. This is usually an element with 2 nodes, which includes 3 degrees of freedom. Each node represents the components of its movement in space. The nodes have no rotations freedom (thus not running); because they have no physical meaning. Indeed, the existence of a rotational freedom degree mean the presence within the bar element (pin-ended element) of bending moment or torsion, which is excluded.

Nonlinear Analysis of truss structures formed by elements bars

We make several working hypotheses, since we consider the truss in the three-dimensional field:

- The connections between the bars are considered as ball joints.

- The loads are applied only at the truss nodes.

- These bar element a subjected only to tensile and compression (it will only axial deformation).

The section is studied under the assumption of small strains and is defined in its main reference.

The section longitudinal normal strain \epsilon (y,z) in the bar element at the coordinate point (y,z) is the strain at its gravity center, is given by:

(1)\epsilon (y,z)={{\displaystyle \epsilon }}_x

The normal strains{{\displaystyle \phi }}_y, {{\displaystyle \phi }}_z (rotations of the section around the axes {{\displaystyle G}}_y and {{\displaystyle G}}_z respectively) and tangent{{\displaystyle \gamma }}_y, {{\displaystyle \gamma }}_z and the torsion angle{{\displaystyle \theta }}_x, as well as their increase{{\displaystyle \Delta \phi }}_y, {{\displaystyle \Delta \phi }}_z,{{\displaystyle \Delta \gamma }}_y, {{\displaystyle \Delta \gamma }}_z, \Delta {{\displaystyle \theta }}_x are neglected.

The material nonlinearity of a bar element is taken into account by the relation linking the normal stress \sigma and the normal strain correspond. The section’s strain increase under normal stresses is given by:

(2)\{\Delta {{\displaystyle \epsilon }}_n\}=\left\{\begin{array}{l}\Delta {{\displaystyle \epsilon }}_x\\ \mathrm{0}\\ \mathrm{0}\end{array}\right\}

The normal internal loads increase in the section is given by:

(3)\{\Delta {{\displaystyle F}}_{\mathrm{mn}}\}={\displaystyle \int {{\displaystyle s}}_{\mathrm{m}}}{{\displaystyle E}}_m(y,z)\left[\begin{array}{l}\mathrm{1}\\ \mathrm{0}\\ \mathrm{0}\end{array}\begin{array}{l}\mathrm{0}\\ \mathrm{0}\\ \mathrm{0}\end{array}\begin{array}{l}\mathrm{0}\\ \mathrm{0}\\ \mathrm{0}\end{array}\right]\mathrm{d}{{\displaystyle S}}_m\{\Delta {{\displaystyle \epsilon }}_n\}

The bar element section is subjected to an external load increase, which is given by:

(4)\{\Delta {{\displaystyle F}}_{sn}\}=\left\{\begin{array}{l}\Delta N\\ \mathrm{0}\\ \mathrm{0}\end{array}\right\}

The section equilibrium is expressed by:

(5)\left\{\begin{array}{l}\Delta {{\displaystyle F}}_{sn}\\ \mathrm{0}\end{array}\right\}=\left\{\begin{array}{l}\Delta {{\displaystyle F}}_{mn}\\ \mathrm{0}\end{array}\right\}

This condition, taking into account the Eq. (3), can be writing as:

(6)\left\{\begin{array}{l}\Delta {{\displaystyle F}}_{sn}\\ \mathrm{0}\end{array}\right\}=\left[\begin{array}{l}[{{\displaystyle k}}_{mn}]\\ O\end{array}\begin{array}{l}O\\ O\end{array}\right]\left\{\begin{array}{l}\Delta {{\displaystyle \epsilon }}_n\\ \mathrm{0}\end{array}\right\}

With:

(7)O=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]

Internal and external tangential loads are zero for the bar element.

For loads acting increase, the resolution of the Eq. (5) is iterative. The solution of this equation is given by:

(8)\left\{\begin{array}{l}\Delta {{\displaystyle \epsilon }}_n\\ \mathrm{0}\end{array}\right\}={[{{\displaystyle k}}_s]}^{-\mathrm{1}}\left\{\begin{array}{l}\Delta {{\displaystyle F}}_{sn}\\ \mathrm{0}\end{array}\right\}

Note. The section stiffness matrix [k_s] is singular. And to simplify the presentation of equilibrium equations of the bar element, we will keep the same dimensions of the matrix and vectors corresponding to the beam element.

Metal profile section’s equilibrium algorithm

The equilibrium algorithm of the metal profile section for the bar element, is described below (Fig. 1):

For a given step r

{\{\Delta {{\displaystyle F}}_s\}}^r: increase efforts to step r.

{{\displaystyle \{\Delta \epsilon \}}}_{\mathrm{0}}: Initial strains increase.

We consider that the section equilibrium is reached when, for two successive iterations, the strain’s Euclidean norm is less than a tolerance fixed:

(9)

T_s: Precision order desired of 0.01

Flowchart of calculation

The flowchart of the equilibrium state search, in the metal profile section is described below (Fig. 2):

Note:

N: Number of iterations set in advance.

N_it: Number of iterations performed by the program.

Stiffness matrix of a nonlinear elastic bar element

The method used in non-linear material calculation consists, at first, to analysis the deformed state of the bar element in the intrinsic coordinate system xyz related to the deformed position of the element on the assumption of small deformations and small displacements. The second order effects are introduced in the transit from the intrinsic system coordinate xyz to the local system coordinate x_oy_oz_o. Then, we establish a stiffness matrix of the element in the local coordinate system, related to the element initial position. Finally, we assemble the structure stiffness matrix from the bar elements stiffness matrix in the absolute system coordinate OXYZ. The calculation process is described by the following formulation.

Formulation of the bar element stiffness matrix in its intrinsic system

The loads on the extremity of the bar element is (Fig. 3):

(10)\{{{\displaystyle F}}_N\}={\{{{\displaystyle N}}_j,\mathrm{0},\mathrm{0},\mathrm{0},\mathrm{0},\mathrm{0}\}}^{\mathrm{T}}

The corresponding displacement is:

(11)\{{{\displaystyle S}}_N\}={\{e,\mathrm{0},\mathrm{0},\mathrm{0},\mathrm{0},\mathrm{0}\}}^{\mathrm{T}}

With:

(12)e=L-{{\displaystyle L}}_{\mathrm{0}}

The section loads in an abscissa can be written as:

(13)\{\begin{array}{l}\begin{array}{l}N(X)=-{{\displaystyle N}}_j\\ {{\displaystyle M}}_y(X)=\mathrm{0}\end{array}\\ {{\displaystyle M}}_z(X)=\mathrm{0}\\ {{\displaystyle T}}_y(X)=\mathrm{0}\\ \begin{array}{l}{{\displaystyle T}}_z(X)=\mathrm{0}\\ {{\displaystyle M}}_x(X)=\mathrm{0}\end{array}\end{array}

The relationship between loads in one section and the loads at the element nodes is given by:

(14)\{{{\displaystyle F}}_S\}=[L(X)]\{{{\displaystyle F}}_N\}

With

(15)\{{{\displaystyle F}}_S\}={\{N(X),\mathrm{0},\mathrm{0},\mathrm{0},\mathrm{0},\mathrm{0}\}}^T

And

(16)[L(X)]=\left[\begin{array}{cccccc}-1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]

The virtual works theorem allows us to write:

(17)\{{{\displaystyle S}}_N\}={\displaystyle {\int }_{\mathrm{0}}^L{[L(x)]}^T}\{\delta (x)\}dx

Where :

(18)\{\delta (x)\}={\{\epsilon (x),\mathrm{0},\mathrm{0},\mathrm{0},\mathrm{0},\mathrm{0}\}}^T

Using the virtual work theorem and neglecting the element length variation in computing of the nodes displacements permits to write:

(19)\{\Delta {{\displaystyle S}}_N\}={\displaystyle {\int }_{\mathrm{0}}^L{[L(x)]}^T}\{\Delta \delta (x)\}dx

By replacing the relations (8) and (17) in Eq. (19), one gets the flexibility matrix of the element {{\displaystyle K}}_N^{-\mathrm{1}} such that:

(20)\{\Delta {{\displaystyle S}}_N\}={[{{\displaystyle K}}_N]}^{-\mathrm{1}}\{\Delta {{\displaystyle F}}_N\}

With :

(21){[{{\displaystyle K}}_N]}^{-\mathrm{1}}={\displaystyle {\int }_{\mathrm{0}}^L{[L(x)]}^T}{[{{\displaystyle K}}_S]}^{-\mathrm{1}}[L(x)]ds

The integration is carried out by the Simpson rule by considering on the element a certain number of calculation sections in which the matrix K_S are determined at each step.

Finally, we get the following relationship:

(22)\{\Delta {{\displaystyle F}}_N\}=[{{\displaystyle K}}_N]\{\Delta {{\displaystyle S}}_N\}

With

(23)[{{\displaystyle K}}_N]=\left[\begin{array}{cccccc}[{{\displaystyle k}}_{mn}]& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]

Formulation of the bar elements stiffness matrix in its local system coordinate

The loads at the element nodes in the local system coordinates x_oy_oz_o, are:

(24)\{{{\displaystyle F}}_L\}={({{\displaystyle F}}_{ixo},{{\displaystyle F}}_{iyo},{{\displaystyle F}}_{izo},\mathrm{0},\mathrm{0},\mathrm{0},{{\displaystyle F}}_{jxo},{{\displaystyle F}}_{jyo},{{\displaystyle F}}_{jzo},\mathrm{0},\mathrm{0},\mathrm{0})}^{\mathrm{T}}

The displacements at the element nodes in the local system coordinates x_oy_oz_o, are:

(25)\{{{\displaystyle s}}_L\}={({{\displaystyle u}}_i,{{\displaystyle v}}_i,{{\displaystyle w}}_i,\mathrm{0},\mathrm{0},\mathrm{0},{{\displaystyle u}}_j,{{\displaystyle v}}_j,{{\displaystyle w}}_j,\mathrm{0},\mathrm{0},\mathrm{0})}^{\mathrm{T}}

The element nodes loads in the intermediate system coordinate can be written as:

(26)\{{{\displaystyle F}}_U\}={({{\displaystyle F}}_{jxo},{{\displaystyle F}}_{jyo},{{\displaystyle F}}_{jzo},\mathrm{0},\mathrm{0},\mathrm{0},\mathrm{0},\mathrm{0},\mathrm{0})}^{\mathrm{T}}

The element nodes displacements in the intermediate system coordinate can be written as:

(27)\{{{\displaystyle S}}_U\}={(u,v,w,\mathrm{0},\mathrm{0},\mathrm{0},\mathrm{0},\mathrm{0},\mathrm{0})}^{\mathrm{T}}

Where:

(28)\{\begin{array}{l}u={{\displaystyle u}}_j-{{\displaystyle u}}_i\\ v={{\displaystyle v}}_j-{{\displaystyle v}}_i\\ w={{\displaystyle w}}_j-{{\displaystyle w}}_i\end{array}

The relationship between the displacements and loads in these two systems are given by:

(29)\{{{\displaystyle S}}_U\}=[{{\displaystyle T}}_{\mathrm{0}}][{{\displaystyle S}}_L]

(30)[{{\displaystyle F}}_L]={[{{\displaystyle T}}_{\mathrm{0}}]}^T\{{{\displaystyle F}}_U\}

(31)\mathrm{W}\mathrm{i}\mathrm{t}\mathrm{h}\hspace{0.17em}[{{\displaystyle T}}_{\mathrm{0}}]=\left[\begin{array}{cccccccccccc}-1& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0& 0\\ 0& -1& 0& 0& 0& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& -1& 0& 0& 0& 0& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]

The displacements increase in the intrinsic system and in the intermediate systems, are linked by:

(32)\{\Delta {{\displaystyle S}}_N\}=[B]\{\Delta {{\displaystyle S}}_U\}

Using the virtual work theorem, we can write that:

(33)\{\Delta {{\displaystyle F}}_U\}={[B]}^T\{\Delta {{\displaystyle F}}_N\}+{[\Delta B]}^T\{{{\displaystyle F}}_N\}

When:

(34)[B]=\left[\begin{array}{ccccccccc}1& \frac{v}{L_0}& \frac{w}{L_0}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]

(35)Aad{[\Delta B]}^T\{{{\displaystyle F}}_N\}=[D]\{\Delta {{\displaystyle S}}_U\}

(36)\mathrm{W}\mathrm{i}\mathrm{t}\mathrm{h}\hspace{0.17em}[D]=\left[\begin{array}{ccccccccc}0& {\displaystyle \frac{-v{{\displaystyle N}}_j}{{{{\displaystyle L}}_{\mathrm{0}}}^{\mathrm{2}}}}& \frac{-w{{\displaystyle N}}_j}{{{{\displaystyle L}}_{\mathrm{0}}}^{\mathrm{2}}}& 0& 0& 0& 0& 0& 0\\ {\displaystyle \frac{-v{{\displaystyle N}}_j}{{{{\displaystyle L}}_{\mathrm{0}}}^{\mathrm{2}}}}& {\displaystyle \frac{{{\displaystyle N}}_j}{{{\displaystyle L}}_{\mathrm{0}}}}& 0& 0& 0& 0& 0& 0& 0\\ {\displaystyle \frac{-w{{\displaystyle N}}_j}{{{{\displaystyle L}}_{\mathrm{0}}}^{\mathrm{2}}}}& 0& {\displaystyle \frac{{{\displaystyle N}}_j}{{{\displaystyle L}}_{\mathrm{0}}}}& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0& 0& 0& 0\end{array}\right]

Finally, the relationship between the nodes loads increase and the nodes displacements increase in the local system coordinate is given by:

(37)\{\Delta {{\displaystyle F}}_L\}=[{{\displaystyle K}}_L]\{\Delta {{\displaystyle S}}_L\}

The stiffness matrix [K_L], of the bar element in the local system coordinate is defined by:

(38)[{{\displaystyle K}}_L]={[{{\displaystyle T}}_{\mathrm{0}}]}^T({[B]}^T[{{\displaystyle K}}_N][B]+[D])[{{\displaystyle T}}_{\mathrm{0}}]

Formulation of the bar element stiffness matrix in the absolute system

The nodes loads increase in the absolute system coordinate are defined as:

(39)\{\Delta {{\displaystyle F}}_X\}={(\Delta {{\displaystyle F}}_{ix},\Delta {{\displaystyle F}}_{iy},\Delta {{\displaystyle F}}_{iz},\mathrm{0},\mathrm{0},\mathrm{0},\Delta {{\displaystyle F}}_{jx},\Delta {{\displaystyle F}}_{jy},\Delta {{\displaystyle F}}_{jz},\mathrm{0},\mathrm{0},\mathrm{0})}^{\mathrm{T}}

The nodes displacements increase in the absolute system coordinate are defined as:

(40)\{\Delta {{\displaystyle S}}_X\}={(\Delta {{\displaystyle X}}_i,\Delta {{\displaystyle Y}}_i,\Delta {{\displaystyle Z}}_i,\mathrm{0},\mathrm{0},\mathrm{0},\Delta {{\displaystyle X}}_j,\Delta {{\displaystyle Y}}_j,\Delta {{\displaystyle Z}}_j,\mathrm{0},\mathrm{0},\mathrm{0})}^{\mathrm{T}}

The relationship linking the nodes displacements increase in the local system coordinate and the nodes displacements increase, in the absolute system coordinate can be written as:

(41)\{\Delta {{\displaystyle S}}_L\}=[RT]\{\Delta {{\displaystyle S}}_X\}

The relationship between the loads increase in the local system coordinate and the loads increase in the absolute system coordinate is:

(42)\{\Delta {{\displaystyle F}}_X\}={[RT]}^{\mathrm{T}}\{\Delta {{\displaystyle F}}_L\}

With

(43)[RT]=\left[\begin{array}{cccc}[{{\displaystyle R}}_{\mathrm{0}}]& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& [{{\displaystyle R}}_{\mathrm{0}}]& 0\\ 0& 0& 0& 0\end{array}\right]

The geometric transformation matrix [R₀] of a three-dimensional bar element; it given by:

(44)[{{\displaystyle R}}_{\mathrm{0}}]=\left[\begin{array}{lll}{{\displaystyle c}}_x& {{\displaystyle c}}_y& {{\displaystyle c}}_z\\ {\displaystyle \frac{-{{\displaystyle c}}_{x}Y}{S}}& {\displaystyle \frac{S}{L}}& {\displaystyle \frac{-{{\displaystyle c}}_{y}Z}{S}}\\ {\displaystyle \frac{-Z}{S}}& \mathrm{0}& {\displaystyle \frac{-X}{S}}\end{array}\right]

(45)\mathrm{W}\mathrm{ith}\{\begin{array}{l}L=\sqrt{{({{\displaystyle x}}_j-{{\displaystyle x}}_i)}^{\mathrm{2}}+{({{\displaystyle y}}_j-{{\displaystyle y}}_i)}^{\mathrm{2}}+{({{\displaystyle z}}_j-{{\displaystyle z}}_i)}^{\mathrm{2}}}\\ S=\sqrt{{({{\displaystyle x}}_j-{{\displaystyle x}}_i)}^{\mathrm{2}}+{({{\displaystyle z}}_j-{{\displaystyle z}}_i)}^{\mathrm{2}}}\\ \cos \left(\overline{x},x\right)={\displaystyle \frac{{{\displaystyle x}}_j-{{\displaystyle x}}_i}{L}}={\displaystyle \frac{X}{L}}\\ \cos \left(\overline{y},y\right)={\displaystyle \frac{{{\displaystyle y}}_j-{{\displaystyle y}}_i}{L}}={\displaystyle \frac{Y}{L}}\\ \cos \left(\overline{z},z\right)={\displaystyle \frac{{{\displaystyle z}}_j-{{\displaystyle z}}_i}{L}}={\displaystyle \frac{Z}{L}}\end{array}

Finally, the relationship between the loads increase and the displacements increase of the bar element nodes in the absolute system coordinate OXYZ, is:

(46)\{\Delta {{\displaystyle F}}_X\}=[{{\displaystyle K}}_X]\{\Delta {{\displaystyle S}}_X\}

The element stiffness matrix in the absolute system coordinate can be written as:

(47)[{{\displaystyle K}}_X]={[RT]}^T[{{\displaystyle K}}_L][RT]

The stiffness matrix [K_S] and [K_N] are singular and cannot remain so, because in the resolution of the general stiffness matrix in the global system coordinate, the null pivots will appear. For this purpose, disrupts the terms of the diagonal corresponding to the shear forces, instead of being zero, are multiplied by a real coefficient between 0 and 1.

After implementing the procedure described above, the other null pivots to appear due to the singularity of the matrix [L(x)] and [B]then to solve this problem we proceed in the same way as for the stiffness matrix (instead of canceled the terms of the diagonal of the matrix, they are taken equal to 10^-4).

Node displacement calculation

The nonlinear problem is solved by using an iterative method [¹⁹,²⁰,²¹], based on the displacements method, in which we recalculate, for each step, the matrix structure’s stiffness connecting it’s displacements increases with loads increases.

The structure stiffness matrix [K] is formed from an element stiffness matrix in the absolute system coordinate [²²,²³].

The nodes displacements increase {DU} linked to the loads increase by the structure equilibrium equation:

(48)\{\Delta P\}=[K]\{\Delta U\}

The nodes displacements increase are obtained by solving the structure equilibrium equation and it can be written as:

(49)\{\Delta U\}={[K]}^{-\mathrm{1}}\{\Delta P\}

The convergence of the nodes displacements is carried out by the following equation.

(50)

At the end of each step; the total deformations of each section, the forces and displacements at the nodes are calculated, accumulating the increments of this step with the values of the previous step.

Before proceeding to a new loading step, one proceeds to the correction of the nodes displacements {U} obtained. This stage is indispensable for the case of structures sensitive to second-order effects [²⁴].

A correction step is carried out in the current step, introducing the effect of a dummy load increase \{\Delta P^{*}\} given by:

(51)\{\Delta P^{*}\}=\{P\}-\{P^{\mathrm{int}}\}

This correction step is characterized by an iterative cycle to balance this dummy load. It is suppressed if the loads Euclidean norm \eta is less than or equal to a certain tolerance T_f fixed a priori:

(52)

After this correction step, the analysis is continued with a new external loading step. The breaking load is obtained when the resolution algorithm diverges for an increase in the absolute value of the load which is less than or equal to an accuracy fixed well before.

Flowchart calculation steps (Fig. 4)

Behavior laws used

In this present study it was used for structural steels natural steels; the perfect elastic-plastic law and the elastic-plastic law with firming for describing steel behavior.

Reliability of structures

Introduction

The safety of a mechanical system is provided by a safety factor: ratio of a variable load and a resistance variable established by a deterministic approach. For a complex structure such efforts are poorly known and its resistance is uncertain, there is still a risk of the ruined structure. This probabilistic approach permit to evaluate the risk in case of reliability analysis methods [²⁵,²⁶,²⁷], developed in recent years.

Research design point

The design point (or the most probable failure) is the point of the limit state surface where U probability density is a maximum; it is also defined as the point of the state limit surface as close to the origin:

(53){{\displaystyle \beta }}_{HL}\underset{g\{{{\displaystyle x}}_i({{\displaystyle u}}_j)\}\le \mathrm{0}}{\min }\sqrt{{\{u\}}^{\mathrm{T}}\{u\}}

Under constraint H(u)\le \mathrm{0}

In this study the constrained minimization problem is solved by using the algorithm of Hasofer-Lind-Rackwitz-Fiessler which is an adaptation of a first order optimization algorithm to the problem of research design point.

Optimization algorithms rackwitz-fiessler

There are many algorithms allow solving the optimization problem [²⁸,²⁹]. In practice, Rackwitz-Fiessler algorithm is the most used because of its simplicity and good results. The algorithm Hasofer-Lind-Rackwitz-Fiessler (HL-RF) is an adaptation of a first order optimization algorithm to the problem of research design point. The algorithm (HL-RF) is an improvement of the Hasofer Lind algorithm, which assumes that the basic variables are Gaussian and uncorrelated. As for him (HL-RF) is more general. It has no restrictions on basic variables. The algorithm to be used in the following is that found in the book of Lemaire [³⁰].

The assumptions considered in the algorithm are:

- The limit state function H has a gradient at the point of coordinate u.

- The gradient \nabla H(u) is not null at any point in the hyper limit state surface.

To determine the design point, we consider a normalized space at a point P^(k) to coordinate {u}^(k), the origin point of the iteration (k). This point does not necessarily belong to constraint and H(u) can be different from zero.

Taylor series development of the state of the limit function H(u) about the point P^(k) gives:

(54)H(u)=H(u^{(k)})+{{\displaystyle \langle \nabla H(u)\rangle }}_{u^{(k)}}(\{u\}-u^{(k)})+O{(\{u\}-u^{(k)})}^{\mathrm{2}}

The equation of the tangent has hyper plan H(u) as {u}^(k):

(55){{\displaystyle \langle \nabla H(u)\rangle }}_{u^{(k)}}\{u\}+c=\mathrm{0}

\nabla H(u^{(k)}) is the gradient H(u) at the point P^(k). Then, we define P^(k+1) by:

(56)H(u^{(k+\mathrm{1})})=H(u^{(k)})+{{\displaystyle \langle \nabla H(u)\rangle }}_{u^{(k)}}({\{u\}}^{(k+\mathrm{1})}-u^{(k)})=\mathrm{0}

Dividing the equation by norm {{\displaystyle \Vert \nabla H(u)\Vert }}_{u^{(k)}} and by introducing the direction cosines of H in P^(k) we get :

(57){\displaystyle \frac{H(u^{(k)})}{{{\displaystyle \Vert \nabla H(u)\Vert }}_{u^{(k)}}}}+{\langle \alpha \rangle }^{(k)}({\{u\}}^{(k+\mathrm{1})}-{\{u\}}^{(k)})=\mathrm{0}

He comes:

(58){\{u\}}^{(k+\mathrm{1})}{\{\alpha \}}^{(k)}={\{u\}}^{(k)}{\{\alpha \}}^{(k)}-{\displaystyle \frac{H(u^{(k)})}{{{\displaystyle \Vert \nabla H(u)\Vert }}_{u^{(k)}}}}

With {\alpha }^{(k)}={\displaystyle \frac{\nabla H(u^{(k)})}{{{\displaystyle \Vert \nabla H(u)\Vert }}_{u^{(k)}}}} is the vector cosine directors (or the vector of the normalized gradient) of H in P^(k).

In the limit when k\to \infty ,d(u^{(k)})=\beta et \{u\}=-\beta \{\alpha \} if the algorithm is convergent. At iteration (k), let:

(59){\{u\}}^{(k+\mathrm{1})}=-{\beta }^{(k)}.{\{\alpha \}}^{(k)}\Rightarrow {\beta }^{(k)}=-{\langle u\rangle }^{(k+\mathrm{1})}.{\{\alpha \}}^{(k)}

This leads to the iterative relationship giving the reliability index:

(60){\beta }^{(k)}={-\langle u\rangle }^{(k)}{\{\alpha \}}^{(k)}+{\displaystyle \frac{H(u^{(k)})}{{{\displaystyle \Vert \nabla H(u)\Vert }}_{u^{(k)}}}}

The search algorithm of reliability index stops when the norm

\Vert {\{u\}}^{(k+\mathrm{1})}-{\{u\}}^{(k)}\Vert \le \epsilon .

And {\{u\}}^{(k+\mathrm{1})} is deducted by substituting Eq. (60) into Eq. (59):

(61)u^{(k+\mathrm{1})}=({\langle u\rangle }^{(k)}{\{\alpha \}}^{(k)}){\{\alpha \}}^{(k)}-{\displaystyle \frac{H(u^{(k)})}{{{\displaystyle \Vert \nabla H(u)\Vert }}_{u^{(k)}}}}{\{\alpha \}}^{(k)}

Summary of the algorithm

The algorithm of Hasofer Lind-Rackwitz-Fiessler (HL-RF) is summarized by the following steps:

1- Choose a starting point {u}⁽⁰⁾;

2- Evaluate the limit state function H(u^(k));

3- Calculate the gradient of the limit state {\{\nabla H(u)\}}^{(k)} and norm {\Vert \nabla H(u)\Vert }^{(k)}, to deduce {\{\alpha \}}^{(k)} by: {\alpha }^{(k)}={\displaystyle \frac{\nabla H(u^{(k)})}{{{\displaystyle \Vert \nabla H(u)\Vert }}_{u^{(k)}}}} ;

4- Calculate the reliability index {\beta }^{(k)};

5- Calculate the coordinate of the next iteration {u}^(k+1);

6- Convergence tests:

If \Vert {\{u\}}^{(k+\mathrm{1})}-{\{u\}}^{(k)}\Vert \le \epsilon, stop calculating;

Else put k= k+ 1 and go 2.

Flowchart Hasofer-Lind-Rackwitz-Fiessler (HL-RF)

The flowchart Hasofer-Lind-Rackwitz-Fiessler (HL-RF) is described below (Fig. 5):

Mechanical-reliability coupling

To realize this coupling, there are three methods of control [³⁰]: The direct coupling, the coupling response surface, the coupling optimization.

In this present study, the mechanical-reliability coupling will be directed by response surface.

Method by analytical response surface

The analytical response surface method is to replace the function of unknown performance g(x) by approximated function \widehat{g}(x). The choice of a high order polynomial is to better represent the model. Very often, a quadratic shape is chosen for writing the substitute \widehat{g}(x) of the performance function g(x). In other words, the substitute can be written as follows:

(62)g(x)\approx \widehat{g}(x)={{\displaystyle c}}_{\mathrm{0}}+{\displaystyle {\sum }_{i=\mathrm{1}}^n{{\displaystyle c}}_i}{{\displaystyle X}}_i+{\displaystyle {\sum }_{i=\mathrm{1}}^n{{\displaystyle c}}_{ii}{{\displaystyle X}}_i^{\mathrm{2}}}

(63)g(x)\approx \widehat{g}(x)={{\displaystyle c}}_{\mathrm{0}}+{\displaystyle {\sum }_{i=\mathrm{1}}^n{{\displaystyle c}}_i}{{\displaystyle X}}_i+{\displaystyle {\sum }_{i=\mathrm{1}}^{n-\mathrm{1}}{\displaystyle {\sum }_{j=i+\mathrm{1}}^n{{\displaystyle c}}_{ij}{{\displaystyle X}}_i}}{{\displaystyle X}}_j

Where: X is the vector n basic variables, and these c_i ,c_ijare the coefficients of the polynomial sought.

Flowchart of coupling reliability-mechanical by response surface

The flowchart of the mechanical-reliability coupling analytical response surface is described below (Fig. 6):

Comparisons with numerical and experimental results

Validation of the non-linear calculation of the truss structures

Plane truss beam

A truss plane in three equal spans has been tested by LOVEGROVE and analyzed by X. SUN and S.L CHAN [³¹] using the finite element method.

The layout and dimensions of the truss are shown in Fig. 7, and sections of the frames. The values of modulus of elasticity and the elastic limit data are respectively equal to 214 GPa, and 285.4 MPa.

The element passing through the point (A) is designed such that the load applied in the vicinity makes it as the critical point.

The evolution of the load versus lateral displacement at mid point (A) is shown in Fig. 8.

The experimental ultimate load is given equal to 88 KN. The calculated ultimate load is 90 KN, a difference of 2.27% (in the sense of an overstatement). By against it is found that the shape of the calculated curve approximates the experimental curve. The agreement between the experiment and our calculation is pretty good.

Bridge trusses

Fig. 9 and 10 shows the bridge Warren truss type of 7.32 meters in height and 36.6 meters in length. The stress-strain relationship was supposed perfectly elastic plastic with a modulus of elasticity of 200000 MPa.

The cross-sectional W8 × 18 with a yield strength of 248 MPa was used for the tension members, including the inferior and diagonal chords. And the cross section of 305 × 305 × 13 mm with a yield strength of 317 MPa was used for flanges subjected to compression, including the upper chords.

This bridge truss (Fig. 11) was the subject of a numerical study by Seung-Eock Kim et al. [³²].

The ultimate load factor (\lambda) found from Seung-Eock Kim et al. [³²]. is equal to 1.06. The ultimate load factor (\lambda) calculated is 1.07, a difference of 0.94%. We find that the truss bridge to cede by rupture rather than buckling, and that the concordance between the calculation of Seung-Eock Kim et al. [³²]. and our calculation is very good.

Validation and evaluation of the reliability of the structures studied

Application to the truss plane

The geometrical and material characteristics of the planar truss studied (scope section, loading, boundary condition, elastic modulus, elastic limit) are described in § 4.1.1.

The procedure to estimate random variables probability distributions selected for this study in order to approximate the statistical law (real). Most often the mean and standard deviation of a random variable are known, however, this information leads to Gaussian distributions that are not adapted to a physical representation of random variables. For example, some random variable cannot be negative, a log-normal or exponential distribution will best represent it.

Figs. 12(a) and 12(b) show the result of estimation of the probability distributions of random variables, by normal distribution, lognormal and exponential.

The results in Fig. 12(a) gives; for a normal distribution a not insignificant probability that the random variable P is negative, and the lognormal probability of a low density, for against the exponential law appears to better represent this parameter compared to the two other types of law. And those in Fig. 12(b) also gives for a normal distribution a not insignificant probability that the random variable is negative, as to the lognormal and exponential laws seem better represent this parameter compared to the normal law except the law lognormal gives a precision unknown phenomena better than the normal distribution and the exponential law, as it well simulates the normal law by a bell curve with slightly picked up results that Act.

Then the random variables used in this study are considered continuous, independent, and they are represented by the vector X, we class as follows:

- The random output variables to limit state (P, \delta), laws of random distributions are modeled by an exponential distribution and lognormal respectively, and whose characteristics are given in Table 1.

- The other parameters (random variables to enter) as the modulus of elasticity E and the yield strength {{\displaystyle \sigma }}_e are considered as deterministic.

The limit state function G (P, \delta) is a nonlinear function implicit (numerically known from our non-linear calculation), the failure of the system is observed when G(x)\ge \mathrm{0} (Fig. 8).

Given the complexity of the finite element model, it is difficult to conduct the study with direct coupling between the non-linear calculation program and the reliability program, then it becomes necessary to construct a response surface (Figs. 13-15).

The method of response surface with the objective of replacing the limit state function obtained in implicit form by another explicit function, in order to apply reliable method for estimating the reliability index of the scenarios and the failure probability of the studied structure. For this, the following steps are considered:

- In a first step we need to transform the limit state function of the physical space to a reduced centered space with zero mean and standard deviation unit under the law of the random variable.

G(P,\delta )\underrightarrow{T(\mu =\mathrm{0},\sigma =\mathrm{1})}H(P,\delta )

- In a second step, it’s to make the choice of a mathematical form to the response surface.

H(P,\delta )\underrightarrow{\mathrm{approximate}\hspace{0.17em}\mathrm{function}}\widehat{H}(P,\delta )

- The third step is the modeling of random variables.

- Finally, the fourth step is to apply the HL-RF method to estimate the reliability index and calculate the probability of failure.

After analysis by reliability-mechanical coupling, the HL-RF method allowed us to obtain the following results (Table 2):

Fig. 14 shows the limit state approach centered in the reduced space \widehat{H}(P,\delta ) et al.so the results of research from the point of conception P*.

The reliability index is found\beta =\mathrm{1.29306}, which corresponds to a probability of failure of the truss estimated{{\displaystyle P}}_f=\varphi (-\beta )=\mathrm{9.853}\%, or reliability of 90.147%. As for the most probable failure point, in physical space, corresponds to the load 28.8314KN and displacement0.0018m.

Then, we proceed to another method; combining between the laws of continuous random variables with the aim of as close as possible real probability distributions of random variables used in this example, it is retained for seven case of combining the distribution laws. The seven cases selected random variables are shown in Table 3 and limit state (response surface) approached in the reduced space centered \widehat{H}(P,\delta ) are shown in Fig. 16.

After analysis by mechanical-reliability coupling seven cases of combination between the laws of continuous random variables, the HL-RF method allowed us to obtain the following results (Table 4).

It is observed for the various random variables selected by using the same distribution law (cases 1, 2, 3), one obtains index much smaller reliability that all the reliability index found using the laws of different distributions (cases 4, 5, 6, 7). Consequently, there is a significant difference between the probability of failure of the random variables in the same distribution law (cases 1, 2, 3) and the probability of failure of random variables act differently distribution (cases 4, 5, 6, 7) it is because in reality the random variables may not follow all the same law.

We also note that if four gives the greatest reliability index \beta =\mathrm{1.29306}, where reliability of 90.20%, which confirms the validity of the first method for estimating probability distributions that can approximate the statistics law real of the random variable.

In our study, we considered that continuous and independent random variables, when in reality it is still not the case for all systems.

Bridge trusses

The geometrical and material characteristics of the truss bridge studied (litters section, loading, boundary condition, elastic modulus, yield point) are described in § 4.1.2.

The procedure to estimate random variables probability distributions selected for this study in order to approximate the statistics law (real).

Figs. 17(a) and 17(b) show the result of estimation of the distribution of the probability distributions of random variables selected by the normal laws, lognormal and exponential.

The results in Fig. 17(a) give; for a normal distribution a not insignificant probability that the random variable \lambda is negative, and the log-normal probability density less than that found for the exponential law; that law seems to best represent this parameter compared to the two other types of laws. And those in Fig. 17(b) also provide for a normal distribution a not insignificant probability that the random variable U/L or negative, about the lognormal and exponential laws seem better represent this parameter compared to the normal law except that the lognormal distribution gives an accuracy of unknown phenomena better than normal or exponential, as it well simulates the normal distribution curve with a more collected results that Act, et al.so because it gives a probability density greater than the other two types of laws.

The random variables used in this study are considered continuous, independent and are represented by the vector X, we class as follows:

- The random variable outputs to limit state (\lambda,U/L ), laws of random distributions are modeled by an exponential distribution and lognormal respectively, whose characteristics are given in Table 5.

- The other parameters (random input variables) as the modulus of elasticity E and the yield stresses ({{\displaystyle \sigma }}_{e\mathrm{1}},{{\displaystyle \sigma }}_{e\mathrm{2}}) are considered as deterministic.

The limit state function G (\lambda,U/L ) is a nonlinear function implicit (numerically known from our non-linear calculation), the failure of the system is observed whenG(x)\ge \mathrm{0} (Figs. 18-20).

After analysis by coupling mechanical reliability engineer, the HL-RF method allowed us to obtain the following results (Table 6):

Fig. 21 shows the state limit approached in the reduced space centered \widehat{H} (\lambda,U/L ) et al.so the results of the research design point P*.

The reliability index is found\beta =\mathrm{1.03}, corresponding to a probability of failure of the truss estimated at{{\displaystyle P}}_f=\varphi (-\beta )=\mathrm{15.15}\%, where reliability of 84.85%.

In the following we proceed to another method; combining between the laws of continuous random variables with the aim of as close as possible the real probability distributions of random variables used in this example, it is retained for seven case of combining the distribution laws. The seven cases selected random variables are shown in Table 7 and approached their limit state (response surface) in the reduced space centered \widehat{H} (\lambda,U/L ) are shown in Fig. 21.

After analysis by a coupling mechanical-reliability seven cases of combination between the continuous random variables laws, the HL-RF method allowed us to obtain the following results (Table 8):

It is observed for the different selected random variables using the same distribution law (cases 1, 2, 3) obtain with much smaller reliability index that all reliability index found by using the laws of different distributions (cases 4, 5, 6, 7). Therefore, there is a gap between the failure probability of the random variables in the same distribution laws (cases 1, 2, 3) and the failure probability of the random variables in different distribution laws (cases 4, 5, 6, 7). In reality, the random variables may not follow all the same law.

There is also the case 4 gives the greatest reliability index \beta =\mathrm{1.03}, where a reliability of 84.85%, which confirms the validity of the first method for estimating probability distributions that can approximate the real statistical law of the random variable.

In our study, we found that continuous and independent random variables, when in reality it is still not the case for all systems.

Conclusion

Steel real behavior laws are used in this present study to treat the structures non-linear calculation. This calculation gives best estimates the real displacement of structures.

The formulation proposed for a bar element, in the case of steel truss, and its implementation allowed to treat the nonlinear material analysis of truss structures under increasing monotonic loading until failure.

Finally, we addressed the reliability problems on some examples to estimate their reliability index and determine their failure probability. The results obtained by combination of the continuous random variables laws show that the method for estimating probability distributions used in the first approach approximate correctly the real statistical distribution of the random variable.

The reliability model coupled to the mechanical model applied to the truss structures; assess the reliability index relating to the ruins of mechanical systems with nonlinear behavior and not normal variables.

References

[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

[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

[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

[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

[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

[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

[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

[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

[29] Low B K. FORM, SORM, and spatial modeling in geotechnical engineering. Structural Safety, 2014, 49: 56–64

[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

[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

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