"Given this evaluation, we can directly construct the load-deflection curve depicted in the right diagram of the above Figure. The appealing feature of this evaluation scheme is the efficiency in comparison to a nonlinear finite-element model."
]
},
{
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"id": "60aab350-850a-479a-bc40-034d6e097ef1",
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...
...
@@ -201,27 +209,26 @@
"id": "5cfbc2a1-76ec-4466-a7bb-d4e75a9976f0",
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"We want to establish the relation between $\\kappa$ and the corresponding bending moment $M$. Therefore, the evaluation starts by prescribing the \n",
"values of $\\varepsilon_\\mathrm{top}$ in a range given by the failure strain of the matrix or of the reinforcement. However, the curvature is given by two values, i.e. $\\varepsilon_\\mathrm{bot}$ and $\\varepsilon_\\mathrm{top}$. What is the value of $\\varepsilon_\\mathrm{bot}$ corresponding to the prescribed value of $\\varepsilon_\\mathrm{top}$. The answer can be found by applying the equilibrium condition requiring that the normal force within the cross section.\n",
"The force in concrete $F_\\mathrm{c}$ is obtained by integrating the stress over the cross sectional height and the force in the reinforcement is obtained by summing up the crack bridging actions into F_\\mathrm{r}. The equilibrium condition then reads \n",
"**Iterative scheme:** We want to establish the relation between $\\kappa$ and the corresponding bending moment $M$. Therefore, the evaluation starts by prescribing the value of $\\kappa$ and finding the corresponding strain $\\varepsilon_\\mathrm{bot}$ for which the equilibrium condition of normal forces in the cross section is satisfied. Note that the strain at any coordinate $z$ is then given by as a linear profile\n",
"Since the strain at each vertical position $z$ is known, we can use the constitutive law of the concrete matrix and of the reinforcement to evaluate the stress profile in concrete $\\sigma_\\mathrm{c}(z)$ over the height, so that the concrete force is given as\n",
" \\varepsilon_\\mathrm{top} = \\kappa z + \\varepsilon_{bot}\n",
"$$\n",
"Thus, by prescribing a fixed value of $\\kappa$ and by choosing a trial value $\\varepsilon_{bot}$, the strain value of each concrete ligament and of each reinforcement layer is known. \n",
"With the strain value at each vertical position $z$ is known, we can use the constitutive law of the concrete matrix and of the reinforcement to evaluate the stress profile in concrete $\\sigma_\\mathrm{c}(z)$ over the height, so that the concrete force is given as\n",
"$$\n",
"F_\\mathrm{c} = \\int_0^h \\sigma_\\mathrm{c}(z) \\cdot b \\, \\mathrm{d}z\n",
"$$\n",
"and the individual crack bridging force contributions of the reinforcement layers $i$ as\n",
"This equation is not fufilled for the trial value of $\\varepsilon_\\mathrm{bot}$. \n",
"Therefore, the evaluation is repeated for the trial values of $\\varepsilon_{bot}$ until a horizontal equilibrium is achieved. Mathematically, the nonlinear equation can be solved using root finding methods provided in mathematical packages. In the present implementation, the method \n",
"[scipy.optimize.root](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.root.html) has been used."
]
},
{
...
...
@@ -235,6 +242,35 @@
"\\end{align}"
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"**Example:** Evaluate the $M(\\kappa)$ relation for a steel-reinforced concrete cross section with the geometry given in [cm] and reinforcement ratio $\\rho = 0.7\\%$. Discuss the correspondence between the stress profile of concrete over the cross section and the constitutive law applied as an input?"
With the coverage of bond and cracking behavior, we can now approach the simulation of a composite as a whole. The described material models can be found and used in general finite tools. It is possible to simulate the material response using 3D solid finite element models. However, in engineering practice, more specialized methods of structural analysis and design are desired which allow us to get the answers to repeating questions. A frequent application of composite materials is provided in form of a beam exposed to bending. Let us show how to utilize the models and theory explained during the previous Tours and show how to apply them to provide an efficient and general method for an assessment of a beam.
One fundamental concept of engineering design is to decompose a complex problem into smaller subproblems that can be described managed in a more transparent way. This concept is relevant also for the description of structural behavior. The present notebook shows such a decomposition for a nonlinear structural analysis of an arbitrarily reinforced beam. It can consider steel, carbon, glass, basalt reinforcement with correspondingly defined constitutive laws. At the same time, various cross sectional shapes can be included as well.
Using the analysis method explained in the present notebook, is is possible to explain the fundamental differences between the design of traditional steel-reinforced concrete structures, and the innovative TRC and carbon concrete structures. Understanding of these differences is crucial for achieving more economic, durable and, thus, sustainable design of structures in the future.
In case of a simply supported beam, the profile of bending moment is known a-priori as it does not depend on the material behavior. In other words, there are no redistribution effects along the beam. Therefore, if we know the relation between the bending moment $M$ and the local curvature $\kappa$ as depicted in the left figure above, we can calculate the deflection of the beam by integrating the curvature to obtain the cross sectional rotation, i.e.
$$
\varphi(x) = \int_0^L \kappa(x) \, \mathrm{d}x + C
$$
and the deflection as
$$
w(x) = \int_0^L \varphi(x) \, \mathrm{d}x + Cx + D
$$
The integration constants must be solved for particular boundary conditions.
Given this evaluation, we can directly construct the load-deflection curve depicted in the right diagram of the above Figure. The appealing feature of this evaluation scheme is the efficiency in comparison to a nonlinear finite-element model.
To accomplish this kind of evaluation, however, we need to derive the $M-\kappa$ relation. This relation is affected by the stress redistribution occurring within a cross section during the cracking. The crack propagation demonstrated in the notebook [7.1](../tour7_cracking/7_1_bending3pt_2d.ipynb#top) has already demonstrated the propagation of the stress peak through the cross section. While the example shown there considered a plain concrete without reinforcement, in the present notebook, we shall consider also the effect of the reinforcement.
We want to establish the relation between $\kappa$ and the corresponding bending moment $M$. Therefore, the evaluation starts by prescribing the
values of $\varepsilon_\mathrm{top}$ in a range given by the failure strain of the matrix or of the reinforcement. However, the curvature is given by two values, i.e. $\varepsilon_\mathrm{bot}$ and $\varepsilon_\mathrm{top}$. What is the value of $\varepsilon_\mathrm{bot}$ corresponding to the prescribed value of $\varepsilon_\mathrm{top}$. The answer can be found by applying the equilibrium condition requiring that the normal force within the cross section.
The force in concrete $F_\mathrm{c}$ is obtained by integrating the stress over the cross sectional height and the force in the reinforcement is obtained by summing up the crack bridging actions into F_\mathrm{r}. The equilibrium condition then reads
**Iterative scheme:** We want to establish the relation between $\kappa$ and the corresponding bending moment $M$. Therefore, the evaluation starts by prescribing the value of $\kappa$ and finding the corresponding strain $\varepsilon_\mathrm{bot}$ for which the equilibrium condition of normal forces in the cross section is satisfied. Note that the strain at any coordinate $z$ is then given by as a linear profile
Since the strain at each vertical position $z$ is known, we can use the constitutive law of the concrete matrix and of the reinforcement to evaluate the stress profile in concrete $\sigma_\mathrm{c}(z)$ over the height, so that the concrete force is given as
Thus, by prescribing a fixed value of $\kappa$ and by choosing a trial value $\varepsilon_{bot}$, the strain value of each concrete ligament and of each reinforcement layer is known.
With the strain value at each vertical position $z$ is known, we can use the constitutive law of the concrete matrix and of the reinforcement to evaluate the stress profile in concrete $\sigma_\mathrm{c}(z)$ over the height, so that the concrete force is given as
$$
F_\mathrm{c} = \int_0^h \sigma_\mathrm{c}(z) \cdot b \, \mathrm{d}z
$$
and the individual crack bridging force contributions of the reinforcement layers $i$ as
This equation is not fufilled for the trial value of $\varepsilon_\mathrm{bot}$.
Therefore, the evaluation is repeated for the trial values of $\varepsilon_{bot}$ until a horizontal equilibrium is achieved. Mathematically, the nonlinear equation can be solved using root finding methods provided in mathematical packages. In the present implementation, the method
[scipy.optimize.root](https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.root.html) has been used.
**Example:** Evaluate the $M(\kappa)$ relation for a steel-reinforced concrete cross section with the geometry given in [cm] and reinforcement ratio $\rho = 0.7\%$. Discuss the correspondence between the stress profile of concrete over the cross section and the constitutive law applied as an input?
The described scheme represents a commonly used concept of interative scheme to identify the $M-\kappa$ relation. The implementation provided here exploits the flexibility of the Python ecosystem of packages for scientific computing, in particular, the `scipy` solvers of nonlinear equations. Moreover, by introducing the integration over the cross section with a variable width, various cross sectional shapes and nonlinear constitutive laws are covered.
The above shown model has been implemented in the model component `MKappa`. covered range of configurations with the specification of inputs and outputs can be summarized as follows
To relate described model to the previously shown description of the bended beam in the notebook [7.2](../tour7_cracking/7_1_bending3pt_2d.ipynb#top), let us first simulate a beam without reinforcement. The model is constructed as follows
By rendering the interactive interface, we can inspect the cross sectional design `cs_design`. The attribute `matrix` specifies the material law of the matrix. The `cross_section_layout` is an empty list, because no reinforcement has been specified. Finally,
The nonlinear moment curvature curve captures the interaction between the tensile and compressive zone of a beam. Depending on the cross-sectional design, the peak load marks either the tensile or compressive failure of the cross section. The `MKappa` model component can be used to verify the cross sectional capacity calculated using the assessment rules in the design codes.
As an example, let us construct a steel reinforced cross section and configure the cross section in such a way either the compressive yielding of concrete or tensile yielding of steel will occur.
