## 336 Displacement Methods

For analysis of a statically determinate or indeterminate structure by any of the displacement methods, independent displacements of the joints, or nodes, are chosen as the unknowns. If the structure is defined in a three-dimensional, orthogonal coordinate system, each of the three translational and three rotational displacement components for a specific node is called a degree of freedom. The displacement associated with each degree of freedom is related to corresponding deformations of members meeting at a node so as to ensure geometric compatibility.

Equilibrium equations relate the unknown displacements A1, A2, . . . , An at degrees of freedom 1,2,..., n, respectively, to the loads Pi on these degrees of freedom in the form

(b)

FIGURE 3.75 Analysis of a continuous beam by the force method. (a) Two-span beam with concentrated and uniform loads. (b) Displacements of beam when supports at B and C are removed. (c) Displacements for unit load at B. (d) Displacements for unit load at C.

Pn = K1A1 + kn2A2 + • • • + knnAn or more compactly as n

Member loads acting between degrees of freedom are converted to equivalent loads acting at these degrees of freedom.

The typical kij coefficient in Eq. (3.J34) is a stiffness coefficient. It represents the resulting force (or moment) at point i in the direction of load Pi when a unit displacement at point j in the direction of Aj is imposed and all other degrees of freedom are restrained against displacement. Pi is the given concentrated load at degree of freedom i in the direction of Ai.

When loads, such as distributed loads, act between nodes, an equivalent force and moment should be determined for these nodes. For example, the nodal forces for one span of a continuous beam are the fixed-end moments and simple-beam reactions, both with signs reversed. Fixed-end moments for several beams under various loads are provided in Fig. 3.76. (See also Arts. 3.37, 3.38, and 3.39.)

(F. Arbabi, Structural Analysis and Behavior, McGraw-Hill, Inc., New York.)

## Renewable Energy 101

Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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