## Calculation of design anchorage length of longitudinal reinforcement according to EN1992-1-1 §8.4

### Ultimate bond stress *f*_{bd}

The design value of ultimate bond stress for ribbed bars is defined in EN1992-1-1 §8.4.2(2):

*f*_{bd} = 2.25 ⋅ *η*_{1} ⋅ *η*_{2} ⋅ *f*_{ctd}

where *f*_{ctd} = *α*_{ct} ⋅ *f*_{ctk,0.05} / *γ*_{c} is the design tensile strength of concrete determined according to EN1992-1-1 §3.1.6(2)P.

The coefficient *η*_{2} takes into account the effect of large bar diameters *Φ* > 32 mm as follows:

*η*_{2} = min[1.0, (132 - *Φ*) / 100 ], where *Φ* in mm

The coefficient *η*_{1} is related to the quality of the bond condition and the position of the bar during concreting.
The coefficient *η*_{1} takes a value of 1.0 when 'good' bond conditions are obtained and a value of 0.7 otherwise i.e. when 'poor' bond conditions exist.
The distinction between 'good' and 'poor' bond conditions is provided in EN1992-1-1 Figure 8.2.

'Good' bond conditions are obtained when any of the following conditions is fulfilled:

- Vertical bars or almost vertical bars inclined at an angle 45° ≤ α ≤ 90° from the horizontal
- Bars that are located up to 250 mm from the bottom of the formwork for elements with height
*h* ≤ 600 mm
- Bars that are located at least 300 mm from the free surface during concreting for elements with height
*h* > 600 mm

'Poor' bond conditions are applicable for all other cases and also for bars in structural elements built with slip-forms, unless it can be shown that 'good' bond conditions exist.

### Basic anchorage length *l*_{b,rqd}

The basic required anchorage length *l*_{b,rqd} for anchoring a straight steel bar with diameter *Φ* under design stress *σ*_{sd} is defined in EN1992-1-1 Eq. (8.3):

*l*_{b,rqd} = (*Φ* / 4) ⋅ (*σ*_{sd} / *f*_{bd})

The maximum value of the design steel stress *σ*_{sd} under ULS loads is equal to the design yield strength of the bar *f*_{yd} = *f*_{yk} / *γ*_{s}.
When the actual design strength of the bar is smaller than *f*_{yd} then the basic required anchorage length is reduced proportionally.

### Minimum anchorage length *l*_{b,min}

When no other limitation is applicable, the provided anchorage length should be at least equal to the minimum value *l*_{b,min} as described in EN1992-1-1 §8.4.4(1):

- For anchorages in tension: *l*_{b,min} ≥ max[0.3⋅*l*_{b,rqd}, 10⋅*Φ*, 100 mm]

- For anchorages in compression: *l*_{b,min} ≥ max[0.6⋅*l*_{b,rqd}, 10⋅*Φ*, 100 mm]

### Design anchorage length *l*_{bd}

The design anchorage length *l*_{bd} is defined in EN1992-1-1 §8.4.4(1) as:

*l*_{bd} = *α*_{1} ⋅ *α*_{2} ⋅ *α*_{3} ⋅ *α*_{4} ⋅ *α*_{5} ⋅ *l*_{b,rqd} ≥ *l*_{b,min}

where the coefficients *α*_{1} to *α*_{5} are defined in EN1992-1-1 Table 8.2 and account for various factors reducing the design anchorage length as follows:

- Coefficient
*α*_{1} accounts for the effect of the shape of the bar (*α*_{1} = 1.0 for straight bars, *α*_{1} = 0.7 for other-than straight bar shapes such as bend, hook, and loop assuming adequate cover *c*_{d} > 3⋅*Φ*, *α*_{1} = 1.0 for other-than straight bars without adequate cover), where *c*_{d} is defined in EN1992-1-1 Figure 8.3.
- Coefficient
*α*_{2} accounts for the effect of minimum concrete cover: a) For straight bars in tension *α*_{2} = 1 - 0.15⋅(*c*_{d} - *Φ*) / *Φ*) and 0.7 ≤ *α*_{2} ≤ 1.0, b) for other-than-straight bars in tension *α*_{2} = 1 - 0.15⋅(*c*_{d} - 3⋅*Φ*) / *Φ*) and 0.7 ≤ *α*_{2} ≤ 1.0, c) for any bar shape in compression *α*_{2} = 1.0
- Coefficient
*α*_{3} accounts for the effect of confinement by transverse reinforcement not welded to main reinforcement. For bars in tension it takes values 0.7 ≤ *α*_{3} ≤ 1.0 depending on the quantity of transverse reinforcement. For bars in compression *α*_{3} = 1.0.
- Coefficient
*α*_{4} accounts for the effect of confinement by welded transverse reinforcement. If the requirements of EN1992-1-1 Table 8.2 are fulfilled then it can take the value *α*_{4} = 0.7.
- Coefficient
*α*_{5} accounts for the effect of confinement by transverse pressure. For bars in tension it takes values 0.7 ≤ *α*_{5} ≤ 1.0 depending on the value of transverse pressure. For bars in compression *α*_{5} is not applicable.
- In any case the lower limit of the product (
*α*_{2}⋅*α*_{3}⋅*α*_{5}) ≥ 0.7 must be observed.

As a simplified and conservative alternative the equivalent anchorage length *l*_{b,eq} may be provided that is *l*_{b,eq} = *α*_{1} ⋅ *l*_{b,rqd} for straight, bend, hook, and loop bar shapes, or *l*_{b,eq} = *α*_{4} ⋅ *l*_{b,rqd} for bars with welded transverse bars.
In the provided tables in this calculation the equivalent anchorage length *l*_{b,eq} is shown.

### Detailing rules for reinforcement anchorage

Standard detailing rules for the anchorage of straight bars and other-than-straight bar shapes (bend, hook, loop) are provided in EN1992-1-1 Figures 8.1 and 8.3.
In general for other-than-straight bars with standard details according to EN1992-1-1 Figure 8.1 the equivalent anchorage length *l*_{b,eq} is measured straight up to the end of the bar shape.
For standard details for the anchorage of links and shear reinforcement see EN1992-1-1 Figure 8.5.

## Calculation of design lap length of longitudinal reinforcement according to EN1992-1-1 §8.7

### Design lap length *l*_{0}

The design lap length *l*_{0} is defined in EN1992-1-1 §8.7.3(1) as:

*l*_{0} = *α*_{1} ⋅ *α*_{2} ⋅ *α*_{3} ⋅ *α*_{5} ⋅ *α*_{6} ⋅ *l*_{b,rqd} ≥ *l*_{0,min}

where the coefficients *α*_{1} to *α*_{5} are as defined above in the estimation of the design anchorage length *l*_{bd}.

The coefficient *α*_{6} accounts for the percentage *ρ*_{l} of reinforcement bars lapped within ±0.65*l*_{0} from the center of the lap length considered.
The coefficient *α*_{6} is defined as:

*α*_{6} = (*ρ*_{l} / 25)^{0.5} and 1.0 ≤ *α*_{6} ≤ 1.5, where *ρ*_{l} expressed in %.

The maximum value of the coefficient *α*_{6} = 1.5 is obtained when the percentage of lapped bars in a section exceeds 50%.
In the provided tables in this calculation the design lap length *l*_{0} is calculated taking into account the value of coefficient *α*_{1} and assuming *α*_{6} = 1.5.
The provided lap lengths are adequate even when more than 50% of the bars are lapped in a section.

### Minimum anchorage length *l*_{0,min}

When no other limitation is applicable, the provided lap length should be at least equal to the minimum value *l*_{0,min} as described in EN1992-1-1 §8.7.3(1):

*l*_{0,min} ≥ max[0.3⋅*α*_{6}⋅*l*_{b,rqd}, 15⋅*Φ*, 200 mm]

### Staggering of laps

According to EN1992-1-1 §8.7.2 laps between bars should normally be staggered and not located in areas of high moments (e.g. plastic hinges).
The required staggering layout is presented in EN1992-1-1 Figure 8.7.
Two adjacent laps are not considered to belong in the same section when the clear distance between the lap ends is ≥ 0.3*l*_{0}.
Staggering is not required for compression reinforcement and secondary (distribution) reinforcement.
It is possible that laps between bars are not staggered because this is not feasible or it is very difficult (e.g. column starter bars).
This case is not covered explicitly in the current version of EN1992-1-1.
The authors of this website recommend to increase the provided lap lengths by a factor of (1.20)^{3/2} = 1.315 when staggering is not possible.
This recommendation is based on the relevant provision of the next issue of EN1992-1-1 that is currently in draft version.

### Detailing rules for reinforcement laps

Standard detailing rules for the arrangement of lapped bars are provided in EN1992-1-1 §8.7.2 and Figure 8.7:

- The lapped bars may be allowed to touch each other.
- The clear distance of lapped bars should be in general not greater than 4
*Φ* or 50 mm. Otherwise the lap length should be increased by a length equal to the clear space where it exceeds 4*Φ* or 50 mm.
- The longitudinal distance between two adjacent laps should not be less than 0.3
*l*_{0}.
- In case of adjacent laps, the clear distance between adjacent lapped bars should not be less than 2
*Φ* or 20 mm.

### Transverse reinforcement in the lap zone

The required transverse reinforcement in the lap zone to resist transverse tension forces is described in EN1992-1-1 §8.7.4.
When the diameter of the bar is *Φ* ≥ 20 mm the total area of the sum of all legs of transverse reinforcement Σ*A*_{st}, placed perpendicular to the direction of lapped bars, should be at least equal to the area *A*_{s} of one lapped bar:

Σ*A*_{st} ≥ 1.0 ⋅ *A*_{s}

Additional detailing rules are presented in EN1992-1-1 Figure 8.9:

- The required transverse reinforcement should be placed at the outer thirds of the lap length
*l*_{0}, i.e. at distance ≤ *l*_{0} / 3 from the lap ends.
- For the case of lapped bars permanently in compression one bar of the transverse reinforcement should be placed outside each end of the lap length and within 4
*Φ* of the end.
- The spacing between transverse reinforcement bars should not exceed 150 mm.
- If more than 50% of the reinforcement is lapped at one point and the distance between adjacent laps at a section is ≤ 10
*Φ* transverse reinforcement should be formed by links or U bars anchored into the body of the section.
- When the diameter of the lapped bars is
*Φ* < 20 mm or the percentage of lapped bars in any section is less than 25%) then any transverse reinforcement or links necessary for other reasons may be assumed sufficient without further justification.