# Validity of Anchorage Zone Design for Post-Tensioned Concrete Members with High-Strength Strands
## 2 Design Method for the Anchorage Zone  
The bursting force equation in ACI 318-19 calculates the bursting force and location of the bursting force as shown in Equations (1) and (2), which are based on the bursting force equation proposed by [PERSON] [10]. It was assumed that the pre-stressing force acting on the bearing plate was applied to the centroid, which was one-quarter of the upper and lower surfaces of the bearing plate, and that the reaction force of the concrete was acting at the centroid of the concrete. The ACI 318-19 design standard estimated that the location of the maximum bursting force would occur at half of the depth from the end when there was no eccentricity, regardless of anchorage shape. In Equations (1) and (2),  
Figure 1: Stress distribution at the anchorage zone in a post-tensioned member. \(\sum P_{pu}\) is the sum of the forces from the individual tendons, \(h\) is the depth of the anchorage device, and \(e_{enc}\) is the eccentricity of the anchorage device. \[F_{burst,ACI}=0.25\sum P_{pu}\bigg{(}1-\frac{h_{anc}}{h}\bigg{)} \tag{1}\]  
\[d_{burst,ACI}=0.5(h-2e_{anc}) \tag{2}\]  
The bearing force was considered in the ACI 318-19 design standard so that the design strength of \(\Phi P_{c}\) exceeded the factored pre-stressing force at the anchorage device, \(P_{pu}\). \(P_{pu}\) and \(\Phi P_{c}\) were then calculated as shown in Equations (3) and (4). \[P_{pu,ACI}=\gamma_{p}\bigg{(}0.80f_{pu}\bigg{)}A_{ps} \tag{3}\]  
\[\Phi P_{c,ACI}=0.85f_{ci}A_{pt} \tag{4}\]  
where \(\gamma_{p}\) is a load factor of 1.2, 0.80\(f_{pu}\) is a permissible tensile stress in the pre-stressed reinforcement, \(\phi\) is a strength reduction factor, \(f_{ci}\) is the specified compressive strength of the concrete at the time of initial pre-stress, and \(A_{pt}\) is the area of the bearing plate. The AASHTO requirements were established based on [PERSON]'s [11] study, which estimated the bursting forces as shown in Equations (5) and (6). [PERSON] proposed a bursting force equation that considered the angle and eccentricity of the strands from [PERSON]'s equation [10], which was based on elastic stress analysis. In Equations (5) and (6), \(\alpha\) is the angle of the strands. \[F_{bst,AASHTO}=0.25\sum P_{pu}\bigg{(}1-\frac{a}{h}\bigg{)}+0.5P_{pu}sin\alpha \tag{5}\]  
\[d_{bst,AASHTO}=0.5(h-2e)+5 esina \tag{6}\]  
The AASHTO states that the bearing force should not exceed the coefficient of ultimate strength calculated using Equation (7). The allowable bearing strength of concrete in Equation (7) was determined to be the smaller of the values calculated from Equations (8) and (9). \[P_{r,AASHTO}=\Phi f_{n}A_{b}{}^{\prime} \tag{7}\]  
\[f_{n}=0.7f_{ci}\sqrt{\frac{A}{A_{g}}} \tag{8}\]  
\[f_{n}=2.25f_{ci} \tag{9}\]  
where \(\Phi\) is the resistance factor, \(f_{n}\) is the allowable bearing strength of the concrete, \(A_{g}\) is the gross area of the bearing plate, \(A_{b}{}^{\prime}\) is the effective net area of the bearing plate (calculated as the area \(A_{g}\) minus the area of the openings in the bearing plate), and A is the maximum area of the portion of the supporting surface that is similar to the loaded area and concentric with it, but does not overlap similar areas for adjacent anchorage devices. The ACI 318-19 and AASHTO codes present the equations for calculating bursting force and bearing force when pre-stressing force is applied as shown in Equations (1)-(9). However, little information is available about a post-tensioned anchorage zone under flexural loading; cracking in fully pre-stressed concrete is not permitted, but on the other hand, in the partially pre-stressed members, the section may be cracked or uncracked depending on the level of the loading and pre-stressing [12]. Furthermore, if a failure occurs in the anchorage zone of post-tensioned concrete members before flexural failure, the member shows brittle fracture behavior. Hence, post-tensioned concrete members should be designed so that a failure of the anchorage zone does not occur until flexural failure.