Usually the chimney breaks near the bottom or near the middle. Why? Where does it break? Can we understand the forces acting on it, and predict how it is going to fall? These were the questions addressed in this experiment. There have been previous attempts in solving this problems, and we have used the existing theory to analyze the effect.
The forces acting on the chimney, at a distance r from the bottom, are the Transverse Shear Stress (S), the Longitudinal Stress Force (P), and the Bending Moment (Nb). The equations for these forces are:
S = (3/4) mg sinq ((r^2/H^2) - (4/3)(r/H) + (1/3))
P = (-1/2) mg (1-(r/H)) [(5 + 3(r/H)) cosq - 3(1 + (r/H))]
Nb = (-1/4) mg sinq H ((r^3/H^3) - 2(r^2/H^2) + (r/H))
where m is the mass of the chimney, H is the total height, and q is the angle of rotation from the vertical direction. These forces can be seen on the following figure:
These forces (and the bending moment) are responsible for the breaking of the toppling chimney. We plot these forces as a function of the height fraction r/H, and for several angles q, to predict where the breaking is more likely to occur.
This is the plot of S vs. r/H for various angles. As it can be seen from this figure, the shear stress is largest at the base of the chimney, it is zero at 1/3 of the chimney's height, and another maximum is present at 2/3 of the chimney's height in the opposite direction, though it is much smaller than the maximum at the base. Therefore, when a chimney's rupture is caused by the shear force, it is most likely to happen near the base (see the photo of the Detroit chimney).
This is the plot of s vs r/H, where sT/L is the stress at the trailing edge (T- dotted curves), or leading edge (L-solid curves) of the falling chimney. sT/L = (P/A)±(Nb l/J), where A is the area of the square cross section, of side a, of the chimney (for simplicity, only square cross sectional chimneys are considered). l=a/2 is the distance of the edges from the longitudinal axis of the chimney, and J=a^4/12 is the moment of inertia of the cross sectional area (see our first paper for details).
The stress s, at the two edges, is therefore due to a combination of P (the longitudinal stress force) and Nb (the bending moment). The stress at the leading edge sL (solid curves) is usually the most intense, and is the primary cause of breaking, apart from the shear stress mentioned above. As you can see from the figure, the portion of the chimney that experiences the biggest stress depends on the angle (see the solid points, representing the maxima of the solid curves). At lower angles, the chimney is more likely to break near the middle, at higher angles the chimney will break at about one third of its height (see the photo of the Glasgow chimney, breaking almost at the middle, for a relatively small angle). The pictures of falling chimneys shown above include the breaking angle, and the position of the rupture.
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