uniform force method design of gussets

huangyhg

uniform force method design of gussets
i have been trying to derive the expression:
? - ?*tan(?) = eb*tan(?) - ec
i understand that this relations is necessary for the force distributions shown in the free body diagrams of figures 13-2b, 13-2c, and 13-2d (page 13-3 of the 13ed of the aisc manual) to remain free of moments on the connection interfaces.
i have taken moments about the cp for the gusset, column, and beam, and then tried to manulipate the three to derive the expression in question to no avail.
i also would like to know the significance of the control point. i see the working point is the intersection of the centerlines but do not see the significance of the control point other than it being the intersection of the top surface of the beam and the centerline of the column.
thanks in advance for any help 鈥?i鈥檝e spent many hours before asking for your assistance and time.
the expression can be derived directly from the geometry shown in figure 13-2 (a) of the 13th ed. manual.  look at it as an upside down triangle formed with the line of force as the hypotenuse, (ec+alpha) as the top horizontal leg length, and (eb+beta) as the left vertical leg length.  tangent of the angle theta then equals opposite over adgacent:  tan theta = (ec+alpha)/(eb+beta).  bringing the denominator over and moving the terms around yields the final equation.  
thank you.  that was just too easy.