Analysing frames
Frames are structures in which at least one member carries loads at three or more points. Unlike trusses, they must be taken apart, member by member, to find the forces inside.
Frames versus trusses
In a truss, every member is a two-force member. A frame contains at least one multi-force member, loaded at three or more points or by forces between its ends, so it can bend as well as stretch or compress. Bicycle frames, building frames, cranes, and furniture are frames.
Taking a frame apart
To find internal forces, draw a separate free-body diagram for each member. Where two members connect at a pin, the forces they exert on each other are equal and opposite by Newton's third law; draw them in opposite directions on the two diagrams, using the same symbols.
Identify two-force members first
Before writing equations, mark any two-force members. Their forces act along the line between their pins, which removes unknowns and tells you directions in advance. Skipping this step is the most common cause of unsolvable frame problems.
Counting equations
Each rigid member in a plane gives three equations. For a frame of n multi-force members, there are 3n equations, enough to find the pin forces and reactions if the frame is statically determinate. Whole-frame equations can be used to find support reactions first, but they are not independent of the member equations.
A bracket example
A horizontal arm ABC, 3 m long, is pinned to a wall at A and carries 600 N at C. A two-force strut joins the arm at B, 2 m from A, and runs down to the wall at 60 degrees to the arm. Moments about A: F sin 60 x 2 = 600 x 3, so the strut force F is about 1,039 N of compression.
Finishing the example
The strut's horizontal component, 1,039 x cos 60, about 520 N, must be balanced by the pin at A, so A pulls the arm toward the wall with 520 N. Vertically, Ay + 900 - 600 = 0, so Ay = -300 N: the pin at A pulls the arm downward with 300 N, while the strut pushes up with 900 N.
Internal bending
Because multi-force members can bend, frames are designed for bending moments as well as axial forces. Mechanics of materials uses the pin forces found in statics to compute the bending stresses inside each member.
Worked example
A folding A-frame stepladder stands on a smooth floor, its two legs pinned at the top and joined by a horizontal tie halfway down. A painter stands on one side. Each leg is a multi-force member, loaded at the top pin, the tie, and the floor, so each leg needs its own free-body diagram; the tie is a two-force member in tension, keeping the legs from spreading on the smooth floor.
Key takeaways
Frames contain multi-force members that can bend, unlike trusses, whose members are all two-force. Analyse frames by drawing a free-body diagram for each member, with equal and opposite pin forces on connected members. Identify two-force members first to fix force directions. Each planar member gives three equations. The bracket example shows a strut in about 1,039 N compression and a pin force pulling the arm toward the wall. Ladder ties and similar links are two-force members in tension.
Flashcard
Structure with a multi-force member
Check your understanding
How does a frame differ from a truss?