Force Distribution at the Tensioning State

𝜅 = 1

𝑆_𝑡

∫︁ 1

−1

𝜅 𝐽_𝑆𝑑𝜌 (6.25)

where the total length of the tendon curve𝑆_𝑡follows from Eq.(6.14/6.15) by inserting 𝜌= 1.

6.3 Force Distribution at the Tensioning State

6.3.1 Loss Due to Friction

𝑆

𝑑𝑆

𝑃 𝑃 +𝑑𝑃

𝑞_𝑠𝑛 𝑞𝑠𝑡

𝑅 = 1/𝜅 𝑑𝜃

Figure 6.2: Forces on an Infinitesimal Tendon Element

Consider equilibrium of an infinitesimal tendon element, as depicted in Fig. 6.2.

Equilibrium in the radial or normal direction leads to the following expression for the contact pressure-force per unit arclength

𝑞_𝑠𝑛 = 𝑃 𝜅 (6.26)

while equilibrium in the circumferential or tangential direction yields the frictional force per unit arclength

𝑞_𝑠𝑡 = −𝑑𝑃

𝑑𝑆 (6.27)

The following approximate friction law will be adopted

𝑞𝑠𝑡 = 𝜇 𝑞𝑠𝑛 + 𝑘 𝑃 (6.28)

where 𝜇 is the ordinary friction coefficient, for tendons usually termed curvature friction coefficient, while 𝑘 is denoted the wobble friction coefficient. Combination of

the three equations yields the following differential equation for the prestressing force 𝑑𝑃

𝑑𝑆 + (𝜇 𝜅 + 𝑘)𝑃 = 0 (6.29)

The solution may be expressed on the form

𝑃 = 𝑃₁𝑒−^(︁𝜇^∫︀₀^𝑆𝜅 𝑑𝑆˜+𝑘𝑆^)︁

(6.30) where𝑃₁ is the applied prestressing force at location𝑆 = 0, i.e. at tendon endpoint 1.

The integral in the exponent may be solved numerically for each new tendon location as explained in conjunction with Eq.(6.11). However, a simplification is to substitute the varying curvature by its mean value along the tendon curve from Eq.(6.25). Then Eq.(6.30) takes the simplified form

𝑃 = 𝑃₁𝑒−(𝜇¯𝜅+𝑘)𝑆 (6.31)

Analogous, if a prestressing force𝑃₂ is applied at tendon endpoint 2, the correspond-ing formula reads

𝑃 = 𝑃₂𝑒−(𝜇¯𝜅+𝑘) (𝑆_𝑡−𝑆) (6.32) where 𝑆_𝑡 is the total length of the tendon curve.

6.3.2 Loss Due to Anchorage Slip

When transferring the tensioning force from the jack to the anchor, a slip motion in the tendon may arise which results in a loss of prestressing. However, due to the frictional force, this loss again will vanish at a certain location, here termed the ‘slip’-location 𝑆_𝑠. Fig. 6.3 illustrates this situation for a tendon jacked from endpoint 1. Mari [23] developed a numerical search procedure for determining 𝑆_𝑠1 and the corresponding loss variation based on the assumptions that the force curves prior to and after the anchorage slip are symmetric about the horizontal dotted line in Fig. 6.3, and further that the area between the two curves is proportional to the amount of slip at the anchor. The same approach was later adopted in [24]-[27].

While the latter of these two assumptions is correct, the first one is not, since the slope of the force curve will always be proportional to the force itself at the same location (ref. Eq.(6.29)). However, a quite simple closed form solution of the problem may be found, which will be covered in the following.

Let the force at the intersection between the initial force curve from endpoint 1 and the curve including anchorage loss be denoted 𝑃_𝑠1. By solving Eq.(6.29) for the latter branch, it follows that this force variation can be expressed by

𝑃_𝑎1 = 𝑃_𝑠1𝑒−(𝜇¯𝜅+𝑘) (𝑆𝑠1−𝑆) (6.33) where subscript ‘𝑎1’ signifies that anchorage loss now is included. 𝑃_𝑠1 itself is given by Eq.(6.31)

𝑃_𝑠1 = 𝑃₁𝑒−(𝜇¯𝜅+𝑘)𝑆𝑠1 (6.34)

𝑃

𝑃₁

𝑃_𝑠1 𝑃₁−Δ𝑃₁

𝑆 𝑆_𝑠1

Eq.(6.31)

Figure 6.3: Loss at Tendon Endpoint 1 and thus

𝑃_𝑎1 = 𝑃₁𝑒−(𝜇¯𝜅+𝑘) (2𝑆_𝑠1 −𝑆) (6.35) The force variation 𝑃_𝑖1 along the initial branch is also expressed by Eq.(6.31). Then the prestressing loss due to anchorage slip becomes

Δ𝑃 = 𝑃_𝑖1 − 𝑃_𝑎1

= 𝑃₁

[︂

𝑒−(𝜇¯𝜅+𝑘)𝑆 − 𝑒−(𝜇¯𝜅+𝑘) (2𝑆_𝑠1−𝑆)^]︂ (6.36) Now, let the slip at the anchor be denoted𝑢𝑠. This slip must be related to the change in strain due to prestressing loss as follows

𝑢_𝑠 =

∫︁ _𝑆𝑠1

0

Δ𝜖_𝑝𝑑𝑆 (6.37)

Since the anchorage loss is an unloading process, a linear stress-strain relationship can be assumed, i.e.

Δ𝜖_𝑝 = Δ𝜎_𝑝

𝐸_𝑝 = Δ𝑃

𝐸_𝑝𝐴_𝑝 (6.38)

where𝐸_𝑝 is the modulus of elasticity of prestressing steel, and𝐴_𝑝 is the cross section area of the tendon. Combination of the last three equations yields the expression

𝑢_𝑠 = 𝑃₁ 𝐸𝑝𝐴𝑝

[︃∫︁ _𝑆𝑠1

0

𝑒−(𝜇¯𝜅+𝑘)𝑆 𝑑𝑆 − 𝑒−2 (𝜇¯𝜅+𝑘)𝑆_𝑠1^{∫︁ 𝑆𝑠1}

0

𝑒(𝜇¯𝜅+𝑘)𝑆 𝑑𝑆^]︃

(6.39) By solving the integrals, the result can be written on the form

𝑒−2 (𝜇¯𝜅+𝑘)𝑆_𝑠1 − 2𝑒−(𝜇¯𝜅+𝑘)𝑆_𝑠1 + 1 − 𝐸_𝑝𝐴_𝑝𝑢_𝑠(𝜇¯𝜅+𝑘)

𝑃₁ = 0 (6.40)

This is a quadratic equation in the term exp{−(𝜇¯𝜅+𝑘)𝑆𝑠1}. The correct solution becomes

𝑒−(𝜇¯𝜅+𝑘)𝑆_𝑠1 = 1 −

√︃𝐸_𝑝𝐴_𝑝𝑢_𝑠(𝜇¯𝜅+𝑘)

𝑃₁ (6.41)

Thus, the inverse relationship yields the solution for the slip-location 𝑆_𝑠1 𝑆𝑠1 = − 1

𝜇¯𝜅+𝑘 ln

⎡

⎣1−

√︃𝐸_𝑝𝐴_𝑝𝑢_𝑠(𝜇¯𝜅+𝑘) 𝑃₁

⎤

⎦ (6.42)

Analogous to Eqs.(6.35,6.42); when jacking the tendon from endpoint 2, the expres-sions for the force variation after anchorage loss 𝑃_𝑎2 and for the slip-location 𝑆_𝑠2 (measured from endpoint 1) become

𝑃_𝑎2 = 𝑃₂ 𝑒−(𝜇¯𝜅+𝑘) (𝑆_𝑡−2𝑆_𝑠2+𝑆) (6.43) 𝑆_𝑠2 = 𝑆_𝑡 + 1

𝜇¯𝜅+𝑘 ln

⎡

⎣1−

√︃𝐸_𝑝𝐴_𝑝𝑢_𝑠(𝜇¯𝜅+𝑘) 𝑃2

⎤

⎦ (6.44)

Finally note that the loss due to anchorage slip is assumed to be a local effect near the anchor. If the amount of slip was sufficiently large (or the friction sufficiently small), so that the loss would affect the prestressing over the entire length of the tendon, the expressions derived in this subsection are not strictly valid. Then Eq.(6.42) (or Eq.(6.44)) would identify a fictitious slip-location behind the opposite end of the tendon, resulting in underestimated loss predictions. In such a case, the force after loss at the opposite end should be treated as the unknown rather than the slip-location.

6.3.3 Force Distribution for a ‘Single-Curve’ Tendon

In this subsection the final force distribution at the tensioning state will be found for a tendon consisting only of one parametrized curve, for short called a ‘single-curve’

tendon.

In general, tendons may be subjected to jacking from both ends, or from either one of the two ends only. When dealing with two-end jacking, the frictional force will change direction somewhere along the tendon. At this location, here termed the

‘reverse’-location 𝑆_𝑟, the prestressing forces arising from jacking from each end are equal. Thus, from Eqs.(6.31,6.32)

𝑃₁𝑒−(𝜇¯𝜅+𝑘)𝑆_𝑟 = 𝑃₂𝑒−(𝜇¯𝜅+𝑘) (𝑆_𝑡−𝑆_𝑟) (6.45) By taking the natural logarithm on both sides, the solution becomes

𝑆_𝑟 = 𝑆_𝑡

2 + 1

2 (𝜇¯𝜅+𝑘) ln𝑃₁

𝑃₂ (6.46)

Now, by also including anchorage losses, the force distribution for a single-curve tendon subjected to two-end jacking, can be summarized as follows

𝑃 =

⎧

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

𝑃₁𝑒−(𝜇¯𝜅+𝑘) (2𝑆𝑠1−𝑆) ; 0≤𝑆 ≤𝑆_𝑠1 𝑃₁𝑒−(𝜇¯𝜅+𝑘)𝑆 ; 𝑆𝑠1 < 𝑆 ≤𝑆𝑟

𝑃₂𝑒−(𝜇¯𝜅+𝑘) (𝑆𝑡−𝑆) ; 𝑆_𝑟 < 𝑆 ≤𝑆_𝑠2 𝑃₂𝑒−(𝜇¯𝜅+𝑘) (𝑆_𝑡−2𝑆_𝑠2+𝑆) ; 𝑆𝑠2 < 𝑆 ≤𝑆𝑡

(6.47)

where the individual force expressions are taken from Eqs.(6.35,6.31,6.32,6.43), in their order of appearance. Furthermore, the expressions for (𝑆_𝑠1, 𝑆_𝑟, 𝑆_𝑠2) are given by Eqs.(6.42,6.46,6.44), respectively, while 𝑆_𝑡 follows from Eq.(6.14/6.15) by inserting 𝜌= 1. The force distribution is depicted in Fig. 6.4.

𝑃

𝑃₁ 𝑃₂

𝑆

𝑆𝑠1 𝑆𝑟 𝑆𝑠2 𝑆𝑡

Figure 6.4: Force Profile for a ‘Single-Curve’ Tendon Jacked from Both Ends The case of one-end jacking from endpoint 1 is covered by the first two formulas of Eq.(6.47), which is obtained by setting 𝑆_𝑟 =𝑆_𝑠2 =𝑆_𝑡. Similarly, one-end jacking from endpoint 2 can be retrieved from Eq.(6.47) by setting 𝑆_𝑠1 =𝑆_𝑟 = 0.

Although the anchorage loss is considered to be a local effect (ref. the closing remark in the preceding subsection), it is necessary in a computer program to foresee all possible situations. When, for two-end jacking, the condition 𝑆_𝑠1 < 𝑆_𝑟 < 𝑆_𝑠2 is not satisfied, the final force distribution will be influenced by the actual tensioning procedure (i.e. which end is jacked first). Since this information is not included as input to the problem, the following interpretations are made for these (rare) situations in the computer program in Chapter 11:

∙ If 𝑆_𝑟 < 𝑆_𝑠1 and 𝑆_𝑟 < 𝑆_𝑠2, then the case is converted to one-end jacking from endpoint 2.

∙ If 𝑆_𝑟 > 𝑆_𝑠1 and 𝑆_𝑟 > 𝑆_𝑠2, then the case is converted to one-end jacking from endpoint 1.

∙ If 𝑆𝑠2 < 𝑆𝑟 < 𝑆𝑠1, then the force distribution is considered undefined (error message).

In addition, since the expressions for (𝑆_𝑠1, 𝑆_𝑟, 𝑆_𝑠2) do not work for zero friction, a separate handling of this simple case is also included. Then the loss due to anchorage slip is uniform along the tendon, given by

Δ𝑃 = 𝐸_𝑝𝐴_𝑝𝑢_𝑠

𝑆_𝑡 (6.48)

6.3.4 Force Distribution for a ‘Multiple-Curve’ Tendon

This subsection deals with the final force distribution at the tensioning state for a tendon composed of more than one parametrized curve, for short called a ‘multiple-curve’ tendon. In the computer program that is reviewed in Chapter 11, this type of tendon may consist of two or three parametrized curves. The procedure for estab-lishing the final force distribution for a multiple-curve tendon is closely related to the concept for a single-curve tendon, but it becomes far more extensive. Nevertheless, the main features can be summarized in the four steps:

∙ First, treat all parametrized curves on an individual basis (i.e. either one-end or trivial ‘zero-zero’ jacking), and determine the quantities (𝑃₁, 𝑃₂, 𝑆_𝑠1, 𝑆_𝑟, 𝑆_𝑠2). If 𝑃₁is the applied force at endpoint 1,𝑃₂ is now the computed force at endpoint 2;

and vice versa.

∙ Next, check the values of the endforces at the continuation points and determine whether the resulting tendon is two-end jacked, or if not; from which end it is one-end jacked.

∙ Then, correct the slip-locations in adjoining parts if initial values indicate cross-ing of continuation points. For two-end jackcross-ing, also determine the reverse-location.

∙ Finally, update the endforces of each part based on the new found information.

The result of this procedure is that each parametrized curve now has an updated set of quantities (𝑃₁, 𝑃₂, 𝑆_𝑠1, 𝑆_𝑟, 𝑆_𝑠2), from which the final force distribution may be retrieved using Eq.(6.47).

Note for two-end jacking, if the condition 𝑆𝑠1 < 𝑆𝑟 < 𝑆𝑠2 now is not satisfied for the resulting tendon, the similar interpretations are made as described for a single-curve tendon. Also the separate case for zero friction is included similarly. Finally, note that the simplification of substituting the varying curvature with its mean value (as basis for the frictional force), now is taken for each parametrized curve separately.

This is a better approximation than applying the mean value for the resulting tendon as a whole. However, if slip-locations are crossing continuation points, the mean curvature is not adjusted accordingly (i.e. the mean curvature from the original end-curve is still retained).

In document Nonlinear analysis of concrete structures based on a 3D shear-beam element formulation (sider 114-120)