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# 10-1 ELASTIC PROPERTIES
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## Flexibility
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Before discussing the elastic-stiffness matrix expressed in Eq. (9-5), it will be useful to define the inverse flexibility relationship. The definition of a flexibility influence coefficient $\widetilde{f}_{i j}$ is
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在讨论由式 (9-5) 表示的弹性刚度矩阵之前，定义逆柔度关系将是有益的。柔度影响系数 $\widetilde{f}_{i j}$ 的定义是
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$$
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\begin{array}{r l}{\widetilde{f}_{i j}=\ }&{\mathrm{deflection\;of\;coordinate\}i\mathrm{~due\;to\;unit\;load}}\\ &{\mathrm{applied\;to\;coordinate\}j}\end{array}
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$$
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For the simple beam shown in Fig. 10-1, the physical significance of some of the flexibility influence coefficients associated with a set of vertical-displacement degrees of freedom is illustrated. Horizontal or rotational degrees of freedom might also have been considered, in which case it would have been necessary to use the corresponding horizontal or rotational unit loads in defining the complete set of influence coefficients; however, it will be convenient to restrict the present discussion to the vertical motions.
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对于图10-1所示的简支梁，阐释了与一组竖向位移自由度相关的某些柔度影响系数的物理意义。也可以考虑水平或转动自由度，在这种情况下，需要使用相应的水平或转动单位载荷来定义完整的影响系数集；然而，为了方便起见，本次讨论将仅限于竖向运动。
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FIGURE 10-1 Definition of flexibility influence coefficients.
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The evaluation of the flexibility influence coefficients for any given system is a standard problem of static structural analysis; any desired method of analysis may be used to compute these deflections resulting from the applied unit loads. When the complete set of influence coefficients has been determined, they are used to calculate the displacement vector resulting from any combination of the applied loads. For example, the deflection at point 1 due to any combination of loads may be expressed
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任何给定系统的柔度影响系数的评估是静力结构分析的一个标准问题；任何期望的分析方法都可以用来计算施加单位载荷所产生的这些变形。当确定了完整的影响系数集后，它们被用来计算由施加的任何载荷组合所产生的位移向量。例如，由于任何载荷组合在点1处的变形可以表示为
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$$
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v_{1}=\widetilde{f}_{11}p_{1}+\widetilde{f}_{12}p_{2}+\widetilde{f}_{13}p_{3}+\ldots+\widetilde{f}_{1N}p_{N}
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$$
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Since similar expressions can be written for each displacement component, the complete set of displacements is expressed
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由于可以为每个位移分量编写类似的表达式，因此完整的位移集表示为
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$$
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\left\{\begin{array}{l}{v_{1}}\\ {v_{2}}\\ {\cdot}\\ {v_{i}}\\ {v_{i}}\\ {\cdot}\end{array}\right\}=\left[\begin{array}{l l l l l l}{\widetilde{f}_{11}}&{\widetilde{f}_{12}}&{\widetilde{f}_{13}}&{\cdots}&{\widetilde{f}_{1i}}&{\cdots}&{\widetilde{f}_{1N}}\\ {\widetilde{f}_{21}}&{\widetilde{f}_{22}}&{\widetilde{f}_{23}}&{\cdots}&{\widetilde{f}_{2i}}&{\cdots}&{\widetilde{f}_{2N}}\\ &{\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}&&\\ &{\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}&&\\ {\widetilde{f}_{i1}}&{\widetilde{f}_{i2}}&{\widetilde{f}_{i3}}&{\cdots}&{\widetilde{f}_{i i}}&{\cdots}&{\widetilde{f}_{i N}}\\ &{\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}&&{.}\end{array}\right]\left\}\begin{array}{l}{p_{1}}\\ {p_{2}}\\ {\cdot}\\ {p_{i}}\\ {p_{i}}\\ {\cdot}\end{array}\right\}
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$$
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or symbolically
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$$
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\mathbf{v}=\widetilde{\mathbf{f}}\,\mathbf{p}
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$$
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in which the matrix of flexibility influence coefficients $\widetilde{\mathbf{f}}$ is called the flexibility matrix of the structure.
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In Eq. (10-4) the deflections are expressed in terms of the vector of externally applied loads p, which are considered positive when acting in the same sense as the positive displacements. The deflection may also be expressed in terms of the elastic forces ${\bf f}_{S}$ which resist the deflections and which are considered positive when acting opposite to the positive displacements. Obviously by statics $\mathbf{f}_{S}=\mathbf{p}$ , and Eq. (10-4) may be revised to read
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其中，柔度影响系数矩阵 $\widetilde{\mathbf{f}}$ 称为结构的柔度矩阵。
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在式 (10-4) 中，变形用外部载荷向量 p 表示，当外部载荷的作用方向与正位移方向相同时，其被认为是正的。变形也可以用抵抗变形的弹性力 ${\bf f}_{S}$ 表示，当弹性力的作用方向与正位移方向相反时，其被认为是正的。显然，根据静力学 $\mathbf{f}_{S}=\mathbf{p}$，式 (10-4) 可以修改为
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$$
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\mathbf{v}=\widetilde{\mathbf{f}}\,\mathbf{f}_{S}
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$$
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## Stiffness
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The physical meaning of the stiffness influence coefficients defined in Eq. (9-4) is illustrated for a few degrees of freedom in Fig. 10-2; they represent the forces developed in the structure when a unit displacement corresponding to one degree of freedom is introduced and no other nodal displacements are permitted. It should be noted that the stiffness influence coefficients in Fig. 10-2 are numerically equal to the applied forces required to maintain the specified displacement condition. They are positive when the sense of the applied force corresponds to a positive displacement and negative otherwise.
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图10-2说明了式(9-4)中定义的刚度影响系数在几个自由度上的物理意义；它们表示当引入对应于一个自由度的单位位移且不允许其他节点位移时，结构中产生的力。需要注意的是，图10-2中的刚度影响系数在数值上等于维持指定位移条件所需的施加力。当施加力的方向与正向位移方向一致时，它们为正，否则为负。
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FIGURE 10-2 Definition of stiffness influence coefficients.
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## Basic Structural Concepts
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### Strain energy
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The strain energy stored in any structure may be expressed conveniently in terms of either the flexibility or the stiffness matrix. The strain energy $U$ is equal to the work done in distorting the system; thus
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任何结构中储存的应变能都可以方便地用柔度矩阵或刚度矩阵来表示。应变能 $U$ 等于使系统变形所做的功；因此
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$$
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U=\frac{1}{2}\:\sum_{i=1}^{N}p_{i}\,v_{i}=\frac{1}{2}\:\mathbf{p}^{T}\,\mathbf{v}
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$$
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where the $\frac{1}{2}$ factor results from the forces which increase linearly with the displacements, and $\mathbf{p}^{T}$ represents the transpose of $\mathbf{p}$ . By substituting Eq. (10-4) this becomes
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其中 $\frac{1}{2}$ 因子源于随位移线性增加的力，且 $\mathbf{p}^{T}$ 表示 $\mathbf{p}$ 的转置。将式 (10-4) 代入后，此式变为
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$$
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U=\frac{1}{2}\:\mathbf{p}^{T}\widetilde{\mathbf{f}}\,\mathbf{p}
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$$
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Alternatively, transposing Eq. (10-6) and substituting Eq. (9-6) leads to the second strain-energy expression (note that $\mathbf{p}=\mathbf{f}_{S}$ ):
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此外，对式(10-6)进行移项并代入式(9-6)，可得到第二个应变能表达式（注意 $\mathbf{p}=\mathbf{f}_{S}$ ）：
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$$
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U=\frac{1}{2}\mathbf{}\mathbf{v}^{T}\,\mathbf{k}\,\mathbf{v}
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$$
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Finally, when it is noted that the strain energy stored in a stable structure during any distortion must always be positive, it is evident that
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最后，当注意到稳定结构在任何变形过程中储存的应变能必须始终为正时，显而易见的是
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$$
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\mathbf{v}^{T}\,\mathbf{k}\,\mathbf{v}>0\qquad\qquad{\mathrm{and}}\qquad\qquad\mathbf{p}^{T}\,\widetilde{\mathbf{f}}\,\mathbf{p}>0
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$$
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Matrices which satisfy this condition, where $\mathbf{v}$ or $\mathbf{p}$ is any arbitrary nonzero vector, are said to be positive definite; positive definite matrices (and consequently the flexibility and stiffness matrices of a stable structure) are nonsingular and can be inverted.
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Inverting the stiffness matrix and premultiplying both sides of Eq. (9-6) by the inverse leads to
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满足此条件的矩阵，其中 $\mathbf{v}$ 或 $\mathbf{p}$ 是任意非零向量，被称为正定矩阵；正定矩阵（因此，稳定结构的柔度矩阵和刚度矩阵）是非奇异的，并且可以求逆。
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对刚度矩阵求逆，并用其逆矩阵左乘式 (9-6) 的两边，得到
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$$
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\mathbf{k}^{-1}\,\mathbf{f}_{S}=\mathbf{v}
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$$
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which upon comparison with Eq. (10-5) demonstrates that the flexibility matrix is the inverse of the stiffness matrix:
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这与式 (10-5) 比较后表明，柔度矩阵是刚度矩阵的逆：
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$$
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\mathbf{k}^{-1}=\widetilde{\mathbf{f}}
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$$
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In practice, the evaluation of stiffness coefficients by direct application of the definition, as implied in Fig. 10-2, may be a tedious computational problem. In many cases, the most convenient procedure for obtaining the stiffness matrix is direct evaluation of the flexibility coefficients and inversion of the flexibility matrix.
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实际上，通过直接应用定义（如图10-2所示）来评估刚度系数可能是一个繁琐的计算问题。在许多情况下，获取刚度矩阵最便捷的方法是直接评估柔度系数并对柔度矩阵进行求逆。
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### Betti’s law
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A property which is very important in structural-dynamics analysis can be derived by applying two sets of loads to a structure in reverse sequence and comparing expressions for the work done in the two cases. Consider, for example, the two different load systems and their resulting displacements shown in Fig. 10-3. If the loads $a$ are applied first followed by loads $b$ , the work done will be as follows:
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结构动力学分析中一个非常重要的性质，可以通过以相反的顺序将两组载荷施加到结构上，并比较两种情况下所做功的表达式来得出。例如，考虑图10-3所示的两种不同的载荷系统及其产生的位移。如果首先施加载荷 $a$，接着施加载荷 $b$，则所做的功将如下所示：
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Case 1:
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$$
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\begin{array}{c}{Loads\ a:\ {W_{a a}=\frac{1}{2}\,\sum p_{i a}{v_{i a}}=\frac{1}{2}\,{\bf p}_{a}{}^{T}{\bf v}_{a}}}\\ Loads\ b:\ {{{ W}_{b b}+W_{a b}=\frac{1}{2}\,{\bf p}_{b}{}^{T}{\bf v}_{b}+{\bf p}_{a}{}^{T}{\bf v}_{b}}}\\ Total:\ {{W_{1}=W_{a a}+W_{b b}+W_{a b}=\frac{1}{2}\,{\bf p}_{a}{}^{T}{\bf v}_{a}+\frac{1}{2}\,{\bf p}_{b}{}^{T}{\bf v}_{b}+{\bf p}_{a}{}^{T}{\bf v}_{b}}}\end{array}
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$$
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Note that the work done by loads $a$ during the application of loads $b$ is not multiplied by $\frac{1}{2}$ ; they act at their full value during the entire displacement $\mathbf{v}_{b}$ . Now if the loads are applied in reverse sequence, the work done is:
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请注意，在施加载荷 $b$ 的过程中，载荷 $a$ 所做的功不乘以 $\frac{1}{2}$；它们在整个位移 $\mathbf{v}_{b}$ 过程中以其完整值作用。现在，如果载荷以相反的顺序施加，所做的功为：
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Load system $a$
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Deflections $a$ :
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Deflections $b$
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FIGURE 10-3 Two independent load systems and resulting deflections.
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Case 2 :
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$$
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\begin{array}{c}Loads\ b:\ {{W_{b b}=\frac{1}{2}\,{\bf p}_{b}^{\ T}{\bf v}_{b}}}\\ Loads\ a:\ {{W_{a a}+W_{b a}=\frac{1}{2}\,{\bf p}_{a}^{\ T}{\bf v}_{a}+{\bf p}_{b}^{\ T}{\bf v}_{a}}}\\ Total:\ {{W_{2}=W_{b b}+W_{a a}+W_{b a}=\frac{1}{2}\,{\bf p}_{b}^{\ T}{\bf v}_{b}+\frac{1}{2}\,{\bf p}_{a}^{\ T}{\bf v}_{a}+{\bf p}_{b}^{\ T}{\bf v}_{a}}}\end{array}
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$$
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The deformation of the structure is independent of the loading sequence, however; therefore the strain energy and hence also the work done by the loads is the same in both these cases; that is, $W_{1}=W_{2}$ . From a comparison of Eqs. (10-11) and (10-12) it may be concluded that $W_{a b}=W_{b a}$ ; thus
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然而，结构的变形与加载顺序无关；因此，应变能以及载荷所做的功在这两种情况下是相同的；即，$W_{1}=W_{2}$。通过比较式(10-11)和式(10-12)，可以得出$W_{a b}=W_{b a}$；因此
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$$
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\mathbf{p}_{a}^{\ T}\mathbf{v}_{b}=\mathbf{p}_{b}^{\ T}\mathbf{v}_{a}
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$$
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Equation (10-13) is an expression of Betti’s law; it states that the work done by one set of loads on the deflections due to a second set of loads is equal to the work of the second set of loads acting on the deflections due to the first.
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If Eq. (10-4) is written for the two sets of forces and displacements and substituted into both sides of Eq. (10-13):
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方程 (10-13) 是贝蒂定律的表达式；它指出一套载荷在由第二套载荷引起的变形上所做的功，等于第二套载荷在由第一套载荷引起的变形上所做的功。
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如果将方程 (10-4) 写成两套力和位移的形式并代入方程 (10-13) 的两边：
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$$
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\mathbf{p}_{a}^{\textit{T}}\widetilde{\mathbf{f}}\,\mathbf{p}_{b}=\mathbf{p}_{b}^{\textit{T}}\widetilde{\mathbf{f}}\,\mathbf{p}_{a}
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$$
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it is evident that
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$$
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\widetilde{\mathbf{f}}=\widetilde{\mathbf{f}}^{T}
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$$
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Thus the flexibility matrix must be symmetric; that is, $\widetilde{f}_{i j}=\widetilde{f}_{j i}$ . This is an expression of Maxwell’s law of reciprocal deflections. Substitutin g sim il arly with Eq. (9-6) (and noting that $\mathbf{p}=\mathbf{f}_{S},$ ) leads to
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因此，柔度矩阵必须是对称的；即，$\widetilde{f}_{i j}=\widetilde{f}_{j i}$。这是麦克斯韦互易变形定律的一种表达。类似地代入式(9-6) (并注意到 $\mathbf{p}=\mathbf{f}_{S},$ )，得到
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$$
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{\bf k}={\bf k}^{T}
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$$
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That is, the stiffness matrix also is symmetric.
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即，刚度矩阵也是对称的。
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## Finite-Element Stiffness
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In principle, the flexibility or stiffness coefficients associated with any prescribed set of nodal displacements can be obtained by direct application of their definitions. In practice, however, the finite-element concept, described in Chapter 1, frequently provides the most convenient means for evaluating the elastic properties. By this approach the structure is assumed to be divided into a system of discrete elements which are interconnected only at a finite number of nodal points. The properties of the complete structure are then found by evaluating the properties of the individual finite elements and superposing them appropriately.
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The problem of defining the stiffness properties of any structure is thus reduced basically to the evaluation of the stiffness of a typical element. Consider, for example, the nonuniform straight-beam segment shown in Fig. 10-4. The two nodal points by which this type of element can be assembled into a structure are located at its ends, and if only transverse plane displacements are considered, it has two degrees of freedom at each node, vertical translation and rotation. The deflected shapes resulting from applying a unit displacement of each type at the left end of the element while constraining the other three nodal displacements are shown in Fig. 10-4. These displacement functions could be taken as any arbitrary shapes which satisfy nodal and internal continuity requirements, but they generally are assumed to be the shapes developed in a uniform beam subjected to these nodal displacements. These are cubic hermitian polynomials which may be expressed as
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原则上，与任何给定节点位移集相关的柔度或刚度系数可以通过直接应用它们的定义来获得。然而，在实践中，第1章中描述的有限元概念经常提供评估弹性特性最方便的方法。通过这种方法，结构被假定划分为一个离散单元系统，这些单元仅在有限数量的节点处相互连接。然后，通过评估各个有限元的特性并适当地叠加它们，可以找到完整结构的特性。
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因此，定义任何结构刚度特性的问题基本上简化为评估典型单元的刚度。例如，考虑图10-4所示的非均匀直梁段。这种单元可以组装成结构的两个节点位于其两端，并且如果只考虑横向平面位移，每个节点有两个自由度，即垂直平移和转动。在单元左端施加每种类型的单位位移，同时约束其他三个节点位移所产生的变形形状如图10-4所示。这些位移函数可以取为满足节点和内部连续性要求的任何任意形状，但它们通常被假定为在承受这些节点位移的均匀梁中产生的形状。这些是三次Hermite多项式，可以表示为
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FIGURE 10-4 Beam deflections due to unit nodal displacements at left end.
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$$
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\begin{array}{l}{\displaystyle{\psi_{1}(x)=1-3\left(\frac{x}{L}\right)^{2}+2\left(\frac{x}{L}\right)^{3}}}\\ {\displaystyle{\psi_{3}(x)=x\left(1-\frac{x}{L}\right)^{2}}}\end{array}
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$$
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The equivalent shape functions for displacements applied at the right end are
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在右端施加的位移的等效形函数是
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$$
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\begin{array}{l}{\displaystyle{\psi_{2}(x)=3\left(\frac{x}{L}\right)^{2}-2\left(\frac{x}{L}\right)^{3}}}\\ {\displaystyle{\psi_{4}(x)=\frac{x^{2}}{L}\,\left(\frac{x}{L}-1\right)}}\end{array}
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$$
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With these four interpolation functions, the deflected shape of the element can now be expressed in terms of its nodal displacements:
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使用这四个插值函数，单元的变形形状现在可以用其节点位移来表示：
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$$
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v(x)=\psi_{1}(x)\,v_{1}+\psi_{2}(x)\,v_{2}+\psi_{3}(x)\,v_{3}+\psi_{4}(x)\,v_{4}
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$$
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where the numbered degrees of freedom are related to those shown in Fig. 10-4 as follows:
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其中，编号的自由度与图10-4中所示的自由度对应关系如下：
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$$
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\left\{\begin{array}{c}{v_{1}}\\ {v_{2}}\\ {v_{3}}\\ {v_{4}}\end{array}\right\}\equiv\left\{\begin{array}{c}{v_{a}}\\ {v_{b}}\\ {v_{b}}\end{array}\right\}
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$$
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It should be noted that both rotations and translations are represented as basic nodal degrees of freedom $v_{i}$ .
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By definition, the stiffness coefficients of the element represent the nodal forces due to unit nodal displacements. The nodal forces associated with any nodaldisplacement component can be determined by the principle of virtual displacements, as described in Section 1-5. Consider, for example, the stiffness coefficient $k_{13}$ for the beam element of Fig. 10-4, that is, the vertical force developed at end $a$ due to a unit rotation applied at that point.
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This force component can be evaluated by introducing a virtual vertical displacement of end $a$ , as shown in Fig. 10-5, while the unit rotation is applied as shown, and equating the work done by the external forces to the work done on the internal forces: $W_{E}\,=\,W_{I}$ . In this case, the external work is done only by the vertical-force component at $a$ because the virtual displacements of all other nodal components vanish; thus
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值得注意的是，旋转和平移都被表示为基本节点自由度$v_{i}$。
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根据定义，单元的刚度系数表示由单位节点位移引起的节点力。与任何节点位移分量相关的节点力可以通过虚位移原理确定，如第1-5节所述。例如，考虑图10-4所示梁单元的刚度系数$k_{13}$，即在端点$a$处由于在该点施加单位旋转而产生的竖向力。
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该力分量可以通过引入端点$a$的虚竖向位移（如图10-5所示），同时按图示施加单位旋转，并将外力所做的功与内力所做的功相等来评估：$W_{E}\,=\,W_{I}$。在这种情况下，外部功仅由$a$处的竖向力分量完成，因为所有其他节点分量的虚位移都为零；因此
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$$
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W_{E}=\delta v_{a}p_{a}=\delta v_{1}k_{13}
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$$
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The internal virtual work is done by the internal moments associated with $\theta_{a}=1$ acting on the virtual curvatures, which are $\partial^{2}/\partial x^{2}[\delta v(x)]=\psi_{1}^{\prime\prime}(x)\,\delta v_{1}$ (neglecting the effects of shear distortion). However, the internal moments due to $\theta_{a}=1$ may be expressed as
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内部虚功由与 $\theta_{a}=1$ 相关的内力矩作用在虚曲率上产生，虚曲率即 $\partial^{2}/\partial x^{2}[\delta v(x)]=\psi_{1}^{\prime\prime}(x)\,\delta v_{1}$ (忽略剪切变形的影响)。然而，由 $\theta_{a}=1$ 引起的内力矩可以表示为
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$$
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{ M}(x)=E{ I}(x)\;\psi_{3}^{\prime\prime}(x)
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$$
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Thus the internal work is given by
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因此，内功由下式给出
|
||||
$$
|
||||
W_{I}=\delta v_{1}\,\int_{0}^{L}E I(x)\,\psi_{1}^{\prime\prime}(x)\,\psi_{3}^{\prime\prime}(x)\,d x
|
||||
$$
|
||||
|
||||
|
||||
FIGURE 10-5 Beam subjected to real rotation and virtual translation of node.
|
||||
|
||||
When the work expressions of Eqs. (10-18) and (10-19) are equated, the expression for this stiffness coefficient is
|
||||
当方程 (10-18) 和 (10-19) 的功表达式相等时，该刚度系数的表达式为
|
||||
$$
|
||||
k_{13}=\int_{0}^{L}E I(x)\,\psi_{1}^{\prime\prime}(x)\,\psi_{3}^{\prime\prime}(x)\,d x
|
||||
$$
|
||||
|
||||
Any stiffness coefficient associated with beam flexure therefore may be written equivalently as
|
||||
任何与梁挠曲相关的刚度系数因此可以等效地写为
|
||||
$$
|
||||
k_{i j}=\int_{0}^{L}E I(x)\,\psi_{i}^{\prime\prime}(x)\,\psi_{j}^{\prime\prime}(x)\,d x
|
||||
$$
|
||||
|
||||
From the form of this expression, the symmetry of the stiffness matrix is evident; that is, $k_{i j}=k_{j i}$ . Its equivalence to the corresponding term in the third of Eqs. (8-18) for the case where $i=j$ should be noted.
|
||||
|
||||
For the special case of a uniform beam segment, the stiffness matrix resulting from Eq. (10-21) when the interpolation functions of Eqs. (10-16) are used may be expressed by
|
||||
从该表达式的形式来看，刚度矩阵的对称性是显而易见的；即 $k_{i j}=k_{j i}$。应该注意，在 $i=j$ 的情况下，它与式(8-18)中第三个方程的对应项是等效的。
|
||||
|
||||
对于均匀梁段的特殊情况，当使用式(10-16)的插值函数时，由式(10-21)得到的刚度矩阵可以表示为
|
||||
$$
|
||||
{\left\{\begin{array}{l}{f_{S1}}\\ {f_{S2}}\\ {f_{S3}}\\ {f_{S4}}\end{array}\right\}}={\frac{2E I}{L^{3}}}{\left[\begin{array}{l l l l}{6}&{-6}&{3L}&{3L}\\ {-6}&{6}&{-3L}&{-3L}\\ {3L}&{-3L}&{2L^{2}}&{L^{2}}\\ {3L}&{-3L}&{L^{2}}&{2L^{2}}\end{array}\right]}{\left\{\begin{array}{l}{v_{1}}\\ {v_{2}}\\ {v_{3}}\\ {v_{4}}\end{array}\right\}}
|
||||
$$
|
||||
|
||||
where the nodal displacements $\mathbf{v}$ are defined by Eq. (10-17a) and $\mathbf{f}_{S}$ is the corresponding vector of nodal forces. These stiffness coefficients are the exact values for a uniform beam without shear distortion because the interpolation functions used in Eq. (10-21) are the true shapes for this case. If the stiffness of the beam is not uniform, applying these shape functions in Eq. (10-21) will provide only an approximation to the true stiffness, but the final result for the complete beam will be very good if it is divided into a sufficient number of finite elements.
|
||||
|
||||
As mentioned earlier, when the stiffness coefficients of all the finite elements in a structure have been evaluated, the stiffness of the complete structure can be obtained by merely adding the element stiffness coefficients appropriately; this is called the direct stiffness method. In effect, any stiffness coefficient $k_{i j}$ of the complete structure can be obtained by adding together the corresponding stiffness coefficients of the elements associated with those nodal points. Thus if elements $m,\,n$ , and $p$ were all attached to nodal point $i$ of the complete structure, the structure stiffness coefficient for this point would be
|
||||
|
||||
其中节点位移 $\mathbf{v}$ 由式 (10-17a) 定义，$\mathbf{f}_{S}$ 是相应的节点力向量。这些刚度系数是均匀梁在没有剪切变形情况下的精确值，因为式 (10-21) 中使用的插值函数是这种情况下的真实形状。如果梁的刚度不均匀，在式 (10-21) 中应用这些形函数将只提供对真实刚度的近似值，但如果将完整梁划分为足够数量的有限元，最终结果将非常好。
|
||||
|
||||
如前所述，当结构中所有有限元的刚度系数都被评估后，仅通过适当叠加单元刚度系数即可获得完整结构的刚度；这被称为直接刚度法。实际上，完整结构的任何刚度系数 $k_{i j}$ 都可以通过叠加与这些节点相关的单元的相应刚度系数来获得。因此，如果单元 $m,\,n$ 和 $p$ 都连接到完整结构的节点 $i$，则该点的结构刚度系数将是
|
||||
$$
|
||||
\hat{\hat{k}}_{i i}=\hat{\hat{k}}_{i i}{}^{(m)}+\hat{\hat{k}}_{i i}{}^{(n)}+\hat{\hat{k}}_{i i}{}^{(p)}
|
||||
$$
|
||||
|
||||
in which the superscripts identify the individual elements. Before the element stiffnesses can be superposed in this fashion, they must be expressed in a common globalcoordinate system which is applied to the entire structure. The double hats are placed over each element stiffness symbol in Eq. (10-23) to indicate that they have been transformed from their local-coordinate form [for example, Eq. (10-22)] to the global coordinates.
|
||||
|
||||
其中上标标识各个单元。在以这种方式叠加单元刚度之前，它们必须表示在一个应用于整个结构的通用全局坐标系中。式(10-23)中每个单元刚度符号上的双帽表示它们已从局部坐标形式[例如，式(10-22)]转换为全局坐标。
|
||||
|
||||
Example E10-1. The evaluation of the structural stiffness matrix is a basic operation of the matrix-displacement method of static structural analysis; althoughageneraldiscussionofthissubjectisbeyondthescopeofthisstructuraldynamics text, it may be useful to apply the procedure to a simple frame structure in order to demonstrate how the element stiffness coefficients of Eq. (10-22) may be used.
|
||||
|
||||
Consider the structure of Fig. E10-1a. If it is assumed that the members do not distort axially, this frame has the three joint degrees of freedom shown. The corresponding stiffness coefficients can be evaluated by successively applying a unit displacement to each degree of freedom while constraining the other two and determining the forces developed in each member by the coefficients of Eq. (10-22).
|
||||
|
||||
|
||||
示例 E10-1。结构刚度矩阵的评估是静力结构分析中矩阵位移法的基本操作；尽管对该主题的普遍讨论超出了本结构动力学教材的范围，但将该程序应用于一个简单的框架结构可能有助于演示如何使用式(10-22)中的单元刚度系数。
|
||||
|
||||
考虑图 E10-1a 所示的结构。如果假设构件不发生轴向变形，则该框架具有所示的三个节点自由度。相应的刚度系数可以通过以下方式评估：依次对每个自由度施加单位位移，同时约束其他两个自由度，并根据式(10-22)的系数确定每个构件中产生的力。
|
||||
|
||||
FIGURE E10-1 Analysis of frame stiffness coefficients: (a) frame properties and degrees of freedom; $(b)$ forces due to displacement $v_{\mathrm{_{1}}}=1$ ; $(c)$ forces due to rotation $v_{2}=1$ .
|
||||
|
||||
When the sidesway displacement shown in Fig. E10-1b is applied, it is clear that only the vertical members are deformed; their end forces are given by elements 1, 3, and 4 in the first column of the stiffness matrix of Eq. (10-22). It will be noted that the structure coefficient $k_{11}$ receives a contribution from each column.
|
||||
|
||||
Considering the joint rotation shown in Fig. E10-1c, both the girder and the left vertical contribute to the structure coefficient $k_{22}$ , the contributions being given by element 3 of column 3 in the stiffness matrix of Eq. (10-22) (taking proper account of the girder properties, of course). Only the left vertical contributes to $k_{12}$ and only the girder to $k_{32}$ . The structure stiffness coefficients due to the right-joint rotation are analogous to these.
|
||||
|
||||
The structure stiffness matrix finally obtained by assembling all these coefficients is
|
||||
当施加图 E10-1b 所示的侧移位移时，显然只有竖向构件发生变形；它们的端部力由式 (10-22) 刚度矩阵第一列中的元素 1、3 和 4 给出。值得注意的是，结构系数 $k_{11}$ 接收到来自每根柱的贡献。
|
||||
|
||||
考虑图 E10-1c 所示的节点转动，梁和左侧竖向构件都对结构系数 $k_{22}$ 有贡献，这些贡献由式 (10-22) 刚度矩阵第 3 列中的元素 3 给出（当然，要适当考虑梁的特性）。只有左侧竖向构件对 $k_{12}$ 有贡献，而只有梁对 $k_{32}$ 有贡献。由于右侧节点转动产生的结构刚度系数与这些系数类似。
|
||||
|
||||
通过组装所有这些系数最终得到的结构刚度矩阵是
|
||||
$$
|
||||
\left\{\begin{array}{c}{{f_{S1}}}\\ {{f_{S2}}}\\ {{f_{S3}}}\end{array}\right\}=\frac{2E I}{L^{3}}\begin{array}{c}{{\left[12\quad3L\quad3L\quad3L\right]}}\\ {{\left[3L\quad6L^{2}\quad2L^{2}\right]}}\\ {{\left[3L\quad2L^{2}\quad6L^{2}\right]}}\end{array}\right\{\begin{array}{c}{{v_{1}}}\\ {{v_{2}}}\\ {{v_{3}}}\end{array}\right\}
|
||||
$$
|
||||
|
||||
# 10-2 MASS PROPERTIES
|
||||
|
||||
# Lumped-Mass Matrix
|
||||
|
||||
The simplest procedure for defining the mass properties of any structure is to assume that the entire mass is concentrated at the points at which the translational displacements are defined. The usual procedure for defining the point mass to be located at each node is to assume that the structure is divided into segments, the nodes serving as connection points. Figure 10-6 illustrates the procedure for a beam-type structure. The mass of each segment is assumed to be concentrated in point masses at each of its nodes, the distribution of the segment mass to these points being determined by statics. The total mass concentrated at any node of the complete structure then is the sum of the nodal contributions from all the segments attached to that node. In the beam system of Fig. 10-6 there are two segments contributing to each node; for example, $m_{1}=m_{1a}+m_{1b}$ .
|
||||
|
||||
For a system in which only translational degrees of freedom are defined, the lumped-mass matrix has a diagonal form; for the system of Fig. 10-6 it would be
|
||||
|
||||
|
||||
|
||||
written
|
||||
|
||||
$$
|
||||
\mathbf{m}={\left[\begin{array}{l l l l l l l}{m_{1}}&{0}&{0}&{\cdots}&{0}&{\cdots}&{0}\\ {}&{m_{2}}&{0}&{\cdots}&{0}&{\cdots}&{0}\\ {0}&{0}&{m_{3}}&{\cdots}&{0}&{\cdots}&{0}\\ &{\cdots\cdots\cdots\cdots\cdots\cdots\cdots}&{}&{}&{}\\ {}&{0}&{0}&{\cdots}&{m_{i}}&{\cdots}&{0}\\ &{\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}&{}&{}\\ {0}&{0}&{0}&{\cdots}&{0}&{\cdots}\end{array}\right]}
|
||||
$$
|
||||
|
||||
in which there are as many terms as there are degrees of freedom. The off-diagonal terms $m_{i j}$ of this matrix vanish because an acceleration of any mass point produces an inertial force at that point only. The inertial force at $i$ due to a unit acceleration of point $i$ is obviously equal to the mass concentrated at that point; thus the mass influence coefficient $m_{i i}=m_{i}$ in a lumped-mass system.
|
||||
|
||||
If more than one translational degree of freedom is specified at any nodal point, the same point mass will be associated with each degree of freedom. On the other hand, the mass associated with any rotational degree of freedom will be zero because of the assumption that the mass is lumped in points which have no rotational inertia. (Of course, if a rigid mass having a finite rotational inertia is associated with a rotational degree of freedom, the diagonal mass coefficient for that degree of freedom would be the rotational inertia of the mass.) Thus the lumped-mass matrix is a diagonal matrix which will include zero diagonal elements for the rotational degrees of freedom, in general.
|
||||
|
||||
# Consistent-Mass Matrix
|
||||
|
||||
Making use of the finite-element concept, it is possible to evaluate mass influence coefficients for each element of a structure by a procedure similar to the analysis of element stiffness coefficients. Consider, for example, the nonuniform beam segment shown in Fig. 10-7, which may be assumed to be the same as that of Fig. 10-4. The degrees of freedom of the segment are the translation and rotation at each end, and it will be assumed that the displacements within the span are defined by the same interpolation functions $\psi_{i}(x)$ used in deriving the element stiffness.
|
||||
|
||||
|
||||
FIGURE 10-7 Node subjected to real angular acceleration and virtual translation.
|
||||
|
||||
If the beam were subjected to a unit angular acceleration of the left end, $\ddot{v}_{3}=$ $\ddot{\theta}_{a}=1$ , accelerations would be developed along its length, as follows:
|
||||
|
||||
$$
|
||||
\ddot{v}(x)=\psi_{3}(x)\,\ddot{v}_{3}
|
||||
$$
|
||||
|
||||
which can be obtained by taking the second derivative of Eq. (10-17). By d’Alembert’s principle, the inertial force resisting this acceleration is
|
||||
|
||||
$$
|
||||
f_{I}(x)=m(x)\;\ddot{v}(x)=m(x)\,\psi_{3}(x)\,\ddot{v}_{3}
|
||||
$$
|
||||
|
||||
Now the mass influence coefficients associated with this acceleration are defined as the nodal inertial forces which it produces; these can be evaluated from the distributed inertial force of Eq. (10-26) by the principle of virtual displacements. For example, the vertical force at the left end can be evaluated by introducing a vertical virtual displacement and equating the work done by the external nodal force $p_{a}$ to the work done on the distributed inertial forces $f_{I}(x)$ . Thus
|
||||
|
||||
$$
|
||||
p_{a}\;\delta v_{a}=\int_{0}^{L}f_{I}(x)\,\delta v(x)\,d x
|
||||
$$
|
||||
|
||||
Expressing the vertical virtual displacement in terms of the interpolation function and substituting Eq. (10-26) lead finally to
|
||||
|
||||
$$
|
||||
m_{13}=\int_{0}^{L}m(x)\,\psi_{1}(x)\,\psi_{3}(x)\,d x
|
||||
$$
|
||||
|
||||
It should be noted in Fig. 10-7 that the mass influence coefficient represents the inertial force opposing the acceleration, but that it is numerically equal to the external force producing the acceleration.
|
||||
|
||||
From Eq. (10-27) it is evident that any mass influence coefficient $m_{i j}$ of an arbitrary beam segment can be evaluated by the equivalent expression
|
||||
|
||||
$$
|
||||
m_{i j}=\int_{0}^{L}m(x)\,\psi_{i}(x)\,\psi_{j}(x)\,d x
|
||||
$$
|
||||
|
||||
The symmetric form of this equation shows that the mass matrix (like the stiffness matrix) is symmetric; that is, $m_{i j}=m_{j i}$ ; also it may be noted that this expression is equivalent to the corresponding term in the first of Eqs. (8-18) in the case where $i=j$ . When the mass coefficients are computed in this way, using the same interpolation functions which are used for calculating the stiffness coefficients, the result is called the consistent-mass matrix. In general, the cubic hermitian polynomials of Eqs. (10- 16) are used for evaluating the mass coefficients of any straight beam segment. In the special case of a beam with uniformly distributed mass the results are
|
||||
|
||||
$$
|
||||
\left\{\begin{array}{c}{f_{I1}}\\ {f_{I2}}\\ {f_{I3}}\\ {f_{I4}}\end{array}\right\}=\frac{\overline{{m}}L}{420}\begin{array}{r l r l}{\mathrm{~156~}}&{54}&{22L}&{-13L}\\ {\mathrm{~54~}}&{156}&{13L}&{-22L}\\ {\mathrm{~22~}}&{13L}&{4L^{2}}&{-3L^{2}}\\ {\mathrm{~}}&{\mathrm{~-22~}}&{-3L^{2}}&{4L^{2}}\end{array}\right\}\begin{array}{c}{\langle\ddot{v}_{1}}\\ {\ddot{v}_{2}}\\ {\ddot{v}_{3}}\\ {\ddot{v}_{4}}\end{array}
|
||||
$$
|
||||
|
||||
When the mass coefficients of the elements of a structure have been evaluated, the mass matrix of the complete element assemblage can be developed by exactly the same type of superposition procedure as that described for developing the stiffness matrix from the element stiffness [Eq. (10-23)]. The resulting mass matrix in general will have the same configuration, that is, arrangement of nonzero terms, as the stiffness matrix.
|
||||
|
||||
The dynamic analysis of a consistent-mass system generally requires considerably more computational effort than a lumped-mass system does, for two reasons: (1) the lumped-mass matrix is diagonal, while the consistent-mass matrix has many off-diagonal terms (leading to what is called mass coupling); (2) the rotational degrees of freedom can be eliminated from a lumped-mass analysis (by static condensation, explained later), whereas all rotational and translational degrees of freedom must be included in a consistent-mass analysis.
|
||||
|
||||
Example E10-2. The structure of Example E10-1, shown again in Fig. E10-2a, will be used to illustrate the evaluation of the structural mass
|
||||
|
||||
|
||||
FIGURE E10-2
|
||||
|
||||
Analysis of lumped- and consistent-mass matrices: (a) uniform mass in members; (b) lumping of mass at member ends; (c) forces due to acceleration $\ddot{v}_{1}=1$ (consistent); $(d)$ forces due to acceleration $\ddot{v}_{2}=1$ (consistent).
|
||||
|
||||
matrix. First the lumped-mass procedure is used: half the mass of each member is lumped at the ends of the members, as shown in Fig. E10-2b. The sum of the four contributions at the girder level then acts in the sidesway degree of freedom $m_{11}$ ; no mass coefficients are associated with the other degrees of freedom because these point masses have no rotational inertia.
|
||||
|
||||
The consistent-mass matrix is obtained by applying unit accelerations to each degree of freedom in succession while constraining the others and determining the resulting inertial forces from the coefficients of Eq. (10-29). Considering first the sidesway acceleration, as shown in Fig. E10-2c, it must be noted that the coefficients of Eq. (10-29) account only for the transverse inertia of the columns. The inertia of the girder due to the acceleration parallel to its axis must be added as a rigid-body mass (3mL), as shown.
|
||||
|
||||
The joint rotational acceleration induces only accelerations transverse to the members, and the resulting girder and column contributions are given by Eq. (10-29), as shown in Fig. E10-2d. The final mass matrices, from the lumpedand consistent-mass formulations, are
|
||||
|
||||
$$
|
||||
\begin{array}{r}{\mathbf{m}=\frac{\overline{{m}}L}{210}\begin{array}{l}{\left[840\quad0\quad0\right]}\\ {\left[\begin{array}{l l l}{0}&{0}&{0}\\ {0}&{0}&{0}\\ {0}&{0}&{0}\end{array}\right]}\end{array}\quad\mathbf{m}=\frac{\overline{{m}}L}{210}\begin{array}{l}{\left[786\quad11L\quad11L\right]}\\ {11L\quad26L^{2}\quad-18L^{2}}\\ {\left[11L\quad-18L^{2}\quad26L^{2}\right]}\end{array}}\end{array}
|
||||
$$
|
||||
|
||||
# 10-3 DAMPING PROPERTIES
|
||||
|
||||
If the various damping forces acting on a structure could be determined quantitatively, the finite-element concept could be used again to define the damping coefficients of the system. For example, the coefficient for any element might be of the form [compare with the corresponding term in the second of Eqs. (8-18) for the case where $i=j]$
|
||||
|
||||
$$
|
||||
c_{i j}=\int_{0}^{L}c(x)\,\psi_{i}(x)\,\psi_{j}(x)\,d x
|
||||
$$
|
||||
|
||||
in which $c(x)$ represents a distributed viscous-damping property. After the element damping influence coefficients were determined, the damping matrix of the complete structure could be obtained by a superposition process equivalent to the direct stiffness method. In practice, however, evaluation of the damping property $c(x)$ (or any other specific damping property) is impracticable. For this reason, the damping is generally expressed in terms of damping ratios established from experiments on similar structures rather than by means of an explicit damping matrix c. If an explicit expression of the damping matrix is needed, it generally will be computed from the specified damping ratios, as described in Chapter 12.
|
||||
|
||||
# 10-4 EXTERNAL LOADING
|
||||
|
||||
If the dynamic loading acting on a structure consists of concentrated forces corresponding with the displacement coordinates, the load vector of Eq. (9-2) can be written directly. In general, however, the load is applied at other points as well as the nodes and may include distributed loadings. In this case, the load terms in Eq. (9-2) are generalized forces associated with the corresponding displacement components.
|
||||
|
||||
Two procedures which can be applied in the evaluation of these generalized forces are described in the following paragraphs.
|
||||
|
||||
# Static Resultants
|
||||
|
||||
The most direct means of determining the effective nodal forces generated by loads distributed between the nodes is by application of the principles of simple statics; in other words, the nodal forces are defined as a set of concentrated loads which are statically equivalent to the distributed loading. In effect, the analysis is made as though the actual loading were applied to the structure through a series of simple beams supported at the nodal points. The reactive forces developed at the supports then become the concentrated nodal forces acting on the structure. In this type of analysis it is evident that generalized forces will be developed corresponding only to the translational degrees of freedom; the rotational nodal forces will be zero unless external moments are applied directly to the joints.
|
||||
|
||||
# Consistent Nodal Loads
|
||||
|
||||
A second procedure which can be used to evaluate nodal forces corresponding to all nodal degrees of freedom can be developed from the finite-element concept. This procedure employs the principle of virtual displacements in the same way as in evaluating the consistent-mass matrix, and the generalized nodal forces which are derived are called the consistent nodal loads. Consider the same beam segment as in the consistent-mass analysis but subjected to the externally applied dynamic loading shown in Fig. 10-8. When a virtual displacement $\delta v_{1}$ is applied, as shown in the sketch, and external and internal work are equated, the generalized force corresponding to $v_{1}$ is
|
||||
|
||||
$$
|
||||
p_{1}(t)=\int_{0}^{L}p(x,t)\,\psi_{1}(x)\,d x
|
||||
$$
|
||||
|
||||
|
||||
FIGURE 10-8 Virtual nodal translation of a laterally loaded beam.
|
||||
|
||||
Thus, the element generalized loads can be expressed in general as
|
||||
|
||||
$$
|
||||
p_{i}(t)=\int_{0}^{L}p(x,t)\,\psi_{i}(x)\,d x
|
||||
$$
|
||||
|
||||
The generalized load $p_{3}$ corresponding to $v_{3}\,=\,\theta_{a}$ is an external moment applied at point $a$ . The positive sense of the generalized loads corresponds to the positive coordinate axes. The equivalence of Eq. (10-32) to the corresponding term in the fourth of Eqs. (8-18) should be noted.
|
||||
|
||||
For the loads to be properly called consistent, the interpolation functions $\psi_{i}(x)$ used in Eq. (10-32) must be the same as those used to define the element stiffness coefficients. If linear interpolation functions
|
||||
|
||||
$$
|
||||
\psi_{1}(x)=1-\frac{x}{L}\qquad\qquad\psi_{2}(x)=\frac{x}{L}
|
||||
$$
|
||||
|
||||
were used instead, Eq. (10-32) would provide the static nodal resultants; in general this is the easiest way to compute the statically equivalent loads.
|
||||
|
||||
In some cases, the applied loading may have the special form
|
||||
|
||||
$$
|
||||
p(x,t)=\chi(x)\,f(t)
|
||||
$$
|
||||
|
||||
that is, the form of load distribution $\chi(x)$ does not change with time; only its amplitude changes. In this case the generalized force becomes
|
||||
|
||||
$$
|
||||
p_{i}(t)=f(t)\,\int_{0}^{L}\chi(x)\,\psi_{i}(x)\,d x
|
||||
$$
|
||||
|
||||
which shows that the generalized force has the same time variation as the applied loading; the integral indicates the extent to which the load participates in developing the generalized force.
|
||||
|
||||
When the generalized forces acting on each element have been evaluated by Eq. (10-32), the total effective load acting at the nodes of the assembled structure can be obtained by a superposition procedure equivalent to the direct stiffness process.
|
||||
|
||||
# 10-5 GEOMETRIC STIFFNESS
|
||||
|
||||
# Linear Approximation
|
||||
|
||||
The geometric-stiffness property represents the tendency toward buckling induced in a structure by axially directed load components; thus it depends not only on the configuration of the structure but also on its condition of loading. In this discussion, it is assumed that the forces tending to cause buckling are constant during the dynamic loading; thus they are assumed to result from an independent static loading and are not significantly affected by the dynamic response of the structure. (When these forces do vary significantly with time, they result in a time-varying stiffness property, and analysis procedures based on superposition are not valid for such a nonlinear system.)
|
||||
|
||||
|
||||
|
||||
In general, two different levels of approximation can be established for the evaluation of geometric-stiffness properties, more or less in parallel with the preceding discussions for mass matrices and load vectors. The simplest approximation is conveniently derived from the physical model illustrated in Fig. 10-9, in which it is assumed that all axial forces are acting in an auxiliary structure consisting of rigid bar segments connected by hinges. The hinges are located at points where the transverse-displacement degrees of freedom of the actual beam are identified, and they are attached to the main beam by links which transmit transverse forces but no axial-force components.
|
||||
|
||||
When the actual beam is deflected by any form of loading, the auxiliary link system is forced to deflect equally, as shown in the sketch. As a result of these deflections and the axial forces in the auxiliary system, forces will be developed in the links coupling it to the main beam. In other words, the resistance of the main beam will be required to stabilize the auxiliary system.
|
||||
|
||||
The forces required for equilibrium in a typical segment $i$ of the auxiliary system are shown in Fig. 10-10. The transverse force components $f_{G i}$ and $f_{G j}$ depend on the value of the axial-force component in the segment $N_{i}$ and on the slope of the segment. They are assumed to be positive when they act in the positive-displacement sense on the main beam. In matrix form, these forces may be expressed
|
||||
|
||||
|
||||
Equilibrium forces due to axial load in auxiliary link.
|
||||
|
||||
$$
|
||||
\left\{\begin{array}{l}{{f_{G i}}}\\ {{f_{G j}}}\end{array}\right\}=\frac{N_{i}}{l_{i}}\;\left[\begin{array}{r r}{{1}}&{{-1}}\\ {{}}&{{}}\\ {{-1}}&{{1}}\end{array}\right]\;\left\{\begin{array}{l}{{v_{i}}}\\ {{v_{j}}}\end{array}\right\}
|
||||
$$
|
||||
|
||||
By combining expressions of this type for all segments, the transverse forces due to axial loads can be written for the beam structure of Fig. 10-9 as follows:
|
||||
|
||||
$$
|
||||
\left\{\begin{array}{c}{f_{G1}}\\ {f_{G2}}\\ {\cdot}\\ {\cdot}\\ {f_{G i}}\\ {\cdot}\\ {\cdot}\\ {f_{G N}}\end{array}\right\}=\left[\begin{array}{c c c c c c}{N_{0}}&{N_{1}}&{-\frac{N_{1}}{l_{1}}}&{0}&{\cdots}&{0}&{\cdots}\\ {-\frac{N_{1}}{l_{1}}}&{\frac{N_{1}}{l_{1}}+\frac{N_{2}}{l_{2}}}&{-\frac{N_{2}}{l_{2}}}&{\cdots}&{0}&{\cdots}\\ {\cdots}&{\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}&{\cdots}\\ {0}&{0}&{0}&{\cdots}&{\frac{N_{i-1}}{l_{i-1}}+\frac{N_{i}}{l_{i}}}&{\cdots}\\ {0}&{\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}&{\cdots}\\ {0}&{0}&{0}&{\cdots}&{0}&{\cdots}\end{array}\right]\left(\begin{array}{c}{v_{1}}\\ {v_{2}}\\ {v_{\cdot}}\\ {v_{i}}\\ {\vdots}\\ {v_{N}}\end{array}\right)
|
||||
$$
|
||||
|
||||
in which it will be noted that magnitude of the axial force may change from segment to segment; for the loading shown in Fig. 10-9 all axial forces would be the same, and the term $N$ could be factored from the matrix.
|
||||
|
||||
Symbolically, Eq. (10-36) may be expressed
|
||||
|
||||
$$
|
||||
\mathbf{f}_{G}=\mathbf{k}_{G}\,\mathbf{v}
|
||||
$$
|
||||
|
||||
where the square symmetric matrix $\mathbf{k}_{G}$ is called the geometric-stiffness matrix of the structure. For this linear approximation of a beam system, the matrix has a tridiagonal
|
||||
|
||||
form, as may be seen in Eq. (10-36), with contributions from two adjacent elements making up the diagonal terms and a single element providing each off-diagonal, or coupling, term.
|
||||
|
||||
# Consistent Geometric Stiffness
|
||||
|
||||
The finite-element concept can be used to obtain a higher-order approximation of the geometric stiffness, as demonstrated for the other physical properties. Consider the same beam element used previously but now subjected to distributed axial loads which result in an arbitrary variation of axial force $N(x)$ , as shown in Fig. 10-11. In the lower sketch, the beam is shown subjected to a unit rotation of the left end $v_{3}=1$ . By definition, the nodal forces associated with this displacement component are the corresponding geometric-stiffness influence coefficients; for example, $k_{G13}$ is the vertical force developed at the left end.
|
||||
|
||||
These coefficients may be evaluated by application of virtual displacements and equating the internal and external work components. The virtual displacement $\delta v_{1}$ required to determine $k_{G13}$ is shown in the sketch. The external virtual work in this case is
|
||||
|
||||
$$
|
||||
W_{E}=f_{G a}\,\delta v_{a}=k_{G13}\,\delta v_{1}
|
||||
$$
|
||||
|
||||
in which it will be noted that the positive sense of the geometric-stiffness coefficient corresponds with the positive displacements. To develop an expression for the internal virtual work, it is necessary to consider a differential segment of length $d x$ , taken from the system of Fig. 10-11 and shown enlarged in Fig. 10-12. The work done in this segment by the axial force $N(x)$ during the virtual displacement is
|
||||
|
||||
$$
|
||||
d W_{I}=N(x)\,d(\delta e)
|
||||
$$
|
||||
|
||||
|
||||
FIGURE 10-11 Axially loaded beam with real rotation and virtual translation of node.
|
||||
|
||||
|
||||
|
||||
where $d(\delta e)$ represents the distance the forces acting on this differential segment move toward each other. By similar triangles it may be seen in the sketch that
|
||||
|
||||
$$
|
||||
d(\delta e)=\frac{d v}{d x}\,d(\delta v)
|
||||
$$
|
||||
|
||||
Interchanging the differentiation and variation symbols on the right side gives
|
||||
|
||||
$$
|
||||
d(\delta e)={\frac{d v}{d x}}\,\delta{\Biggr(}{\frac{d v}{d x}}\,d x{\Biggr)}
|
||||
$$
|
||||
|
||||
and hence introducing this into Eq. (10-39) leads to
|
||||
|
||||
$$
|
||||
d W_{I}=N(x)\,{\frac{d v}{d x}}\,\delta\!\left({\frac{d v}{d x}}\right)d x
|
||||
$$
|
||||
|
||||
Expressing the lateral displacements in terms of interpolation functions and integrating finally gives
|
||||
|
||||
$$
|
||||
W_{I}=\delta v_{1}\,\int_{0}^{L}N(x)\,{\frac{d\psi_{3}(x)}{d x}}\,{\frac{d\psi_{1}(x)}{d x}}\,d x
|
||||
$$
|
||||
|
||||
Hence, by equating internal to external work, this geometric-stiffness coefficient is found to be
|
||||
|
||||
$$
|
||||
k_{G13}=\int_{0}^{L}{N(x)\,\psi_{3}^{\prime}(x)\,\psi_{1}^{\prime}(x)\,d x}
|
||||
$$
|
||||
|
||||
or in general the element geometric-stiffness influence coefficients are
|
||||
|
||||
$$
|
||||
k_{G i j}=\int_{0}^{L}{N(x)\,\psi_{i}^{\prime}(x)\,\psi_{j}^{\prime}(x)\,d x}
|
||||
$$
|
||||
|
||||
The equivalence of this equation to the last term in the third of Eqs. (8-18) should be noted; also its symmetry is apparent, that is, $k_{G i j}=k_{G j i}$ .
|
||||
|
||||
If the hermitian interpolation functions [Eqs. (10-16)] are used in deriving the geometric-stiffness coefficients, the result is called the consistent geometric-stiffness
|
||||
|
||||
matrix. In the special case where the axial force is constant through the length of the element, the consistent geometric-stiffness matrix is
|
||||
|
||||
$$
|
||||
\left\{\begin{array}{c}{{f_{G1}}}\\ {{f_{G2}}}\\ {{f_{G3}}}\\ {{f_{G4}}}\end{array}\right\}=\frac{N}{30L}\begin{array}{c c c c}{{\left[\begin{array}{c c c c}{{36}}&{{-36}}&{{3L}}&{{3L}}\\ {{-36}}&{{36}}&{{-3L}}&{{-3L}}\\ {{3L}}&{{-3L}}&{{4L^{2}}}&{{-L^{2}}}\\ {{3L}}&{{-3L}}&{{-L^{2}}}&{{4L^{2}}}\end{array}\right]}}\\ {{\left[\begin{array}{c}{{\left(v_{1}\right)}}\\ {{f_{G4}}}\end{array}\right]}}\end{array}\right.
|
||||
$$
|
||||
|
||||
On the other hand, if linear-interpolation functions [Eq. (10-33)] are used in Eq. (10- 42), and if the axial force is constant through the element, its geometric stiffness will be as derived earlier in Eq. (10-35).
|
||||
|
||||
The assembly of the element geometric-stiffness coefficients to obtain the structure geometric-stiffness matrix can be carried out exactly as for the elastic-stiffness matrix, and the result will have a similar configuration (positions of the nonzero terms). Thus the consistent geometric-stiffness matrix represents rotational as well as translational degrees of freedom, whereas the linear approximation [Eq. (10-35)] is concerned only with the translational displacements. However, either type of relationship may be represented symbolically by Eq. (10-37).
|
||||
|
||||
# 10-6 CHOICE OF PROPERTY FORMULATION
|
||||
|
||||
In the preceding discussion, two different levels of approximation have been considered for the evaluation of the mass, elastic-stiffness, geometric-stiffness, and external-load properties: (1) an elementary approach taking account only of the translational degrees of freedom of the structure and (2) a “consistent” approach, which accounts for the rotational as well as translational displacements. The elementary approach is considerably easier to apply; not only are the element properties defined more simply but the number of coordinates to be considered in the analysis is much less for a given structural assemblage. In principle, the consistent approach should lead to greater accuracy in the results, but in practice the improvement is often slight. Apparently the rotational degrees of freedom are much less significant in the analysis than the translational terms. The principal advantage of the consistent approach is that all the energy contributions to the response of the structure are evaluated in a consistent manner, which makes it possible to draw certain conclusions regarding bounds on the vibration frequency; however, this advantage seldom outweighs the additional effort required.
|
||||
|
||||
The elementary lumped-mass approach presents a special problem when the elastic-stiffness matrix has been formulated by the finite-element approach or by any other procedure which includes the rotational degrees of freedom in the matrix. If the evaluation of all the other properties has excluded these degrees of freedom, it is necessary to exclude them also from the stiffness matrix before the equations of motion can be written.
|
||||
|
||||
The process of eliminating these unwanted degrees of freedom from the stiffness matrix is called static condensation. For the purpose of this discussion, assume that the rotational and translational degrees of freedom have been segregated, so that Eq. (9-5) can be written in partitioned form
|
||||
|
||||
$$
|
||||
\left[\mathbf{k}_{t t}\quad\mathbf{k}_{t\theta}\right]\;\left\{\mathbf{v}_{t}\atop\mathbf{v}_{\theta}\right\}=\left\{\mathbf{f}_{\mathrm{St}}\atop\mathbf{f}_{S\theta}\right\}=\left\{\begin{array}{l l}{\mathbf{f}_{\mathrm{St}}}\\ {\mathbf{f}_{\mathrm{S}}}\end{array}\right\}=\left\{\begin{array}{l l}{\mathbf{f}_{\mathrm{St}}}\\ {\mathbf{0}}\end{array}\right\}
|
||||
$$
|
||||
|
||||
where $\mathbf{v}_{t}$ represents the translations and $\mathbf{v}_{\theta}$ the rotations, with corresponding subscripts to identify the submatrices of stiffness coefficients. Now, if none of the other force vectors acting in the structure include any rotational components, it is evident that the elastic rotational forces also must vanish, that is, $\mathbf{f}_{S\theta}=\mathbf{0}$ . When this static constraint is introduced into Eq. (10-44), it is possible to express the rotational displacements in terms of the translations by means of the second submatrix equation, with the result
|
||||
|
||||
$$
|
||||
\mathbf{v}_{\theta}=-\mathbf{k}_{\theta\theta}^{\phantom{\theta}-1}\,\mathbf{k}_{\theta t}\,\mathbf{v}_{t}
|
||||
$$
|
||||
|
||||
Substituting this into the first of the submatrix equations of Eq. (10-44) leads to
|
||||
|
||||
$$
|
||||
\left(\mathbf{k}_{t t}-\mathbf{k}_{t\theta}\mathbf{\Deltak}_{\theta\theta}^{\prime}^{-1}\mathbf{\Deltak}_{\theta t}\right)\mathbf{\Deltav}_{t}=\mathbf{f}_{\mathrm{St}}
|
||||
$$
|
||||
|
||||
or
|
||||
|
||||
$$
|
||||
\mathbf{k}_{t}\ \mathbf{v}_{t}=\mathbf{f}_{\mathrm{St}}
|
||||
$$
|
||||
|
||||
where
|
||||
|
||||
$$
|
||||
\mathbf{k}_{t}=\mathbf{k}_{t t}-\mathbf{k}_{t\theta}\mathbf{\Deltak}_{\theta\theta}^{\phantom{}}-1\mathbf{\Deltak}_{\theta t}
|
||||
$$
|
||||
|
||||
is the translational elastic stiffness. This stiffness matrix is suitable for use with the other elementary property expressions; in other words, it is the type of stiffness matrix implied in Fig. 10-2.
|
||||
|
||||
Example E10-3. To demonstrate the use of the static-condensation procedure, the two rotational degrees of freedom will be eliminated from the stiffness matrix evaluated in Example E10-1. The resulting condensed stiffness matrix will retain only the translational degree of freedom of the frame and thus will be compatible with the lumped-mass matrix derived in Example E10-2.
|
||||
|
||||
The stiffness submatrix associated with the rotational degrees of freedom of Example E10-1 is
|
||||
|
||||
$$
|
||||
\mathbf{k}_{\theta\theta}={\frac{2E I}{L^{3}}}\left[{\begin{array}{l l}{6L^{2}}&{2L^{2}}\\ {2L^{2}}&{6L^{2}}\end{array}}\right]={\frac{4E I}{L}}\left[{\begin{array}{l l}{3}&{1}\\ {1}&{3}\end{array}}\right]
|
||||
$$
|
||||
|
||||
and its inverse is
|
||||
|
||||
$$
|
||||
{\bf k}_{\theta\theta}{}^{-1}={\frac{L}{32E I}}\left[{\begin{array}{c c}{{3}}&{{-1}}\\ {{}}&{{}}\\ {{-1}}&{{3}}\end{array}}\right]
|
||||
$$
|
||||
|
||||
When this is used in Eq. (10-45), the rotational degrees of freedom can be expressed in terms of the translation:
|
||||
|
||||
$$
|
||||
\left\{{\begin{array}{c}{{v_{2}}}\\ {{v_{3}}}\end{array}}\right\}=-{\frac{L}{32E I}}\,\left[{\begin{array}{r r}{{3}}&{{-1}}\\ {{}}&{{}}\\ {{-1}}&{{3}}\end{array}}\right]\,{\frac{2E I}{L^{3}}}\,\left\{{\begin{array}{c}{{3L}}\\ {{3L}}\end{array}}\right\}\,v_{1}=-{\frac{3}{8L}}\,\left\{{\begin{array}{c}{{1}}\\ {{1}}\end{array}}\right\}\,v_{1}
|
||||
$$
|
||||
|
||||
The condensed stiffness given by Eq. (10-47) then is
|
||||
|
||||
$$
|
||||
\mathbf{k}_{t}=\frac{2E I}{L^{3}}\left(12-<3L\quad3L>\left\{\begin{array}{l}{\frac{3}{8L}}\\ {\frac{3}{8L}}\end{array}\right\}\right)=\frac{2E I}{L^{3}}\:\frac{39}{4}
|
||||
$$
|
||||
|
||||
# PROBLEMS
|
||||
|
||||
10-1. Using the hermitian polynomials, Eq. (10-16), as shape functions $\psi_{i}(x)$ , evaluate by means of Eq. (10-21) the finite-element stiffness coefficient $k_{23}$ for a beam having the following variation of flexural rigidity: $E I(x)=E I_{0}(1+x/L)$ .
|
||||
|
||||
10-2. Making use of Eq. (10-28), compute the consistent mass coefficient $m_{23}$ for a beam with the following nonuniform mass distribution: $m(x)=\overline{{m}}(1+x/L)$ . Assume the shape functions of Eq. (10-16) and evaluate the integral by Simpson’s rule, dividing the beam into four segments of equal length.
|
||||
|
||||
10-3. The distributed load applied to a certain beam may be expressed as
|
||||
|
||||
$$
|
||||
p(x,t)=\overline{{p}}\Big(2+\frac{x}{L}\Big)\sin\overline{{\omega}}t
|
||||
$$
|
||||
|
||||
Making use of Eq. (10-34a), write an expression for the time variation of the consistent load component $p_{2}(t)$ based on the shape function of Eq. (10-16).
|
||||
|
||||
10-4. Using Eq. (10-42), evaluate the consistent geometric stiffness coefficient $k_{G24}$ for a beam having the following distribution of axial force: $N(x)=N_{0}(2\!-\!x/L)$ . Make use of the shape functions of Eq. (10-16) and evaluate the integral by Simpson’s rule using $\triangle x=L/4$ .
|
||||
|
||||
10-5. The plane frame of Fig. P10-1 is formed of uniform members, with the properties of each as shown. Assemble the stiffness matrix defined for the three DOFs indicated, evaluating the member stiffness coefficients by means of Eq. (10-22).
|
||||
|
||||
FIGURE P10-1
|
||||
|
||||
10-6. Assemble the mass matrix for the structure of Prob. 10-5, evaluating the individual member mass coefficients by means of Eq. (10-29).
|
||||
10-7. Assemble the load vector for the structure of Prob. 10-5, evaluating the individual member nodal loads by Eq. (10-32).
|
||||
10-8. For a plane frame of the same general form as that of Prob. 10-5, but having different member lengths and physical properties, the stiffness and lumped mass matrices are as follows:
|
||||
|
||||
$$
|
||||
\mathbf{k}={\frac{E I}{L^{3}}}{\left[\begin{array}{l l l}{\;\;20}&{-10L}&{-5L}\\ {-10L}&{\;\;15L^{2}}&{\;\;8L^{2}}\\ {-5L}&{\;\;8L^{2}}&{12L^{2}}\end{array}\right]}\qquad\mathbf{m}={\overline{{m}}}L{\left[\begin{array}{l l l}{30}&{0}&{0}\\ {0}&{0}&{0}\\ {0}&{0}&{0}\end{array}\right]}
|
||||
$$
|
||||
|
||||
$(a)$ Using static condensation, eliminate the two rotational degrees of freedom from the stiffness matrix.
|
||||
$(b)$ Using the condensed stiffness matrix, write the SDOF equation of motion for undamped free vibrations.
|
||||
@ -22,30 +22,31 @@ The motion of this structure will be assumed to be defined by the displacements
|
||||
该结构的运动将假定由梁上离散点集 $v_{1}(t),v_{2}(t),...,v_{i}(t),...,v_{N}(t)$ 的位移来定义。原则上，这些点可以任意放置在结构上；实际上，它们应与可能重要的物理特性的特定特征相关联，并应分布以提供对变形形状的良好定义。要考虑的自由度（位移分量）的数量由分析师自行决定；数量越多，对真实动力行为的近似越好，但在许多情况下，仅用两到三个自由度就能获得出色的结果。在图9-1的梁中，每个节点只关联了一个位移分量。然而，应该注意的是，每个点可以识别出几个位移分量；例如，转动 $\partial v/\partial x$ 和纵向运动可以作为每个点的附加自由度。
|
||||
|
||||
|
||||
## 9-2 DYNAMIC-EQUILIBRIUM CONDITION
|
||||
## 9-2 DYNAMIC-EQUILIBRIUM CONDITION动态平衡条件
|
||||
|
||||
The equation of motion of the system of Fig. 9-1 can be formulated by expressing the equilibrium of the effective forces associated with each of its degrees of freedom. In general four types of forces will be involved at any point $i$ : the externally applied load $p_{i}(t)$ and the forces resulting from the motion, that is, inertia $f_{I i}$ , damping $f_{D i}$ , and elastic $f_{S i}$ . Thus for each of the several degrees of freedom the dynamic equilibrium may be expressed as
|
||||
|
||||
图9-1所示系统的运动方程可以通过表述与其每个自由度相关的有效力的平衡来建立。通常，在任意点 $i$ 将涉及四种类型的力：外部施加的载荷 $p_{i}(t)$ 以及由运动产生的力，即惯性力 $f_{I i}$、阻尼力 $f_{D i}$ 和弹性力 $f_{S i}$。因此，对于若干个自由度中的每一个，动力平衡可以表示为
|
||||
$$
|
||||
\begin{array}{c}{f_{I1}+f_{D1}+f_{S1}=p_{1}(t)}\\ {f_{I2}+f_{D2}+f_{S2}=p_{2}(t)}\\ {f_{I3}+f_{D3}+f_{S3}=p_{3}(t)}\end{array}
|
||||
$$
|
||||
|
||||
or when the force vectors are represented in matrix form,
|
||||
|
||||
或者当力矢量以矩阵形式表示时，
|
||||
$$
|
||||
\mathbf{f}_{I}+\mathbf{f}_{D}+\mathbf{f}_{S}=\mathbf{p}(t)
|
||||
$$
|
||||
|
||||
which is the MDOF equivalent of the SDOF equation (2-1).
|
||||
这就是 SDOF 方程 (2-1) 的 MDOF 等效形式。
|
||||
|
||||
Each of the resisting forces is expressed most conveniently by means of an appropriate set of influence coefficients. Consider, for example, the elastic-force component developed at point 1; this depends in general upon the displacement components developed at all points of the structure:
|
||||
|
||||
每一个阻力都可以通过一组合适的**影响系数**最方便地表示。例如，考虑在点1处产生的弹性力分量；这通常取决于在结构所有点处产生的位移分量：
|
||||
$$
|
||||
f_{S1}=k_{11}v_{1}+k_{12}v_{2}+k_{13}v_{3}+\cdot\cdot\cdot+k_{1N}v_{N}
|
||||
$$
|
||||
|
||||
Similarly, the elastic force corresponding to the degree of freedom $v_{2}$ is
|
||||
|
||||
类似地，对应于自由度 $v_{2}$ 的弹性力是
|
||||
$$
|
||||
f_{S2}=k_{21}v_{1}+k_{22}v_{2}+k_{23}v_{3}+\cdot\cdot\cdot+k_{2N}v_{N}
|
||||
$$
|
||||
@ -57,13 +58,13 @@ f_{S i}=k_{i1}v_{1}+k_{i2}v_{2}+k_{i3}v_{3}+\cdot\cdot\cdot+k_{i N}v_{N}
|
||||
$$
|
||||
|
||||
In these expressions it has been tacitly assumed that the structural behavior is linear, so that the principle of superposition applies. The coefficients $k_{i j}$ are called stiffness influence coefficients, defined as follows:
|
||||
|
||||
在这些表达式中，已经默认假设结构行为是线性的，从而叠加原理适用。系数 $k_{i j}$ 被称为刚度影响系数，定义如下：
|
||||
$$
|
||||
\begin{array}{r}{k_{i j}=\mathrm{~force~corresponding~to~coordinate~}i\mathrm{~d}\mathrm{t}}\\ {\mathrm{~a~unit~}d i s p l a c e m e n t\;\mathrm{of~coordinate~}j\quad\quad}\end{array}
|
||||
$$
|
||||
|
||||
In matrix form, the complete set of elastic-force relationships may be written
|
||||
|
||||
以矩阵形式，完整的弹性力关系可以写成
|
||||
$$
|
||||
\left\{\begin{array}{l}{f_{S1}}\\ {f_{S2}}\\ {\cdot}\\ {\cdot}\\ {f_{S i}}\\ {\cdot}\\ {\cdot}\end{array}\right\}=\left[\begin{array}{l l l l l l l}{k_{11}}&{k_{12}}&{k_{13}}&{\cdots}&{k_{1i}}&{\cdots}&{k_{1N}}\\ {k_{21}}&{k_{22}}&{k_{23}}&{\cdots}&{k_{2i}}&{\cdots}&{k_{2N}}\\ {\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}\\ {k_{i1}}&{k_{i2}}&{k_{i3}}&{\cdots}&{k_{i i}}&{\cdots}&{k_{i N}}\\ {\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}\end{array}\right]\left\{\begin{array}{l}{v_{1}}\\ {v_{2}}\\ {v_{3}}\\ {\cdot}\\ {v_{i}}\\ {\cdot}\\ {\cdot}\end{array}\right\}
|
||||
$$
|
||||
@ -77,13 +78,14 @@ $$
|
||||
in which the matrix of stiffness coefficients $\mathbf{k}$ is called the stiffness matrix of the structure (for the specified set of displacement coordinates) and $\mathbf{v}$ is the displacement vector representing the displaced shape of the structure.
|
||||
|
||||
If it is assumed that the damping depends on the velocity, that is, the viscous type, the damping forces corresponding to the selected degrees of freedom may be expressed by means of damping influence coefficients in similar fashion. By analogy with Eq. (9-5), the complete set of damping forces is given by
|
||||
|
||||
其中刚度系数矩阵 $\mathbf{k}$ 称为结构的刚度矩阵 (对于指定的一组位移坐标)，且 $\mathbf{v}$ 是表示结构变形形状的位移向量。
|
||||
如果假设阻尼取决于速度，即黏性类型，则对应于所选自由度的阻尼力可以类似地通过阻尼影响系数来表示。参照式 (9-5)，完整的阻尼力集由下式给出
|
||||
$$
|
||||
\left\{\begin{array}{c}{f_{D1}}\\ {f_{D2}}\\ {\cdot}\\ {\cdot}\\ {f_{D i}}\\ {\cdot}\end{array}\right\}=\left[\begin{array}{c c c c c c c}{c_{11}}&{c_{12}}&{c_{13}}&{\cdots}&{c_{1i}}&{\cdots}&{c_{1N}}\\ {c_{21}}&{c_{22}}&{c_{23}}&{\cdots}&{c_{2i}}&{\cdots}&{c_{2N}}\\ {\cdots}&{\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}\\ {c_{i1}}&{c_{i2}}&{c_{i3}}&{\cdots}&{c_{i i}}&{\cdots}&{c_{i N}}\\ {\cdots}&{\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}\end{array}\right]\,\left\{\begin{array}{c}{\dot{v}_{1}}\\ {\dot{v}_{2}}\\ {\dot{v}_{3}}\\ {\cdot}\\ {\dot{v}_{i}}\\ {\dot{v}_{i}}\\ {\cdot}\end{array}\right\}\,\,
|
||||
$$
|
||||
|
||||
in which $\dot{v}_{i}$ represents the time rate of change (velocity) of the $i$ displacement coordinate and the coefficients $c_{i j}$ are called damping influence coefficients. The definition of these coefficients is exactly parallel to Eq. (9-4):
|
||||
|
||||
其中 $\dot{v}_{i}$ 表示第 $i$ 个位移坐标的时间变化率（速度），且系数 $c_{i j}$ 称为阻尼影响系数。这些系数的定义与式 (9-4) 完全平行。
|
||||
$$
|
||||
\begin{array}{c}{{c_{i j}=\mathrm{force\;corresponding\;to\;coordinate\;}i\mathrm{\;due\;to\;unit}}}\\ {{{\nu e l o c i t y\;\mathrm{of\;coordinate\;}j}}}\end{array}
|
||||
$$
|
||||
@ -97,13 +99,15 @@ $$
|
||||
in which the matrix of damping coefficients $\mathbf{c}$ is called the damping matrix of the structure (for the specified degrees of freedom) and $\dot{\mathbf{v}}$ is the velocity vector.
|
||||
|
||||
The inertial forces may be expressed similarly by a set of influence coefficients called the mass coefficients. These represent the relationship between the accelerations of the degrees of freedom and the resulting inertial forces; by analogy with Eq. (9-5), the inertial forces may be expressed as
|
||||
其中阻尼系数矩阵 $\mathbf{c}$ 称为结构（对于指定的自由度）的阻尼矩阵，且 $\dot{\mathbf{v}}$ 是速度向量。
|
||||
|
||||
惯性力可以类似地通过一组称为质量系数的影响系数来表示。这些表示自由度加速度与由此产生的惯性力之间的关系；参照式 (9-5)，惯性力可以表示为
|
||||
$$
|
||||
\left\{\begin{array}{l}{f_{I1}}\\ {f_{I2}}\\ {\cdot}\\ {\cdot}\\ {f_{I i}}\\ {\cdot}\\ {\cdot}\end{array}\right\}=\left[\begin{array}{l l l l l l l}{m_{11}}&{m_{12}}&{m_{13}}&{\cdots}&{m_{1i}}&{\cdots}&{m_{1N}}\\ {m_{21}}&{m_{22}}&{m_{23}}&{\cdots}&{m_{2i}}&{\cdots}&{m_{2N}}\\ {\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}\\ {m_{i1}}&{m_{i2}}&{m_{i3}}&{\cdots}&{m_{i i}}&{\cdots}&{m_{i N}}\\ {\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}\\ {\cdot}&{\cdot}&{\cdot}&{\cdot}&{\cdot}&{\cdot}\\ {m_{i1}}&{m_{i2}}&{m_{i3}}&{\cdots}&{\cdots}&{m_{i i}}&{\cdots}&{\cdots}\\ {\cdots}&{\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}\end{array}\right]=\left\{\begin{array}{l}{\ddot{v_{1}}}\\ {\ddot{v_{2}}}\\ {\ddot{v_{2}}}\\ {\cdot}\\ {\dot{v_{i}}}\\ {\dot{v_{i}}}\\ {\cdot}\end{array}\right\}
|
||||
$$
|
||||
|
||||
where $\ddot{v}_{i}$ is the acceleration of the $i$ displacement coordinate and the coefficients $m_{i j}$ are the mass influence coefficients, defined as follows:
|
||||
|
||||
其中 $\ddot{v}_{i}$ 是第 $i$ 个位移坐标的加速度，且系数 $m_{i j}$ 是质量影响系数，定义如下：
|
||||
$$
|
||||
m_{i j}=\mathrm{force\corresponding\to\coordinate\}i\ \mathrm{due}
|
||||
$$
|
||||
@ -117,27 +121,30 @@ $$
|
||||
in which the matrix of mass coefficients $\mathbf{m}$ is called the mass matrix of the structure and $\ddot{\mathbf{v}}$ is its acceleration vector, both defined for the specified set of displacement coordinates.
|
||||
|
||||
Substituting Eqs. (9-6), (9-9), and (9-12) into Eq. (9-2) gives the complete dynamic equilibrium of the structure, considering all degrees of freedom:
|
||||
其中，质量系数矩阵 $\mathbf{m}$ 被称为结构的质量矩阵，且 $\ddot{\mathbf{v}}$ 是其加速度向量，两者都针对指定的位移坐标集定义。
|
||||
|
||||
将式 (9-6)、(9-9) 和 (9-12) 代入式 (9-2)，得到考虑所有自由度的结构完整动力平衡方程：
|
||||
$$
|
||||
\mathbf{m}\,\ddot{\mathbf{v}}(t)+\mathbf{c}\,\dot{\mathbf{v}}(t)+\mathbf{k}\,\mathbf{v}(t)=\mathbf{p}(t)
|
||||
$$
|
||||
|
||||
This equation is the MDOF equivalent of Eq. (2-3); each term of the SDOF equation is represented by a matrix in Eq. (9-13), the order of the matrix corresponding to the number of degrees of freedom used in describing the displacements of the structure. Thus, Eq. (9-13) expresses the $N$ equations of motion which serve to define the response of the MDOF system.
|
||||
这个方程是方程(2-3)的多自由度（MDOF）等效形式；单自由度（SDOF）方程的每个项在方程(9-13)中由一个矩阵表示，该矩阵的阶数对应于用于描述结构位移的自由度数量。因此，方程(9-13)表达了用于定义多自由度（MDOF）系统响应的$N$个运动方程。
|
||||
|
||||
## 9-3 AXIAL-FORCE EFFECTS
|
||||
|
||||
It was observed in the discussion of SDOF systems that axial forces or any load which may tend to cause buckling of a structure may have a significant effect on the stiffness of the structure. Similar effects may be observed in MDOF systems; the force component acting parallel to the original axis of the members leads to additional load components which act in the direction (and sense) of the nodal displacements and which will be denoted by $\mathbf{f}_{G}$ . When these forces are included, the dynamic-equilibrium expression, Eq. (9-2), becomes
|
||||
|
||||
在单自由度（SDOF）系统讨论中观察到，轴向力或任何可能导致结构屈曲的载荷可能会对结构的刚度产生显著影响。在多自由度（MDOF）系统中也可能观察到类似效应；作用于构件原始轴线平行的力分量会产生额外的载荷分量，这些分量作用于节点位移的方向（和指向），并用 $\mathbf{f}_{G}$ 表示。当包含这些力时，动力平衡表达式，即式（9-2），变为
|
||||
$$
|
||||
\mathbf{f}_{I}+\mathbf{f}_{D}+\mathbf{f}_{S}-\mathbf{f}_{G}=\mathbf{p}(t)
|
||||
$$
|
||||
|
||||
in which the negative sign results from the fact that the forces $\mathbf{f}_{G}$ are assumed to contribute to the deflection rather than oppose it.
|
||||
|
||||
These forces resulting from axial loads depend on the displacements of the structure and may be expressed by influence coefficients, called the geometric-stiffness
|
||||
|
||||
coefficients, as follows:
|
||||
These forces resulting from axial loads depend on the displacements of the structure and may be expressed by influence coefficients, called the geometric-stiffness coefficients, as follows:
|
||||
其中，负号是由于假设力 $\mathbf{f}_{G}$ 促进变形而不是阻碍变形这一事实造成的。
|
||||
|
||||
这些由轴向载荷产生的力取决于结构的位移，并且可以通过影响系数（称为几何刚度系数）表示，如下所示：
|
||||
$$
|
||||
\left\{\begin{array}{l}{f_{G1}}\\ {f_{G2}}\\ {\cdot}\\ {\cdot}\\ {f_{G i}}\\ {\cdot}\\ {\cdot}\end{array}\right\}=\left[\begin{array}{c c c c c c c}{k_{G11}}&{k_{G12}}&{k_{G13}}&{\cdot\cdot}&{k_{G1i}}&{\cdot\cdot}&{k_{G1N}}\\ {k_{G21}}&{k_{G22}}&{k_{G23}}&{\cdot\cdot}&{k_{G2i}}&{\cdot\cdot}&{k_{G2N}}\\ {\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}\\ {\vdots}&{k_{G i1}}&{k_{G i2}}&{k_{G i3}}&{\cdot\cdot}&{k_{G i i}}&{\cdot\cdot}&{k_{G i N}}\\ {\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots}\end{array}\right]\left\{\begin{array}{l}{v_{1}}\\ {v_{2}}\\ {v_{2}}\\ {\cdot}\\ {v_{i}}\\ {\cdot}\\ {\cdot}\end{array}\right\}
|
||||
$$
|
||||
@ -155,13 +162,15 @@ $$
|
||||
where $\mathbf{k}_{G}$ is called the geometric-stiffness matrix of the structure.
|
||||
|
||||
When this expression is introduced, the equation of dynamic equilibrium of the structure [given by Eq. (9-13) without axial-force effects] becomes
|
||||
其中，$\mathbf{k}_{G}$ 称为结构的几何刚度矩阵。
|
||||
|
||||
当引入此表达式时，结构的动力平衡方程[由不考虑轴向力效应的式 (9-13) 给出]变为
|
||||
$$
|
||||
\mathbf{m}\,\ddot{\mathbf{v}}(t)+\mathbf{c}\,\dot{\mathbf{v}}(t)+\mathbf{k}\,\mathbf{v}(t)-\mathbf{k}_{G}\,\mathbf{v}(t)=\mathbf{p}(t)
|
||||
$$
|
||||
|
||||
or when it is noted that both the elastic stiffness and the geometric stiffness are multiplied by the displacement vector, the combined stiffness effect can be expressed by a single symbol and Eq. (9-18) written
|
||||
|
||||
或者当注意到弹性刚度和几何刚度都乘以位移向量时，组合刚度效应可以用一个符号表示，并且式 (9-18) 写成
|
||||
$$
|
||||
\mathbf{m}\,\ddot{\mathbf{v}}(t)+\mathbf{c}\,\dot{\mathbf{v}}(t)+\overline{{\mathbf{k}}}\,\mathbf{v}(t)=\mathbf{p}(t)
|
||||
$$
|
||||
@ -172,4 +181,5 @@ $$
|
||||
\overline{{\mathbf{k}}}=\mathbf{k}-\mathbf{k}_{G}
|
||||
$$
|
||||
|
||||
is called the combined stiffness matrix, which includes both elastic and geometric effects. The dynamic properties of the structure are expressed completely by the four influence-coefficient matrices of Eq. (9-18), while the dynamic loading is fully defined by the load vector. The evaluation of these physical-property matrices and the evaluation of the load vector resulting from externally applied forces will be discussed in detail in the following chapter. The effective-load vector resulting from support excitation will be discussed in connection with earthquake-response analysis in Chapter 26.
|
||||
is called the combined stiffness matrix, which includes both elastic and geometric effects. The dynamic properties of the structure are expressed completely by the four influence-coefficient matrices of Eq. (9-18), while the dynamic loading is fully defined by the load vector. The evaluation of these physical-property matrices and the evaluation of the load vector resulting from externally applied forces will be discussed in detail in the following chapter. The effective-load vector resulting from support excitation will be discussed in connection with earthquake-response analysis in Chapter 26.
|
||||
称为组合刚度矩阵，其包含弹性效应和几何效应。结构的动力特性完全由式(9-18)的四个影响系数矩阵表示，而动力载荷则完全由载荷向量定义。这些物理特性矩阵的评估以及由外部施加力产生的载荷向量的评估将在下一章详细讨论。由支座激励产生的有效载荷向量将在第26章结合地震响应分析进行讨论。
|
||||
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