EP3425120B1 - Stage anchor and method for anchoring a stage anchor in a ground or component - Google Patents

Description

The present invention relates to a step anchor for anchoring in a subsurface or component, with a plurality of stretchable tension members, at least two of the tension members having a different definable free anchor length L _tf .

Furthermore, the present invention relates to a method for anchoring a step anchor in a subsurface or component, wherein the step anchor has a plurality of stretchable tension members and at least two of the tension members have a different definable free anchor length L _tf .

A step anchor and a method for anchoring such a step anchor in an underground are, for example, from US Pat EP 1 707 684 B1 known. The known step anchor has several tension members that can be tensioned, each having a different free anchor length. The respective anchoring lengths are connected at different depths in a borehole to a grouting body that connects the tension members to the surrounding subsoil for load transfer into the subsoil. The individual tension members each have essentially the same anchoring length. Through this staggered or staggered arrangement of the anchoring lengths along the depth of the borehole, the total load capacity of the stepped anchor is staggered or staggered. This can be described as an axial staggering of the anchoring length of the step anchor in the borehole.

In classic grout anchors without such a stepped or staggered load entry, all tension members of the anchor are stretched by the same amount or elongation for DIN-compliant testing and fixing of the anchors. Applying this test method to a step anchor with different free anchor lengths of the individual tension members results in different forces and stresses in the individual tension members with the same expansion path. More precisely, given a uniform elongation and the same elongation path of all tension members, different tension states result in the individual tension members due to their different free anchor lengths. Tension members with shorter free anchor lengths are more stressed than tension members with longer free anchor lengths, so that, for example, a test load for tension members with shorter free anchor lengths already occurs with an elongation in which tension members with longer free anchor lengths are still far below this test load. Since the DIN specifications for testing and dimensioning grouted anchors are basically a uniform force respectively Assuming the same free anchor length within the entire bundle of tension members of a grouted anchor, different test methods have been developed that take account of this problem of the different free anchor lengths and the resulting different force and stress distributions.

One solution to the problem is to stretch or tension the individual tension members with different free anchor lengths using individual presses, so that each tension member is subjected to the same load or tension at all times of stretching or tensioning. A simple tensioning can then be carried out using a conventional bundle press, which in turn tensions each tension member simultaneously by the same expansion path. Here, however, it is problematic that in this case too, different tensions in the individual tension members have to be accepted from the use of the bundle press due to their different free anchor lengths. However, this method is already better than a method in which such a bundle press and all tension members are tensioned by the same extension path from the start. However, due to the use of single presses as well as bundle presses and the several clamping steps, this process is very complex and ultimately leads to time-shifted and unfavorable shear stresses within the pressing body. Furthermore, the previously known methods for testing and tensioning conventional anchors in step anchors in various ways ultimately lose reference to the design specifications and test criteria of DIN.

Further tie rods and procedures for anchoring them are in GB 2 356 884 A and EP 2 609 258 A1 described.

The present invention is therefore based on the object of designing and developing a step anchor and a method for anchoring a step anchor of the type mentioned at the outset in such a way that safe testing and safe use of the step anchor are made possible with structurally simple means.

According to the invention, the above object is achieved by a step anchor with the features of claim 1. The step anchor is designed and developed in such a way that respective tensile stiffnesses of the at least two tension members can be defined in the area of their respective free anchor length Ltf in such a way that the at least two tension members have a predeterminable stress state or a predefinable tension after stretching by the same expansion path per tension member.

Furthermore, the above object is achieved by a method for anchoring a step anchor with the features of claim 11. The method is designed and developed in such a way that respective tensile stiffnesses of the at least two tension members are defined in the area of their respective free anchor length Ltf in such a way that the at least two tension members have a predeterminable stress state or a predeterminable tension after stretching by the same extension path per tension member.

According to the invention, it was first recognized that the above object is achieved in a surprisingly simple manner by cleverly designing the step anchor. In a further manner according to the invention, it was then recognized that a specific specification of the respective tensile stiffness of tension members with different free anchor lengths enables a significant simplification of test procedures for the step anchor and thus a safe use of such step anchors. The respective tensile stiffnesses of the tension members are defined in the area of their respective free anchor lengths in such a way that these tension members have a predeterminable stress state or a predeterminable tension after tensioning by the same expansion path per tension member. The definition of the respective expansion stiffnesses defines the tension generated in the respective tension members after tensioning by the same expansion path. As a result, the tension or state of tension can be specified individually for each tension element as a function of individual operating conditions, namely via the definition and specification of the respective individual tensile stiffnesses of the individual tension elements. The stresses or stress states in the individual tension members can be preselected in such a way that simple and known standard test methods can be used, for example testing with a bundle press, whereby the tension states or stresses in the individual tension members can vary within possible tolerance ranges of the tension. In such a case, the same stresses or stress states are not absolutely necessary and deliberately different stresses - within the range of predefinable limits - can be tolerated if this should be useful for testing and / or using the step anchor.

Consequently, with the step anchor according to the invention and the method according to the invention for anchoring a step anchor, a step anchor and a corresponding method are specified, according to which safe testing and safe use of the step anchor are made possible with structurally simple means.

Basically, it should be noted at this point that the step anchor according to the invention - regardless of the type of load application and the time of the load application - is pretensioned or tensioned or stretched, which also creates a free stretchable length through a relative displacement of the two ends can be - both a so-called prestressed step anchor and a non-prestressed step anchor. The definition of the tensile stiffness in the respective tension members and the resulting stress conditions in the tension members when the tension members are stretched are always of great advantage when testing the step anchors in the case of non-prestressed step anchors. In this respect, the present invention is not restricted to prestressed step anchors.

With regard to a particularly safe test and a particularly safe use of the step anchor, the predeterminable tension states or tensions in the tension members can be essentially the same after tensioning. Shear stresses within a grouting body can be avoided as far as possible and the reference to the design specifications and testing criteria of the DIN is particularly promoted.

The definition of the respective expansion stiffnesses of the at least two tension members can advantageously be carried out in very different ways. The tensile stiffness can be definable by specifying the number of support members or strands of the respective tension members, provided that the individual tension members are made up of several individual support members or strands. It should be noted here that a step anchor according to the present invention is constructed from a plurality of tension members, which can each be constructed from a plurality of support members or strands. When specifying the number of support members or strands, the idea is that if all tension members are simultaneously stretched by the same extension path, support members or strands with different free anchor lengths - with the same one respective tensile stiffness - different forces result, ie a greater force with a short free anchor length than with a long free anchor length. In this respect, the tensile stiffness can be defined concretely in that a tension member with a shorter free anchor length has a larger number of support members or strands than a tension member with a greater free anchor length. The definition of the expansion stiffness also specifies the respective spring stiffnesses or spring constants, which are defined as the expansion stiffness per length. By choosing the number of support links or strands of the respective tension members, a very simple definition of the tensile stiffness of the respective tension members is possible.

As an alternative or in addition to specifying the number of support members or strands of the respective tension members, the tensile stiffness can be defined by specifying a cross-sectional area and / or an elastic modulus of individual support members or strands of the respective tension members. A number of support members or strands in a tension member could remain the same if support members or strands with a larger or smaller cross-sectional area are used at the same time, as required. Here, support members or strands with a larger cross-sectional area have greater stiffness than support members or strands with a smaller cross-sectional area. Further alternatively or in addition to this, the respective modulus of elasticity of an individual support member or an individual strand could be definable. For example, different materials - for example steel and / or glass fiber material and / or carbon material - could be used to adjust the respective tensile stiffness of the tension members by adapting the elasticity module of support members or strands of the respective tension members. All of the aforementioned alternatives - number of support members or strands, specification of a cross-sectional area and specification of an elasticity module - can be realized in an alternative or in a combined manner in order to define the respective expansion stiffnesses in the required manner.

If tension members of the step anchor do not have individual support members or strands, but are each formed by a tension element, the expansion stiffness can be definable by specifying a cross-sectional area and / or an elastic modulus of the respective tension members or tension elements. In this case, the alternative of specifying the number of support members or strands as a variation option Otherwise, in an alternative or combined manner, cross-sectional areas and elasticity modules can be adapted or varied in a defined manner, for example using different materials, in order to define the required required tensile stiffnesses of the respective tension members.

With a view to a very simple definition of the respective elongation stiffness by specifying the number of support members or strands of the respective tension members, the number of support members or strands in the at least two tension members can be defined by the following formula in order to achieve approximately equal stress states or tensions in the tension members be: n _{long (right)} = n _short * (L _{tf, long} + L _e ) / (L _{tf, short} + L _e ), where n _{long (right) is} the arithmetically determined number of support links or strands in the tension link with a longer free one Anchor _length L _{tf, long} , n _{short is} the number of support members or strands in the tension member with a shorter free anchor length L _{tf, short} and L _{e is} the length of an external overhang. In this way, the ratio of the number of support members or strands n _{long (right)} in the tension member with the longer free anchor _length to the number of support members or strands n _short in the tension member with the shorter free anchor length can be calculated. One of the two number of support members or strands can be specified or the other number of support members or strands can be calculated. In the present case, this is exemplified by a specification of the number of support members or strands for the tension member with the short free anchor length and a calculation of the number of support members or strands for the tension member with the longer free anchor length. However, the reverse procedure is also conceivable, in which case the number of support members or strands n is _briefly calculated. It should be taken into account that this formula can result in a value that is not an integer if the parameters listed on the right side of the formula and the number of support links or strands for the tension link with the longer free anchor length are calculated. This deviation from an integer can be compensated with a correction value. Ultimately, the odd number must be rounded up or down to an even number, so that desired tension conditions or tension states result in the tension members after they have been stretched by the same extension path.

Depending on the embodiment, the respective anchoring lengths L _{tb of} the at least two tension members can be essentially the same size. Ultimately, this leads to an essentially uniform application of force into the substrate or the component per tension member. Depending on the requirements, however, different anchoring lengths L _tb can also be selected. This is based on the respective application.

The respective anchoring lengths L _{tb of} the at least two tension members can be arranged essentially one behind the other in the direction of tension. This leads to a particularly even application of force into the substrate or into the component. Alternatively or in addition to this, the respective anchoring lengths L _tb can be arranged in a common pressing or composite body. Such a grouting or composite body also serves to load evenly into a substrate or into a component.

With regard to a particularly safe use and a particularly uniform force input and to reduce unfavorable shear stresses, tension members with different definable free anchor lengths L _tf can be arranged essentially alternately next to one another or evenly distributed in the step anchor. For example, if tension members with two different free anchor lengths are present, a shorter one next to a longer tension member or a tension member with a shorter free anchor length could always be arranged next to a tension member with a longer free anchor length in order to create essentially balanced or approximately symmetrical load conditions. When using a perforated disk for separating individual tension members, the tension members of different lengths could be distributed substantially uniformly in the perforated disk.

In a particularly advantageous manner, a bundle press for generating the tension can be assigned to the step anchor. In this respect, a system consisting of step anchors and a bundle press is technically and economically advantageous.

With regard to the advantages of the claimed method for anchoring a step anchor in a subsurface or component, reference can be made to the above-mentioned advantages of a step anchor in order to avoid repetition, the advantages mentioned for the step anchor also analogously applying to the method.

In an advantageous embodiment of the method, the tensioning, pretensioning or partial pretensioning or stretching of the at least two tension members can be carried out simultaneously, preferably by means of a bundle press. As an alternative or in addition to this, the predefinable tension states or tensions in the tension members after a load has been applied, for example stretching or tensioning, can be substantially the same.

The stretching of the tension members can in any case alternatively be generated by a time-delayed and / or alternating displacement of two anchoring points against each other. The anchoring points can be moved independently of one another relative to one another.

Furthermore, individual presses or combinations of individual presses can also be used in principle to generate the stretching of the tension members, which can also be used at different times.

In the following, advantageous aspects of exemplary embodiments of the step anchor according to the invention are explained again for a particularly good understanding of the claimed teaching:
In the step anchor according to the invention and the method according to the invention, the tensile stiffness of tension members is defined. The expansion stiffness is the product of the elastic modulus E of a material in the direction of loading and the cross-sectional area A perpendicular to the direction of loading - regardless of the shape of the cross-section: Expansion stiffness = E * A. The extent to which the absolute change in length ΔL of a component is given the tensile force depends not only on the tensile stiffness but also on its length.

For a uniform cross section, the spring stiffness or spring constant c is equal to the stiffness of the spring cross section divided by the length of the spring: c = E * A / L, which represents the stiffness per length.

Different components can be used as tension members. This can be a support member or a strand or an arrangement of support members or strands per tension member or a rope or an arrangement of Ropes per tension member or a bar or an arrangement of bars per tension member. Furthermore, different materials can be used for a tension member, for example steel or a carbon or glass fiber component. With the same materials, the modulus of elasticity E is the same. When using different materials in the tension members, depending on the material, there is a different modulus of elasticity with regard to the different tension members.

A spring constant c = E * A / L is made up of the sum of the respective spring constants belonging to the individual tension elements in a step anchor with several tension elements. It should be taken into account that each tension member has an individual modulus of elasticity and possibly an individual cross section and an individual free anchor length Ltf depending on the material. An equivalent spring constant c * can be assigned to such a step anchor, which is composed of the individual spring constants of the individual tension members. This can be represented as c * = (E * A) * / L *, whereby the product of the modulus of elasticity and cross-sectional area - the expansion stiffness - can be described as the equivalent expansion stiffness and L * as the equivalent expansion length.

Ultimately, a replacement anchor consisting of several partial anchors formed by individual tension members can be designed by specifying the dimensioning and calculation based on the principle of resulting elongation stiffness and / or spring stiffness in such a way that, for example, by calculating and selecting a different number of support links or strands per sub-bundle with the same elongation and known different free expansion lengths at the level of the working load, the individual tension members or sub-bundles receive approximately the same force.

Fig. 1

in einer schematischen Darstellung ein Ausführungsbeispiel eines erfindungsgemäßen Stufenankers zur Verankerung in einem Untergrund oder Bauteil.

There are now various possibilities for advantageously designing and developing the teaching of the present invention. For this purpose, reference is made on the one hand to the claims subordinate to claims 1 and 11 and on the other hand to the following explanation of preferred exemplary embodiments of the invention with reference to the drawing. In connection with the explanation of the preferred exemplary embodiments of the invention with reference to the drawing, generally preferred refinements and developments of the teaching are also explained. In the drawing shows

Fig. 1

a schematic representation of an embodiment of a step anchor according to the invention for anchoring in a substrate or component.

Fig. 1 shows a schematic representation of an embodiment of a step anchor according to the invention for anchoring in a subsurface or component, wherein the step anchor has five tensile tension members 1, 2, two tension members 1 having a shorter free anchor length L _{tf, short} than three tension members 2, which are longer have free anchor _length L _{tf, long} . The areas of the free anchor _lengths L _{tf, long} and L _{tf, in short} the tension members 1, 2 are each arranged in a cladding tube 3 and are freely movable therein.

With regard to a safe test and a safe use of the step anchor with structurally simple means, the respective tensile stiffness of the tension members 1 and 2 in the area of their respective free anchor _lengths L _{tf, long} and L _tf, can be defined in such a _short way that the tension members 1, 2 follow a stretching by the same stretching path per tension member 1, 2 has a predeterminable stress state or a predeterminable stress, which can be essentially the same in a preferred exemplary embodiment.

The tensile stiffnesses are defined in the exemplary embodiment shown here by specifying the number n _short and n _long on support members or strands of tension members 1 and 2, respectively. This setting of the number of support members or strands based on the idea that by increasing the Tragglied- or of strands _long n of the longer tension member 2 relative to the tension member 1 with a shorter free anchor length L _tf, can be set _short in substantially the same Dehnsteifigkeiten or spring constants. The ratio EA _short / (L _{tf, short} + Le) = EA _long / (L _tf , _long + Le) applies. A _short or A _long denotes the total cross-sectional area of the short tension member 1 or long tension member 2, this cross-sectional area A _short or A _{long being} composed of the cross sections of all the individual strands if the tension member has individual strands.

Both for the embodiment according to Fig. 1 As well as for the following explanations for a test of the step anchor, the following explanation of the abbreviations used applies: L _{tb, long} [mm] Anchorage length of the long strands L _{tf, long} [mm] free anchor length of the long strands L _{tb, short} [mm] Anchorage length of the short strands L _{tf, short} [mm] free anchor length of the short strands L _e [mm] Strand protrusion for tensioning L _app, * [mm] theoretical value for free ( replacement ) expansion length of entire step anchor ( incl. L _e ) at the border line "c" L _tf, * [mm] theoretical value for free (replacement) anchor length of entire step anchor L _{app, a *} [mm] Theoretical value for free (replacement) expansion length of entire relay anchor ( incl. L _e ) at the border line "a" Ltb * [mm] theoretical value for ( replacement ) anchorage length of entire step anchor n _long [-] Number of long strands (selected) n _{long, right} [-] arithmetically required number of long strands n _short [-] Number of short strands (selected) P _short [N] Power of the short partial anchor P _{short, perm} [N] permissible maximum test force of the short partial anchor P _long [N] Force of the long part anchor P _{long, perm} [N] permissible maximum test force of the long partial anchor P _{P, perm.} [N] permissible test force of the anchor (entire anchor) P _P [N] selected test force of the anchor (entire anchor) P _a [N] Preload of the anchor (entire anchor) A _strand [mm ² ] Cross-sectional area per strand E _t [N / mm ² ] Modulus of elasticity of the tension member f _{t, k} [N / mm ² ] char. Tensile strength prestressing steel or tension member material f _{t, 0.1, k} [N / mm ² ] char. Tension at 0.1% permanent elongation Δel, _PP [mm] total elastic stretch (incl. stretch from preload) Δs _el [mm] elastic strain theoretical limit line (entire anchor) Δs _max [mm] elastic stretch upper limit line (entire anchor) Δs _min [mm] elastic elongation lower limit line (entire anchor)

The basics and requirements for a test of the step anchor are given below:
Ultimately, the present invention also relates to a test method for testing a step anchor according to the invention, the step anchor being definable in order to simplify the test in the manner according to the invention with regard to the respective stiffness of the at least two tension members 1, 2.

In one embodiment of such a test method, all tension members and / or strands of the anchor can be checked and / or fixed together and simultaneously. In many cases, proficiency tests have to be carried out.

For tensile members and / or strands with different free lengths - with the same elongation stiffness - the common elongation of a conventional anchor results in differently high forces or stress states, with a short free length experiencing a greater force and a long free length experiencing a smaller force or tension. Through the choice according to the invention of a correspondingly different number of support members or strands per tension member, an adjustment of partial stiffnesses of the respective tension members is sought and an excessive difference in the partial forces or partial stresses is prevented. In a particularly advantageous case, the forces in the short tension member are just as large as the forces in the long tension member with P _short = P _long .

A "greasing" of strands, preferably in the factory, within cladding tubes 3 in the region of the free anchor lengths is advantageous for controlled load introduction and in many cases even necessary.

• Es ist grundsätzlich nachzuweisen, dass die Zugglieder, Tragglieder oder Litzen mit der kürzeren freien Ankerlänge nicht überlastet werden, siehe unten "zulässige Prüfkraft P_{p, zul}" > P_{max, vorh}.
• Zur Erreichung von näherungsweise gleichen Teilkräften P_kurz = P_lang wird die Tragglied- oder Litzenanzahl (n_lang; n_kurz) der beiden Zugglieder oder Teilanker ermittelt zu: $${\mathrm{n}}_{\mathrm{lang}\left(\mathrm{rech}\right)}={\mathrm{n}}_{\mathrm{kurz}}\frac{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{lang}}+\mathrm{Le}\right)}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{kurz}}+\mathrm{Le}\right)}$$
  
• Es werden in vorteilhafter Weise nur Tragglieder, Litzen oder Zugglieder des gleichen Typs verwendet.
• Die Grenzlinien werden mit einer theoretischen, errechneten freien Ersatzdehnlänge L_app* und einer theoretischen, errechneten Ersatzverankerungslänge L_tb* für den gesamten Stufenanker ermittelt.
• Die gleichlangen Verankerungslängen liegen bei diesem Ausführungsbeispiel hintereinander in einem gemeinsamen Verpresskörper und überschneiden sich nicht oder nur mit einer konstruktiven Überlappung von etwa 0,3m. Die Länge der Überlappung kann jedoch auch andere Werte annehmen.
• Es liegt der gleiche Baugrund im gesamten Krafteintragungsbereich des gemeinsamen Verpresskörpers vor.
• Der Nachweis der tiefen Gleitfuge muss oder soll für die Mitte der (Teil-) Verankerungslänge des kurzen Ankers geführt werden!
• Die kurzen Zugglieder, Tragglieder oder Litzen und die langen Zugglieder, Tragglieder oder Litzen sollten gekennzeichnet werden. Es ist empfehlenswert, jedoch nicht zwingend erforderlich, die kurzen und langen Zugglieder, Tragglieder oder Litzen möglichst alternierend/abwechselnd in der Verpressstrecke und/oder in der Lochscheibe anzuordnen. Es soll vermieden werden, dass alle kurzen Zugglieder, Tragglieder oder Litzen auf einer Seite und alle langen Zugglieder, Tragglieder oder Litzen auf der anderen Seite zu liegen kommen.

With regard to advantageous configurations of such an examination procedure, the following aspects are essential:

• It must always be demonstrated that the tension members, support members or strands with the shorter free anchor length are not overloaded, see below "permissible test force P _{p, perm} "> P _{max, existing} .
• To achieve approximately equal partial forces P _short = P _long , the number of supporting links or strands (n _long ; n _short ) of the two tension links or partial anchors is determined as follows: $${\mathrm{n}}_{\mathrm{long}\left(\mathrm{right}\right)}={\mathrm{n}}_{\mathrm{short}}\frac{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{long}}+\mathrm{Le}\right)}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{short}}+\mathrm{Le}\right)}$$
  
• Only support members, strands or tension members of the same type are advantageously used.
• The boundary lines are determined with a theoretical, calculated free equivalent extension length L _app * and a theoretical, calculated equivalent anchorage length L _tb * for the entire step anchor.
• In this exemplary embodiment, the anchoring lengths of the same length lie one behind the other in a common pressing body and do not overlap or only overlap by about 0.3 m. However, the length of the overlap can also take other values.
• The same building ground is present in the entire force application area of the common pressing body.
• The verification of the deep sliding joint must or should be performed for the middle of the (partial) anchoring length of the short anchor!
• The short tension links, support links or strands and the long tension links, support links or strands should be marked. It is recommended, but not absolutely necessary, to arrange the short and long tension members, support members or strands alternately / alternately in the pressing section and / or in the perforated disk. It should be avoided that all short tension members, support members or strands come to rest on one side and all long tension members, support members or strands on the other side.

In the following, important formulas and definitions are given for the realization of the step anchor and / or for the implementation of the examination procedure:

Determination of the static quantities

For the sake of simplicity, only the term "stranded wire" is used below. However, this also means, alternatively and in general, a “support member” which can also be designed differently than a strand.

From the common elongation Δs _{el of} the entire anchor, which can be referred to as "BBV multibond anchor", the strands of the partial anchors with different free stretchable lengths L _{tf, long} > L _{tf, short} , with the same tensile stiffness (E * A) different forces
[short free length = greater force, long free length = smaller force] $$\begin{array}{cc}{\mathit{P}}_{\mathit{short}}={\mathrm{\Delta s}}_{\mathrm{el}}\frac{{\mathrm{EA}}_{\mathrm{short}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{short}}+\mathrm{Le}\right)}& {\mathit{P}}_{\mathit{long}}={\mathrm{\Delta s}}_{\mathrm{el}}\frac{{\mathrm{EA}}_{\mathrm{long}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{long}}+\mathrm{Le}\right)}\end{array}$$

By choosing different numbers of strands "n" / per partial anchor, an attempt is made to match the partial stiffnesses of the two anchor parts and to prevent a large difference in the partial forces.

Note: In principle, it is also possible to match the partial forces by choosing different materials (steel, plastics, e.g. CFRP ...) with different E-modules and / or different cross-sections (e.g. rods with different diameters), but they can be used as grouted anchors In the following, only tension members (strands / rods) with the same cross-sections are considered as a bundle.

Theoretical goal: partial forces P _short = P _long $$\begin{array}{c}{\mathit{P}}_{\mathit{short}}={\mathrm{\Delta s}}_{\mathrm{el}}\frac{{\mathrm{EA}}_{\mathrm{short}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{short}}+\mathrm{Le}\right)}={\mathrm{\Delta s}}_{\mathrm{el}}\frac{{\mathrm{EA}}_{\mathrm{long}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{long}}+\mathrm{Le}\right)}={\mathit{P}}_{\mathit{long}}\\ \frac{{\mathrm{A}}_{\mathrm{short}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{short}}+\mathrm{Le}\right)}=\frac{{\mathrm{A}}_{\mathrm{long}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{long}}+\mathrm{Le}\right)}\\ \frac{{\mathrm{EA}}_{\mathrm{strand}}{\mathrm{n}}_{\mathrm{short}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{short}}+\mathrm{Le}\right)}=\frac{{\mathrm{EA}}_{\mathrm{strand}}{\mathrm{n}}_{\mathit{long}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{long}}+\mathrm{Le}\right)}\\ \frac{{\mathbf{n}}_{\mathit{short}}}{\left({\mathbf{L}}_{\mathbf{tf},\mathbf{short}}+\mathbf{Le}\right)}=\frac{{\mathbf{n}}_{\mathit{long}}}{\left({\mathbf{L}}_{\mathbf{tf},\mathbf{long}}+\mathbf{Le}\right)}\end{array}$$

vorgegeben (!): n_kurz -> berechnet: n_lang(rech) --> gewählt: n_{lang(gewählt)} The number of strands of the short partial anchor n is _briefly known from the choice of the “experience anchor”. Formation after n _long :
The number of strands of the long part anchor n _long calculated as: $${\mathbf{n}}_{\mathbf{long}\left(\mathbf{right}\right)}={\mathbf{n}}_{\mathit{short}}\frac{\left({\mathbf{L}}_{\mathbf{tf},\mathbf{long}}+{\mathbf{L}}_e\right)}{\left({\mathrm{L}}_{\mathbf{tf},\mathbf{short}}+{\mathbf{L}}_e\right)}$$

predefined (!): n _short -> calculated: n _{long (right)} -> selected: n _{long (selected)}

Ermittlung eines Korrekturbeiwerts ε_S $${\mathrm{\epsilon }}_{\mathrm{S}}=\frac{P_{\mathit{kurz}}}{P_{\mathit{lang}}}=\frac{n_{\mathrm{kurz}}\left[L_{\mathit{tf},\mathit{lang}}+\mathit{Le}\right]}{n_{\mathrm{lang}\left(\mathrm{gewählt}\right)}\left[L_{\mathit{tf},\mathit{kurz}}+\mathit{Le}\right]}$$

Since only complete strands of the same type should be used (see note above)
n _{long (selected)} mostly not equal to n _{long (right)} .
For example: n _{long (arithmetic)} = 4.78 not equal to n _{long (selected)} = 5.
From (n _{long, not} equal to n _{long, selected} ) results (P _short not equal to P _long )

Determination of a correction coefficient ε _S $${\mathrm{\epsilon }}_{\mathrm{S}}=\frac{P_{\mathit{short}}}{P_{\mathit{long}}}=\frac{n_{\mathrm{short}}\left[L_{\mathit{tf},\mathit{long}}+\mathit{Le}\right]}{n_{\mathrm{long}\left(\mathrm{chosen}\right)}\left[L_{\mathit{tf},\mathit{short}}+\mathit{Le}\right]}$$

n _long is not arbitrarily calculable / selectable!

• n _long depends on the influencing parameters: n _short ; L _e ; L _{tf , short} ; L _{tb , short}
• (L _{tf, long} is not a free parameter, since: L _{tf, long} = L _{tf, short} + L _tb )

Choice of n _long depending on the influencing parameters:

• L_tf,kurz + L_e > l_tb,kurz
• L_tf,kurz + L_e > 5m

Avoid too short free lengths L _{tf, short} + L _e !

• L _{tf, short} + L _e > l _{tb, short}
• L _{tf, short} + L _e > 5m

• Verbundstrecke L_tb,kurz < 8m
• (L_tf,kurz + L_e + L_tb)/ (L_tf,kurz + L_e) < 1,5
• sinnvoller Weise sollte hier n_lang < 1,5 n_kurz betragen

An excessive difference in the free partial lengths (eg by choosing a "too long" link section L _{tb, short} ) must be avoided!
L _{app, long} = L _{tf, long} + L _e = L _{tf, short} + L _e + L _tb must not be much larger than L _{tf, short} + L _e = L _{app, short}

• Compound section L _{tb, short} <8m
• (L _{tf, short} + L _e + L _tb ) / (L _{tf, short} + L _e ) <1.5
• It makes sense here to be n _long <1.5 n _short

proof

• It must be demonstrated that the strands of the partial anchors are not overloaded. "permissible test force P _p.zul "> P _{max, existing} $${\mathbf{P}}_{\mathrm{p},\mathrm{perm}\mathrm{.},\mathrm{Strand}}=\mathit{min}\left\{\begin{array}{l}0.80\times {\mathit{A}}_{\mathit{Strand}}\times {\mathit{f}}_{\mathit{t},\mathit{k}}\\ 0.90\times {\mathit{A}}_{\mathit{Strand}}\times {\mathit{f}}_{\mathit{t},0.1\mathit{k}}\end{array}\right\}$$
  
  under special conditions 0.95 x A _strand x f _{t, 0.1, k} also applies
• Determination of the permissible test force of the multibond anchor using the correction factor based on the partial anchor with the lowest number of strands and the shortest free length $$\begin{array}{l}{\mathrm{P}}_{\mathrm{p},\mathrm{perm},\mathbf{multibond}}={\mathrm{P}}_{\mathrm{p},\mathrm{perm},\mathrm{short}}+{\mathrm{P}}_{\mathrm{p},\mathrm{perm}\mathrm{long}\mathrm{With}}{\mathrm{\epsilon }}_{\mathbf{S}}=\frac{{\mathit{P}}_{\mathit{short}}}{{\mathit{P}}_{\mathit{long}}}\\ ={\mathrm{P}}_{\mathrm{p},\mathrm{perm},\mathrm{short}}+{\mathrm{P}}_{\mathrm{p},\mathrm{perm}\mathrm{short}}\frac{1}{{\mathrm{\epsilon }}_{\mathbf{S}}}={\mathrm{P}}_{\mathrm{p},\mathrm{perm}\mathrm{short}}\left(1+\frac{1}{{\mathrm{\epsilon }}_{\mathbf{S}}}\right)\end{array}$$
  
  $${\mathbf{P}}_{\mathrm{p},\mathrm{perm},\mathrm{multibond}}=\mathit{min}\left\{\begin{array}{l}0.80\times {\mathit{A}}_{\mathit{Strand}}\times {\mathit{f}}_{\mathit{t},\mathit{k}}\\ 0.90\times {\mathit{A}}_{\mathit{Strand}}\times {\mathit{f}}_{\mathit{t},0.1,\mathit{k}}\end{array}\right\}{\mathrm{n}}_{\mathrm{short}}\left(1+\frac{1}{{\epsilon }_{\mathit{S}}}\right)$$
  
  With $${\mathrm{\epsilon }}_{\mathrm{S}}=\frac{P_{\mathit{short}}}{P_{\mathit{long}}}=\frac{n_{\mathrm{short}}\left[L_{\mathit{tf},\mathit{long}}+\mathit{Le}\right]}{n_{\mathrm{long}\left(\mathrm{chosen}\right)}\left[L_{\mathit{tf},\mathit{short}}+\mathit{Le}\right]}$$
  
• The resulting elastic elongation of the entire anchor through the test force P _{p, perm} is to be calculated.
  • Elastic elongation Δ _{el, PP} with test force (entire anchor): [incl. Elongation from preload] $$\mathrm{\Delta }\mathit{el},\mathit{PP}=\frac{\mathrm{Pp}\times {\mathrm{10th}}^6}{\mathrm{Et}\times \mathrm{ALitze}\times \left[\frac{\mathrm{nlang}}{\left(\mathrm{Ltf},\mathrm{long}+\mathrm{Le}\right)}+\frac{\mathrm{n\; short}}{\left(\mathrm{Ltf},\mathrm{short}+\mathrm{Le}\right)}\right]}$$
    
  • Elastic elongation Δ _{el, PP} with test force (entire anchor): [without elongation from preload] $$\mathrm{\Delta }\mathit{el},\mathit{PP}=\frac{\left(\mathrm{Pp}-\mathrm{Pa}\right)\times {\mathrm{10th}}^6}{\mathrm{Et}\times \mathrm{ALitze}\times \left[\frac{\mathrm{nlang}}{\left(\mathrm{Ltf},\mathrm{long}+\mathrm{Le}\right)}+\frac{\mathrm{n\; short}}{\left(\mathrm{Ltf},\mathrm{short}+\mathrm{Le}\right)}\right]}$$
    
• The partial forces result in the short and long anchor part with test force
  • Part anchor force short part - anchor $$\mathrm{PP},\mathrm{short}=\frac{\mathrm{\Delta }\mathrm{el},\mathrm{PP}\times \mathrm{Et}\times \mathrm{ALitze}}{\left(\mathrm{Ltf},\mathrm{short}+\mathrm{Le}\right)}\times \mathrm{n\; short}$$
    
  • Partial anchor force long part - anchor $$\mathrm{PP},\mathrm{long}=\frac{\mathrm{\Delta }\mathrm{el},\mathrm{PP}\times \mathrm{Et}\times \mathrm{ALitze}}{\left(\mathrm{Ltf},\mathrm{long}+\mathrm{Le}\right)}\times \mathrm{nlang}$$
    

A theoretical replacement anchor is determined for the joint testing of the partial anchors as the overall anchor BBV-multibond, consisting of a free replacement steel length Lapp * (= L _tf * + L _e ) and a replacement composite section L _tb *.

ATTENTION!

These sizes are theoretical substitute sizes, do not correspond to the real sizes of the manufactured tension member and may therefore not be used for the manufacture / installation of the tension member!

d.h. für den Multibond Ersatzanker gilt: $${\mathrm{P}}_{{\mathrm{p}}^{\mathit{multibond}}}=\frac{\mathrm{\Delta }{\mathrm{el}}_{\mathrm{PP}}\times {\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Litze}\ast }\times \left({\mathrm{n}}_{\mathrm{lang}}+\mathrm{nkurz}\right)}{{\mathit{L}}_{\mathit{app},}}$$

When considering elasticity, the following applies: (Hook's law) $$\mathrm{F}=\frac{\mathrm{\Delta }\mathrm{el}}{\mathrm{L}}\ast \mathrm{E}\ast \mathrm{A}$$

ie for the Multibond replacement anchor: $${\mathrm{P}}_{{\mathrm{p}}^{\mathit{multibond}}}=\frac{\mathrm{\Delta }{\mathrm{el}}_{\mathrm{PP}}\times {\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Strand}\ast }\times \left({\mathrm{n}}_{\mathrm{long}}+\mathrm{n\; short}\right)}{{\mathit{L}}_{\mathit{app},}}$$

The replacement length L _app * for the total anchor is unknown!
the following also applies: $${\mathrm{P}}_{{\mathrm{p}}^{\mathit{multibond}}}={\mathrm{P}}_{\mathrm{p},\mathrm{short}}+{\mathrm{P}}_{\mathrm{p},\mathrm{long}}$$

The partial lengths of the partial anchor bundles are known!

Determination of the free (replacement) steel length Lapp * for the replacement anchor BBV multibond ( replacement length including Le)
when determining the free substitute steel length Lapp * (= Ltf * + Le) as substitute length at the theoretical transition L _tf to L _tb elastic (boundary line "c"): $${\mathrm{P}}_{{\mathrm{p}}^{\mathbf{multibond}}}=\frac{\mathrm{\Delta }\mathrm{el},\mathrm{PP}\times \mathrm{Et}\times \mathrm{ALitze}}{\left(\mathrm{Ltf},\mathrm{short}+\mathrm{Le}\right)}\times \mathrm{n\; short}+\frac{\mathrm{\Delta }\mathrm{el},\mathrm{PP}\times \mathrm{Et}\times \mathrm{ALitze}}{\left(\mathrm{Ltf},\mathrm{long}+\mathrm{Le}\right)}\times \mathrm{nlang}$$

$$\frac{\left({\mathbf{n}}_{\mathrm{lang}}+\mathbf{n}\mathrm{kurz}\right)}{\mathit{Lapp},\ast }=\frac{\mathit{n}\mathit{kurz}}{\left(\mathit{Ltfkurz}+\mathit{Le}\right)}+\frac{\mathit{n}\mathit{lang}}{\left(\mathit{ltflang}+\mathit{Le}\right)}$$

By inserting the known sizes and reducing the factor present on both sides of the equations (Δel, Pp x Et x ALitze), it follows that: $${\mathrm{P}}_{{\mathrm{p}}^{\mathbf{multibond}}}={\mathrm{P}}_{\mathrm{p},\mathrm{short}}+{\mathrm{P}}_{\mathrm{p},\mathrm{long}}$$

$$\frac{\left({\mathbf{n}}_{\mathrm{long}}+\mathbf{n}\mathrm{short}\right)}{\mathit{Lapp},\ast }=\frac{\mathit{n}\mathit{short}}{\left(\mathit{Ltf\; short}+\mathit{Le}\right)}+\frac{\mathit{n}\mathit{long}}{\left(\mathit{ltflang}+\mathit{Le}\right)}$$

$${\mathrm{L}}_{\mathrm{app}}*={\mathrm{L}}_{\mathrm{tf}}*+{\mathrm{L}}_{\mathrm{e}}$$

L_app* kann berechnet werden!The changeover gives the free substitute _{steel length} L _app * (= L _tf * + L _e ) as a substitute length at the theoretical transition from L _tf to L _tb elastic (boundary line "c"). $${\mathbf{L}}_{\mathbf{app},\ast }=\frac{\left({\mathbf{L}}_{\mathbf{tf},\mathbf{long}}+{\mathbf{L}}_e\right)\times \left({\mathbf{L}}_{\mathbf{tf},\mathbf{short}}+{\mathbf{L}}_e\right)\times \left({\mathbf{n}}_{\mathbf{long}}+{\mathbf{n}}_{\mathbf{short}}\right)}{{\mathbf{n}}_{\mathbf{long}}\left({\mathbf{L}}_{\mathbf{tf},\mathbf{short}}+{\mathbf{L}}_e\right)+{\mathbf{n}}_{\mathbf{short}}\left({\mathbf{L}}_{\mathbf{tf},\mathbf{long}}+{\mathbf{L}}_e\right)}$$

$${\mathrm{L}}_{\mathrm{app}}*={\mathrm{L}}_{\mathrm{tf}}*+{\mathrm{L}}_{\mathrm{e}}$$

L _app * can be calculated!

Determination of the free (substitute) steel length L _{app, a} * for the replacement anchor BBV multibond

By determining the free substitute steel length L _app * as the substitute length L _{app, a} * = L _tf * + L _e + 0.5L _tb * (at the limit line "a") for the test load: $$\begin{array}{l}{\mathbf{L}}_{\mathbf{app},\mathbf{a},\ast }=\mathrm{Lapp}\ast +0.5\mathrm{Ltb}\ast \\ =\frac{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{long}+}{\mathrm{L}}_{\mathrm{e}+0.5}{\mathrm{L}}_{\mathrm{tblang}}\right)\times \left({\mathrm{L}}_{\mathrm{tf},\mathrm{short}+}{\mathrm{L}}_{\mathrm{e}+0.5}{\mathrm{L}}_{\mathrm{tb\; short}}\right)\times \left({\mathrm{n}}_{\mathrm{long}}+{\mathrm{n}}_{\mathrm{short}}\right)}{{\mathrm{n}}_{\mathrm{long}}\left({\mathrm{L}}_{\mathrm{tf},\mathrm{short}+}{\mathrm{L}}_{\mathrm{e}+0.5}{\mathrm{L}}_{\mathrm{tb\; short}}\right)+{\mathrm{n}}_{\mathrm{short}}\left({\mathrm{L}}_{\mathrm{tf},\mathrm{long}}+{\mathrm{L}}_{\mathrm{e}+0.5}{\mathrm{L}}_{\mathrm{tblang}}\right)}\end{array}$$

$${\mathrm{L}}_{\mathrm{tb}}*=2\left({\mathrm{L}}_{\mathrm{app},\mathrm{a}}*-{\mathrm{L}}_{\mathrm{app}}*\right)$$

näherungsweise gilt: L_tb* = L_tb,kurz = L_tb,lang Since L _{tb short} , = L _{tb, long} = L _{tb is} defined and selected by default, the alternative composite route Ltb * of the Multibond replacement anchor is calculated as: $$0.5{\mathrm{L}}_{\mathrm{tb}}*={\mathrm{L}}_{\mathrm{app},\mathrm{a}}*-{\mathrm{L}}_{\mathrm{app}}*$$

$${\mathrm{L}}_{\mathrm{tb}}*=\mathrm{2nd}\left({\mathrm{L}}_{\mathrm{app},\mathrm{a}}*-{\mathrm{L}}_{\mathrm{app}}*\right)$$

approximately: L _tb * = L _{tb, short} = L _{tb, long}

und $${\mathrm{L}}_{\mathrm{tb}}*={\mathrm{L}}_{\mathrm{tb},\mathrm{kurz}}={\mathrm{L}}_{\mathrm{tb},\mathrm{lang}}$$

können die Grenzlinien für die Prüfung des Ankers berechnet werden.Using replacement lengths $${\mathrm{L}}_{\mathrm{app}}*={\mathrm{L}}_{\mathrm{tf}}*+{\mathrm{L}}_{\mathrm{e}}$$

and $${\mathrm{L}}_{\mathrm{tb}}*={\mathrm{L}}_{\mathrm{tb},\mathrm{short}}={\mathrm{L}}_{\mathrm{tb},\mathrm{long}}$$

the boundary lines for testing the anchor can be calculated.

Strain calculation for boundary lines (entire relay anchor):

$$\mathrm{\Delta }{\mathbf{s}}_{\mathit{min}}=\frac{\left({\mathbf{P}}_{\mathrm{P}}-{\mathbf{P}}_{\mathrm{a}}\right)\times {10}^6}{{\mathbf{E}}_{\mathrm{t}}\times {\mathbf{A}}_{\mathrm{Litze}}\times \left[{\mathbf{n}}_{\mathrm{lang}}+{\mathbf{n}}_{\mathrm{kurz}}\right]}\times \left[\mathrm{0,8}\mathrm{Ltf}\ast +\mathrm{Le}\right]$$

mit L _{app ∗} = Ltf ∗ + Le $${\mathrm{\Delta s}}_{\mathrm{el}}=\frac{\left({\mathrm{P}}_{\mathrm{P}}-{\mathrm{P}}_{\mathrm{a}}\right)\times {10}^6}{{\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Litze}}\times \left[\frac{{\mathrm{n}}_{\mathrm{lang}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{lang}}+{\mathrm{L}}_{\mathrm{e}}\right)}+\frac{{\mathrm{n}}_{\mathrm{kurz}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{kurz}}+{\mathrm{L}}_{\mathrm{e}}\right)}\right]}=$$

$$\mathrm{\Delta }{\mathbf{s}}_{\mathbf{el}}=\frac{\left({\mathbf{P}}_{\mathbf{P}}-{\mathbf{P}}_{\mathbf{a}}\right)\times {10}^6}{{\mathbf{E}}_t\times {\mathbf{A}}_{\mathit{Litze}}\times \left[{\mathbf{n}}_{\mathit{lang}}+{\mathbf{n}}_{\mathit{kurz}}\right]}\times {\mathbf{L}}_{\mathbf{app}\ast }$$

mit L_app ∗ = Ltf ∗ + Le

$${\mathrm{\Delta s}}_{\max }=\frac{\left({\mathrm{P}}_{\mathrm{P}}-{\mathrm{P}}_{\mathrm{a}}\right)\times {10}^6}{{\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Litze}}\times \left[\frac{{\mathrm{n}}_{\mathrm{lang}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{lang}}+{\mathrm{L}}_{\mathrm{e}}+\frac{{\mathrm{L}}_{\mathrm{tb},\mathrm{lang}}}{2}\right)}+\frac{{\mathrm{n}}_{\mathrm{kurz}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{kurz}}+{\mathrm{L}}_{\mathrm{e}}+\frac{{\mathrm{L}}_{\mathrm{tb},\mathrm{kurz}}}{2}\right)}\right]}$$

$$\mathrm{\Delta }{\mathbf{s}}_{\mathit{max}}=\frac{\left({\mathbf{P}}_{\mathbf{P}}-{\mathbf{P}}_{\mathbf{a}}\right)\times {10}^6}{{\mathbf{E}}_t\times {\mathbf{A}}_{\mathit{Litze}}\times \left[{\mathbf{n}}_{\mathit{lang}}+{\mathbf{n}}_{\mathit{kurz}}\right]}\times \left({\mathbf{L}}_{\mathbf{app}\ast }+\mathrm{0,5}{\mathbf{L}}_{\mathbf{tb}\ast }\right)$$

mit L_tb * = 2(L_app,a* - L_app*)
Wahl: Anzahl der Litzen je Ankerteil $${\mathbf{n}}_{\mathit{lang}\left(\mathbf{rech}\right)}={\mathbf{n}}_{\mathit{kurz}}\frac{\left({\mathbf{L}}_{\mathbf{tf},\mathbf{lang}}+{\mathbf{L}}_e\right)}{\left({\mathbf{L}}_{\mathbf{tf},\mathbf{kurz}}+{\mathbf{L}}_e\right)}$$

vorgegeben (!): nkurz -> berechnet: nlang(rech) --> gewählt: nlang(gewählt) $$\mathrm{Korrekturfaktor}{\mathrm{\epsilon }}_{\mathrm{S}}=\frac{P_{\mathit{kurz}}}{P_{\mathit{lang}}}=\frac{n_{\mathrm{kurz}}\left[L_{\mathit{tf},\mathit{lang}}+\mathit{Le}\right]}{n_{\mathrm{lang}\left(\mathrm{gewählt}\right)}\left[L_{\mathit{tf},\mathit{kurz}}+\mathit{Le}\right]}$$

zulässige Prüfkraft P _P,zul. (gesamter Anker): $${\mathrm{P}}_{\mathrm{p},\mathrm{zul},\mathrm{multibond}}=\min \left\{\begin{array}{l}\mathrm{0,80}\times {\mathrm{A}}_{\mathrm{Litze}}\times {\mathrm{f}}_{\mathrm{t},\mathrm{k}}\\ \mathrm{0,90}\times {\mathrm{A}}_{\mathrm{Litze}}\times {\mathrm{f}}_{\mathrm{t},0.1,\mathrm{k}}\end{array}\right\}{\mathrm{n}}_{\mathrm{kurz}}\left(1+\frac{1}{{\mathrm{\epsilon }}_{\mathbf{S}}}\right)$$

elastische Dehnung Δ _el,PP bei Prüfkraft (gesamter Anker): [inkl. Dehnung aus Vorlast] $$\mathrm{\Delta }\mathbf{el},\mathbf{Pp}=\frac{{\mathrm{P}}_{\mathrm{P}}\times {10}^6}{{\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Litze}}\times \left[\frac{{\mathrm{n}}_{\mathrm{lang}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{lang}}+{\mathrm{L}}_{\mathrm{e}}\right)}+\frac{{\mathrm{n}}_{\mathrm{kurz}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{kurz}}+{\mathrm{L}}_{\mathrm{e}}\right)}\right]}$$

elastische Dehnung Δ _el,PP bei Prüfkraft (gesamter Anker): [ohne Dehnung aus Vorlast] $$\mathrm{\Delta }\mathbf{el},\mathbf{Pp}=\frac{\left({\mathrm{P}}_{\mathrm{P}-\mathrm{Pa}}\right)\times {10}^6}{{\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Litze}}\times \left[\frac{{\mathrm{n}}_{\mathrm{lang}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{lang}}+{\mathrm{L}}_{\mathrm{e}}\right)}+\frac{{\mathrm{n}}_{\mathrm{kurz}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{kurz}}+{\mathrm{L}}_{\mathrm{e}}\right)}\right]}$$

$${\mathrm{\Delta s}}_{\min }=\frac{\left({\mathrm{P}}_{\mathrm{P}}-{\mathrm{P}}_{\mathrm{a}}\right)\times {\mathrm{10th}}^6}{{\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Strand}}\times \left[\frac{{\mathrm{n}}_{\mathrm{long}}}{\left(0.8\times {\mathrm{L}}_{\mathrm{tf},\mathrm{long}}+{\mathrm{L}}_{\mathrm{e}}\right)}+\frac{{\mathrm{n}}_{\mathrm{short}}}{\left(0.8\times {\mathrm{L}}_{\mathrm{tf},\mathrm{short}}+{\mathrm{L}}_{\mathrm{e}}\right)}\right]}$$

$$\mathrm{\Delta }{\mathbf{s}}_{\mathit{min}}=\frac{\left({\mathbf{P}}_{\mathrm{P}}-{\mathbf{P}}_{\mathrm{a}}\right)\times {\mathrm{10th}}^6}{{\mathbf{E}}_{\mathrm{t}}\times {\mathbf{A}}_{\mathrm{Strand}}\times \left[{\mathbf{n}}_{\mathrm{long}}+{\mathbf{n}}_{\mathrm{short}}\right]}\times \left[0.8\mathrm{Ltf}\ast +\mathrm{Le}\right]$$

with L _{app ∗} = Ltf ∗ + Le $${\mathrm{\Delta s}}_{\mathrm{el}}=\frac{\left({\mathrm{P}}_{\mathrm{P}}-{\mathrm{P}}_{\mathrm{a}}\right)\times {\mathrm{10th}}^6}{{\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Strand}}\times \left[\frac{{\mathrm{n}}_{\mathrm{long}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{long}}+{\mathrm{L}}_{\mathrm{e}}\right)}+\frac{{\mathrm{n}}_{\mathrm{short}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{short}}+{\mathrm{L}}_{\mathrm{e}}\right)}\right]}=$$

$$\mathrm{\Delta }{\mathbf{s}}_{\mathbf{el}}=\frac{\left({\mathbf{P}}_{\mathbf{P}}-{\mathbf{P}}_{\mathbf{a}}\right)\times {\mathrm{10th}}^6}{{\mathbf{E}}_t\times {\mathbf{A}}_{\mathit{Strand}}\times \left[{\mathbf{n}}_{\mathit{long}}+{\mathbf{n}}_{\mathit{short}}\right]}\times {\mathbf{L}}_{\mathbf{app}\ast }$$

with L _app ∗ = Ltf ∗ + Le

$${\mathrm{\Delta s}}_{\mathrm{Max}}=\frac{\left({\mathrm{P}}_{\mathrm{P}}-{\mathrm{P}}_{\mathrm{a}}\right)\times {\mathrm{10th}}^6}{{\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Strand}}\times \left[\frac{{\mathrm{n}}_{\mathrm{long}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{long}}+{\mathrm{L}}_{\mathrm{e}}+\frac{{\mathrm{L}}_{\mathrm{tb},\mathrm{long}}}{\mathrm{2nd}}\right)}+\frac{{\mathrm{n}}_{\mathrm{short}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{short}}+{\mathrm{L}}_{\mathrm{e}}+\frac{{\mathrm{L}}_{\mathrm{tb},\mathrm{short}}}{\mathrm{2nd}}\right)}\right]}$$

$$\mathrm{\Delta }{\mathbf{s}}_{\mathit{Max}}=\frac{\left({\mathbf{P}}_{\mathbf{P}}-{\mathbf{P}}_{\mathbf{a}}\right)\times {\mathrm{10th}}^6}{{\mathbf{E}}_t\times {\mathbf{A}}_{\mathit{Strand}}\times \left[{\mathbf{n}}_{\mathit{long}}+{\mathbf{n}}_{\mathit{short}}\right]}\times \left({\mathbf{L}}_{\mathbf{app}\ast }+0.5{\mathbf{L}}_{\mathbf{tb}\ast }\right)$$

with L _tb * = 2 (L _{app, a} * - L _app *)
Choice: Number of strands per anchor part $${\mathbf{n}}_{\mathit{long}\left(\mathbf{right}\right)}={\mathbf{n}}_{\mathit{short}}\frac{\left({\mathbf{L}}_{\mathbf{tf},\mathbf{long}}+{\mathbf{L}}_e\right)}{\left({\mathbf{L}}_{\mathbf{tf},\mathbf{short}}+{\mathbf{L}}_e\right)}$$

given (!): nshort -> calculated: nlang (right) -> selected: nlang (selected) $$\mathrm{Correction\; factor}{\mathrm{\epsilon }}_{\mathrm{S}}=\frac{P_{\mathit{short}}}{P_{\mathit{long}}}=\frac{n_{\mathrm{short}}\left[L_{\mathit{tf},\mathit{long}}+\mathit{Le}\right]}{n_{\mathrm{long}\left(\mathrm{chosen}\right)}\left[L_{\mathit{tf},\mathit{short}}+\mathit{Le}\right]}$$

permissible test force P _{P, perm.} (entire anchor): $${\mathrm{P}}_{\mathrm{p},\mathrm{perm},\mathrm{multibond}}=\min \left\{\begin{array}{l}0.80\times {\mathrm{A}}_{\mathrm{Strand}}\times {\mathrm{f}}_{\mathrm{t},\mathrm{k}}\\ 0.90\times {\mathrm{A}}_{\mathrm{Strand}}\times {\mathrm{f}}_{\mathrm{t},0.1,\mathrm{k}}\end{array}\right\}{\mathrm{n}}_{\mathrm{short}}\left(1+\frac{1}{{\mathrm{\epsilon }}_{\mathbf{S}}}\right)$$

elastic elongation Δ _{el, PP} with test force (entire anchor): [incl. Elongation from preload] $$\mathrm{\Delta }\mathbf{el},\mathbf{Pp}=\frac{{\mathrm{P}}_{\mathrm{P}}\times {\mathrm{10th}}^6}{{\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Strand}}\times \left[\frac{{\mathrm{n}}_{\mathrm{long}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{long}}+{\mathrm{L}}_{\mathrm{e}}\right)}+\frac{{\mathrm{n}}_{\mathrm{short}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{short}}+{\mathrm{L}}_{\mathrm{e}}\right)}\right]}$$

elastic strain Δ _{el, PP} with test force (entire anchor): [without strain from preload] $$\mathrm{\Delta }\mathbf{el},\mathbf{Pp}=\frac{\left({\mathrm{P}}_{\mathrm{P}-\mathrm{Pa}}\right)\times {\mathrm{10th}}^6}{{\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Strand}}\times \left[\frac{{\mathrm{n}}_{\mathrm{long}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{long}}+{\mathrm{L}}_{\mathrm{e}}\right)}+\frac{{\mathrm{n}}_{\mathrm{short}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{short}}+{\mathrm{L}}_{\mathrm{e}}\right)}\right]}$$

$${\mathrm{P}}_{\mathrm{P},\mathrm{lange}\mathrm{Litze}}=\frac{{\mathrm{\Delta }}_{\mathrm{el},\mathrm{PP}}\times {\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Litze}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{lang}}+{\mathrm{L}}_{\mathrm{e}}\right)}$$

(für je eine Litze) Teilkraft in kurzem bzw. langem Ankerteil bei Prüfkraft: $${\mathrm{P}}_{\mathrm{P},\mathrm{kurz}}=\frac{{\mathrm{\Delta }}_{\mathrm{el},\mathrm{PP}}\times {\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Litze}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{kurz}}+{\mathrm{L}}_{\mathrm{e}}\right)}\times {\mathrm{n}}_{\mathrm{kurz}}$$

(Teilankerkraft kurzer Teil - Anker) $${\mathrm{P}}_{\mathrm{P},\mathrm{lang}}=\frac{{\mathrm{\Delta }}_{\mathrm{el},\mathrm{PP}}\times {\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Litze}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{lang}}+{\mathrm{L}}_{\mathrm{e}}\right)}\times {\mathrm{n}}_{\mathrm{lang}}$$

(Teilankerkraft langer Teil - Anker) theoretische, freie Ersatz-Stahllänge L _app, * (gesamter Ersatzanker) : $$\begin{array}{l}{\mathbf{L}}_{\mathbf{app},\ast }=\frac{{\mathrm{\Delta }}_{\mathrm{el},\mathrm{PP}}\times {\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Litze}}\times \left({\mathrm{n}}_{\mathrm{lant}}+{\mathrm{n}}_{\mathrm{kurz}}\right)}{{\mathrm{P}}_{\mathrm{P}}}\\ =\frac{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{lang}}+{\mathrm{L}}_{\mathrm{e}}\right)\times \left({\mathrm{L}}_{\mathrm{tf},\mathrm{kurz}}+{\mathrm{L}}_{\mathrm{e}}\right)\times \left({\mathrm{n}}_{\mathrm{lang}}+{\mathrm{n}}_{\mathrm{kurz}}\right)}{{\mathrm{n}}_{\mathrm{lang}}\left({\mathrm{L}}_{\mathrm{tf},\mathrm{kurz}}+{\mathrm{L}}_{\mathrm{e}}\right)+{\mathrm{n}}_{\mathrm{kurz}}\left({\mathrm{L}}_{\mathrm{tf},\mathrm{lang}}+{\mathrm{L}}_{\mathrm{e}}\right)}\end{array}$$

(Ersatzlänge an der "c"-Linie) $${\mathbf{L}}_{\mathbf{app},\ast }=\mathrm{Ltf}\ast +\mathrm{Le}$$

theoretische, freie Ersatz-Stahllänge L _app,a * (gesamter Ersatzanker) an der Grenzlinie "a": $$\begin{array}{l}{\mathbf{L}}_{\mathbf{app},\mathbf{a},\ast }=\mathrm{Lapp}\ast +\mathrm{0,5}\mathrm{Ltb}\ast \\ =\frac{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{lang}+}{\mathrm{L}}_{\mathrm{e}+\mathrm{0,5}}{\mathrm{L}}_{\mathrm{tblang}}\right)\times \left({\mathrm{L}}_{\mathrm{tf},\mathrm{kurz}+}{\mathrm{L}}_{\mathrm{e}+\mathrm{0,5}}{\mathrm{L}}_{\mathrm{tbkurz}}\right)\times \left({\mathrm{n}}_{\mathrm{lang}}+{\mathrm{n}}_{\mathrm{kurz}}\right)}{{\mathrm{n}}_{\mathrm{lang}}\left({\mathrm{L}}_{\mathrm{tf},\mathrm{kurz}+}{\mathrm{L}}_{\mathrm{e}+\mathrm{0,5}}{\mathrm{L}}_{\mathrm{tbkurz}}\right)+{\mathrm{n}}_{\mathrm{kurz}}\left({\mathrm{L}}_{\mathrm{tf},\mathrm{lang}}+{\mathrm{L}}_{\mathrm{e}+\mathrm{0,5}}{\mathrm{L}}_{\mathrm{tblang}}\right)}\end{array}$$

$${\mathbf{L}}_{\mathbf{app},\ast }=\mathrm{Ltf}\ast +\mathrm{Le}$$

theoretische, Ersatz-Verankerungslänge Ltb,* (gesamter Staffelanker) :
L_tb* = 2 [L _app,a * - L_app*] (theoretische, Ersatz- Verankerungslänge L_tb* gesamter Staf-felanker)
näherungsweise L_tb* = L_tb,kurz = L_tb,lang
Dehnungsberechnung für Grenzlinien (gesamter Staffelanker):
untere Grenzlinie "b" : $${\mathrm{\Delta s}}_{\min }=\frac{\left({\mathrm{P}}_{\mathrm{P}}-{\mathrm{P}}_{\mathrm{a}}\right)\times {10}^6}{{\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Litze}}\times \left[\frac{{\mathrm{n}}_{\mathrm{lang}}}{\left(\mathrm{0,8}\times {\mathrm{L}}_{\mathrm{tf},\mathrm{lang}}+{\mathrm{L}}_{\mathrm{e}}\right)}+\frac{{\mathrm{n}}_{\mathrm{kurz}}}{\left(\mathrm{0,8}\times {\mathrm{L}}_{\mathrm{tf},\mathrm{kurz}}+{\mathrm{L}}_{\mathrm{e}}\right)}\right]}$$

$$\mathrm{\Delta }{\mathbf{s}}_{\mathit{min}}=\frac{\left({\mathbf{P}}_{\mathrm{P}}-{\mathbf{P}}_{\mathrm{a}}\right)\times {10}^6}{{\mathbf{E}}_{\mathrm{t}}\times {\mathbf{A}}_{\mathrm{Litze}}\times \left[{\mathbf{n}}_{\mathrm{lang}}+{\mathbf{n}}_{\mathrm{kurz}}\right]}\times \left[\mathrm{0,8}\mathrm{Ltf}\ast +\mathrm{Le}\right]$$

mit Lapp * = Ltf * + Le theoretische Grenzlinie "c" : $${\mathrm{\Delta s}}_{\mathrm{el}}=\frac{\left({\mathrm{P}}_{\mathrm{P}}-{\mathrm{P}}_{\mathrm{a}}\right)\times {10}^6}{{\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Litze}}\times \left[\frac{{\mathrm{n}}_{\mathrm{lang}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{lang}}+{\mathrm{L}}_{\mathrm{e}}\right)}+\frac{{\mathrm{n}}_{\mathrm{kurz}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{kurz}}+{\mathrm{L}}_{\mathrm{e}}\right)}\right]}=$$

$$\mathrm{\Delta }{\mathbf{s}}_{\mathbf{el}}=\frac{\left({\mathbf{P}}_{\mathbf{P}}-{\mathbf{P}}_{\mathbf{a}}\right)\times {10}^6}{{\mathbf{E}}_t\times {\mathbf{A}}_{\mathit{Litze}}\times \left[{\mathbf{n}}_{\mathit{lang}}+{\mathbf{n}}_{\mathit{kurz}}\right]}\times {\mathbf{L}}_{\mathbf{app}\ast }$$

mit Lapp * = Ltf * + Le
obere Grenzlinie "a" : $${\mathrm{\Delta s}}_{\max }=\frac{\left({\mathrm{P}}_{\mathrm{P}}-{\mathrm{P}}_{\mathrm{a}}\right)\times {10}^6}{{\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Litze}}\times \left[\frac{{\mathrm{n}}_{\mathrm{lang}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{lang}}+{\mathrm{L}}_{\mathrm{e}}+\frac{{\mathrm{L}}_{\mathrm{tb},\mathrm{lang}}}{2}\right)}+\frac{{\mathrm{n}}_{\mathrm{kurz}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{kurz}}+{\mathrm{L}}_{\mathrm{e}}+\frac{{\mathrm{L}}_{\mathrm{tb},\mathrm{kurz}}}{2}\right)}\right]}$$

$$\mathrm{\Delta }{\mathbf{s}}_{\mathit{max}}=\frac{\left({\mathbf{P}}_P-{\mathbf{P}}_a\right)\times {10}^6}{{\mathbf{E}}_t\times {\mathbf{A}}_{\mathbf{Litze}}\times \left[{\mathbf{n}}_{\mathbf{lang}}+{\mathbf{n}}_{\mathbf{kurz}}\right]}\times \left({\mathbf{L}}_{\mathbf{app}\ast }+\mathrm{0,5}{\mathbf{L}}_{\mathbf{tb}\ast }\right)$$

mit L_tb* = 2 [L_app,a* - L_app*] näherungsweise L_tb,* = L_tb,kurz = L_tb,lang Single force in strand with test force: $${\mathrm{P}}_{\mathrm{P},\mathrm{short}\mathrm{Strand}}=\frac{{\mathrm{\Delta }}_{\mathrm{el},\mathrm{PP}}\times {\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Strand}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{short}}+{\mathrm{L}}_{\mathrm{e}}\right)}$$

$${\mathrm{P}}_{\mathrm{P},\mathrm{Long}\mathrm{Strand}}=\frac{{\mathrm{\Delta }}_{\mathrm{el},\mathrm{PP}}\times {\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Strand}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{long}}+{\mathrm{L}}_{\mathrm{e}}\right)}$$

(for one strand each) partial force in short or long anchor part with test force: $${\mathrm{P}}_{\mathrm{P},\mathrm{short}}=\frac{{\mathrm{\Delta }}_{\mathrm{el},\mathrm{PP}}\times {\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Strand}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{short}}+{\mathrm{L}}_{\mathrm{e}}\right)}\times {\mathrm{n}}_{\mathrm{short}}$$

(Partial anchor force short part - anchor) $${\mathrm{P}}_{\mathrm{P},\mathrm{long}}=\frac{{\mathrm{\Delta }}_{\mathrm{el},\mathrm{PP}}\times {\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Strand}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{long}}+{\mathrm{L}}_{\mathrm{e}}\right)}\times {\mathrm{n}}_{\mathrm{long}}$$

(Partial anchor force long part - anchor) theoretical, free replacement steel length L _app, * (entire replacement anchor): $$\begin{array}{l}{\mathbf{L}}_{\mathbf{app},\ast }=\frac{{\mathrm{\Delta }}_{\mathrm{el},\mathrm{PP}}\times {\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Strand}}\times \left({\mathrm{n}}_{\mathrm{lant}}+{\mathrm{n}}_{\mathrm{short}}\right)}{{\mathrm{P}}_{\mathrm{P}}}\\ =\frac{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{long}}+{\mathrm{L}}_{\mathrm{e}}\right)\times \left({\mathrm{L}}_{\mathrm{tf},\mathrm{short}}+{\mathrm{L}}_{\mathrm{e}}\right)\times \left({\mathrm{n}}_{\mathrm{long}}+{\mathrm{n}}_{\mathrm{short}}\right)}{{\mathrm{n}}_{\mathrm{long}}\left({\mathrm{L}}_{\mathrm{tf},\mathrm{short}}+{\mathrm{L}}_{\mathrm{e}}\right)+{\mathrm{n}}_{\mathrm{short}}\left({\mathrm{L}}_{\mathrm{tf},\mathrm{long}}+{\mathrm{L}}_{\mathrm{e}}\right)}\end{array}$$

(Replacement length on the "c" line) $${\mathbf{L}}_{\mathbf{app},\ast }=\mathrm{Ltf}\ast +\mathrm{Le}$$

theoretical, free replacement steel length L _{app, a} * (entire replacement anchor) at the border line "a": $$\begin{array}{l}{\mathbf{L}}_{\mathbf{app},\mathbf{a},\ast }=\mathrm{Lapp}\ast +0.5\mathrm{Ltb}\ast \\ =\frac{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{long}+}{\mathrm{L}}_{\mathrm{e}+0.5}{\mathrm{L}}_{\mathrm{tblang}}\right)\times \left({\mathrm{L}}_{\mathrm{tf},\mathrm{short}+}{\mathrm{L}}_{\mathrm{e}+0.5}{\mathrm{L}}_{\mathrm{tb\; short}}\right)\times \left({\mathrm{n}}_{\mathrm{long}}+{\mathrm{n}}_{\mathrm{short}}\right)}{{\mathrm{n}}_{\mathrm{long}}\left({\mathrm{L}}_{\mathrm{tf},\mathrm{short}+}{\mathrm{L}}_{\mathrm{e}+0.5}{\mathrm{L}}_{\mathrm{tb\; short}}\right)+{\mathrm{n}}_{\mathrm{short}}\left({\mathrm{L}}_{\mathrm{tf},\mathrm{long}}+{\mathrm{L}}_{\mathrm{e}+0.5}{\mathrm{L}}_{\mathrm{tblang}}\right)}\end{array}$$

$${\mathbf{L}}_{\mathbf{app},\ast }=\mathrm{Ltf}\ast +\mathrm{Le}$$

theoretical, replacement anchorage length Ltb, * (entire relay anchor):
L _tb * = 2 [ L _{app, a} * - L _app *] (theoretical, substitute anchoring length L _tb * entire stake anchor)
approximately L _tb * = L _{tb, short} = L _{tb, long}
Strain calculation for boundary lines (entire relay anchor):
lower limit line "b" : $${\mathrm{\Delta s}}_{\min }=\frac{\left({\mathrm{P}}_{\mathrm{P}}-{\mathrm{P}}_{\mathrm{a}}\right)\times {\mathrm{10th}}^6}{{\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Strand}}\times \left[\frac{{\mathrm{n}}_{\mathrm{long}}}{\left(0.8\times {\mathrm{L}}_{\mathrm{tf},\mathrm{long}}+{\mathrm{L}}_{\mathrm{e}}\right)}+\frac{{\mathrm{n}}_{\mathrm{short}}}{\left(0.8\times {\mathrm{L}}_{\mathrm{tf},\mathrm{short}}+{\mathrm{L}}_{\mathrm{e}}\right)}\right]}$$

$$\mathrm{\Delta }{\mathbf{s}}_{\mathit{min}}=\frac{\left({\mathbf{P}}_{\mathrm{P}}-{\mathbf{P}}_{\mathrm{a}}\right)\times {\mathrm{10th}}^6}{{\mathbf{E}}_{\mathrm{t}}\times {\mathbf{A}}_{\mathrm{Strand}}\times \left[{\mathbf{n}}_{\mathrm{long}}+{\mathbf{n}}_{\mathrm{short}}\right]}\times \left[0.8\mathrm{Ltf}\ast +\mathrm{Le}\right]$$

with Lapp * = Ltf * + Le theoretical boundary line "c" : $${\mathrm{\Delta s}}_{\mathrm{el}}=\frac{\left({\mathrm{P}}_{\mathrm{P}}-{\mathrm{P}}_{\mathrm{a}}\right)\times {\mathrm{10th}}^6}{{\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Strand}}\times \left[\frac{{\mathrm{n}}_{\mathrm{long}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{long}}+{\mathrm{L}}_{\mathrm{e}}\right)}+\frac{{\mathrm{n}}_{\mathrm{short}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{short}}+{\mathrm{L}}_{\mathrm{e}}\right)}\right]}=$$

$$\mathrm{\Delta }{\mathbf{s}}_{\mathbf{el}}=\frac{\left({\mathbf{P}}_{\mathbf{P}}-{\mathbf{P}}_{\mathbf{a}}\right)\times {\mathrm{10th}}^6}{{\mathbf{E}}_t\times {\mathbf{A}}_{\mathit{Strand}}\times \left[{\mathbf{n}}_{\mathit{long}}+{\mathbf{n}}_{\mathit{short}}\right]}\times {\mathbf{L}}_{\mathbf{app}\ast }$$

with Lapp * = Ltf * + Le
upper limit line "a" : $${\mathrm{\Delta s}}_{\mathrm{Max}}=\frac{\left({\mathrm{P}}_{\mathrm{P}}-{\mathrm{P}}_{\mathrm{a}}\right)\times {\mathrm{10th}}^6}{{\mathrm{E}}_{\mathrm{t}}\times {\mathrm{A}}_{\mathrm{Strand}}\times \left[\frac{{\mathrm{n}}_{\mathrm{long}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{long}}+{\mathrm{L}}_{\mathrm{e}}+\frac{{\mathrm{L}}_{\mathrm{tb},\mathrm{long}}}{\mathrm{2nd}}\right)}+\frac{{\mathrm{n}}_{\mathrm{short}}}{\left({\mathrm{L}}_{\mathrm{tf},\mathrm{short}}+{\mathrm{L}}_{\mathrm{e}}+\frac{{\mathrm{L}}_{\mathrm{tb},\mathrm{short}}}{\mathrm{2nd}}\right)}\right]}$$

$$\mathrm{\Delta }{\mathbf{s}}_{\mathit{Max}}=\frac{\left({\mathbf{P}}_P-{\mathbf{P}}_a\right)\times {\mathrm{10th}}^6}{{\mathbf{E}}_t\times {\mathbf{A}}_{\mathbf{Strand}}\times \left[{\mathbf{n}}_{\mathbf{long}}+{\mathbf{n}}_{\mathbf{short}}\right]}\times \left({\mathbf{L}}_{\mathbf{app}\ast }+0.5{\mathbf{L}}_{\mathbf{tb}\ast }\right)$$

with L _tb * = 2 [L _{app, a} * - L _app *] approximately L _tb, * = L _{tb, short} = L _{tb, long}

With regard to further advantageous embodiments of the teaching according to the invention, reference is made to the general part of the description and to the appended claims in order to avoid repetition.

Finally, it should be expressly pointed out that the exemplary embodiments described above are only used to discuss the claimed teaching, but do not restrict them to the exemplary embodiments.

Reference list

1

Tension member, short

2nd

Tension member, long

3rd

Cladding tube

Claims (11)

1. Stepped anchor for anchoring in a base or a component, having a plurality of expandable tension members (1, 2), wherein at least two of the tension members (1, 2) have a different defined free anchor length Ltf,
   characterised in that the respective expansion rigidities of the at least two tension members (1, 2) in the region of the respective free anchor length Ltf thereof are defined in such a manner that the at least two tension members (1, 2) after a simultaneous common expansion by the same expansion path for each tension member (1, 2) have a substantially identical predetermined tension state or a substantially identical predetermined tension.
2. Stepped anchor according to claim 1, characterised in that the expansion rigidities are defined by predetermining the number n of carrier members or strands of the respective tension members (1, 2).
3. Stepped anchor according to claim 1 or claim 2,
   characterised in that the expansion rigidities are defined by predetermining a cross-sectional surface-area A and/or an elasticity modulus E of carrier members or strands of the respective tension members (1, 2).
4. Stepped anchor according to claim 1, characterised in that the expansion rigidities are defined by predetermining a cross-sectional surface-area A and/or an elasticity modulus E of the respective tension members (1, 2).
5. Stepped anchor according to any one of claims 1 to 3, characterised in that the number of carrier members or strands n in the at least two tension members (1, 2) in order to achieve substantially identical tension states or tensions in the tension members (1, 2) is defined by the following formula: $${\mathrm{n}}_{\mathrm{lang}\left(\mathrm{rech}\right)}={\mathrm{n}}_{\mathrm{kurz}}*\left({\mathrm{L}}_{\mathrm{tf},\mathrm{lang}}+{\mathrm{L}}_{\mathrm{e}}\right)/\left({\mathrm{L}}_{\mathrm{tf},\mathrm{kurz}}+{\mathrm{L}}_{\mathrm{e}}\right)$$

   

   where n_lang(rech) is the number of carrier members or strands in the tension member (2) established by calculation with a longer free anchor length L_{tf, lang}, n_kurz is the number of carrier members or strands in the tension member (1) with a shorter free anchor length L_{tf, kurz} and L_e is the length of an external overhang.
6. Stepped anchor according to any one of claims 1 to 5, characterised in that respective anchoring lengths L_tb of the at least two tension members (1, 2) are substantially of the same size.
7. Stepped anchor according to any one of claims 1 to 6, characterised in that respective anchoring lengths L_tb of the at least two tension members (1, 2) in the pulling direction are arranged substantially one behind the other and/or in a common pressing member.
8. Stepped anchor according to any one of claims 1 to 7, characterised in that tension members (1, 2) with a different defined free anchor length L_tf are arranged in a substantially alternating manner beside each other or in a state distributed in a uniform manner in the stepped anchor.
9. Stepped anchor according to any one of claims 1 to 8, characterised in that a bundle press for producing the tension, pretension or partial pretension or expansion is associated with the stepped anchor.
10. Method for anchoring a stepped anchor in a base or component, in particular a stepped anchor according to any one of claims 1 to 9, wherein the stepped anchor has a plurality of expandable tension members (1, 2),
    wherein at least two of the tension members (1, 2) have a different defined free anchor length Ltf, and wherein the at least two tension members (1, 2) are expanded simultaneously and together by the same expansion path for each tension member (1, 2),
    characterised in that respective expansion rigidities of the at least two tension members (1, 2) in the region of the respective free anchor length Ltf thereof are defined in such a manner that the at least two tension members (1, 2) after the simultaneous common expansion by the same expansion path for each tension member (1, 2) have a substantially identical predetermined tension state or a substantially identical predetermined tension.
11. Method according to claim 10, characterised in that the tensioning, pretensioning or partial pretensioning or expansion of the at least two tension members (1, 2) is carried out at the same time, preferably by means of a bundle press.

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