![]() All bodies to be studied in this book are rigid, except for springs. Most bodies encountered in engineering work can be considered rigid from the mechanical analysis point of view becase the deformations that take place within these bodies under the action of loads can be neglected when compared to other effects produced by the loads. A body is said to be rigid when the relative positions of its particles are always fixed and do not change when the body is acted upon by any load (whether a force or a couple). The size of a body affects the results of any mechanical analysis on it. A body is formed by a group of particles. The size of a particle is very small compared to the size of the system being analysed. A particle is a body whose size does not have any effect on the results of mechanical analyses on it and, therefore, its dimensions can be neglected. The ability to understand mechanics and many other engineering disiciplines is dependent on mastering the concept of equilibrium. The concept must be really understood by every student. The concept of equilibrium is the most basic and most important concept in engineering analysis. In statics, the concept of equilibrium is usually used in the analysis of a body which is stationary, or is said to be in the state of static equilibrium. There is no unbalanced force or unbalanced couple acting on it. A body under such a state is acted upon by balanced forces and balanced couples only. The concept of equilibrium is introduced to describe a body which is stationary or which is moving with a constant velocity. The following section describes the simplest form of kinematic and constitutive equations.3.3 EQUILIBRIUM EQUATIONS FOR A RIGID BODY constitutive equations, that relate strains to stresses.kinematic equations, that relate displacements to strains, and.We need equations that relate displacement to stresses. The solution of such problem requires knowledge of the material properties. The solution of a general problem with arbitrary boundary conditions requires more equations to have a determined problem (as many equations as unknowns). 3 Continuum mechanics solution of an arbitrary problemįigure 3.6 shows an example of an arbitrary shaped continuous solid subjected to external stresses, external forces, body forces, and displacement constraints (bottom fixture).Īs highlighted before, notice that there are 6 unknowns (9 unknowns if displacements are included) and 3 equations in Cauchy's equations of equilibrium (Eq. The solution to this problem will be developed in section 3.3.4.ģ. The horizontal stresses cannot be determined with the current equations. Is positive if after a displacement, points in opposite direction to Read the values as the stress on face perpendicular to base vector However, the numbers that represent the value of a vector (such as velocity or force ) or a tensor depend on the coordinate system.Ī tensor, like stress, also depends on the coordinate system used to express its numerical values. Is independent of the coordinate system orientation and origin (Figure 3.1). The number that represents the value of a scalar (such as temperature or pore pressure ) at a given point ![]() Is your middle finger, and the third element of the base Is your index finger, the second element of the base In a right-handed coordinate system, the first element of the base 3 Continuum mechanics solution of an arbitrary problemĬonsider a 3D space with a given right-handed orthogonal coordinate system 2 Application of Cauchy's equations for total vertical stress calculation 1 Cauchy's equations of stress equilibrium Next: 3.2 Kinematic equations: displacements Up: 3.
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