Engineering Statics

C-18


For the frame and loading shown, determine the reactions at points A and C. (Note: AB is a link connected at points A and B.)

Member AB is a two force member and the support at A is a roller, hence the force at B (a pin connection) must pass through point A.
Member BCD is a three-force members.  The three forces pass 
through one point.  Since the force at D is vertical and 
we know that the force at B makes a 45 degree angle (it must 
to pass through point A) the location of the point can be 
found.  The distance from the point to C can then be found 
using the Pythagorean theorem.
For member BCD we know that only forces at B and C have components 
in the x-direction.  Hence the x-components are equal.  Furthermore, 
since B is at a 45 degree angle, the x and y components of B are 
equal.
The force equation in the y direction gives a relationship 
between the y components of C and B and the 40 lb force at D. 
Notice that a substitution has been made for B<sub>y</sub> 
in terms of B<sub>x</sub> and then C<sub>x</sub>.
But the two components of C are related by the geometric 
relationship defined by the triangle in the FBD.
Substituting these relationships in gives an equation 
for C.
Solving for C allows us to use the geometric proportionality 
again to solve for the two components of C.
Since there are only two forces at A, both at 45 degrees, 
and the x-component of C equals the x component at B we 
can find the force at B (60/cos45) which in turn is equal 
to the force at A.