Yesterday, 25th March 2010. In the roof of the Chapter House at Westminster Abbey. There are tie bars between the centre of the vault and the roof.
T is the tension in N
µ is the unit weight of the bar Kg/m
L is the length between supports
r is the radius of the bar
The measured frequencies were about 3.5Hz. Unfortunately, we don't have good figures for the length of the bars or for the stiffness of the ends. It looks, though, as though there is very little tension in the bars.
Question: are these bars carrying load. Try measuring the frequency.
The frequency turns out to be not that of a simple "string" but modified substantially by the stiffness of the bars. The formulae below give approximate relationships between tension, stiffness and frequency.
Sorry about the layout of the formulae. I will try to find a way of presenting them sensibly.
f^0=1/2L √(T/μ) (the fundamental frequency of a string of the same length and weight)
β=r^2/L √((π.E)/T) (A function of the stiffness of the bar and the tension)T is the tension in N
µ is the unit weight of the bar Kg/m
L is the length between supports
r is the radius of the bar
E is Young’s modulus for the material
The measured frequencies were about 3.5Hz. Unfortunately, we don't have good figures for the length of the bars or for the stiffness of the ends. It looks, though, as though there is very little tension in the bars.