The DEFI_CABLE_BP operator ========================= Assessment of cable layout characteristics ---------------------------------------- Cubic spline interpolation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Here we present the method used to obtain geometric interpolation of cables, which is essential to accurately calculate the curvilinear abscissa and the angle :math:`\alpha` used in prestress loss formulas. We start by constructing an interpolation of the trajectory of the cable (in fact an interpolation of the two projections of the trajectory in the two planes :math:`\mathit{Oxy}` and :math:`\mathit{Oxz}`), then from these interpolations, we estimate the curvilinear abscissa, and the cumulative angular deviation, according to the formulas: .. _RefEquation 2.1-1: :math:`s(x)\mathrm{=}\underset{0}{\overset{x}{\mathrm{\int }}}\sqrt{1+{y}^{\text{'}2}(x)+{z}^{\text{'}2}(x)}\text{dx}` eq 2.1-1 :math:`\alpha (x)\mathrm{=}\underset{0}{\overset{x}{\mathrm{\int }}}\frac{\sqrt{{y}^{\text{'}\text{'}2}(x)+{z}^{\text{'}\text{'}2}(x)+{\left[{y}^{\text{'}\text{'}}(x){z}^{\text{'}}(x)\mathrm{-}{y}^{\text{'}}(x){z}^{\text{'}\text{'}}(x)\right]}^{2}}}{1+{y}^{\text{'}2}(x)+{z}^{\text{'}2}(x)}\text{dx}` eq 2.1-2 In order to maintain the topology of the cable (and in particular the ordering of the nodes that compose it) the operator DEFI_CABLE_BP works from meshes and groups of meshes, (rather than nodes and groups of nodes), in order to be able to calculate the quantities by following the sequence of nodes along the cable. The interpolation used for the calculation of prestress in concrete will be a cubic Spline interpolation performed in parallel on the three spatial coordinates according to the curvilinear abscissa. The coordinates of the cable nodes are the "real" coordinates, that is, the coordinates defined by the cable mesh. All calculations presented in the context of operator DEFI_CABLE_BP are defined based on the real geometry of the structures and the real positions of the nodes. Node tension calculations will be performed from node to node, in the order given by the mesh topology, using the formulas mentioned above [:ref:`éq 2.1-1 <éq 2.1-1>`] and [:ref:`éq 2.1-2 <éq 2.1-2>`]. The calculation of the cumulative angular deviation and the curvilinear abscissa requires the precise calculation of the derivatives of the cable trajectory defined in the operator in a discrete manner by the position of the nodes of the cable mesh. Lagrange polynomials present instabilities, in particular for irregular meshes. In addition, a large number of discretization points will lead to polynomials of high degrees. Moreover, a small uncertainty in the interpolation coefficients will result in a significant error in the results, in terms of derivatives. By choosing a low-degree polynomial interpolation, we will obtain zero or non-continuous second derivatives (depending on the degree). The advantage of a cubic Spline interpolation is to obtain continuous second drifts and calculation costs of order :math:`n`, if :math:`n` is the number of points of the tabulated function to be interpolated, with polynomials of low degree. The principle of this interpolation method is described exclusively in the case of a function of the form :math:`x\to f(x)`. We assume that we perform an interpolation of the tabulated function, based on the values of the function at discretization points :math:`{x}_{1},{x}_{2},\mathrm{...},{x}_{n}`, and its second derivative. We can thus build a polynomial of order 3, on each of the intervals :math:`{x}_{i},{x}_{i+1}`, whose polynomial expression is as follows: :math:`y\mathrm{=}\frac{{x}_{j+1}\mathrm{-}x}{{x}_{j+1}\mathrm{-}{x}_{j}}{y}_{j}+\frac{x\mathrm{-}{x}_{j}}{{x}_{j+1}\mathrm{-}{x}_{j}}{y}_{j+1}+{{\text{Cy}}_{j}}^{\text{'}\text{'}}+{{\text{Dy}}_{j+1}}^{\text{'}\text{'}}` with: :math:`C=\frac{1}{6}\left[{(\frac{{x}_{j+1}-x}{{x}_{j+1}-{x}_{j}})}^{3}-(\frac{{x}_{j+1}-x}{{x}_{j+1}-{x}_{j}})\right]{({x}_{j+1}-{x}_{j})}^{2}` :math:`D=\frac{1}{6}\left[{(\frac{x-{x}_{j}}{{x}_{j+1}-{x}_{j}})}^{3}-(\frac{x-{x}_{j}}{{x}_{j+1}-{x}_{j}})\right]{({x}_{j+1}-{x}_{j})}^{2}` It can easily be verified that: :math:`\begin{array}{c}\begin{array}{ccc}y({x}_{j})={y}_{j}& \text{et}& {y}^{\text{'}\text{'}}({x}_{j})={{y}_{j}}^{\text{'}\text{'}}\end{array}\\ \begin{array}{ccc}y({x}_{j+1})={y}_{j+1}& \text{et}& {y}^{\text{'}\text{'}}({x}_{j+1})={{y}_{j+1}}^{\text{'}\text{'}}\end{array}\end{array}` It is then necessary to estimate the values of the second derivative at the interpolation points. By writing the equality of the interpolations on the intervals :math:`\mathrm{[}{x}_{i\mathrm{-}1},{x}_{i}\mathrm{]}`, and :math:`\mathrm{[}{x}_{i},{x}_{i+1}\mathrm{]}` of the derivative of order one, at point :math:`{x}_{i}`, we obtain the following expression: :math:`\frac{{x}_{j}-{x}_{j-1}}{6}{{y}_{j-1}}^{\text{'}\text{'}}+\frac{{x}_{j+1}-{x}_{j-1}}{3}{{y}_{j}}^{\text{'}\text{'}}+\frac{{x}_{j+1}-{x}_{j}}{6}{{y}_{j+1}}^{\text{'}\text{'}}=\frac{{y}_{j+1}-{y}_{j}}{{x}_{j+1}-{x}_{j}}-\frac{{y}_{j}-{y}_{j-1}}{{x}_{j}-{x}_{j-1}}` We thus obtain :math:`(n\mathrm{-}2)` equations relating the values of the second derivatives to the discretization points :math:`{x}_{1},{x}_{2},\mathrm{...},{x}_{n}`. By writing the boundary conditions in :math:`{x}_{1}` and :math:`{x}_{n}` on the values of the second derivatives, we obtain a system :math:`(n,n)` from which we can uniquely determine the value of all the derivatives, and thus obtain the interpolation function. Two solutions then arise for establishing boundary conditions: * set the value of the second derivative arbitrarily at points :math:`{x}_{1}`, and :math:`{x}_{n}`, to zero for example, * assign the real values of the second derivative at these points, if this data is available. We obtain a system of equations whose unknowns are the :math:`n` second derivatives of the tabulated function to be interpolated. This linear system has the particularity of being tri-diagonal, which means that the resolution is of the order of :math:`O(n)`. In practice, interpolation is divided into two steps: * the first consists in calculating the estimated values of the second derivative at the points, an operation which is performed only once, * the second consists in calculating, for a given value of :math:`x`, the value of the interpolated function, an operation that can be repeated as many times as desired. Tests carried out on the sine function, over three periods, show that the results are highly dependent on the number of points, as well as on the distribution of the points of the curve to be interpolated, (expected result), but that even in delicate situations (few points and a very irregular curve), the interpolation does not diverge. In other words, even if the correlation concerning the trajectory of the cable is not very good (interpolation with very few points) the interpolation will be approximately in a range close to the real trajectory. This case will not occur in practice, but makes it possible to verify the stability of the interpolation method. For the problem we are considering here, we cannot always write the trajectory of the cable in the form :math:`\mathrm{[}y(x)\mathrm{]},\mathrm{[}z(x)\mathrm{]}`, in cases where this curve is not bijective, in particular when the projection of the trajectory in one of the two planes :math:`\mathit{Oxy}` or :math:`\mathit{Oxz}` is cyclic or closed (case of a circular concrete structure). By taking an intermediate variable of the type :math:`u\mathrm{=}\mathrm{\int }\mathrm{\mid }{x}^{\text{'}}\mathrm{\mid }`, a parameter that is always increasing and increasing in absolute value identical to that of the variable :math:`x`, we can reduce ourselves to expressions :math:`\mathrm{[}y(u)\mathrm{]}` bijective functions of the variable :math:`u`. The cubic Spline interpolation described above is then applicable to the function :math:`y(u)` (as well as to the function :math:`z(u)`). In practice, however, this leads to problems with tangent connections (angular points) at the points where the variable :math:`x` changes direction of variation, and to point irregularities. The cable trajectory is described as a parametric curve. Knowing a set of points on the curve, the most easily accessible parameter is then the curvilinear abscissa. We write the trajectory of the cable in the form :math:`\mathrm{[}x(p),y(p)\mathrm{]}`, in the :math:`\mathit{Oxy}` plane, (respectively :math:`\mathrm{[}x(p),y(p),z(p)\mathrm{]}` in a three-dimensional space). The cumulative chord :math:`p` discretized at the tabulated points of the function we are interpolating :math:`{P}^{1},{P}^{\mathrm{2,}}\mathrm{...},{P}^{n}` is calculated as follows: :math:`p(1)\mathrm{=}0` at point :math:`{P}^{1}`, :math:`p(k)\mathrm{=}p(k\mathrm{-}1)` + distance :math:`({P}^{k\mathrm{-}1}{P}^{k})` to point :math:`{P}^{k}` There are thus two curves defined by a set of pairs [x (i), p (i)] and [y (i), p (i)] to which the cubic Spline interpolation presented above can be directly applied, and which makes it possible to overcome the difficulties encountered previously. The interpolation is done for the two coordinates, (or three coordinates, in dimension 3), independently of each other. Method without interpolation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ It is possible to simply calculate the curvilinear abscissa and the angle :math:`\alpha` without interpolation. This method is obviously less accurate but it is very robust. Moreover, the finer the mesh, the more its precision increases. However, the problems associated with spline interpolation occur precisely when the mesh is too fine in relation to the irregularities it contains. This method without interpolation is used when spline interpolation has failed (see :ref:`2.1.3 `). Calculating curvilinear abscissa is very simple. It consists in summing the length of the cable meshes. .. image:: images/100000000000018E000000C74122F335D05CD8DA.png :width: 3.3291in :height: 1.3535in .. _RefImage_100000000000018E000000C74122F335D05CD8DA.png: **Calculation of the cumulative angular deviation:** The following example suffices to describe the calculation of the angular deviation by this method. The knitwear :math:`{M}_{1}` , :math:`{M}_{2}` and :math:`{M}_{3}` make up a cable. :math:`{N}_{1}` is the first node in the cable, the value of its cumulative angular deviation is :math:`\alpha \mathrm{=}0`. Angle :math:`\mathit{theta}` is the angular deviation between cells :math:`{M}_{1}` and :math:`{M}_{2}`. At any point in :math:`\mathrm{]}{N}_{1}{N}_{2}\mathrm{[}` the cumulative angular deviation is always zero because the tangent vector to the curve at these points is :math:`\overrightarrow{{N}_{1}{N}_{2}}`. At any point in :math:`\mathrm{]}{N}_{2}{N}_{3}\mathrm{[}`, the cumulative angular deviation is equal to :math:`\mathit{theta}` because the tangent vector at each of these points is :math:`\overrightarrow{{N}_{2}{N}_{3}}` (:math:`\mathit{theta}` is the angle between :math:`\overrightarrow{{N}_{1}{N}_{2}}` and :math:`\overrightarrow{{N}_{2}{N}_{3}}`). It was decided, that the tangent vector to the curve in :math:`{N}_{2}` is the average of :math:`\overrightarrow{{N}_{1}{N}_{2}}` and :math:`\overrightarrow{{N}_{2}{N}_{3}}`. This means that the cumulative angular deviation in :math:`{N}_{2}` is :math:`\alpha \mathrm{=}\frac{\mathit{theta}}{2}`. With the same logic, the cumulative angular deviation in :math:`{N}_{3}` is :math:`\alpha \mathrm{=}\mathit{theta}+\frac{\mathit{beta}}{2}`, and :math:`\mathit{theta}+\mathit{beta}` in :math:`{N}_{4}`. .. _RefNumPara__1971_1362436802: Spline interpolation control ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In order to check whether the spline interpolations for the three coordinates of space are correct, the number of changes in the variation of the first derivative and the number of changes in the sign of the second derivative are calculated. If the number of sign changes is smaller than the number of changes in variation (+ an integer constant set to 10), the interpolation is considered to be of good quality. If not, we switch to the method without interpolation. Determination of the voltage profile in the cable according to BPEL 91 ---------------------------------------------------------------- General formula ~~~~~~~~~~~~~~~~~ The operator DEFI_CABLE_BP makes it possible to calculate the voltage :math:`F(s)` along the curvilinear abscissa :math:`s` of the cable. This is determined based on the rules of BPEL 91 [:ref:`bib1 `]. Overall, the following formulation is obtained: :math:`F(s)\mathrm{=}\tilde{F}(s)\mathrm{-}\left\{{x}_{\text{flu}}\mathrm{\times }{F}_{0}+{x}_{\text{ret}}\mathrm{\times }{F}_{0}+r(j)\mathrm{\times }\frac{5}{\text{100}}\mathrm{\times }{\rho }_{\text{1000}}\left[\frac{\tilde{F}(s)}{{S}_{a}\mathrm{\times }{\sigma }_{y}}\mathrm{-}{\mu }_{0}\right]\mathrm{\times }\tilde{F}(s)\right\}` eq 2.2.1-1 where :math:`s` refers to the curvilinear abscissa along the cable. The parameters introduced in this expression are: * :math:`{F}_{0}` initial voltage, * :math:`{x}_{\mathrm{flu}}` flat rate of tension loss due to concrete creep, compared to the initial tension, * :math:`{x}_{\mathrm{ret}}` flat rate of tension loss due to concrete shrinkage, compared to the initial tension, * :math:`{\rho }_{1000}` relaxation of steel at 1000 hours, expressed in%, * :math:`{S}_{a}` area of the straight section of the cable, * :math:`{\sigma }_{y}` elastic limit stress of steel, * :math:`{\mu }_{0}` dimensionless relaxation coefficient of prestressed steel. In this formula, :math:`{F}_{0}` refers to the initial tension at the anchors (before recoil), :math:`\tilde{F}(s)` represents the tension after taking into account friction and anchor recoil losses, :math:`{x}_{\text{flu}}\times {F}_{0}` represents the tension loss due to concrete creep, :math:`{x}_{\text{ret}}\times {F}_{0}` the tension loss due to concrete shrinkage, the tension loss due to concrete shrinkage, :math:`r(j)\times \frac{5}{\text{100}}\times {\mathrm{\rho }}_{\text{1000}}\left[\frac{\tilde{F}(s)}{{S}_{a}\times {s}_{y}}-{m}_{0}\right]\times \tilde{F}(s)` the relaxation losses of steels. **Note:** *The introduction of voltage losses into these elements is optional. Thus, if one intends to calculate creep and/or shrinkage of concrete using an appropriate law with* *STAT_NON_LINE, one should not enter these elements into the losses calculated by* *DEFI_CABLE_BP.* The evaluation of losses requires knowledge of the curvilinear abscissa :math:`s` and the cumulative angular deviation :math:`\alpha` calculated from the first and second derivatives of the cable trajectory. The precise calculation of these derivatives requires interpolation between the cable passage points. This interpolation is carried out using Splines, which are better than Lagrange polynomials, which present instabilities, in particular for irregular meshes (cf. previous paragraph). In what follows, each of the mechanisms involved in calculating the voltage is detailed. Loss of tension due to friction ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ We start by calculating the tension along the cable taking into account the contact losses between the cable and the concrete: :math:`{F}_{c}(s)={F}_{0}\text{exp}\left(-f\alpha -\phi s\right)` where :math:`\alpha` designates the cumulative angular deviation and the parameters introduced are: * :math:`f` coefficient of friction of the cable on the concrete partly curved, in :math:`{\mathit{rad}}^{\mathrm{-}1}`, * :math:`\phi` coefficient of friction per unit length, in :math:`{m}^{\mathrm{-}1}`, * :math:`{F}_{0}` voltage applied to one or both ends of the cable. * .. _RefNumPara__64223_128794603: Loss of tension due to anchor recoil ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ To take into account the anchoring setback, we make the following reasoning: the tension along the cable is affected by the anchor recoil over a distance :math:`d` that is calculated by solving a problem with two unknowns: the function :math:`{F}^{\ast }(s)` which represents the force after the anchor recoil and the scalar :math:`d`: +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ |:math:`\Delta =\frac{1}{{E}_{a}{S}_{a}}\underset{0}{\overset{d}{\int }}\left[F(s)-{F}^{\ast }(s)\right]\text{ds}`, :math:`F(s)` is equal to :math:`{F}_{0}{e}^{(-\mathrm{f\alpha }-\phis )}` :math:`\Delta` is the value of the anchor recoil (it is a given)| | + + .. image:: images/10000000000003C500000247E5A3ED94766C2591.jpg + | | :width: 3.2047in | + + :height: 2.0516in + | | | + + + | | | +-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------+ :math:`{F}^{\ast }(s)`, the force after the anchor recoils, is determined using the formula [:ref:`bib1 `]: :math:`\left[F(s)\text{.}{F}^{\ast }(s)\right]={\left[F(d)\right]}^{2}`, The length :math:`d` will be determined iteratively using the previous integral. Other authors use different relationships such as: :math:`\left[F(s)-F(d)\right]=\left[F(d)-{F}^{\ast }(s)\right]` For the calculation of :math:`d`, three special cases may occur: 1. This loss by recoil of the anchor is localized in the zone of the anchor. If the cable is curved, and the length of the cable is short enough, it may happen that :math:`d` is longer than the length of the cable. In this case, the loss of prestress due to the setback of the anchor applies everywhere. It is necessary to calculate the area between the two curves :math:`F(s)` and :math:`{F}^{\ast }(s)`, which must be equal to :math:`{E}_{a}{S}_{a}\Delta`, and which thus makes it possible to calculate :math:`{F}^{\ast }(s)`. 2. In the case where tension is applied to each of the two ends of the cable, let's call :math:`{F}_{1}(s)` the initial tension distribution calculated as if the tension were only applied to the first anchor, and :math:`{F}_{2}(s)` the initial tension distribution calculated as if the tension were only applied to the second anchor. The value that should be retained at all points on the cable as the initial tension is :math:`F(s)=\mathrm{Max}({F}_{1}(s),{F}_{2}(s))`. 3. Finally, in the case where *d* is greater than the length of the cable, and when tension is applied to each of the two ends of the cable (superposition of the two previous cases), the following procedure must be applied: * calculation of :math:`{F}_{1}(s)` initial tension calculated as if the tension were only applied to the first anchor and taking into account the anchor recoil (as in the particular case1), * calculation of :math:`{F}_{2}(s)` initial tension calculated as if the tension were only applied to the second anchor and taking into account the anchor recoil (as in the particular case1), * calculation of :math:`F(s)=\text{Min}({F}_{1}(s),{F}_{2}(s))`. Delayed deformations of steel ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The relaxation loss of steel, for an infinite time, is expressed as follows: :math:`r(j)\times \frac{5}{\text{100}}\times {\mathrm{\rho }}_{\text{1000}}\left[\frac{\tilde{F}(s)}{{S}_{a}\times {\mathrm{\sigma }}_{y}}-{\mathrm{\mu }}_{0}\right]\times \tilde{F}(s)` (:math:`{\rho }_{\text{1000}}` the relaxation at 1000 hours in%; :math:`{\mu }_{0}` the relaxation coefficient of prestressed steel and :math:`{\sigma }_{y}` the guaranteed value of the maximum load at breakage of the cable). This relationship expresses the loss due to relaxation of the cables for an infinite amount of time. BPEL 91 proposes the following formula: :math:`r\left(j\right)=\frac{j}{j+9\text{.}{r}_{m}^{0}}` where :math:`j` indicates the age of the structure in days and :math:`{r}_{m}^{0}` indicates the age of the structure in days and a characteristic radius obtained by the ratio of the section of the concrete structure, in :math:`{m}^{2}`, by the perimeter of the section (in meters) of concrete. Loss of tension due to instantaneous deformations of concrete ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Instant losses are not taken into account in the [:ref:`éq 2.2.1-1 <éq 2.2.1-1>`] formula used in*Code_Aster*. What BPEL calls instantaneous voltage loss is in fact the voltage loss induced in cables already laid by the installation of a new group of cables. To model this phenomenon, it is necessary to represent the prestress phasing in the Code_Aster calculation, i.e. not to stretch all the cables at the same time but in succession by chaining the CALC_PRECONT together (see test SSNV164). Determination of the voltage profile in the cable according to ETC -C ---------------------------------------------------------------- General formula ~~~~~~~~~~~~~~~~~ The operator DEFI_CABLE_BP makes it possible to calculate the voltage :math:`F(s)` along the curvilinear abscissa :math:`s` of the cable according to the rules of ETC -C [:ref:`bib4 `]. The theoretical formula is as follows: :math:`F(s)={F}_{0}-\Delta {F}_{\mu }-\Delta {F}_{\mathit{anc}}-\Delta {F}_{\mathit{el}}-\Delta {F}_{r}-\Delta {ϵ}_{\mathit{cs}}-\Delta {ϵ}_{\mathit{cc}}` where: * :math:`{F}_{0}` is the initial voltage applied to the cable * :math:`\Delta {F}_{\mu }` are tension losses due to friction, * :math:`\Delta {F}_{\mathit{anc}}` are the tension losses due to anchor recoil, * :math:`\Delta {F}_{\mathit{el}}` are the tension losses due to the elastic deformation of concrete, * :math:`\Delta {F}_{r}` are the tension losses due to the relaxation of the steels, * :math:`\Delta {F}_{\mathit{cs}}` are the tension losses due to the shrinkage of the concrete, * :math:`\Delta {F}_{\mathit{cc}}` are tension losses due to concrete creep. Losses due to elastic deformation are estimated according to ETC -C at: :math:`\Delta {F}_{\mathit{el}}(s)=\frac{{A}_{p}{E}_{p}\Delta {\sigma }_{c}(x)}{2E}` with :math:`E` Young's modulus of concrete, :math:`{A}_{p}` and :math:`{E}_{p}` the cross section and Young's modulus of steel, and :math:`\Delta {\sigma }_{c}(x)` the stress induced in concrete by prestress. They can be estimated by simulating the phasing of the prestress using the operator CALC_PRECONT. These steps are not taken into account in operator DEFI_CABLE_BP. The tension losses due to the shrinkage of concrete :math:`\Delta {F}_{\mathit{cc}}` and to the creep of concrete :math:`\Delta {F}_{\mathit{cs}}` can be obtained by imposing an equivalent deformation field after the cables have been tensioned. Again, they are therefore not taken into account in the DEFI_CABLE_BP operator. In the end, the formula implemented in DEFI_CABLE_BP is as follows: :math:`F(s)={F}_{0}-\Delta {F}_{\mu }-\Delta {F}_{\mathit{anc}}-\Delta {F}_{r}` eq. 2.3.1-1 The 3 types of loss are detailed in the paragraphs below. Tensile losses due to friction ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In accordance with ETC -C, friction losses are estimated by the following formula: :math:`{F}_{c}(s)={F}_{0}\left(1-{e}^{-\mu (\alpha +k\text{}s)}\right)` [:external:ref:`éq. 2.3.2-1 <éq. 2.3.2-1>`] where :math:`\alpha` designates the cumulative angular deviation and the parameters introduced are: * :math:`\mu` coefficient of friction of the cable on concrete e * :math:`k` the online loss coefficient :math:`[{m}^{-1}]` * :math:`{F}_{0}` voltage applied to one or both ends of the cable. * Tension losses due to anchor recoil ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The formula is the same as BPEL. Refer to § :ref:`2.2.3 `. Losses due to the relaxation of steel ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The formula given by the ETC -C is as follows: .. _RefEquation 23.4-1: :math:`\Delta {F}_{r}(s)\mathrm{=}\mathrm{0,8}\mathrm{\times }\mathrm{0,66}{\rho }_{1000}\text{.}{\mathrm{exp}}^{\mathrm{9,1}\tilde{F}(s)\mathrm{/}{F}_{\mathit{prg}}}\text{.}{(\frac{\mathit{nh}}{1000})}^{\mathrm{0,75}(1\mathrm{-}\tilde{F}(s))\mathrm{/}{F}_{\mathit{prg}}}\text{.}{10}^{\mathrm{-}5}\tilde{F}(s)` eq 23.4-1 where: * :math:`s` refers to the curvilinear abscissa along the cable. * :math:`{\rho }_{1000}` relaxation of steel at 1000 hours, expressed in%, * :math:`{F}_{\mathit{prg}}` breaking stress in steel, * :math:`\mathit{nh}` the number of hours after prestressing corresponding to the date when the relaxation losses of the steel are calculated. In this formula, :math:`\tilde{F}(s)` represents the tension after taking into account the losses by friction and by anchoring recoil as well as normally after taking into account the elastic losses. Two calculation options are proposed corresponding to the choice TYPE_RELAX =' ETCC_DIRECT 'or TYPE_RELAX =' ETCC_REPRISE'. If the user chooses option TYPE_RELAX =' ETCC_DIRECT ', then the tension used to calculate the loss due to the relaxation of steels does not take into account elastic losses but only losses due to friction and anchor recoil. If the user chooses option TYPE_RELAX =' ETCC_REPRISE ', then the voltage used to calculate the loss due to the relaxation of steels takes the 3 types of prestress losses into account. This voltage must be provided by the user (keyword TENSION under MODI_CABLE_ETCC). It will have been obtained during a first calculation that can be qualified as a "short-term" state by modeling the losses, by friction, by anchoring recoil and the elastic losses by modeling the phasing (cf. test SSNV229B for example of implementation). Determination of kinematic relationships between steel and concrete ------------------------------------------------------------- Since the nodes of the cable mesh do not necessarily coincide with the nodes of the concrete mesh, it is necessary to define kinematic relationships that model the perfect adhesion between the cables and the concrete. The following paragraphs describe in order the spatial geometric considerations for defining the concept of neighborhood between the nodes of cable and concrete elements, and then the method for calculating the coefficients of kinematic relationships. Definition of neighboring nodes ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ The calculation of the coefficients of the kinematic relationships requires determining the "neighboring" nodes of each of the nodes of the cable mesh. The following diagram symbolizes a cable knot and a concrete mesh: .. image:: images/10000D0A000069D5000035895F67186A7B745950.svg :width: 460 :height: 233 .. _RefImage_10000D0A000069D5000035895F67186A7B745950.svg: .. _1022611795: .. _1022610231: The mesh defined by nodes :math:`\mathrm{1,}\mathrm{2,}\mathrm{3,}4` contains the cable node. So the neighboring nodes are vertices :math:`\mathrm{1,}\mathrm{2,}\mathrm{3,}4`. If the cable node is located inside an element with :math:`p` nodes :math:`{P}_{1},{P}_{2},\mathrm{...},{P}_{n}`, then the nodes :math:`{P}_{1},{P}_{2},\mathrm{...},{P}_{n}` are called "neighboring nodes" of the cable node. Plate elements without eccentricity and massive elements are treated in the same way. The calculation of the eccentricity of each node of the cable mesh is necessary for the calculation of the coefficients of the kinematic relationships. In the case of plate elements, when the cable node is characterized by a non-zero eccentricity, neighboring nodes are defined as the set of vertex nodes of the element that contains the projection of the cable node in the tangential plane to the concrete mesh. In the case where the cable node (or its projection in the tangential plane to the concrete mesh) belongs to a border of an element, it is the vertices of this border that form the set of neighboring nodes. Calculation of the coefficients of kinematic relationships ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ In all of the descriptions that follow, the quantities are systematically expressed in the global coordinate system of the mesh. Kinematic relationships are therefore expressed as a function of the degrees of freedom expressed in this base. Rotation normals and vectors are expressed in the global coordinate system, unless explicitly stated otherwise. In finite element modeling of the cable-concrete structure, the displacement of a material point of the concrete structure can be easily expressed using the shape functions of the element or of the concrete mesh whose vertices form the neighboring nodes, according to the displacements of the neighboring nodes of the "concrete" discretization. Likewise, a quantity or a displacement of a point on the cable, (or of its projection on the tangential plane of the concrete grid) is identical to the value of this quantity at the material point of the concrete structure which occupies this same position (perfect connection between concrete and steel), and is therefore expressed as a function of the value of this same quantity at the vertices of the element, using form functions. If :math:`(x,y,z)` are the coordinates of the cable node, or those of its projection, and :math:`{N}_{1},{N}_{2},\mathrm{...},{N}_{n}` are the shape functions associated with the concrete nodes :math:`{P}_{1},{P}_{2},\mathrm{...},{P}_{n}` vertices of a concrete mesh element (or vertices of a border of a concrete mesh element), and :math:`({x}_{i},{y}_{i},{z}_{i})` the coordinates of the concrete mesh element), and the coordinates of the node :math:`i`, then the interpolation of a variable :math:`u` on the element is written: :math:`u(x,y,z)=\sum _{i=1}^{n}{N}_{i}(x,y,z)\text{.}u({x}_{i},{y}_{i},{z}_{i})=\sum _{i=1}^{n}{N}_{i}(x,y,z)\text{.}{u}_{i}` :math:`u` can be a coordinate, or any other nodal data. Kinematic links make it possible to express the identity of movement between the cable mesh node and the concrete material point that occupies the same position. This corresponds to the hypothesis of a perfect connection between concrete and cable. Case where concrete is modelled by massive finite elements ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ By using the previous notations and considering :math:`{\text{dx}}^{c}\text{,}{\text{dy}}^{c},{\text{dz}}^{c}` the movements of the cable node, and :math:`{\text{dx}}_{j}^{b},{\text{dy}}_{j}^{b},{\text{dz}}_{j}^{b}` the movements of the nodes :math:`j` (:math:`j\mathrm{=}\mathrm{1,}n`) of the concrete structure neighboring the cable node, we obtain the following relationships: :math:`\{\begin{array}{c}{\text{dx}}^{c}=\sum _{i=1}^{n}{N}_{i}({x}^{c},{y}^{c},{z}^{c}){\text{dx}}_{{i}^{b}}\\ {\text{dy}}^{c}=\sum _{i=1}^{n}{N}_{i}({x}^{c},{y}^{c},{z}^{c}){\text{dy}}_{{i}^{b}}\\ {\text{dz}}^{c}=\sum _{i=1}^{n}{N}_{i}({x}^{c},{y}^{c},{z}^{c}){\text{dz}}_{{i}^{b}}\end{array}` :math:`n` being the number of nodes of the concrete element neighboring the cable node, or that of one of its borders. For each cable node, 3 kinematic relationships are obtained between the movements of the nodes of the two cable and concrete meshes. Case where concrete is modelled by plate finite elements ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ .. image:: images/1000102400002113000021136F8325A8F33AA93B.svg :width: 460 :height: 233 .. _RefImage_1000102400002113000021136F8325A8F33AA93B.svg: Let :math:`{P}_{0}^{c}` be the initial position of a cable point in the undeformed geometry and let :math:`{P}^{c}` be the position of this same point after deformation. Let's call :math:`{P}_{0}^{p}` the projection of :math:`{P}_{0}^{c}` onto the surface of the middle sheet of the undeformed concrete shell and :math:`{P}^{p}` the projection of :math:`{P}^{c}` onto the surface of the middle sheet of the deformed concrete shell. Let :math:`\overrightarrow{{n}_{0}}` be normal to the average plane of the concrete shell in :math:`{P}_{0}^{p}` and :math:`\overrightarrow{n}` the normal one in :math:`{P}^{p}`. :math:`{P}_{O}^{p}` is given by: :math:`\left\{\begin{array}{c}{x}_{0}^{p}\\ {y}_{0}^{p}\\ {z}_{0}^{p}\end{array}\right\}=\left\{\begin{array}{c}{x}_{0}^{c}\\ {y}_{0}^{c}\\ {z}_{0}^{c}\end{array}\right\}-\left[\left\{\begin{array}{c}{x}_{0}^{c}-{x}_{0}^{b}\\ {y}_{0}^{c}-{y}_{0}^{b}\\ {z}_{0}^{p}-{y}_{0}^{b}\end{array}\right\}\cdot \left\{\begin{array}{c}{n}_{\mathrm{0x}}\\ {n}_{\mathrm{0y}}\\ {n}_{\mathrm{0z}}\end{array}\right\}\right]\cdot \left\{\begin{array}{c}{n}_{\mathrm{0x}}\\ {n}_{\mathrm{0y}}\\ {n}_{\mathrm{0z}}\end{array}\right\}` :math:`{P}^{p}` is given by: :math:`\left\{\begin{array}{c}{x}_{}^{p}\\ {y}_{}^{p}\\ {z}_{}^{p}\end{array}\right\}=\left\{\begin{array}{c}{x}_{}^{c}\\ {y}_{}^{c}\\ {z}_{}^{c}\end{array}\right\}-\left[\left\{\begin{array}{c}{x}_{}^{c}-{x}_{}^{b}\\ {y}_{}^{c}-{y}_{}^{b}\\ {z}_{}^{p}-{y}_{}^{b}\end{array}\right\}\cdot \left\{\begin{array}{c}{n}_{}x\\ {n}_{}y\\ {n}_{}z\end{array}\right\}\right]\cdot \left\{\begin{array}{c}{n}_{}x\\ {n}_{}y\\ {n}_{}z\end{array}\right\}` Point :math:`{P}_{O}^{p}` belongs to a concrete plate mesh whose nodes are noted :math:`{P}_{1}^{b},{P}_{2}^{b}\text{et}{P}_{3}^{b}`. We define the eccentricity of the cable with respect to the concrete shell as the distance :math:`e=\parallel \overrightarrow{{P}_{0}^{p}{P}_{0}^{c}}\parallel` and we make the assumption that this eccentricity does not vary when the structure deforms: :math:`e=\parallel \overrightarrow{{P}_{0}^{p}{P}_{0}^{c}}\parallel =\parallel \overrightarrow{{P}^{p}{P}^{c}}\parallel` The movements of the points of the cable and its projection are introduced: :math:`\overrightarrow{{P}_{0}^{c}{P}^{c}}=\{\begin{array}{c}d{x}^{c}\\ d{y}^{c}\\ d{z}^{c}\end{array}` :math:`\overrightarrow{{P}_{0}^{p}{P}^{p}}=\{\begin{array}{c}{\text{dx}}^{p}=\sum _{i=1}^{n}{N}_{i}({x}_{0}^{p},{y}_{0}^{p},{z}_{0}^{p}){\text{dx}}_{{i}^{b}}\\ {\text{dy}}^{p}=\sum _{i=1}^{n}{N}_{i}({x}_{0}^{p},{y}_{0}^{p},{z}_{0}^{p}){\text{dy}}_{{i}^{b}}\\ {\text{dz}}^{p}=\sum _{i=1}^{n}{N}_{i}({x}_{0}^{p},{y}_{0}^{p},{z}_{0}^{p}){\text{dz}}_{{i}^{b}}\end{array}` We introduce the "rotation" vector :math:`\overrightarrow{\theta }` of the plate at point :math:`{P}^{p}` and the degrees of freedom of rotation of the nodes of the plate are introduced: :math:`\overrightarrow{\mathrm{\theta }}=\{\begin{array}{c}{\text{drx}}^{b}=\sum _{i=1}^{n}{N}_{i}({x}_{0}^{p},{y}_{0}^{p},{z}_{0}^{p}){\text{drx}}_{{i}^{b}}\\ {\text{dry}}^{b}=\sum _{i=1}^{n}{N}_{i}({x}_{0}^{p},{y}_{0}^{p},{z}_{0}^{p}){\text{dry}}_{{i}^{b}}\\ {\text{drz}}^{b}=\sum _{i=1}^{n}{N}_{i}({x}_{0}^{p},{y}_{0}^{p},{z}_{0}^{p}){\text{drz}}_{{i}^{b}}\end{array}` By definition of :math:`\overrightarrow{\theta }`, we have: :math:`\overrightarrow{n}-\overrightarrow{{n}_{0}}=\overrightarrow{\theta }\wedge \overrightarrow{{n}_{0}}` We can then write: :math:`` :math:`\overrightarrow{{P}^{p}{P}^{c}}=e\overrightarrow{n}` Subtracting these two equations, taking into account the definition of :math:`\overrightarrow{\theta }`, we find: :math:`\{\begin{array}{c}{\text{dx}}^{c}-{\text{dx}}^{p}=e\text{.}({\text{dry}}^{p}\text{.}{n}_{\mathrm{0z}}-{\text{drz}}^{p}\text{.}{n}_{\mathrm{0y}})\\ {\text{dy}}^{c}-{\text{dy}}^{p}=e\text{.}({\text{drz}}^{p}\text{.}{n}_{\mathrm{0x}}-{\text{drx}}^{p}\text{.}{n}_{\mathrm{0z}})\\ {\text{dz}}^{c}-{\text{dz}}^{p}=e\text{.}({\text{drx}}^{p}\text{.}{n}_{\mathrm{0y}}-{\text{dry}}^{p}\text{.}{n}_{\mathrm{0x}})\end{array}` By injecting the form functions into this last equation, we finally have: :math:`\{\begin{array}{c}{\text{dx}}^{c}-(\sum _{i=1}^{n}{N}_{i}({x}^{p},{y}^{p},{z}^{p}){\text{dx}}_{i}^{b})=e\text{.}((\sum _{i=1}^{n}{N}_{i}({x}^{p},{y}^{p},{z}^{p}){\text{dry}}_{i}^{b})\text{.}{n}_{\mathrm{0z}}-(\sum _{i=1}^{n}{N}_{i}({x}^{p},{y}^{p},{z}^{p}){\text{drz}}_{i}^{b})\text{.}{n}_{\mathrm{0y}})\\ {\text{dy}}^{c}-(\sum _{i=1}^{n}{N}_{i}({x}^{p},{y}^{p},{z}^{p}){\text{dx}}_{i}^{b})=e\text{.}((\sum _{i=1}^{n}{N}_{i}({x}^{p},{y}^{p},{z}^{p}){\text{drz}}_{i}^{b})\text{.}{n}_{\mathrm{0x}}-(\sum _{i=1}^{n}{N}_{i}({x}^{p},{y}^{p},{z}^{p}){\text{drx}}_{i}^{b})\text{.}{n}_{\mathrm{0z}})\\ {\text{dz}}^{c}-(\sum _{i=1}^{n}{N}_{i}({x}^{p},{y}^{p},{z}^{p}){\text{dx}}_{i}^{b})=e\text{.}((\sum _{i=1}^{n}{N}_{i}({x}^{p},{y}^{p},{z}^{p}){\text{drx}}_{i}^{b})\text{.}{n}_{\mathrm{0y}}-(\sum _{i=1}^{n}{N}_{i}({x}^{p},{y}^{p},{z}^{p}){\text{dry}}_{i}^{b})\text{.}{n}_{\mathrm{0x}})\end{array}` Case where the cable node is projected onto a concrete mesh node ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ The distance between the projection :math:`{P}_{0}^{p}` of the cable node :math:`{P}_{0}^{c}` and a concrete node :math:`{P}_{i}^{b}` is given by: :math:`d=\parallel {P}_{0}^{P}{P}_{i}^{b}\parallel =∥(\begin{array}{}{x}^{c}-{x}_{i}^{b}\\ {y}^{c}-{y}_{i}^{b}\\ {z}^{c}-{z}_{i}^{b}\end{array})-\left[(\begin{array}{}{x}^{c}-{x}_{i}^{b}\\ {y}^{c}-{y}_{i}^{b}\\ {z}^{c}-{z}_{i}^{b}\end{array})\text{.}{\overrightarrow{n}}_{0}\right]\text{.}{\overrightarrow{n}}_{0}∥` If this distance happens to be zero (in practice less than 10-5), it is because the cable node projects to the top of a concrete mesh, and then the kinematic relationships are simplified: :math:`\{\begin{array}{c}{\text{dx}}^{c}-{\text{dx}}_{i}^{p}=e\text{.}({\text{dry}}_{i}^{p}\text{.}{n}_{\mathrm{0z}}-{\text{drz}}_{i}^{p}\text{.}{n}_{\mathrm{0y}})\\ {\text{dy}}^{c}-{\text{dy}}_{i}^{p}=e\text{.}({\text{drz}}_{i}^{p}\text{.}{n}_{\mathrm{0x}}-{\text{drx}}_{i}^{p}\text{.}{n}_{\mathrm{0z}})\\ {\text{dz}}^{c}-{\text{dz}}_{i}^{p}=e\text{.}({\text{drx}}_{i}^{p}\text{.}{n}_{\mathrm{0y}}-{\text{dry}}_{i}^{p}\text{.}{n}_{\mathrm{0x}})\end{array}` These relationships are the general relationships in which: :math:`{N}_{j}({x}^{p},{y}^{p},{z}^{p})=0` if :math:`j\ne i`. Treatment of cable end areas ----------------------------------------- The modeling of a prestress cable as it is done in *Code_Aster* consists in representing the entire cable, passage duct, and all the anchor parts, only thanks to a series of bar elements. The link between the cable elements and the concrete medium is ensured by kinematic conditions on the degrees of freedom of each of the cable nodes, and those of the concrete elements crossed. When cable tension is applied, it is observed that the reactions generated at the ends of the cables on the concrete create stress levels that are much higher than reality, and cause damage to the concrete. For example, in some studies, compressive stresses of more than :math:`200\mathit{MPa}` have been observed, which greatly exceeds the observed experimental value (:math:`40\mathit{MPa}`). In reality, this phenomenon is not observed thanks to the establishment of a stress diffusion cone (see drawing below) which distributes the prestress force over a large area of the concrete. In the case of the finite element model, this surface does not exist, since the force is directly taken up by a node. .. _OLE_LINK1: +-------------------------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------+ | | | + .. image:: images/10055F360000307900002447FE8413C38C64A32B.svg + .. image:: images/1004AC7800001DAA00002461DFCC84359ED21171.svg + | :width: 460 | :width: 402 | + :height: 233 + :height: 184 + | | | + + + | | | +-------------------------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------+ |Real situation |EF model without cone | +-------------------------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------------------------+ This way of modeling has several drawbacks: * the concentration of this effort crushes the concrete, * the spatial discretization of the model changes the results. To remedy this problem, the keyword CONEde the DEFI_CABLE_BPpermet operator no longer distributes this prestress force over a node, but over all the nodes contained in a volume (all the nodes of this volume are linked together to form a rigid solid) delimited by a cylinder of radius :math:`R` and length :math:`L`, representing the equivalent of the zone of influence of the expansion cone of an anchor (see figure below). .. image:: images/1006464A0000205900002957AF7399D11C980FF9.svg :width: 460 :height: 233 .. _RefImage_1006464A0000205900002957AF7399D11C980FF9.svg: The identification and creation of kinematic relationships between the concrete and cable nodes is done automatically by the command DEFI_CABLE_BP, where the new data :math:`R` and :math:`L` will be provided by the user. Note: Calculation of cable tension as a mechanical load ------------------------------------------------------------------------- We made the choice to leave the cable elements in the mechanical model to support finite element calculation (linear or not). As a result, there is no equivalent force calculation to be carried over to the nodes of the mesh. We are simply saying that pretension cables have a non-zero initial stress state. This stress state is the one deduced from the tension as calculated by DEFI_CABLE_BP. For reasons of simplicity, the computer object created by the operator DEFI_CABLE_BP is a table storing values at the nodes of the cable. Let us then consider two related elements of the cable: :math:`\mathrm{e1}` of summits :math:`\mathit{N1}` and :math:`\mathit{N2}`, and :math:`\mathrm{e2}` of summit :math:`\mathit{N2}` and :math:`\mathit{N3}`. We assume that :math:`{l}_{1}` and :math:`{S}_{1}` are the length and section of a :math:`\mathrm{e1}` element and that :math:`{l}_{2}` and :math:`{S}_{2}` are the length and section of the :math:`\mathrm{e2}` element. .. image:: images/1000042E000069D50000314DD3809EA87F9F1CB4.svg :width: 460 :height: 233 .. _RefImage_1000042E000069D50000314DD3809EA87F9F1CB4.svg: DEFI_CABLE_BP will calculate a voltage :math:`{T}_{{N}_{2}}` at node :math:`\mathit{N2}` defined by: :math:`{T}_{{N}_{2}}=\frac{1}{2}(\frac{\underset{{e}_{1}}{\int }T(s)\text{ds}}{{l}_{1}}+\frac{\underset{{e}_{2}}{\int }T(s)\text{ds}}{{l}_{2}})` Conversely, for finite element calculation, operator STAT_NON_LINE will assume that the initial constraint in element :math:`\mathit{e1}` is :math:`{\sigma }_{0}^{\mathrm{e1}}=\frac{{T}_{{N}_{1}}+{T}_{{N}_{2}}}{{\mathrm{2S}}_{1}}` **Note:** It will always be considered that the law of cable behavior is of an incremental type.