Adisorn Owatsiriwong

1. Spring Model Representation:

• k_s : Represents the stiffness of the anchor.

• k_c1 : Represents the stiffness of the concrete directly behind the anchor (local zone).

• k_c2 : Represents the stiffness of the surrounding concrete (general zone).

Forces in local zone (spring with k_c1) and at surrounding concrete (general zone with spring k_c2) can be computed from this given objective function.

```
% Objective function calculation with stress penalty and constraint penalties
function [area, q1, q2] = calculateObjective(x, k_s, k_c1, k_c2, P, q_max)
% Define the stiffness matrix K
K = [k_c2 + k_s, -k_s;
-k_s, k_s + k_c1];
% Define the force vector F
F = [P; 0];
% Solve for the displacements u1 and u2
u = K \ F; % u = [u1; u2]
u1 = u(1);
u2 = u(2);
% Force in k_c1 (bearing stress behind the block)
P_c1 = k_c1 * u2; % Force in kc1 is due to displacement u2
% Force in k_c2 (bearing stress in surrounding concrete)
P_c2 = k_c2 * u1; % Force in kc2 is due to displacement u1
.
.
.
end
```

2. Failure of k_c1 (Concrete Behind the Anchor):

• Localized Bearing Failure: If k_c1 , representing the concrete directly behind the anchor, fails (e.g., due to crushing or splitting), this indicates a local failure in the concrete immediately behind the anchor. This failure would mean that the concrete in the local zone is no longer able to effectively resist the applied prestressing force.

• Load Redistribution: With the failure of k_c1, the load that was originally resisted by the concrete behind the anchor must be redistributed. This load will be taken up by the surrounding concrete represented by k_c2.

3. Role of k_c2 (Surrounding Concrete):

• Continued Load Resistance: The surrounding concrete, represented by k_c2, is still capable of resisting the load. In the proposed spring model, this means that even if the local zone (represented by k_c1 ) fails, the general zone (represented by k_c2) can still bear the load.

• Increased Stress in k_c2: However, because the load has to be redistributed to k_c2, the stress in the surrounding concrete increases. This increased stress can lead to additional concerns, particularly regarding tensile splitting.

4. Tensile Splitting in the General Zone:

• Stress Redistribution Consequences: As the load redistributes to the surrounding concrete after the failure of k_c1, the general zone (represented by k_c2 ) experiences higher tensile stresses. These stresses can cause vertical tensile splitting, particularly if the concrete in the general zone is not adequately reinforced.

• Splitting in General Zone: This splitting occurs because the force is now spread out over a larger area, but the concentrated force still needs to be accommodated. The redistribution of the load creates tensile forces perpendicular to the direction of the applied load, leading to cracking in the general zone.

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