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.
The answer is that C/T=[n-sin(30 deg)]/n
Given the formulation of the problem, one can only assume that each member of the truss is a two-force member and can only support forces along its length. If one then considers the entire structure, and determines the equilibrium force at the wall attachments, at C there can only exist a horizontal force, acting to the right on the truss. From a summation of moments, it can be shown that the magnitude of this force must be W*n/cos(30 deg) tension at A. For equilibrium to exist, the force at B on the truss must be and equal and opposite force to that at A acting to the left of magnitude W*n/cos(30deg) and a vertical force of W acting upwards. Since all the truss segments are two force members, only the angular truss (i.e. not the horizontal member) at B can act to support the vertical force W. Since it is placed at an angle of 30 degrees to the vertical, the total force in this member is therefore W/cos(30 deg) compression. It follows then that the horizontal member at B must support the balance of the reaction to the equal and opposite horizontal force of W*n/cos(30 deg) (compression) which equals W*tan(30) in the angular member and W*(n/cos(30)-tan(30)) in the horizontal member (compression). This latter value for the horizontal member at B is a larger compression force than the value in the angular member. Therefore the ratio of Cmax/Tmax = [n/cos30-tan(30)]/[n/cos(30)], which simplifies to the above answer.
Posted by Kenny M
on 2010-02-26 22:39:04