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
(**N****b**).
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**)±(**N****b
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
**N****b**
(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.

Please turn to our next page to see our **Toy
Models** in action.