Fundamentals of Structural Mechanics by Keith D. Hjelmstad

By Keith D. Hjelmstad

A superior creation to uncomplicated continuum mechanics, emphasizing variational formulations and numeric computation. The booklet deals a whole dialogue of numerical procedure options utilized in the examine of structural mechanics.

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A vectorfieldis a function v(x) that assigns a vector to each point x in a particular domain. The displacement of a body is a vector field. Each point of the body moves by some amount in some direction. The force induced by gravitational attraction is a vector field. Figure 21 shows two examples of vector fields. The pictures show the vectors at only enough points to get the idea of how the vectors are oriented and sized. The second vector field shown in the figure can be expressed in functional form as v(x) = Xiei+jCjCs (76) 40 Fundamentals of Structural Mechanics The vectors point in the radial direction, and their length is equal to the distance of the point of action to the origin.

We avoid the temptation to use the notation T = [ ] because we do not want to give the impression that we are setting a tensor equal to a matrix of its components. If there is any question as to what the basis is, then this abbreviated notation does not make sense, and should not be used. The reason this 20 Fundamentals of Structural Mechanics Ci v--- gi Figure 12 Components of a vector in different coordinate systems notation is useful is that tensor multiplication is the same as matrix multiplication if the components are stored in the manner shown.

59)c can be true only if n, • Tij = 0, that is, if they are orthogonal. Q Notice that orthogonality does not hold if the eigenvalues are repeated because Eqn. (59)c is satisfied even if n, • n^ r^ 0. We will see the ramification of this observation in the following examination of the special cases. Special cases. There are two special cases that deserve mention. Both correspond to repeated roots of the characteristic equation. The main concern is how to find the eigenvectors associated with repeated roots.

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