 # Advanced Mechanics of Materials and Applied Elasticity: Analysis of Stress

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## 1.4 Conditions of Equilibrium

A structure is a unit consisting of interconnected members supported in such a way that it is capable of carrying loads in static equilibrium. Structures are of four general types: frames, trusses, machines, and thin-walled (plate and shell) structures. Frames and machines are structures containing multiforce members. The former support loads and are usually stationary, fully restrained structures. The latter transmit and modify forces (or power) and always contain moving parts. The truss provides both a practical and economical solution, particularly in the design of bridges and buildings. When the truss is loaded at its joints, the only force in each member is an axial force, either tensile or compressive.

The analysis and design of structural and machine components require a knowledge of the distribution of forces within such members. Fundamental concepts and conditions of static equilibrium provide the necessary background for the determination of internal as well as external forces. In Section 1.6, we shall see that components of internal-forces resultants have special meaning in terms of the type of deformations they cause, as applied, for example, to slender members. We note that surface forces that develop at support points of a structure are called reactions. They equilibrate the effects of the applied loads on the structures.

The equilibrium of forces is the state in which the forces applied on a body are in balance. Newton's first law states that if the resultant force acting on a particle (the simplest body) is zero, the particle will remain at rest or will move with constant velocity. Statics is concerned essentially with the case where the particle or body remains at rest. A complete free-body diagram is essential in the solution of problems concerning the equilibrium.

Let us consider the equilibrium of a body in space. In this three-dimensional case, the conditions of equilibrium require the satisfaction of the following equations of statics:

Equation 1.2 The foregoing state that the sum of all forces acting on a body in any direction must be zero; the sum of all moments about any axis must be zero.

In a planar problem, where all forces act in a single (xy) plane, there are only three independent equations of statics:

Equation 1.3 That is, the sum of all forces in any (x, y) directions must be zero, and the resultant moment about axis z or any point A in the plane must be zero. By replacing a force summation with an equivalent moment summation in Eqs. (1.3), the following alternative sets of conditions are obtained:

Equation 1.4a provided that the line connecting the points A and B is not perpendicular to the x axis, or

Equation 1.4b Here points A, B, and C are not collinear. Clearly, the judicious selection of points for taking moments can often simplify the algebraic computations.

A structure is statically determinate when all forces on its members can be found by using only the conditions of equilibrium. If there are more unknowns than available equations of statics, the problem is called statically indeterminate. The degree of static indeterminacy is equal to the difference between the number of unknown forces and the number of relevant equilibrium conditions. Any reaction that is in excess of those that can be obtained by statics alone is termed redundant. The number of redundants is therefore the same as the degree of indeterminacy.