![]() ![]() Many more examples are available at this site. Source: Engineering Mechanics, Jacob Moore, et al. In instances where you have more unknowns than equations, the problem is known as a statically indeterminate problem and you will need additional information to solve for the given unknowns. The number of unknowns that you will be able to solve for will be the number of equilibrium equations that you have. Once you have your equilibrium equations, you can solve them for unknowns using algebra. Collectively these are known as the equilibrium 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 if you need more guidance on this). ![]() 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 chose the x, y, and z axes. It is also useful to label all forces, key dimensions, and angles. This is done by removing everything but the body and drawing in all forces acting on the body. The first step in equilibrium analysis is drawing a free body diagram. In the free body diagram, provide values for any of the know magnitudes or directions for the force vectors and provide variable names for any unknowns (either magnitudes or directions). ![]() This diagram should show all the known and unknown force vectors acting on the body. The first step in finding the equilibrium equations is to draw a free body diagram of the body being analyzed. Since it is a particle, there are no moments involved like there is when it comes to rigid bodies. The equations used when dealing with particles in equilibrium are: Individual forces acting on the object, represented by force vectors, may not have zero magnitude but the sum of all the force vectors will always be equal to zero for objects in equilibrium. Therefore, if we know that the acceleration of an object is equal to zero, then we can assume that the sum of all forces acting on the object is zero. Newton’s Second Law states that the force exerted on an object is equal to the mass of the object times the acceleration it experiences. These objects may be stationary, or they may have a constant velocity. The stress components on each side of the cube is a function of the position since we have a non uniform but continuous stress field.Objects in static equilibrium are objects that are not accelerating (either linear acceleration or angular acceleration). The stresses acting on the opposite sides of the cube are slightly different. A fixed support, limits the motion in all directions, so there woul. Figure 1: Infinitesimal parallelepiped representing a point in a body under static equilibrium. Statics Practice Problem 5-64: Example of a 3D Rigid Body Equilibrium with Fixed Support. We will assume that the stress field is continuous and differentiable inside the whole body. We cut an infinitesimal parallelepiped inside the body and we analyze the forces that act on it as shown in Fig. Surface and body forces act on this body. This approach may be found in international bibliography.Ĭonsider a solid body in static equilibrium that neither moves nor rotates. ![]() A more elegant solution may be derived by using Gauss's theorem and Cauchy's formula. In this article we will prove the equilibrium equations by calculating the resultant force and moment on each axis. This can be expressed by the equilibrium equations. 17, 2020Ī solid body is in static equilibrium when the resultant force and moment on each axis is equal to zero. ![]()
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