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

doi:10.1007/s11709-017-0433-7

Front. Struct. Civ. Eng.

2018, Vol. 12

Issue (4) : 439-453 https://doi.org/10.1007/s11709-017-0433-7

RESEARCH ARTICLE

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

Fatiha IGUETOULENE(

), Youcef BOUAFIA, Mohand Said KACHI

University Mouloud Mammeri of Tizi-Ouzou, 15000, Algeria

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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

Corresponding Author(s): Fatiha IGUETOULENE

Online First Date: 31 October 2017 Issue Date: 20 November 2018

Cite this article:

Fatiha IGUETOULENE,Youcef BOUAFIA,Mohand Said KACHI. Non linear modeling of three-dimensional reinforced and fiber concrete structures[J]. Front. Struct. Civ. Eng., 2018, 12(4): 439-453.

URL:

https://academic.hep.com.cn/fsce/EN/10.1007/s11709-017-0433-7
https://academic.hep.com.cn/fsce/EN/Y2018/V12/I4/439

Fig.1 Uniaxial constitutive law of the concrete compression

Fig.2 Stress-Strain behavior of tensile concrete

Fig.3 Stress-strain of fiber concrete [¹³]

Fig.4 Discretization of the concrete section into trapezoidal tables

Fig.5 Flow chart of the procedure for the research of equilibrium section

Fig.6 Axis system of an element in 3D- initial and deformed state of the element

Fig.7 Flow chart of the procedure for the research of equilibrium structure

Fig.8 The portico modeling of Vecchio and Emara

Fig.9 Evolution lateral displacement according to lateral load

Fig.10 Composed bending test for reinforced concrete pile with circular section

Index of the test piece	$f c j$ (MPa)	$f t j$ (MPa)	$E b 0$ (GPa)	$R b$	$R c$	$ε r t$ (‰)	$ε 0$ (‰)
BA	41.42	3.08	34.40	1.6	0.7	–20	2.1
BT	44.69	2.94	34.40	1.6	0.7	–5	2.1

Tab.1 Mechanical properties (measuring cylinders (ZHAN)) of the reinforced concrete (BA), measured at 350 kg/m³ of cement; and control concrete (BT), measured at 400 kg/m³ of cement

Index of the test piece	$f c j$ (MPa)	$f t j$ (MPa)	$E b 0$ (GPa)	$R b$	$R c$	$ε r t$ (‰)	$ε 0$ (‰)	$ε c u$ (‰)
BFAC	47.6	3.37	38.18	1.6	0.7	–50	2.1	3.5
Index of the test piece	$E a$ (GPa)	$l f$ (mm)	$ω$ (%)	$φ$ (mm)	$ε u$ (‰)	$τ u$ (Mpa)
BFAC	200	60	0.31	1	–0.74	7

Tab.2 Mechanical properties of the composite (reinforced concrete with steel fibers). Measuring cylinders (ZHAN)

Fig.11 Comparison of the load-deflection curve between simulation and experiment (BT)

Fig.12 Comparison of results calculated with the result experimental BA pile

Fig.13 Comparison of results calculated with the result experimental BFAC pile

Fig.14 OG3 beam

Fig.15 Detail of reinforcement for beam OG3 and the Section A-A of the OG3 beam

Reference	Diameter $φ$ (mm)	Module $E a$ (MPa)	Elasticity limit $f e$ (MPa)	Stress of rupture $f r$
OG3	16	2.05×10⁵	575	700
OG3	6	2.05×10⁵	215	700

Tab.3 Mechanical characteristics of the sample

Mixed	Age j (day)	Compression				Traction (Bending)
Mixed	Age j (day)	$f c j$ (Mpa)	$E i j$ (MPa)	$E d$ (MPa)	$ε b$ (‰)	$f t j$ (MPa)
OG3	44	51.3	41600	42750	1.58	2.8

Tab.4 Mechanical characteristics of concrete measured on specimens

Fig.16 Load-deflection curve for the beam OG3

$[K]$	Stiffness matrix
${Δ U}$	Vector of nodes displacements increase
${Δ F}$	Vector nodes forces increase
${Δ P}$	Vector of applied loads increase
${Δ S}$	Vector of nodes displacements increase
[Ks]	Sections Stiffness matrix
[Ss]	Sections flexibility matrix
[B], [D]	Matrix of geometrical transformations
[RT]	Matrix linking the local coordinates and the absolute coordinates
e	Longitudinal stretching of the element
$E a$	Elastic modulus of passive reinforcement
$E b 0$	Elastic modulus of concrete
$f c j$	Concrete compressive strength at j day
$f t j$	Concrete tensile strength at j day
$ε b 0$	Peak of the strains corresponding to $f c j$
$G$	Shear modulus
$K b$ and $K ′ b$	Dimensionless parameters of the Sargin low
Lo	Bar length before deformation
L	Bar length after deformation
M	Bending moment
N	Normal load
T	Shear
u, v	Longitudinal displacements of the nodes
q, z	Rotations of the elements
$ε$	Longitudinal deformation
$σ e$	Elastic stress from passive steel
$σ r$	Tensile strength of steel reinforcement
$E c t$	Initial modulus of the composite in tension
$f f t$	Tensile strength of the composite
$ε u$	Ultimate strain of the composite
$ε r$	Fracture strain of the composite
$ε s u$	Ultimate steel strain
$σ u c$	Residual stress
$ω ¯$	Percentage of fibers
$θ o$	Fiber orientation coefficient
$l f$	Length of the fiber
$τ u$	Ultimate bounding stress of the fiber
$φ$	Diameter of the fiber
G	Shear modulus
$A y$ , $A z$	Reduced sections
$I x$	Torsion inertia of section
$γ y$ , $γ z$	The shear strains in the plane xy and xz
$θ x$	The angle of torsion
$α$	Angle between the center line of the reinforcement and the axis G_x
$β$	Angle between the reinforcement, projection in the plane of the section and the axis G_y
( $y f$ , $z f$ )	The reinforcement crossing point in the coordinate G_yz axis system
${Δ ε n}$	Normal strains increase
${Δ ε t}$	Shear strains increase
$E m (y, z)$	Concrete modulus
$[S s]$	The flexibility matrix of the section, $β i$ The angle between the projection of the reinforcement bare i in the plan $G y z$ , and the axis $G y$
$(y f i, z f i)$	The coordinate of the reinforcement bare i in the axis system $G y z$
$α i$	the angle between of the reinforcement bare i and the axis $G x$
$s f i$	The cross-section of the reinforcement bare i
$E f i$	The elastic modulus of the reinforcement

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