The front trapezoid cannot be handled in this way because it is an unstable pinned structure. This instability can be overcome by adding a tube between the bottom bracket and the top of the head tube as shown in the diagram. With the additional tube AF, the front section is also a fully triangulated structure and the pin joints don't transfer moments.

The calculated forces shown on the diagram indicate that the added tube, AF, carries a compressive load of 26.8 lbf, comparable to that carried by the seat tube. The tube doen tube, AE, carries a tensile force of 93.5 lbf. The geometry of the front section can be arranged so that the added tube carries no load. This is shown in (a) opposite. For this geometry there is no moment about the point of intersection of the two tubes and the line of the vertical force at the front axle. Drawing (b) shows how these forces intersect in the usual frame geometry and give rise to the bending moment on the head tube. This is counteracted in a real bike by the weld joints between the frame tubes.

Diagrams from: McMahon & Graham, "The Bicycle & the Walkman," Merion (1992)