Determine the shape factor, F12, for the rectangles shown.
6 m (2 0.5 mI 2 1 m
(a) Perpendicular rectangles without a common edge
(b) Parallel rectangles of unequal areas.
The correct answer and explanation is :
To determine the shape factor ( F_{12} ) for the given rectangles, we need to understand the context of the shape factor, which is used in engineering and structural analysis. The shape factor is often used to describe the interaction or relative configuration between two areas or sections in terms of their moments of inertia and distances between centroids.
We will need to calculate this for both of the following scenarios:
Scenario (a): Perpendicular rectangles without a common edge
In this case, we have two rectangles arranged at a right angle (perpendicular) to each other, and they do not share a common edge. The shape factor ( F_{12} ) for two perpendicular areas is given by:
[
F_{12} = \frac{I_{12}}{I_1 + I_2}
]
Where:
- ( I_1 ) and ( I_2 ) are the moments of inertia of the individual rectangles.
- ( I_{12} ) is the moment of inertia of the system (combined areas), calculated considering the distance between the centroids of the two areas.
Given:
- Rectangle 1: Dimensions 6 m x 2.5 m
- Rectangle 2: Dimensions 6 m x 1 m
To calculate the moments of inertia, use the formula for a rectangle’s moment of inertia about its centroid:
[
I = \frac{b h^3}{12}
]
Where ( b ) is the base (width) and ( h ) is the height (thickness).
Then, calculate ( I_{12} ) considering the parallel axis theorem (since the centroids are offset from each other). The final shape factor would result from this calculation.
Scenario (b): Parallel rectangles of unequal areas
In this scenario, two rectangles are arranged parallel to each other, but they have unequal areas. For parallel rectangles, the shape factor can also be determined by considering the combined moments of inertia and the relative distance between their centroids.
The formula for ( F_{12} ) is similar to the one above:
[
F_{12} = \frac{I_{12}}{I_1 + I_2}
]
The only difference is that, since the rectangles are parallel, the distance between their centroids is constant along the axis of symmetry.
Again, using the formula for the moment of inertia, you can compute the individual moments and the total combined moment, ( I_{12} ), adjusting for the distance between the centroids.
Explanation:
The shape factor helps quantify how the two rectangles (or areas) interact in terms of their resistance to bending or other structural behavior, depending on their relative orientation.
For perpendicular areas, the distance between centroids contributes to the calculation of the total moment of inertia, whereas for parallel areas, the relative distance between centroids simplifies the calculation.
The exact numerical answer will depend on the specific calculation of ( I_1 ), ( I_2 ), and ( I_{12} ) based on the provided dimensions, which would require further detailed computation.