Final answer:
The uncracked transformed moment of inertia for a beam is found by calculating the centroidal moment of inertia and then using the parallel axis theorem to adjust for the beam's actual point of rotation.
Step-by-step explanation:
Finding the uncracked transformed moment of inertia for a beam involves calculating the moment of inertia of the cross-sectional area about a neutral axis. This is commonly done in structural engineering when analyzing beams for bending stresses.
Firstly, one must find the moment of inertia for the centroidal axis (ICM) of the beam's cross-section. This step involves integrating the squared distance of each differential area from the centroidal axis, multiplied by the area itself.
Then the parallel axis theorem is used to find the moment of inertia about the actual point of rotation, which is not through the center of mass. The formula I = ICM + Ad2 is applied, where 'I' is the moment of inertia about the point of rotation, 'ICM' is the centroidal moment of inertia, 'A' is the area of the cross-section, and 'd' is the distance between the centroidal axis and the axis of rotation.
This approach can be extended to composite sections and used to calculate the moment of inertia for more complex shapes — a critical step in structural design for ensuring the stability and integrity of constructions under various loading conditions.