Abstract
Composite steel–concrete structures are commonly used in the field of bridges, where the steel frame provides great ease of installation, and concrete provides useful strength at low cost. This construction system makes it possible to seek to use each material to the best of its ability, so as to provide the entire construction system with greater savings. The purpose of this article is to be able to perform the simulation and the non-linear elastic calculation of composite steel–concrete beams through a calculation approach based on a matrix method of displacements. The numerical calculation model developed is based on taking into account the non-linearity of materials, or a set of laws allowing the modeling of the nonlinear behaviors of materials under an instantaneous and monotonous loading increasing until the ruin; the concrete is represented in its post-elastic part by a softening branch in compression and the contribution of the concrete stretched between two successive cracks is taken into account. Steel is represented by a perfect elastoplastic law or an elastoplastic law with firming. The proposed approach has been implemented on the Fortran programming language, where our procedure of numerical modeling of the mechanical behavior seems capable of correctly simulating the three-dimensional nonlinear behavior of isostatic and hyperstatic composite steel–concrete beams, under monotonous (increasing) static loading until ruin. It was validated by comparing the results of our calculations to experimental results or to analytical solutions.
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Abbreviations
- \(\varphi \left( \varepsilon \right)\) :
-
The actual behavior of the materials,
- \({\text{E}}_{{{\text{b}}0}}\) :
-
Concrete modulus at the origin,
- \({\upvarepsilon }_{{{\text{b}}0}}\) :
-
Peak strain corresponding to \({\text{f}}_{{{\text{cj}}}}\),
- \({\text{f}}_{{{\text{cj}}}}\) :
-
Concrete compressive strength at the age j,
- \({\text{f}}_{{{\text{cc}}}}\) :
-
Minimum characteristic resistance of concrete on cube,
- \({\text{k}}_{{\text{b}}}\) :
-
Dimensionless parameters, sargin law,
- \(\grave{\hbox{k}}_{{\text{b}}}\) :
-
Dimensionless parameters, sargin law,
- \({\text{f}}_{{{\text{tj}}}}\) :
-
Concrete tensile strength,
- \({\upvarepsilon }_{{{\text{rt}}}}\) :
-
Steel ultimate strain,
- \({\upvarepsilon }_{{{\text{bt}}}}\) :
-
Concrete fiber tensile strain,
- \({\upvarepsilon }_{{{\text{ft}}}}\) :
-
Tensile strain corresponding to \({\text{ f}}_{{\text{tj }}}\),
- \({\upvarepsilon }_{{{\text{s}}1}}\) :
-
Strain corresponding to the end of the plastic bearing,
- \({\upvarepsilon }_{{{\text{s}}2}}\) :
-
Strain corresponding to the end of the firming,
- \({\upvarepsilon }_{{{\text{su}}}}\) :
-
Breaking strain,
- \({\text{E}}_{{\text{a}}}\) :
-
Steel longitudinal modulus,
- \({\upvarepsilon }_{{\text{e}}}\) :
-
Limit elastic strain of the steel,
- \({\upsigma }_{{\text{e}}}\) :
-
Elastic yield stress of steel,
- \({\upvarepsilon }_{{\text{u}}}\) :
-
Ultimate strain of steel,
- \({\text{E}}_{{\text{a}}}\) :
-
Young's modulus of steel at the origin,
- \({\upsigma }_{{\text{e}}} { }\) :
-
Conventional elastic limit at 2‰,
- \({\upsigma }_{{\text{p}}} { }\) :
-
Stress in the prestressing steel,
- \({\upvarepsilon }_{{\text{p}}} { }\) :
-
Strain in the prestressing steel,
- \({\text{E}}_{{\text{p}}} { }\) :
-
Young's modulus at the origin in the prestressing steel,
- fpeg :
-
The conventional elastic limit at 0.1%
- \({\upvarepsilon }_{{{\text{su}}}}\) :
-
Failure strain
- \({\upvarepsilon }_{{\text{e}}}\) :
-
Steel yield strain
- \({\upsigma }_{{\text{e}}}\) :
-
Steel yield stress,
- \({\upvarepsilon }_{{\text{u}}}\) :
-
Steel ultimate strain,
- \({\text{E}}_{{\text{a}}}\) :
-
Steel modulus at the origin,
- \({\upvarepsilon }_{{\text{x}}}\) :
-
Gravity center strain,
- \({\text{N}}_{{{\text{btr}}}}\) :
-
Number of concrete trapezoid,
- \({\text{N}}_{{{\text{am}}}}\) :
-
Number of steel profile trapezoid,
- \({\text{N}}_{{\text{s}}}\) :
-
Number of passive reinforcing beds,
- \({\text{A}}_{{{\text{ai}}}}\) :
-
Passive steel bed area,
- \({\text{y}}_{{{\text{ai}}}}\) :
-
Passive steel bed ordinate,
- \({\text{y}}_{{\text{i}}}\) :
-
Lower trapezoid ordinate,
- \({\text{y}}_{{{\text{i}} + 1}}\) :
-
Upper trapezoid ordinate,
- \({\text{b}}_{{\text{i}}}\) :
-
The lower abscissa of trapezoid along the x axis,
- \({\text{b}}_{{{\text{i}} + 1}}\) :
-
The upper abscissa of trapezoid along the x axis,
- \(\left\{ {{\text{ F}}_{{{\text{mn}}}} } \right\}\) :
-
Internal loads vector,
- \({\text{E}}_{{\text{m}}} \left( {{\text{y}},{\text{z}}} \right)\) :
-
The longitudinal elastic modulus at a current point of the section,
- \({\text{S}}_{{\text{b}}}\) :
-
The concrete cross-section,
- \({\text{S}}_{{\text{p}}}\) :
-
The metal profile’s cross-section,
- \(\Delta {\upsigma }_{{\text{m}}} \left( {{\text{y}},{\text{z}}} \right)\) :
-
Normal stress increase in a current point,
- \({\text{E}}_{{{\text{ai}}}}\) :
-
The passive reinforcement elastic modulus,
- \(\left[ {{\text{k}}_{{{\text{mn}}}} } \right]{ }\) :
-
Section stiffness matrix,
- \(\left\{ {{\text{ F}}_{{{\text{sn}}}} } \right\}\) :
-
Section load vector,
- \({\text{e}}\) :
-
Element length increase,
- \({\text{L}}_{0}\) :
-
Element initially length,
- L:
-
Element length after deformation,
- \(\left[ {\text{B}} \right]{ }\) :
-
Geometric transformation matrix,
- \(\left[ {{\text{K}}_{{\text{S}}} } \right]{ }\) :
-
Global stiffness matrix of the composite cross section,
- \(\left[ {{\text{K}}_{{\text{L}}} } \right]\) :
-
Element stiffness matrix in the local coordinate,
- \(\left[ {{\text{K}}_{{\text{X}}} } \right]{ }\) :
-
Element stiffness matrix in the absolute coordinates,
- \(\left[ {{\text{K}}_{{\text{N}}} } \right]{ }\) :
-
Element stiffness matrix in the intrinsic system coordinates,
- \(\left[ {{\text{K}}_{{\text{U}}} } \right]{ }\) :
-
Element Stiffness matrix in the intermediate local system coordinate,
- \(\left\{ {{\text{F}}_{{\text{X}}} } \right\}{ }\) :
-
The nodes load vector in the absolute system coordinate,
- \(\left\{ {{\text{S}}_{{\text{X}}} } \right\}{ }\) :
-
The nodes displacement vector in the absolute coordinate system,
- \(\left\{ {{\text{F}}_{{\text{L}}} } \right\}\) :
-
The nodes load in the local coordinate system,
- \(\left\{ {{\text{S}}_{{\text{L}}} } \right\}\) :
-
The nodes displacement vector in the local system coordinate,
- \(\left\{ {{\text{F}}_{{\text{U}}} } \right\}\) :
-
The nodes load vector in the intermediate system coordinate,
- \(\left\{ {{\text{S}}_{{\text{U}}} } \right\}\) :
-
The nodes displacement vector in the intermediate system coordinate,
- \({\text{u}}_{{\text{i}}}\), \({\text{v}}_{{\text{i}}}\), \({\text{w}}_{{\text{i}}}\) :
-
Components of the displacement vector in the local coordinate system,
- \(\left[ {{\text{S}}_{{\text{S}}} } \right]_{{{\text{i}} - 1}} { }\) :
-
Sections flexibility matrix of the iteration (i-1),
- \(\varepsilon_{{\text{s}}} { }\) :
-
Strains balanced in the previous step,
- \(\left\{ {\Delta {\text{F}}_{{\text{s}}} } \right\}{ }^{{\text{r}}}\) :
-
Loads increase in the step r,
- \(\left\{ {\Delta {\upvarepsilon }} \right\}{ }_{0} { }\) :
-
Initial strains increase,
- \(\left[ {\text{K}} \right]_{{\text{i}}} { }\) :
-
Structure stiffness matrix at the iteration (i),
- \(\left\{ {{\text{ U}}_{{\text{s}}} } \right\}{ }\) :
-
Node displacement vector at the latest stable step,
- \(\left\{ {{\Delta P}} \right\}{ }^{{\text{r}}}\) :
-
Applied load increase in the r step,
- \(\left\{ {\text{P}} \right\}{ }\) :
-
External structures applied loads,
- \(\left\{ {{\text{P}}^{{{\text{int}}}} } \right\}{ }\) :
-
Internal structures applied loads,
- \({\text{G}}.{\text{A}}_{{\text{y}}}\) :
-
The rigidity of the section with the shear force in the xy plane,
- \({\text{G}}.{\text{A}}_{{\text{z}}}\) :
-
The rigidity of the section with the shear force in the xz plane,
- \({\text{G}}.{\text{J}}\) :
-
The torsional stiffness of the section,
- \(\rho\) :
-
Density of the concrete
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Benyahi, K., Bouafia, Y., Oudjene, M. et al. Numerical Procedure for the Three-Dimensional Nonlinear Modelling of Composite Steel–Concrete Beams. Int J Steel Struct 21, 1063–1081 (2021). https://doi.org/10.1007/s13296-021-00490-1
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DOI: https://doi.org/10.1007/s13296-021-00490-1