To calculate the stresses at the top and bottom of the beam at mid-span, we need to consider both the prestress release and service conditions. Let's analyze each condition separately and then compare the changes.
1. Prestress Release Condition:
In this condition, the beam is subjected to an initial prestress force, Pi. Due to losses, the effective prestress force, Pe, is 20% less than Pi. Additionally, we assume that the beam's own weight is acting on it.
To calculate the stresses at mid-span, we can use the following equations:
Stress at top (fi) = (Pe + beam's own weight) / Area of cross-section
Stress at bottom (fz) = (Pe - beam's own weight) / Area of cross-section
2. Service Condition:
In this condition, a uniform load of 2 k/ft has been added to the beam above its own weight. The prestress force remains at Pe.
To calculate the stresses at mid-span under service conditions, we can use similar equations as before:
Stress at top (fi) = (Pe + additional load + beam's own weight) / Area of cross-section
Stress at bottom (fz) = (Pe - additional load - beam's own weight) / Area of cross-section
By comparing the stress distributions between the prestress release and service conditions, we can observe the following changes:
1. Magnitude of Stresses: The addition of the uniform load in service conditions increases both the compressive stress at the top and tensile stress at the bottom of the beam compared to the prestress release condition.
2. Redistribution of Stresses: In prestress release conditions, most of the stress is carried by the prestressing tendons. However, in service conditions, a significant portion of the load is transferred to the concrete due to additional external loads. This redistribution leads to a decrease in the prestress force and an increase in the stresses at the top and bottom of the beam.
3. Change in Stress Distribution Shape: The stress distribution shape may change between the two conditions due to the redistribution of forces. In prestress release conditions, the stress distribution is typically more uniform along the cross-section. However, in service conditions, the stress distribution may become non-uniform, with higher stresses closer to the top and bottom surfaces of the beam.
In summary, when comparing the prestress release and service conditions, we observe changes in stress magnitude, redistribution of stresses, and potential changes in stress distribution shape.