(a) To determine the maximum horizontal acceleration that the chimney can tolerate in an earthquake, we need to consider the stability and structural integrity of the chimney.
Given:
Height of the chimney (h) = 35 m
Base diameter of the chimney (d) = 2.5 m
Assuming the chimney is a solid cylinder, we can calculate its critical buckling load using the following formula:
P_critical = (π^2 * E * I) / (h^2)
Where:
P_critical is the critical buckling load
E is the modulus of elasticity of the material (assumed to be the bricks)
I is the moment of inertia of the cross-sectional shape of the chimney
To find the moment of inertia (I) of a solid cylinder, we can use the formula:
I = (π * d^4) / 64
Substituting this value of I into the first formula and rearranging, we can solve for E:
E = (P_critical * h^2 * 64) / (π * d^4)
Since the shear strength of the cement is assumed to be very large, it does not affect the calculation of maximum horizontal acceleration directly.
Hence, the maximum horizontal acceleration (a_max) that the chimney can tolerate can be calculated using the formula:
a_max = g * (P_critical / (m * A))
Where:
g is the acceleration due to gravity (approximately 9.8 m/s^2)
m is the mass of the chimney
A is the area of the base of the chimney
To find the mass of the chimney (m), we can calculate its volume and multiply it by the density of the material (assumed to be the bricks). The volume (V) of the chimney is given by:
V = π * (d / 2)^2 * h
Hence, the mass (m) is given by:
m = V * density
To calculate the area of the base of the chimney (A), we can use the formula for the area of a circle:
A = π * (d / 2)^2
(b) In the case where the shear strength of the cement is zero and the tensile strength is large, the stability and structural integrity of the chimney will be significantly affected.