The cantilever structure shown in the figure consists of 4n-1 struts
of the same length plus one that is half that length. Each strut can
handle a maximum tension force T before it will snap and a maximum
compression force C before it will buckle. The structure is connected
to a wall at points B and C. A weight W is attached at point A.
The weight W is increased until two struts fail - one from tension
and the other from compression.
What is the value of the ratio C/T if n = 25?
Consider the struts as weightless.
|
|
Submitted by Bractals
|
|
Rating: 3.5000 (2 votes)
|
|
|
Solution:
|
(Hide)
|
Label the joints from A to B as A=J1, J3, J5, ... ,J2n+1=B.
Label the joints from D to C as D=J2, J4, J6, ... ,J2n+2=C.
If we look at the joints in the order J1, J2, J3, ... ,J2n we can easily calculate
the forces as
Strut Force Type
J2k-1J2k+1 (2k-1)S Compression
J2k-1J2k 2S Tension
J2kJ2k+1 2S Compression
J2kJ2k+2 2kS Tension
where S = W√3/3 and k = 1, 2, 3, ... , n.
The largest compression and tension forces are in the horizontal struts nearest the wall.
Therefore, for these to fail at the same time, we must have
C = (2n-1)S and T = 2nS
or
C/T = (2n-1)/(2n)
Thus, for n = 25, C/T = 0.98
|
