## 1. Introduction

In Prestressed concrete structures, the prestressing force is a crucial variable type. The behaviors of pre-stressed concrete structures depend on the effective prestress because it provides compressive stresses to counteract the tensile stresses that develop in the concrete due to loads. However, the prestressing force does not remain constant over time due to various factors that cause prestress losses. These losses can occur during the transfer of prestress from the tendons to the concrete member or over the service life of the structure.

In the case of **pre-tensioning**, tension losses are attributed to tendon relaxation, elastic shortening, creep, and shrinkage.

In the case of **post-tensioning**, tension losses are attributed to frictions between tendons and sheaths, anchorage slip, tendon relaxation, elastic shortening, creep, and shrinkage.

Overall, it is crucial to consider the various types of prestress losses when designing prestressed concrete structures to ensure that the final prestressing force is sufficient to withstand the design loads over the service life of the structure.

The losses are mainly classified into two categories:

- Immediate losses (Instantaneous losses upon release)
- Anchorage Slip
- Elastic shortening of concrete
- Friction between PS tendon and sheaths
- Time-dependent losses after the release
- Creep in concrete
- Shrinkage in concrete
- Relaxation of PS tendons

Figure 1. Types of Losses

## 2. Losses in Prestress

The various types of prestress losses are as follows:

### 2.1 Losses due to Anchorage slip

When a PS tendon is tensioned and released, the pre-stress is transferred to the anchorage. The anchorage typically consists of a series of wedges that grip the strands of the tendon to hold them in place. During tensioning, the wedges are pushed into the anchorage, causing the strands to be compressed together and tightly held in place. However, after the tensioning force is released, the wedges can slip back slightly, causing the strands to loosen slightly and resulting in a small amount of slack in the tendon. This movement is known as anchorage take-up and can result in tension loss in the tendon in the vicinity of the anchorage.

The effect of anchorage slip is present up to a certain length (*lset*) due to the setting of the anchorage block, where reverse friction occurs as the tendon shortens. Beyond this setting length, the effect of anchorage slip is absent as shown in Figure 2.1.

Figure 2.1 Effect on the pre-stressing force due to anchorage slip

The loss due to anchorage slip can be calculated using the following equation,

where,

Δl is the anchorage slip

l is the length of the cable

Ep is the modulus of elasticity of prestressing material

### 2.2 Losses due to Elastic Shortening of Concrete

When a pre-stress force is applied to a concrete member, the concrete undergoes compression, which leads to a reduction in its length. This compression is due to the transfer of the pre-stress force from the tendons to the concrete. As a result, the tendon shortens by the same amount as the concrete, which reduces the tension force in the tendon.

The elastic shortening that occurs during pre-tensioning and post-tensioning is similar in principle, but there are some differences. During pre-tensioning, the concrete member is cast with the tendons already tensioned, and the concrete is allowed to gain strength. During post-tensioning, the concrete member is first cast without any pre-stress force and then the tendons are tensioned after the concrete has gained sufficient strength.

Figure 2.2 Prestress loss due to elastic shortening (pre-tensioned member)

In pre-tensioning, the pre-stress force is applied to the tendon before casting the concrete member, and the tendon is anchored to the end abutments. Once the concrete has hardened, the tendon is released from the end abutments, which results in an instantaneous elastic shortening of the tendon due to its elongation during pre-stressing. This instantaneous elastic shortening causes a loss of pre-tension in the tendon, which is reflected in the difference between the pre-tension (Pj) in the tendon and the pre-stressing force (Pi) applied to the concrete member as shown in Figure 2.2.

In post-tensioning, the pre-stress force is directly applied to the concrete member through the tendon after the concrete has hardened. The elastic shortening that occurs due to the pre-stress force is similar to pre-tensioning, but in post-tensioning, the tension force in the tendon is measured after the elastic shortening has taken place. This means there is no pre-stress loss due to elastic shortening in post-tensioning.

**Note:** MIDAS/CIVIL does not consider prestress loss due to elastic shortening. Therefore, when specifying a pre-stress force in a concrete member that uses pre-tensioning, the pre-stress load (Pi) should be entered instead of the jacking load (Pj) to account for the loss of pre-stress due to elastic shortening.

In post-tensioned members, multiple tendons are usually placed, stressed, and anchored in a pre-defined sequence. A series of concrete elastic shortenings takes place in the same member, and the pre-stress loss in each tendon changes as the pre-stressing sequence progresses. There is no tension loss in the first tendon being stressed as shown in Figure 2.3(b). When the second tendon is tensioned as shown in Figure 2.3(c), a tension loss is observed in the first tendon due to the subsequent shortening from the second tensioning. MIDAS/CIVIL software takes into account pre-stress loss due to elastic shortening at each construction stage and reflects all pre-stress losses due to elastic shortenings caused by external forces. This ensures accurate and reliable analysis and design of post-tensioned members.

Figure 2.3 Prestress losses in multi-tendons due to sequential tensioning (post-tensioned member)

The loss due to elastic shortening can be calculated using the following equation,

EN 1992-1-1 (2004), Clause 5.10.5.1 (2)

where,

ΔPel is loss due to elastic deformation of concrete

Ap is the area of prestressing material

Ep is the modulus of elasticity of prestressing material

Δσc is the variation of stress at the center of gravity of the tendons applied at time t

j is a coefficient equal to

(n-1)/2n where n is the number of identical tendons successively prestressed.

As an approximation, it can be taken as 0.5. Where the stress varies in the tendons due to variations of permanent actions applied after prestressing, j = 1 as all tendons are affected similarly

Ecm is the modulus of elasticity of concrete at time t

### 2.3 Losses due to friction between PS tendons and sheaths

In post-tensioning, friction exists between the prestressed steel tendon and its sheathing or duct. The frictional force causes a reduction in the pre-stressing force in the tendon as it moves away from the jacking ends. The frictional force can be classified into two types: length effect and curvature effect. The length effect, also known as the wobbling effect of the duct, is caused by friction stemming from an imperfect linear alignment of the duct. This effect depends on the length and stress of the tendon and can be expressed using a frictional coefficient, k (/m), per unit length of the duct. The curvature effect, on the other hand, results from the intended curvature of the tendon in addition to the unintended wobbling of the duct. This effect is expressed using a frictional coefficient, μ (/radian), per unit angle of curvature.

If a pre-stressing force P0 is applied at the jacking end, the tendon force Px at a location, *l* away from the end with the angular change α can be expressed as follows:

EN 1992-1-1 (2004), Clause 5.10.5.2 (1)

### 2.4 Time-dependent Losses

Pre-stress losses can occur over time due to various factors, including concrete creep, shrinkage, and tendon relaxation. These effects can cause a gradual reduction in the pre-stressing force in the tendons. EN 1992 gives the following simplified expression to evaluate the time-dependent loss.

EN 1992-1-1 (2004), Clause 5.10.6 (2)

where,

Δ*P*c+s+r is the absolute value of the variation of stress in the tendons due to creep,

shrinkage and relaxation at location x, at time t

*εcs* is the estimated shrinkage stain at time t

Ep is the modulus of elasticity of prestressing material

Ecm is the short-term modulus of elasticity of concrete

Δσpr is the absolute value of the variation of stress in the tendons due to creep,

shrinkage and relaxation at location x, at time t

Φ(t,t0) is the creep coefficient at a time t for initial load application at time t0

σc, QP is the stress in the concrete adjacent to the tendons due to self-weight, initial

prestress and all other quasi-permanent actions where relevant

*A*p is the area of all the prestressing tendons at the section being considered

*A*c is the area of the concrete section

*I*c is the second moment of area of the concrete section

*Z*cp is the eccentricity of the tendons, i.e. the distance between the centroid of the

tendons and the centroid of the concrete section

Accurately accounting for creep, shrinkage, and relaxation losses usually requires a software program because the losses produced in an interval of time affect the state of stress and the creep and relaxation losses over the next interval of time. However, to verify the losses individually the equation can be derived as follows using the above equation.

**Shrinkage of concrete**

Designer Guide to EN 1992-2, D5.10-9

**Creep of concrete**

Designer Guide to EN 1992-2, D5.10-11

**Relaxation loss**

Relaxation loss refers to the decrease in stress with time under constant strain as steel material experiences creep. The amount of relaxation depends on the time, temperature, and level of stress. As per EN 1991, the relaxation loss for the prestress material should be based on the ρ1000 value, the relaxation loss (in %) at 1000 hours after tensioning and at a mean temperature of 20 **°**C. The relaxation loss can be calculated using the following equation. and the value of ρ1000 can be assumed equal to 8% for class 1, 2.5% for class 2, and 4% for class 3.

EN 1992-1-1 (2004), Clause 3.3.2 (5)

where,

Δσpr is the absolute value of the relaxation losses of the prestress

σpi is the absolute value of the initial prestress taken as the prestress value applied to

the concrete immediately after tensioning and anchoring (post-tensioning) or

after prestressing (pre-tensioning) by subtracting the immediate losses

fpk is the characteristic value of the tensile strength of the tendon

µ = σpi / fpk

ρ1000 is the value of relaxation loss (%)

t is the time after tensioning (in hrs)

## 3. Prestress Loss Verification in MIDAS CIVIL

### 3.1 Problem statement

### 3.1.1 Section properties

To verify the prestress loss, let’s consider the PSC I-section as shown in the below image of 28m length. and the material and section properties are as follows.

Figure 3.1 Cross-section details

1 | Grade of Concrete | C40/50 |
---|---|---|

2 | Modulus of elasticity (Concrete),Ec | 35220 MPa |

3 | Modulus of elasticity (Steel),Es | 205000 MPa |

4 | Cross Section Area, Ac | 790000 mm2 |

5 | Moment of Inertia, Iy | 280500000000 mm4 |

### 3.1.2 Tendon properties

Two tendon profiles are defined as shown in the below image, and a tendon prestress load of **1400 MPa** is applied at the beginning point.

Figure 3.2 Cable 1, Tendon profile details

Figure 3.3 Cable 2, Tendon profile details

Tendon properties are as follows:

1 | Modulus of elasticity (Steel), Es | 205000 MPa |
---|---|---|

2 | Tendon type | Internal(Post-Tension) |

3 | Area of one tendon, (Ap) | 1480.65 mm2 |

4 | Duct diameter | 100mm |

5 | Relaxation coefficient | European (Hot rolled) |

6 | Curvature friction factor, (μ) | 0.3 |

7 | Wobble friction factor, (k) | 0.0066 /m |

8 | Anchorage slip (one side), (Δs) | 6 mm |

### 3.1.3 Time-dependent material properties

To verify the time-dependent loss, Creep, and Shrinkage parameters are defined as per ES 1991.

Figure 3.4 Creep and Shrinkage parameter definition in MIDAS CIVIL

### 3.1.4 Tendon loss verification

To verify the prestress losses, separate MIDAS CIVIL models are created to analyze only one loss variable at a time and the prestress losses are calculated using the above equation as mentioned in part 2 of the section. So, let’s compare the results obtained from MIDAS CIVIL and manual calculation.

**A. Anchorage Slip**

To check the anchorage slip loss, for the tendon property definition in MIDAS CIVIL only the parameter for anchorage slip is defined as 6mm (one-sided).

Figure 3.5 Tendon property definition (Anchorage slip loss)

The results obtained from MIDAS CIVIL and manual calculation for anchorage slip loss are shown in the figure below.**For cable 1,**Anchorage slip, ΔL = 6mm

Length of cable 1, L1 = 28009.17 mm

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