![]() ![]() ![]() ![]() ∑ F x and ∑ F y are the summation of the x and y components of all the forces acting on the structure, and ∑ M z is the summation of the couple moments and the moments of all the forces about an axis z, perpendicular to the plane xy of the action of the forces.Ī structure in three dimensions, that is, in a space, must satisfy the following six requirements to remain in equilibrium when acted upon by external forces:ģ.2 Types of Supports and Their Characteristics The above three conditions are commonly referred to as the equations of equilibrium for planar structures. The equilibrium requirements for structures in two and three dimensions are stated below.įor a structure subjected to a system of forces and couples which are lying in the xy plane to remain at rest, it must satisfy the following three equilibrium conditions: The number of unknowns that you will be able to solve for will again be the number of equations that you have.\)Įquilibrium Structures, Support Reactions, Determinacy and Stability of Beams and FramesĮngineering structures must remain in equilibrium both externally and internally when subjected to a system of forces. Once you have your equilibrium equations, you can solve these formulas for unknowns. All moments will be about the \(z\) axis for two-dimensional problems, though moments can be about the \(x\), \(y\) and \(z\) axes for three-dimensional problems. To write out the moment equations, simply sum the moments exerted by each force (adding in pure moments shown in the diagram) about the given point and the given axis, and set that sum equal to zero. Remember that any force vector that travels through a given point will exert no moment about that point. Any point should work, but it is usually advantageous to choose a point that will decrease the number of unknowns in the equation. To do this you will need to choose a point to take the moments about. Next you will need to come up with the the moment equations. ![]() Your first equation will be the sum of the magnitudes of the components in the \(x\) direction being equal to zero, the second equation will be the sum of the magnitudes of the components in the \(y\) direction being equal to zero, and the third (if you have a 3D problem) will be the sum of the magnitudes in the \(z\) direction being equal to zero. Once you have chosen axes, you need to break down all of the force vectors into components along the \(x\), \(y\) and \(z\) directions (see the vectors page in Appendix 1 page for more details on this process). If you choose coordinate axes that line up with some of your force vectors you will simplify later analysis. These axes do need to be perpendicular to one another, but they do not necessarily have to be horizontal or vertical. Next you will need to choose the \(x\), \(y\), and \(z\) axes. In the free body diagram, provide values for any of the known magnitudes, directions, and points of application for the force vectors and provide variable names for any unknowns (either magnitudes, directions, or distances). This diagram should show all the force vectors acting on the body. \Īs with particles, the first step in finding the equilibrium equations is to draw a free body diagram of the body being analyzed. ![]()
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