Non linear modeling of three-dimensional reinforced and fiber concrete structures

Abstract

Under the effect of the ascending loading, the behavior of reinforced concrete structures is rather non linear. Research in industry and science aims to extend forward the use of non-linear calculation of fiber concrete for structural parts such as columns, veils and pious, as the fiber concrete is more ductile behavior then the classical concrete behavior. The formulation of the element has been established for modeling the nonlinear behavior of elastic structures in three dimensions, based on the displacement method. For the behavior of concrete and fiber concrete compressive and tensile strength (stress-strain) the uniaxial formulation is used. For steel bi-linear relationship is used. The approach is based on the discretization of the cross section trapezoidal tables. Forming the stiffness matrix of the section, the integral of the surface is calculated as the sum of the integrals on each of the cutting trapezoids. To integrate on the trapeze we have adopted the type of Simpson integration scheme.

Keywords

numerical modeling / column and beam / nonlinear analysis / fibers / pious / reinforcement / 3D formulation / response load-deflection

Introduction

Research on the non-linear calculation of structures has progressed considerably in recent years; it is necessary for the prediction of the actual response of the structures. The methods based on non-linear analysis have made it possible to better estimate a domain of security. The study of the capacity of reinforced concrete and fiber concrete structures requires realistic models of materials, finite element discretization, and the search for a non-linear response.

The model based on endochronic theory has been adapted to concrete behavior by Bažant and Bhat [¹–²], Bažant [³], this theory in which special unloading properties are included have been suggested by Bažant [⁴]. The model based on the theory of cracking band for the rupture of concrete is presented by Bažant and Oh [⁵]; the author presents continuum theory for strain-soffening [⁶]. A general model of microplane that is applicable both to tensile and compression, and both to fragile cracking and plastic response that characterize damage in concrete is developed by Bažant and Prat [⁷].

A method for treating crack growth by particle methods, the modeling of a crack was performed as a set of cracked particles, this method described in detail by Rabczuk et al. [⁸]. Another approach based on the modeling of cracks in meshfree methods where the crack can be arbitrarily oriented is described in detail by Rabczuk and Belytschko [⁹]. An approach for the modeling discrete cracks in meshfree particle methods in three-dimensions were the cracks can be arbitrarily oriented, but their growth is represented discretely, the detail given by Rabczuk and Belytschko [¹⁰]. A procedure applicable to non-linear problems, moderate cracking problems, applications to the static response of reinforced concrete structures where concrete is discretized with particles and steels with elements is described by Rabczuk and Belytschko [¹¹]. A two dimensional approach to the cracking of reinforced concrete structures under increasing static load conditions presented by Rabczuk et al. [¹²]. Fiber concrete model is described by Bouafia et al. [¹³].

Since 1975, several approaches and researches have been developed within the framework of the finite elements of displacement types. These are based on the choice of section deformation, which allows direct evaluation of the elementary stiffness matrix, although most approaches do not consider concrete strength to be tensile in 3D, and the sections are decomposed into rectangles.

For reinforced concrete structures, geometric non-linearity and the formulation of the finite element are accompanied by the development and progress of concrete constituents and their interactions with steel [¹⁴,¹⁵]. The method of three dimensional cohesive cracks for reinforced concrete structures. The cracking in the concrete is modeled with an extended element- free Galerkin method that is coupled to finite elements for the reinforcement according to the general formulation of geometrically nonlinear problems given by Rabczcuk et al. [¹⁵]. A response based on the geometric nonlinear formulation and the tangent matrix formulation, this approach does not present material nonlinearity given by Izzuddin [¹⁶]. An analytical model for modeling reinforced concrete structures in nonlinear elasticity is presented by Adjrad et al. [¹⁷]. An approach for the modeling of semi-rigid joints for columns based on Eurocode 3 given by Fang et al. [¹⁸]. The numerical implementation of the model in the finite element program described by Marante and Flórez-López [¹⁹], the model has been developed within the framework of cumulative damage mechanics for the analysis of reinforced concrete frames.

Other studies [²⁰–²⁵] have integrated models of fiber concrete.

In this study, the desired response is obtained by the displacement method which takes account of nonlinear phenomena [²⁶,²⁷] using the theory of beams. The contribution of taut concrete between two successive cracks has been taken account into. This method allows us to describe, in a manner close to reality, the behavior until the ruin of the structures undergoing considerable displacements or redistributions of effort. The 3D modeling is based on the discretization of the finite element structure “beam” and each finite element consists of several straight sections themselves decomposed into trapezoids.

The equilibrium of the structure is solved using iterative methods using the displacement method in an incremental variable stiffness formulation. We will establish the rope stiffness matrix of a beam element, taking into account the nonlinearity of the materials and Second-order effects due to node displacements. The model presented focuses on the introduction of torsional stiffness and shear forces in the linear domain. Another object of our work is the simulation by computer program of the three-dimensional structures subjected to an increasing loading until the ruin.

The paper is organized as follows: in the first place we give the different laws of behavior of materials (concrete, steel, fiber concrete), then; we describe the equilibrium of the section, as well as the establishment of the stiffness matrix. Finally, verification of the reliability of theoretical developments and the developed software is done by comparing numerical results with experimental results published in the literature.

Calculation assumptions

This method admits that the displacements are small, the forces are conservative, the element is only loaded at its ends, assumed short, so that the second order effects due to deformations in the intrinsic axis system are negligible, the torsional and shear stiffness are calculated by linear elasticity, normal stresses due to torsion are neglected (there is no effect of bi- moment) and each element contains two nodes with 6 degrees of freedom for each node. For each load step, the problem consists in searching increased nodes displacements.

The straight sections before deformation remain straight after deformation. The adhesion of steel to concrete is perfect.

Behavior of materials

Concrete behavior in compression and traction

For concrete in compression of the law Sargin [²⁸] was adopted (see Fig. 1), the tensile behavior will be modeled by the law of Grelat [²⁸] (Fig. 2).

(1)\sigma =f_{cj}\frac{k_b.\overline{\epsilon }+\left(k_b^{\text{'}}-\mathrm{1}\right){\overline{\epsilon }}^{\mathrm{2}}}{\mathrm{1}+\left(k_b-\mathrm{2}\right)\overline{\epsilon }+k_b^{\text{'}}{\overline{\epsilon }}^{\mathrm{2}}},

with

(2)\overline{\epsilon }={\displaystyle \frac{\epsilon }{{\epsilon }_{b\mathrm{0}}}},\hspace{0.17em}\hspace{0.17em}{k}_b={\displaystyle \frac{E_{b\mathrm{0}}·{\epsilon }_{b\mathrm{0}}}{f_{cj}}},

\left|{\epsilon }_{bt}\right|\le {\epsilon }_{ft}\mathrm{;}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\sigma }_{bt}=E_{b\mathrm{0}}·{\epsilon }_{bt}

(3)

\left|{\epsilon }_{bt}\right|>{\epsilon }_r\mathrm{;}\hspace{0.17em}\hspace{0.17em}{\sigma }_{bt}=\mathrm{0.}

Behavior of fiber concrete

We used the model of Bouafia et al. [¹³] to model the fiber concrete tensile behavior. A graphic illustration of the model is presented in Fig. 3.

(4)\{\begin{array}{ccc}\hfill \sigma =E_{ct}\epsilon \hfill & \hfill \mathrm{if}\hfill & \hfill \mathrm{0}\le \epsilon \le {\epsilon }_U\hfill \\ \hfill \sigma ={\sigma }_{uc}-\left[{\sigma }_{uc}-f_{ft}\right]{\displaystyle \frac{{\left(\epsilon -{\epsilon }_u\right)}^{\mathrm{6}}}{{\left({\epsilon }_{ft}-{\epsilon }_u\right)}^{\mathrm{6}}}}\hfill & \hfill \mathrm{if}\hfill & \hfill {\epsilon }_{ft}\le \epsilon \le {\epsilon }_u\hfill \\ \hfill \sigma ={\sigma }_{uc}\left[\mathrm{1}-{\displaystyle \frac{{\left(\epsilon -{\epsilon }_u\right)}^{\mathrm{6}}}{{\left({\epsilon }_r-{\epsilon }_u\right)}^{\mathrm{6}}}}\right]\hfill & \hfill \mathrm{if}\hfill & \hfill {\epsilon }_u\le \epsilon \le {\epsilon }_r\hfill \end{array}.

Behavior of reinforcing steel

The reinforcing steel will be modeled by the elastoplastic [²⁹]:

(5)\{\begin{array}{ccc}\hfill \sigma =E_a{\epsilon }_S\hfill & \hfill \mathrm{if}\hfill & \hfill {\epsilon }_s\le {\epsilon }_e\hfill \\ \hfill \sigma =f_e\hfill & \hfill \mathrm{if}\hfill & \hfill {\epsilon }_e\le {\epsilon }_s\le {\epsilon }_{su}\hfill \\ \hfill \sigma =\mathrm{0}\hfill & \hfill \mathrm{if}\hfill & \hfill {\epsilon }_s\ge {\epsilon }_{su}\hfill \end{array}.

Study of the element cross section

To better approximate the area delimited by the contour of the cross section of any shape, the concrete section is defined by the series of trapezoidal table (Fig. 4).

The strains

The concrete normal strains

The section is subjected to the deformation state, characterized by {\epsilon }_x ,{\phi }_y and {\phi }_z, the normal strain in the coordinate point \left(y,z\right) of the concrete section is given by:

(6)\epsilon \left(y,z\right)={\epsilon }_x+{\phi }_{y}Z+{\phi }_{z}Y,

where \epsilon \left(y,z\right) is the normal strain; {\epsilon }_xis the deformation of the sections gravity center; {\phi }_yand {\phi }_zare the sections rotations around the axes G_yand G_z, respectively.

The normal strains will be designated by the \left\{{\epsilon }_n\right\} vector as:

(7)\left\{{\epsilon }_n\right\}={\left({\epsilon }_x,{\phi }_y,{\phi }_z\right)}^{\mathrm{T}}.

Shear strains

The tangents deformations can be written as the vector \left\{{\epsilon }_t\right\}defined by:

(8)\left\{{\epsilon }_t\right\}={\left({\gamma }_y,{\gamma }_z,{\theta }_x\right)}^{\mathrm{T}}.

Reinforcements strains

For increased strains \left\{\mathrm{\Delta }{\epsilon }_n\right\} and \left\{\mathrm{\Delta }{\epsilon }_t\right\} of the cross section it corresponds, for a concrete-inclined reinforcement, the strains such as:

(9)\left\{\mathrm{\Delta }{\epsilon }_f\right\}={\cos }^{\mathrm{2}}\alpha \left(\mathrm{1},z_f,y_f\right)\left\{\mathrm{\Delta }{\epsilon }_n\right\}+\sin \alpha ·\cos \alpha ·{\left[{\mathit{R}}_a\right]}^{\mathrm{T}}·\left\{\mathrm{\Delta }{\epsilon }_t\right\},

where \left[{\mathit{R}}_a\right] is the column matrix defined as follows:

(10)\left[{\mathit{R}}_a\right]=\left[\begin{array}{c}\hfill \cos \beta \hfill \\ \hfill \sin \beta \hfill \\ \hfill \left(z_f\right)·\cos \beta -\left(y_f\right)·\sin \beta \hfill \end{array}\right].

Calculation of internal forces

Normal forces due to the concrete

The internal forces increase due to the sections normal stresses increase are given by:

(11)\left\{\mathrm{\Delta }{\mathit{F}}_{mn}\right\}=\int \mathrm{\Delta }{\sigma }_m\left(y,z\right)·\left\{\begin{array}{c}\hfill \mathrm{1}\hfill \\ \hfill z\hfill \\ \hfill y\hfill \end{array}\right\}·\mathrm{d}{s}_m,

(12)\left\{\mathrm{\Delta }\mathit{\sigma }\left(y,z\right)\right\}=E_m\left(y,z\right)·\left\{\mathrm{\Delta }\epsilon \left(y,z\right)\right\}.

Given Eqs. (7), (11), (12) and after development we can write:

(13)\mathrm{\Delta }{\mathit{F}}_{mn}=\int E_m\left(y,z\right)·\left[\begin{array}{ccc}\hfill \mathrm{1}\hfill & \hfill z\hfill & \hfill y\hfill \\ \hfill z\hfill & \hfill z^{\mathrm{2}}\hfill & \hfill yz\hfill \\ \hfill y\hfill & \hfill yz\hfill & \hfill y^{\mathrm{2}}\hfill \end{array}\right]·\mathrm{d}{S}_m·\left\{\mathrm{\Delta }{\epsilon }_n\right\}.

That can be written in the matrix form

(14)\left\{\mathrm{\Delta }{\mathit{F}}_{mn}\right\}=\left[{\mathit{K}}_{mn}\right]·\left\{\mathrm{\Delta }{\mathit{\epsilon }}_n\right\},

where \left[{\mathit{K}}_{mn}\right]is cord stiffness matrix linked the increase of the normal forces and the sections normal strains increase.

Shear forces due to the concrete

In to the axis system OXYZ, We can also write

(15)\left\{{\mathit{F}}_{mt}\right\}=\left\{\begin{array}{c}\hfill T_{my}\hfill \\ \hfill T_{mz}\hfill \\ \hfill M_{mcx}\hfill \end{array}\right\},

where T_{my}and T_{mz} are shear forces and M_{mcx}is twisting moment around the torsion center.

We note \left[{\mathit{K}}_{mt}\right] the shear stiffness matrix linking the shear forces to the shear strains of the section and assuming this linear relationship.

We can write

(16)\left\{{\mathit{F}}_{mt}\right\}=\left[{\mathit{K}}_{mt}\right]·\left\{{\mathit{\epsilon }}_t\right\}.

Shear and torsion stiffness are assumed constant, the relationship is linear and is then valid for the forces increase. It can be written as follows:

(17)\left\{\mathrm{\Delta }{\mathit{F}}_{mt}\right\}=\left[{\mathit{K}}_{mt}\right]·\left\{\mathrm{\Delta }{\mathit{\epsilon }}_t\right\}.

\left[{\mathit{K}}_{mt}\right]: cord stiffness matrix given by

(18)\left[{\mathit{K}}_{mt}\right]=\left[\begin{array}{ccc}\hfill G·A_y\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill G·A_z\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill G·I_x\hfill \end{array}\right].

Finally, we can write Eqs. (14) and (17) in matrix form:

(19)\left(\begin{array}{c}\hfill \left\{\mathrm{\Delta }{\mathit{F}}_{mn}\right\}\hfill \\ \hfill \left\{\mathrm{\Delta }{\mathit{F}}_{mt}\right\}\hfill \end{array}\right)=\left[\begin{array}{cc}\hfill \left[{\mathit{K}}_{mn}\right]\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill \left[{\mathit{K}}_{mt}\right]\hfill \end{array}\right]·\left\{\begin{array}{c}\hfill \left\{\mathrm{\Delta }{\mathit{\epsilon }}_n\right\}\hfill \\ \hfill \left\{\mathrm{\Delta }{\mathit{\epsilon }}_t\right\}\hfill \end{array}\right\}.

Forces due to the reinforcements

1) Normal forces due to the reinforcement

The normal forces increase due to the reinforcement is given by Eq. (20):

(20)\mathrm{\Delta }{\mathit{F}}_{fn}=\left[{\mathit{K}}_{fn}\right]·\mathrm{\Delta }{\mathit{\epsilon }}_n+\left[{\mathit{K}}_{fnt}\right]·\mathrm{\Delta }{\mathit{\epsilon }}_t.

2) Tangential forces increase due to reinforcement

The tangential forces increase due to the reinforcement is given by the Eq. (21):

(21)\mathrm{\Delta }{\mathit{F}}_{ft}=\left[{\mathit{K}}_{ftn}\right]·\mathrm{\Delta }{\mathit{\epsilon }}_n+\left[{\mathit{K}}_{ft}\right]·\mathrm{\Delta }{\mathit{\epsilon }}_t.

We can write the Eqs. (20) and (21) in matrix form as follows:

(22)\left(\begin{array}{c}\hfill \mathrm{\Delta }{\mathit{F}}_{fn}\hfill \\ \hfill \mathrm{\Delta }{\mathit{F}}_{ft}\hfill \end{array}\right)=\left[\begin{array}{cc}\hfill {\mathit{K}}_{fn}\hfill & \hfill {\mathit{K}}_{fnt}\hfill \\ \hfill {}^t{\mathit{K}}_{fnt}\hfill & \hfill {\mathit{K}}_{ft}\hfill \end{array}\right].\left(\begin{array}{c}\hfill \mathrm{\Delta }{\mathit{\epsilon }}_n\hfill \\ \hfill \mathrm{\Delta }{\mathit{\epsilon }}_t\hfill \end{array}\right),

with

(23)\left[{\mathit{K}}_{fn}\right]={\displaystyle \sum }_{i=\mathrm{1}}^{n_f}{E}_{fi}·{\cos }^{\mathrm{3}}\alpha i·s_{fi}·\left[\begin{array}{ccc}\hfill \mathrm{1}\hfill & \hfill z_{fi}\hfill & \hfill y_{fi}\hfill \\ \hfill z_{fi}\hfill & \hfill z_{fi}^{\mathrm{2}}\hfill & \hfill y_{fi}{z}_{fi}\hfill \\ \hfill y_{fi}\hfill & \hfill y_{fi}{z}_{fi}\hfill & \hfill y_{fi}^{\mathrm{2}}\hfill \end{array}\right],

(24)\left[{\mathit{K}}_{fnt}\right]={\displaystyle \sum }_{i=\mathrm{1}}^{n_f}{E}_{fi}·s_{fi}·\sin \alpha i·{\cos }^{\mathrm{2}}\alpha i·\left[\begin{array}{ccc}\hfill \cos \beta i\hfill & \hfill \sin \beta i\hfill & \hfill y_{fi}\sin \beta i-z_{fi}\cos \beta i\hfill \\ \hfill z_{fi}\cos \beta i\hfill & \hfill z_{fi}\sin \beta i\hfill & \hfill z_{fi}·y_{fi}\sin \beta i-z_{fi}^{\mathrm{2}}\cos \beta i\hfill \\ \hfill y_{fi}\cos \beta i\hfill & \hfill y_{fi}\sin \beta i\hfill & \hfill y_{fi}^{\mathrm{2}}\sin \beta i-y_{fi}·z_{fi}\cos \beta i\hfill \end{array}\right],

(25)\left[{\mathit{K}}_{ftn}\right]={\left[{\mathit{K}}_{fnt}\right]}^t,

(26)\left[{\mathit{K}}_{ft}\right]={\displaystyle \sum }_{i=\mathrm{1}}^{n_f}{E}_{fi}·{\sin }^{\mathrm{2}}\alpha i·s_{fi}·V_{fti}·{}^t{V}_{fti},

With

(27)V_{fti}=\left(\begin{array}{c}\hfill \cos {\beta }_i\hfill \\ \hfill \sin B_i\hfill \\ \hfill y_{fi}\sin \beta i-z_{fi}\cos \beta i\hfill \end{array}\right).

External forces

The section of the beam element is subject to the external forces increase, which is given by:

(28)\mathrm{\Delta }{\mathit{F}}_{sn}=\left(\begin{array}{c}\hfill N\hfill \\ \hfill M_y\hfill \\ \hfill M_z\hfill \end{array}\right)\hspace{0.17em}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\mathrm{\Delta }{\mathit{F}}_{st}=\left(\begin{array}{c}\hfill T_y\hfill \\ \hfill T_z\hfill \\ \hfill M_{cx}\hfill \end{array}\right).

Equilibrium of the section

The equilibrium of the beam element section is expressed by the following equation:

(29)\left(\begin{array}{c}\hfill \mathrm{\Delta }{\mathit{F}}_{sn}\hfill \\ \hfill \mathrm{\Delta }{\mathit{F}}_{st}\hfill \end{array}\right)=\left[{\mathit{K}}_s\right].\left(\begin{array}{c}\hfill \mathrm{\Delta }{\mathit{\epsilon }}_n\hfill \\ \hfill \mathrm{\Delta }{\mathit{\epsilon }}_t\hfill \end{array}\right),

where \mathrm{\Delta }{\mathit{F}}_{sn} and \mathrm{\Delta }{\mathit{F}}_{st}are normal forces increase and shear forces increase.

\left[{\mathit{K}}_s\right]: cord sections stiffness matrix given by:

(30)\left[{\mathit{K}}_s\right]=\left[\begin{array}{cc}\hfill \left[{\mathit{K}}_{mn}\right]+\left[{\mathit{K}}_{fn}\right]\hfill & \hfill \left[{\mathit{K}}_{fnt}\right]\hfill \\ \hfill {}_{}{}^t\left[{\mathit{K}}_{fnt}\right]\hfill & \hfill \left[{\mathit{K}}_{mt}\right]+\left[{\mathit{K}}_{ft}\right]\hfill \end{array}\right].

For a given external forces increase \left\{{\mathit{F}}_s\right\} defined by Eq. (31), the resolution in the strains of the Eq. (29) is iterative, this method is described in detail in Refs. [³⁰,³¹]. At the section equilibrium, the strains increase \left\{\mathrm{\Delta }{\mathit{\epsilon }}_s\right\} defined by Eq. (32) is given by the Eq. (33):

(31)\left\{\mathrm{\Delta }{\mathit{F}}_s\right\}={\left(\mathrm{\Delta }N,\mathrm{\Delta }{M}_y,\mathrm{\Delta }{M}_z,\mathrm{\Delta }{T}_y,\mathrm{\Delta }{T}_Z,\mathrm{\Delta }{M}_c\right)}^{\mathrm{T}},

(32)\left\{\mathrm{\Delta }{\mathit{\epsilon }}_s\right\}=\left(\mathrm{\Delta }{\epsilon }_x,\mathrm{\Delta }{\phi }_y,\mathrm{\Delta }{\phi }_z,\mathrm{\Delta }{\gamma }_y,\mathrm{\Delta }{\gamma }_z,\mathrm{\Delta }{\theta }_x\right),

(33)\left\{\mathrm{\Delta }{\mathit{\epsilon }}_s\right\}=\left[{\mathit{S}}_s\right]·\left\{\mathrm{\Delta }{\mathit{F}}_s\right\},

(34)\left[{\mathit{S}}_s\right]={\left[{\mathit{K}}_s\right]}^{-\mathrm{1}}.

Flow chart for the research of the equilibrium section is illustrated in Fig. 5.

The structure’s stiffness matrix

The structure’s stiffness matrix [K] is formed from the element’s stiffness matrix [K_X] in the absolute axis system. The element stiffness matrix [K_X] is given from the section stiffness matrix in the calculated in the intrinsic axis system. For each loading step, the problem is to calculate the nodes displacement increase {ΔU} by solving of the following nonlinear system which describe the structure equilibrium:

(35)\mathrm{\Delta }P=\left[\mathit{K}\right]\left\{\mathrm{\Delta }U\right\}.

Key concepts

In the axis system XYZ, one positions of the local axis system x_{\mathrm{0}}{y}_{\mathrm{0}}{z}_{\mathrm{0}} of the element associated with here initial position, at the increasing of the loading, I_{\mathrm{0}} and j_{\mathrm{0}} nodes of the element are moved in I and J, respectively, then introduced the concept of intrinsic axis system, noted xyz. We can note \left\{{\mathit{F}}_L\right\} and \left\{{\mathit{S}}_L\right\} respectively the column matrix (12×1) containing the nodes forces, and the nodes displacements in the local axis system x_{\mathrm{0}}{y}_{\mathrm{0}}{z}_{\mathrm{0}} (Fig. 6).

\left\{{\mathit{F}}_L\right\}=

(36){\left(F_{ixo},F_{iyo},F_{izo},M_{ixo},M_{iyo},M_{izo},F_{jxo},F_{jyo},F_{jzo},M_{jxo},M_{jyo},M_{jzo}\right)}^{\mathrm{T}},

(37)\left\{{\mathit{S}}_L\right\}={\left(u_i,v_i,w_i,{\theta }_{ixo},{\theta }_{iyo},{\theta }_{izo},u_j,v_j,w_j,{\theta }_{ixo},{\theta }_{iyo},{\theta }_{izo}\right)}^{\mathrm{T}}.

The nodes forces and displacements in the intermediate axis system are:

(38)\left\{{\mathit{F}}_u\right\}={\left(F_{jxo},F_{jyo},F_{jzo},M_{ixo},M_{iyo},M_{izo},M_{jxo},M_{jyo},M_{jzo}\right)}^{\mathrm{T}},

(39)\left\{{\mathit{S}}_u\right\}={\left(u,v,w,{\theta }_{ixo},{\theta }_{iyo},{\theta }_{izo},{\theta }_{jxo},{\theta }_{jyo},{\theta }_{jzo}\right)}^{\mathrm{T}},

where

\begin{array}{c}\hfill u=u_j-u_i\hfill \\ \hfill v=v_j-v_i\hfill \\ \hfill w=w_j-w_i\hfill \end{array}.

The relation between the nodes forces and nodes displacements into the intermediate axis system and into the local axis is given by:

(40)\left\{{\mathit{S}}_u\right\}=\left[{\mathit{T}}_{\mathrm{0}}\right]·\left\{{\mathit{S}}_L\right\},

(41)\left\{{\mathit{F}}_L\right\}={\left[{\mathit{T}}_{\mathrm{0}}\right]}^{\mathrm{T}}·\left\{{\mathit{F}}_u\right\}.

With

\left[{\mathit{T}}_{\mathrm{0}}\right]=\left[\begin{array}{cccccccccccc}\hfill -\mathrm{1}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{1}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill -\mathrm{1}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{1}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill -\mathrm{1}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{1}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{1}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{1}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{1}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{1}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{1}\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{1}\hfill \end{array}\right].

Stiffness matrix of the element in the intrinsic axis system

\left\{{\mathit{F}}_n\right\}the column matrix (6×1) containing independent loads (internal loads at the ends of the element) in the intrinsic axis system

(42)\left\{{\mathit{F}}_n\right\}={(N_j,M_{iy},M_{iz},M_{jx},M_{jy},M_{jz})}^{\mathrm{T}},

(43)\{\begin{array}{c}\hfill N\left(x\right)=-N_j\hfill \\ \hfill M_y\left(x\right)=\left(\mathrm{1}-{\displaystyle \frac{x}{L}}\right)·M_{iy}-{\displaystyle \frac{x}{L}}{M}_{jy}\hfill \\ \hfill M_z\left(x\right)=-\left(\mathrm{1}-{\displaystyle \frac{x}{L}}\right)·M_{iz}+{\displaystyle \frac{x}{L}}{M}_{jz}\hfill \\ \hfill T_y\left(x\right)={\displaystyle \frac{\mathrm{1}}{L}}\left(M_{iz}+M_{jz}\right)\hfill \\ \hfill T_z\left(x\right)=-{\displaystyle \frac{\mathrm{1}}{L}}\left(M_{iy}+M_{jy}\right)\hfill \\ \hfill M_x\left(x\right)=-M_{jx}\hfill \end{array}.

The relation linking the sections loads to the nodes forces in the intrinsic axis system:

(44)\left\{{\mathit{F}}_s\right\}=\left[\mathit{L}\left(X\right)\right]·\left\{{\mathit{F}}_n\right\},

with

\left\{{\mathit{F}}_s\right\}={\left(N\left(x\right),M_y\left(x\right),M_z\left(x\right),T_y\left(x\right),T_z\left(x\right),M_c\left(x\right)\right)}^{\mathrm{T}},

\left[\mathit{L}\left(X\right)\right]=\left[\begin{array}{cccccc}\hfill -\mathrm{1}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill \left(\mathrm{1}-{\displaystyle \frac{X}{L}}\right)\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill -X/L\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill -\left(\mathrm{1}-{\displaystyle \frac{X}{L}}\right)\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill X/L\hfill \\ \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{1}/L\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{1}/L\hfill \\ \hfill \mathrm{0}\hfill & \hfill -\mathrm{1}/L\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill -\mathrm{1}/L\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill y_c/L\hfill & \hfill z_c/L\hfill & \hfill -\mathrm{1}\hfill & \hfill y_c/L\hfill & \hfill z_c/L\hfill \end{array}\right].

We can write considering the Eq. (44):

(45)\left\{\mathrm{\Delta }{F}_s\right\}=\left[\mathit{L}\left(X\right)\right]·\left\{\mathrm{\Delta }{\mathit{F}}_n\right\}.

Relationship between the nodes displacements and the sections strains

Any nodes forces increase \left\{\delta {\mathit{F}}_n\right\} generates:

- An external virtual work:

(46)\delta W_e={\left\{\delta {\mathit{F}}_n\right\}}^{\mathrm{T}}·\left\{{\mathit{S}}_n\right\}.

- A virtual work of the strains \delta W_isuch as:

(47)\delta W_i=\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}{\left\{\delta {\mathit{F}}_s\right\}}^{\mathrm{T}}·\epsilon \left(x\right)·\mathrm{d}x,

where \left\{\epsilon \left(x\right)\right\} represents the column matrix (6×1) containing the section strains in the abscissa x.

According to the complementary virtual work theorem:

(48)\delta W_i=\delta W_e,

after development we have:

(49)\left\{S_n\right\}={\displaystyle {\int }_{\mathrm{0}}^L{\left[\mathit{L}\left(X\right)\right]}^{\mathrm{T}}·\left\{\epsilon \left(x\right)\right\}·\mathrm{d}x}.

We can write from the Eq. (49):

(50)\left\{\mathrm{\Delta }{\mathit{S}}_n\right\}={\displaystyle {\int }_{\mathrm{0}}^L{\left[\mathit{L}\left(X\right)\right]}^{\mathrm{T}}·\left\{\mathrm{\Delta }\epsilon \left(x\right)\right\}·\mathrm{d}x}.

Relationship between the nodes forces and the nodes displacements

Using Eq. (33) linking the increase of the deformations with the increases of the forces in a cross-section, Eq. (50) becomes:

(51)\left\{\mathrm{\Delta }{\mathit{S}}_n\right\}=\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}{\left[\mathit{L}\left(x\right)\right]}^{\mathrm{T}}·\left(\left[{\mathit{S}}_s\right]·\left\{\mathrm{\Delta }{\mathit{F}}_s\right\}\right)·\mathrm{d}x.

Taking into account Eq. (45), we arrive at:

(52)\left\{\mathrm{\Delta }{\mathit{S}}_n\right\}={\left[{\mathit{K}}_n\right]}^{-\mathrm{1}}·\left\{\mathrm{\Delta }{\mathit{F}}_s\right\},

where {\left[{\mathit{K}}_n\right]}^{-\mathrm{1}}is reduced flexibility matrix (6×6) of the element in the intrinsic axis system xyz

(53){\left[{\mathit{K}}_n\right]}^{-\mathrm{1}}=\underset{\mathrm{0}}{\overset{L}{{\displaystyle \int }}}{\left[\mathit{L}\left(x\right)\right]}^{\mathrm{T}}·\left[{\mathit{S}}_s\right]·\left(\left[\mathit{L}\left(x\right)\right]\right)·\mathrm{d}x.

From Eq. (51), we come to the matrix relationship between the nodes forces increase and the nodes displacements increase, in the intrinsic axis system (x, y, z).

(54)\left\{\mathrm{\Delta }{\mathit{F}}_n\right\}=\left[{\mathit{K}}_n\right]·\left\{\mathrm{\Delta }{\mathit{S}}_n\right\}.

The matrix \left[{\mathit{K}}_n\right] is obtained from the section stiffness matrix in the intrinsic axis system. The matrix \left[{\mathit{K}}_n\right] takes into account the materials nonlinearity.

The integration of the terms of the matrix {\left[{\mathit{K}}_n\right]}^{-\mathrm{1}} is carried out numerically in the case of a nonlinear material behavior analysis by Simpson’s Rule [²⁸], considering the item an odd number of calculation sections. By cons if the behavior of materials constituting the element is linear, the compliance matrix is considerably reduced.

Stiffness matrix of an element in its local axis system

Relationship between the nodes displacements in the local axis system and the intrinsic axis system

For any time t, we can write according to the following geometric relationships

(55)\{\begin{array}{c}\hfill e=\sqrt{{\left(L_{\mathrm{0}}+u\right)}^{\mathrm{2}}+v^{\mathrm{2}}+w^{\mathrm{2}}}-L_{\mathrm{0}}\hfill \\ \hfill {\theta }_{iz}={\theta }_{iz\mathrm{0}}-{\theta }_v\hfill \\ \hfill {\theta }_{iy}={\theta }_{iy\mathrm{0}}+{\theta }_w\hfill \\ \hfill {\theta }_x={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}·\left({\theta }_{jx\mathrm{0}}-{\theta }_{ix\mathrm{0}}\right)\hfill \\ \hfill {\theta }_{jz}={\theta }_{jz\mathrm{0}}-{\theta }_v\hfill \\ \hfill {\theta }_{iy}={\theta }_{iy\mathrm{0}}-{\theta }_w\hfill \end{array},

where

(56)\{\begin{array}{c}\hfill {\theta }_V=\mathrm{Arcsin}{\displaystyle \frac{v}{\sqrt{{\left(L_{\mathrm{0}}+u\right)}^{\mathrm{2}}+v^{\mathrm{2}}+w^{\mathrm{2}}}}}\hfill \\ \hfill {\theta }_w=\mathrm{Arcsin}{\displaystyle \frac{w}{\sqrt{{\left(L_{\mathrm{0}}+u\right)}^{\mathrm{2}}+v^{\mathrm{2}}+w^{\mathrm{2}}}}}\hfill \end{array},

u, v, w, relative translation displacements of the nodes I and J expressed in the local axis system are given in the subsection 5.1.

The differentiation of expressions e,{\theta }_{iz}, {\theta }_{iy},{\theta }_{jz}{\theta }_x,, {\theta }_{jy} makes it possible to establish:

(57)\left\{\mathrm{\Delta }{\mathit{S}}_n\right\}=\left[\mathit{B}\right].\left\{\mathrm{\Delta }{\mathit{S}}_u\right\},

where [B] is matrix of geometric transformation.

Under the assumption of small displacements, we can content ourselves with the first order terms in the series expansion of the function’s partial derivatives.

(58)\left[\mathit{B}\right]=\left[\begin{array}{ccccccccc}\hfill \mathrm{1}\hfill & \hfill v/L_{\mathrm{0}}\hfill & \hfill w/L_{\mathrm{0}}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill \\ \hfill -w/L_{\mathrm{0}}^{\mathrm{2}}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{1}/L_{\mathrm{0}}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{1}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill \\ \hfill v/L_{\mathrm{0}}^{\mathrm{2}}\hfill & \hfill -\mathrm{1}/L_{\mathrm{0}}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{1}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill -\mathrm{1}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{1}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill \\ \hfill -w/L_{\mathrm{0}}^{\mathrm{2}}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{1}/L_{\mathrm{0}}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{1}\hfill & \hfill \mathrm{0}\hfill \\ \hfill v/L_{\mathrm{0}}^{\mathrm{2}}\hfill & \hfill -\mathrm{1}/L_{\mathrm{0}}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{1}\hfill \end{array}\right].

Relationship between the nodes forces and displacements in the local axis system

According to the theorem of virtual work, taking into account that the work of the forces are zero in the rigid body displacement of the intrinsic axis system in balance local axis system, we can say that the virtual work of the forces applied to the nodes \left\{{\mathit{F}}_n\right\} in the intrinsic axis system due to the virtual displacement \left\{\mathrm{\Delta }{\mathit{S}}_n\right\} is equal to the work of the forces applied to the nodes in the intermediate axis system, \left\{{\mathit{F}}_u\right\} due to a shift, that is to say:

(59){\left\{{\mathit{F}}_n\right\}}^{\mathrm{T}}·\left\{\mathrm{\Delta }{\mathit{S}}_n\right\}={\left\{{\mathit{F}}_u\right\}}^{\mathrm{T}}·\left\{\mathrm{\Delta }{\mathit{S}}_u\right\}\forall \left\{\mathrm{\Delta }{\mathit{S}}_u\right\},\hspace{0.17em}\forall \left\{\mathrm{\Delta }{\mathit{S}}_n\right\}.

Taking into account Eq. (57) we will have:

(60){\left\{{\mathit{F}}_n\right\}}^{\mathrm{T}}·\left[\mathit{B}\right]·\left\{\mathrm{\Delta }{\mathit{S}}_u\right\}={\left\{{\mathit{F}}_u\right\}}^{\mathrm{T}}·\left\{\mathrm{\Delta }{\mathit{S}}_u\right\}.

What gives

(61)\left\{{\mathit{F}}_u\right\}={\left[\mathit{B}\right]}^{\mathrm{T}}·\left\{{\mathit{F}}_n\right\}.

Differentiation of the Eq. (61), leads us to:

(62)\left\{\mathrm{\Delta }{\mathit{F}}_u\right\}={\left[\mathit{B}\right]}^{\mathrm{T}}·\left\{\mathrm{\Delta }{\mathit{F}}_n\right\}+{\left[\mathrm{\Delta }\mathit{B}\right]}^{\mathrm{T}}·\left\{{\mathit{F}}_n\right\}.

Considering the relations (54) and (57), Eq. (62) becomes:

(63)\left\{\mathrm{\Delta }{\mathit{F}}_u\right\}={\left[\mathit{B}\right]}^{\mathrm{T}}·\left[{\mathit{K}}_n\right]·\left[\mathit{B}\right]·\left\{\mathrm{\Delta }{\mathit{S}}_u\right\}+{\left[\mathrm{\Delta }\mathit{B}\right]}^{\mathrm{T}}·\left\{{\mathit{F}}_n\right\}.

Considering the term: {\left[\mathrm{\Delta }\mathit{B}\right]}^{\mathrm{T}}·\left\{{\mathit{F}}_n\right\}

The terms of the matrix [B] depend only on the partial derivatives of the functions e and q which are a function of relative displacements u, v, w of the nodes i and j.

After a development, we arrive at the following equation:

(64){\left[\mathrm{\Delta }\mathit{B}\right]}^{\mathrm{T}}·\left\{{\mathit{F}}_n\right\}=\left[\mathit{D}\right]·\left\{\mathrm{\Delta }{\mathit{S}}_u\right\}.

with \left[\mathit{D}\right]: geometric transformation matrix.

Given the hypothesis of small displacements, we can content ourselves with the first-order terms in the series development of the terms of the matrix [D]:

(65)\{\begin{array}{c}\hfill D_{\mathrm{11}}={\displaystyle \frac{-\mathrm{2}v\left(M_{iz}+M_{jz}\right)}{L_{\mathrm{0}}^{\mathrm{3}}}}+{\displaystyle \frac{\mathrm{2}w\left(M_{iy}+M_{jy}\right)}{L_{\mathrm{0}}^{\mathrm{3}}}}\hfill \\ \hfill D_{\mathrm{12}}=D_{\mathrm{21}}={\displaystyle \frac{-v{N}_j}{L_{\mathrm{0}}^{\mathrm{2}}}}+{\displaystyle \frac{\left(M_{iz}+M_{jz}\right)}{L_{\mathrm{0}}^{\mathrm{2}}}}\hfill \\ \hfill D_{\mathrm{13}}=D_{\mathrm{31}}={\displaystyle \frac{-w{N}_j}{L_{\mathrm{0}}^{\mathrm{2}}}}+{\displaystyle \frac{\left(M_{iy}+M_{jy}\right)}{L_{\mathrm{0}}^{\mathrm{2}}}}\hfill \\ \hfill D_{\mathrm{22}}={\displaystyle \frac{N_j}{L_{\mathrm{0}}}}+{\displaystyle \frac{\mathrm{2}v\left(M_{iz}+M_{jz}\right)}{L_{\mathrm{0}}^{\mathrm{3}}}}\hfill \\ \hfill D_{\mathrm{23}}=D_{\mathrm{32}}=\mathrm{0}\hfill \\ \hfill D_{\mathrm{33}}={\displaystyle \frac{N_j}{L_{\mathrm{0}}}}-{\displaystyle \frac{\mathrm{2}w\left(M_{iy}+M_{jy}\right)}{L_{\mathrm{0}}^{\mathrm{3}}}}\hfill \end{array}.

After development, the following relationship is obtained as follows:

(66)\left\{\mathrm{\Delta }{\mathit{F}}_u\right\}=\left[{\mathit{K}}_u\right]·\left\{\mathrm{\Delta }{\mathit{S}}_u\right\}.

by virtue the Eqs. (40) and (41) of the section 5.1.

The relationship between the forces increase and the displacement increase of the element in the local axis system is thus written

(67)\left\{\mathrm{\Delta }{\mathit{F}}_L\right\}=\left[{\mathit{K}}_L\right].\left\{\mathrm{\Delta }{\mathit{S}}_L\right\},

with \left[{\mathit{K}}_L\right]: the elements stiffness matrix in the local axis system, defined by:

(68)\left[{\mathit{K}}_L\right]={\left[{\mathit{T}}_{\mathrm{0}}\right]}^{\mathrm{T}}·\left({\left[\mathit{B}\right]}^{\mathrm{T}}·\left[{\mathit{K}}_n\right]·\left[\mathit{B}\right]+\left[\mathit{D}\right]\right)·\left[{\mathit{T}}_{\mathrm{0}}\right].

Stiffness matrix of an element in the absolute axis system

Either an element of the structure, initially between the node i_o and the node j_o linked to the local axis system x_oy_oz_o, and OXYZ the absolute axis system connected to the structure (see Fig. 6).

\left\{{\mathit{F}}_X\right\}and \left\{{\mathit{S}}_X\right\}are the columns matrices respectively (12×1) containing the nodes forces and the nodes displacements, in the absolute axis system OXYZ.

\left\{\mathrm{\Delta }{\mathit{F}}_X\right\}and \left\{\mathrm{\Delta }{\mathit{S}}_X\right\}: are the increments of the nodes forces and the nodes displacements in the absolute coordinate OXYZ

(69)\left\{\mathrm{\Delta }{\mathit{F}}_X\right\}={\left(\mathrm{\Delta }{F}_{iX},\mathrm{\Delta }{F}_{iY},\mathrm{\Delta }{F}_{iZ},\mathrm{\Delta }{M}_{iX},\mathrm{\Delta }{M}_{iY},\mathrm{\Delta }{M}_{iZ},\mathrm{\Delta }{F}_{jX},\mathrm{\Delta }{F}_{jY},\mathrm{\Delta }{F}_{jZ},\mathrm{\Delta }{M}_{jX},\mathrm{\Delta }{M}_{jY},\mathrm{\Delta }{M}_{jZ}\right)}^{\mathrm{T}},

(70)\{\mathrm{\Delta }{\mathit{S}}_X\}={(\mathrm{\Delta }{X}_i,\mathrm{\Delta }{Y}_i,\mathrm{\Delta }{Z}_i,\mathrm{\Delta }{\zeta }_{iX},\mathrm{\Delta }{\zeta }_{iY},\mathrm{\Delta }{\zeta }_{iZ},\mathrm{\Delta }{X}_j,\mathrm{\Delta }{Y}_j,\mathrm{\Delta }{Z}_j,\mathrm{\Delta }{\zeta }_{jX},\mathrm{\Delta }{\zeta }_{jY},\mathrm{\Delta }{\zeta }_{jZ})}^{\mathrm{T}}.

The matrix relationship between node displacements \left\{{\mathit{S}}_L\right\} in the local axis system and the nodes displacements \left\{{\mathit{S}}_X\right\} in the absolute axis system is written:

(71)\left\{{\mathit{S}}_L\right\}=\left[\mathit{R}\mathit{T}\right]·\left\{{\mathit{S}}_X\right\}.

In the case of the forces we can write:

(72)\left\{{\mathit{F}}_X\right\}={\left[\mathit{R}\mathit{T}\right]}^{\mathrm{T}}·\left\{{\mathit{F}}_L\right\}.

The differentiation of the Eq. (72) leads to:

(73)\left\{\mathrm{\Delta }{\mathit{F}}_X\right\}={\left[\mathit{R}\mathit{T}\right]}^{\mathrm{T}}·\left\{\mathrm{\Delta }{\mathit{F}}_L\right\}.

By substituting for Eq. (73) and Eq. (67), it comes

(74)\left\{\mathrm{\Delta }{\mathit{F}}_X\right\}={\left[\mathit{R}\mathit{T}\right]}^{\mathrm{T}}·\left[{\mathit{K}}_L\right]·\left\{\mathrm{\Delta }{\mathit{S}}_L\right\}.

The differentiation of Eq. (71) leads to:

(75)\left\{\mathrm{\Delta }{\mathit{S}}_L\right\}=\left[\mathit{R}\mathit{T}\right]·\left\{\mathrm{\Delta }{\mathit{S}}_X\right\}.

By substituting Eq. (74) for Eq. (75), we arrive at the matrix relation linking the increases of forces and the increments of displacements at the extremities of an element in the absolute axis system OXYZ, linked to the structure

(76)\left\{\mathrm{\Delta }{\mathit{F}}_X\right\}=\left[{\mathit{K}}_X\right]·\left\{\mathrm{\Delta }{\mathit{S}}_X\right\}.

\left[{\mathit{K}}_X\right]: stiffness matrix (12×12) of the element in the absolute coordinate OXZ given by:

(77)\left[{\mathit{K}}_X\right]={\left[\mathit{R}\mathit{T}\right]}^{\mathrm{T}}·\left[{\mathit{K}}_L\right]·\left[\mathit{R}T\right],

\left[\mathit{R}\mathit{T}\right]is orthogonal rotation matrix.

The relation between the increase of forces and displacements in the absolute axis system is obtained by means of the orthogonal rotation matrix \left[\mathit{R}\mathit{T}\right], such as:

(78)\left[\mathit{R}\mathit{T}\right]=\left[\begin{array}{cccc}\hfill \left[{\mathit{R}}_{\mathrm{0}}\right]\hfill & \hfill o\hfill & \hfill o\hfill & \hfill o\hfill \\ \hfill o\hfill & \hfill \left[{\mathit{R}}_{\mathrm{0}}\right]\hfill & \hfill o\hfill & \hfill o\hfill \\ \hfill o\hfill & \hfill o\hfill & \hfill \left[{\mathit{R}}_{\mathrm{0}}\right]\hfill & \hfill o\hfill \\ \hfill o\hfill & \hfill o\hfill & \hfill o\hfill & \hfill \left[{\mathit{R}}_{\mathrm{0}}\right]\hfill \end{array}\right],

(79)\left[{\mathit{R}}_{\mathrm{0}}\right]=\left[{\mathit{R}}_{\gamma }\right]·\left[{\mathit{R}}_{\beta }\right]·\left[{\mathit{R}}_{\alpha }\right],

where

\left[{\mathit{R}}_{\alpha }\right]=\left[\begin{array}{ccc}\hfill \cos \alpha \hfill & \hfill \mathrm{0}\hfill & \hfill \sin \alpha \hfill \\ \hfill \mathrm{0}\hfill & \hfill \mathrm{1}\hfill & \hfill \mathrm{0}\hfill \\ \hfill -\sin \alpha \hfill & \hfill \mathrm{0}\hfill & \hfill \cos \alpha \hfill \end{array}\right],

\left[{\mathit{R}}_{\beta }\right]=\left[\begin{array}{ccc}\hfill \cos \beta \hfill & \hfill \sin \beta \hfill & \hfill \mathrm{0}\hfill \\ \hfill -\sin \beta \hfill & \hfill \cos \beta \hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{1}\hfill \end{array}\right],

\left[{\mathit{R}}_{\gamma }\right]=\left[\begin{array}{ccc}\hfill \mathrm{1}\hfill & \hfill \mathrm{0}\hfill & \hfill \mathrm{0}\hfill \\ \hfill \mathrm{0}\hfill & \hfill \cos \gamma \hfill & \hfill \sin \gamma \hfill \\ \hfill \mathrm{0}\hfill & \hfill -\sin \gamma \hfill & \hfill \cos \gamma \hfill \end{array}\right].

Angles \alpha, \betaare according to the coordinates of the nodes given in the absolute axis system, linked to the structure.

Model validation

Software was developed in FORTRAN language; it is capable of solving the equilibrium of the structure in the three dimensions, taking into account the real behavior of materials.

This program was used to calculate several structures to make the comparison of results with those found experimentally. Flow chart for the research of the equilibrium structure is illustrated in Fig. 7.

The portico of Vecchio and Emara [³²]

This test is performed by Vecchio and Emara in 1992. The portico is recessed at the feet, consists in first applying a total vertical load of 700 kN to the each column, this load was maintained throughout the test. The lateral load (F) was applied to ruin portico. The resistance of concrete to compression is taken equal to 30 MPa, and the longitudinal modulus of concrete is taken equal 26 GPa. The yield strength of the steel is taken to be 418 MPa, and the plastic limit of the steel is taken to be 596 MPa. The longitudinal modulus of steel is taken as equal to 192.5 GPa. The dimensions of the chosen gantry, are referenced in the article [³²], see Fig. 8.

Figure 9 shows the evolution of the arrow according to the applied force F. The calculated curve approximates the experimental curve, the calculated maximum deflection is however lower; this is probably due to a numerical problem (divergence from the calculation due to the zero tangent).

Bending composite on the piles

We have studied the experimental tests conducted by Zhan [³³], in the laboratory CEBTP. These piles of diameter 500 mm of the control concrete, reinforced concrete and fiber concrete. The reinforcement of reinforced concrete pile consists of five steel bar HA diameter 16 mm (0.5% by volume); on scale test.

The piles are subjected to combine bending. The normal compression force is 1370 kN (applied using external prestressing), the geometric characteristics of the pile are shown in Fig. 10.

The pile is reinforced by the fibers at 25 kg/m² (corresponding to 0.31% by volume), it is noted BFAC. The characteristics of piles subjected tocombined bending are given in Tables 1 and 2.

Figures 11, 12, and 13 shows the comparison of results calculated with the result experimental.

Reinforced concrete pile (BA): experimentally the breaking strength of the pile is 410 kN, numerically: 434.44 kN. The numerical curve is very close to the experimental curve which explains the modeling done correctly predicted the behavior of structures.

Reinforced concrete pile with the steel fiber at 25 kg/m³ (BFAC): experimentally the force at break pile is 390 kN, numerically 400.15 kN, in the linear and nonlinear part (the phase of the development of cracks with participation concrete fiber) numerical calculation is good, deformations stop at a value less than those of the experimental for the third phase (post cracking with participation of fiber concrete) only small deformations were considered.

Pile control concrete (BT) experimentally breaking pile to 372 kN, numerically 375.37 kN. Results show that after reaching the breaking strength, the numerical calculation of deformations approach the experimental.

We also found an improvement of the arrow at break with respect to the reinforced concrete of a significant percentage for the fiber concrete pile.

FOURE test

This is a series of tests performed by Fourré [³⁴] (OG1, OG2, OG3, OG4,...).

We have studied the OG3 beam, which rests on two simple supports located at a distance of 3 m between supports and 0.5 m between loads (Fig. 14). Details of sections and loading are shown in Fig. 15. The experiment in the laboratory is described in detail in Ref. [³⁴]. The mechanical characteristics measured on samples are presented in the Tables 3 and 4.

As can be seen in Fig. 16 the good correlation between the calculation and the experiment. The calculation ruin load is 75.89 kN and a real ruin load 78.04 kN.

Conclusions

Material nonlinearity defines loads curves- arrows anywhere structures, verifying the ability of structures until failure (arrows, forces, cracking). They become more convenient because the non linear behavior is determined. The results obtained are encouraging. The numerical method used seems to give satisfactory accuracy of results compared with experimental results, for a reasonable calculation time. This model can be simulating the global behavior of the structures in the case of the reinforced concrete and fibber reinforced concrete. So, this model based on the tangent stiffness matrix can only simulate the structures behavior until ultimate force, it can’t carry out the softening brunch after this ultimate force. In the case of the fibber reinforced concrete, we can see that this model can be simulating the structure behavior seemly with an excellent correlation until ultimate force. It can be able to taking account the materials real behavior in compression and in traction. It can equally predict correctly the deflexion corresponding to the maximal loads in the case of the reinforced concrete and fiber reinforced concrete. It is interesting to perform a method able to taking account a non linear shear behavior for a better estimation of the structures deformability subjected to high value of shear force in order also to be able to predict the structures shear failure.

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