On the calculation of stresses in braced frameworks

1. Bending and twisting actions involve stresses which may range between wide limits, but in a straight bar subjected to tension or compression the stress has practically the same intensity at every point. Accordingly, in civil engineering, economy in material is attained by the use of skeletal or “framed” structures, built of straight members connected at their ends, and designed so that external forces (other than those arising from the weights of the members themselves) are applied only at the joints. Under these conditions, to a close approximation, every member is subjected to simple tension or compression. “Simple” and “Redundant” Frameworks . 2. In estimating the actions of the constituent members, it is customary to neglect entirely the effects of fixity at the joints, and to substitute for the actual framework a “skeleton diagram” in which every member is replaced by a line of thrust or tension. The problem then presented may be soluble by purely statical methods, or it may involve the elastic properties of the members, according as the number of these (m ) is related to the number (j ) of the joints. In a “plane frame” (where the external forces, as well as the lines of thrust or tension, are coplanar) the actions will be statically determinate ifm = 2j — 3, (1) —as may be seen from the consideration that two equations of equilibrium can be written down for each joint, but that of the resulting 2j equations only (2j — 3) are really independent, because three relations between the external forces are imposed by the conditions for equilibrium of the framework as a whole. In a “space frame,” similar considerations show that the actions will be statically determinate ifm = 3j — 6. (2)