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6. STRUCTURES IN EQUILIBRIUM 6.1 TRUSSES TRUSSES - structures supported and loaded at their joints (places where the bars are pinned together), with bar weights neglected - each bar is a two-force member, so the forces at the ends of a member must be equal in magnitude, opposite in direction, and directed along the line between the joints (the force is called the axial force in the member); when the forces are directed away from each other, the member is in tension, when the forces are directed toward each other, the member is in compression - examples: structures used to support bridges and the roofs of houses 6.2 THE METHOD OF JOINTS - preferred approach when the axial forces in all members are to be determined - involves drawing free-body diagrams of the joints of a truss one by one, and using the equilibrium equations to determine the axial forces in the members. Procedure: 1. Draw a free-body diagram of the entire truss (treat the truss as a single object) and determine the reactions at its supports. 2. Choose a joint (isolate it by “cutting” members pinned at considering joint) and draw its free-body diagram (suppose the directions of unknown axial forces as tension forces for a member). 3. Use the equilibrium equations to determine the unknown forces (if the result/force is positive, the member is in tension, and if negative – in compression); for 2-D problems, there are only 2 independent equilibrium equations (the forces are concurrent and summing the moments about a point does not result in an additional independent equation). 4. Repeat 2. and 3. for other joints PARTICULAR TYPES OF JOINTS: TRUSS JOINTS WITH TWO COLLINEAR MEMBERS AND NO LOAD The sum of forces must equal zero T1 = T2 (the axial forces are equal) TRUSS JOINTS WITH TWO NONCOLLINEAR MEMBERS AND NO LOAD From the sum of the forces for the joint T2 = 0 = T1 (the axial forces are zero) TRUSS JOINTS WITH THREE MEMBERS, TWO OF WHICH ARE COLLINEAR, AND NO LOAD From the sum of the forces in the x-direction T3 = 0, and from the sum of the forces in the y-direction T1 = T2 (the axial forces in the collinear members are equal, and the axial force in the third member is zero) 6.3 THE METHOD OF SECTIONS - preferred approach when the axial forces only in a few members are to be determined - involves drawing a single free-body diagram of a section of the truss, and using the equilibrium equations to determine the axial forces in specific members Procedure: 1. Draw a free-body diagram of the entire truss (treat the truss as a single object) and determine the reactions at its supports. 2. Choose a section - cut several members (usually no more then 3) including member(s) whose axial force(s) are to be determined, and draw a free-body diagram of a section (suppose the directions of unknown axial forces as tension forces for a member). 3. Use the equilibrium equations to determine the unknown forces (if the result is positive, the member is in tension, and if negative – in compression); for 2-D problems, there are 3 independent equilibrium equations (the forces are not usually concurrent and summing the moments about a point does result in an additional independent equation). 6.4 SPACE TRUSSES SUPPORTS 6.5 FRAMES AND MACHINES ANALYZING THE ENTIRE STRUCTURE - objects subjected to a system of forces and ANALYZING THE MEMBERS - objects subjected to a system of forces and moments TWO-FORCE MEMBERS LOADS APPLIED AT JOINTS