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59
书籍/力学书籍/力学/Dynamics of Structures (Ray Clough, Joseph Penzien) (Z-Library)/auto/Chap 10 EVALUATION OF STRUCTURALPROPERTY MATRICES.md
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@ -19,7 +19,7 @@ FIGURE 10-1 Definition of flexibility influence coefficients.
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 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
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 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
$$ $$
v_{1}=\widetilde{f}_{11}p_{1}+\widetilde{f}_{12}p_{2}+\widetilde{f}_{13}p_{3}+\ldots+\widetilde{f}_{1N}p_{N} v_{1}=\widetilde{f}_{11}p_{1}+\widetilde{f}_{12}p_{2}+\widetilde{f}_{13}p_{3}+\ldots+\widetilde{f}_{1N}p_{N}\tag{10-2}
$$ $$
Since similar expressions can be written for each displacement component, the complete set of displacements is expressed Since similar expressions can be written for each displacement component, the complete set of displacements is expressed
@ -53,7 +53,7 @@ $$
or symbolically or symbolically
$$ $$
\mathbf{v}=\widetilde{\mathbf{f}}\,\mathbf{p} \mathbf{v}=\widetilde{\mathbf{f}}\,\mathbf{p}\tag{10-4}
$$ $$
in which the matrix of flexibility influence coefficients $\widetilde{\mathbf{f}}$ is called the flexibility matrix of the structure. in which the matrix of flexibility influence coefficients $\widetilde{\mathbf{f}}$ is called the flexibility matrix of the structure.
@ -63,7 +63,7 @@ In Eq. (10-4) the deflections are expressed in terms of the vector of externally
在式 (10-4) 中，变形用外部载荷向量 p 表示，当外部载荷的作用方向与正位移方向相同时，其被认为是正的。变形也可以用抵抗变形的弹性力 ${\bf f}_{S}$ 表示，当弹性力的作用方向与正位移方向相反时，其被认为是正的。显然，根据静力学 $\mathbf{f}_{S}=\mathbf{p}$，式 (10-4) 可以修改为 在式 (10-4) 中，变形用外部载荷向量 p 表示，当外部载荷的作用方向与正位移方向相同时，其被认为是正的。变形也可以用抵抗变形的弹性力 ${\bf f}_{S}$ 表示，当弹性力的作用方向与正位移方向相反时，其被认为是正的。显然，根据静力学 $\mathbf{f}_{S}=\mathbf{p}$，式 (10-4) 可以修改为
$$ $$
\mathbf{v}=\widetilde{\mathbf{f}}\,\mathbf{f}_{S} \mathbf{v}=\widetilde{\mathbf{f}}\,\mathbf{f}_{S}\tag{10-5}
$$ $$
## Stiffness ## Stiffness
@ -79,13 +79,13 @@ FIGURE 10-2 Definition of stiffness influence coefficients.
The strain energy stored in any structure may be expressed conveniently in terms of either the flexibility or the stiffness matrix. The strain energy $U$ is equal to the work done in distorting the system; thus The strain energy stored in any structure may be expressed conveniently in terms of either the flexibility or the stiffness matrix. The strain energy $U$ is equal to the work done in distorting the system; thus
任何结构中储存的应变能都可以方便地用柔度矩阵或刚度矩阵来表示。应变能 $U$ 等于使系统变形所做的功；因此 任何结构中储存的应变能都可以方便地用柔度矩阵或刚度矩阵来表示。应变能 $U$ 等于使系统变形所做的功；因此
$$ $$
U=\frac{1}{2}\:\sum_{i=1}^{N}p_{i}\,v_{i}=\frac{1}{2}\:\mathbf{p}^{T}\,\mathbf{v} U=\frac{1}{2}\:\sum_{i=1}^{N}p_{i}\,v_{i}=\frac{1}{2}\:\mathbf{p}^{T}\,\mathbf{v}\tag{10-6}
$$ $$
where the $\frac{1}{2}$ factor results from the forces which increase linearly with the displacements, and $\mathbf{p}^{T}$ represents the transpose of $\mathbf{p}$ . By substituting Eq. (10-4) this becomes where the $\frac{1}{2}$ factor results from the forces which increase linearly with the displacements, and $\mathbf{p}^{T}$ represents the transpose of $\mathbf{p}$ . By substituting Eq. (10-4) this becomes
其中 $\frac{1}{2}$ 因子源于随位移线性增加的力，且 $\mathbf{p}^{T}$ 表示 $\mathbf{p}$ 的转置。将式 (10-4) 代入后，此式变为 其中 $\frac{1}{2}$ 因子源于随位移线性增加的力，且 $\mathbf{p}^{T}$ 表示 $\mathbf{p}$ 的转置。将式 (10-4) 代入后，此式变为
$$ $$
U=\frac{1}{2}\:\mathbf{p}^{T}\widetilde{\mathbf{f}}\,\mathbf{p} U=\frac{1}{2}\:\mathbf{p}^{T}\widetilde{\mathbf{f}}\,\mathbf{p}\tag{10-7}
$$ $$
Alternatively, transposing Eq. (10-6) and substituting Eq. (9-6) leads to the second strain-energy expression (note that $\mathbf{p}=\mathbf{f}_{S}$ ): Alternatively, transposing Eq. (10-6) and substituting Eq. (9-6) leads to the second strain-energy expression (note that $\mathbf{p}=\mathbf{f}_{S}$ ):
@ -94,13 +94,13 @@ $$
\mathbf{f}_{S}=\mathbf{k}\ \mathbf{v}\tag{9-6} \mathbf{f}_{S}=\mathbf{k}\ \mathbf{v}\tag{9-6}
$$ $$
$$ $$
U=\frac{1}{2}\mathbf{}\mathbf{v}^{T}\,\mathbf{k}\,\mathbf{v} U=\frac{1}{2}\mathbf{}\mathbf{v}^{T}\,\mathbf{k}\,\mathbf{v}\tag{10-8}
$$ $$
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 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
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 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
$$ $$
\mathbf{v}^{T}\,\mathbf{k}\,\mathbf{v}>0\qquad\qquad{\mathrm{and}}\qquad\qquad\mathbf{p}^{T}\,\widetilde{\mathbf{f}}\,\mathbf{p}>0 \mathbf{v}^{T}\,\mathbf{k}\,\mathbf{v}>0\qquad\qquad{\mathrm{and}}\qquad\qquad\mathbf{p}^{T}\,\widetilde{\mathbf{f}}\,\mathbf{p}>0\tag{10-9}
$$ $$
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. 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.
@ -116,7 +116,7 @@ $$
which upon comparison with Eq. (10-5) demonstrates that the flexibility matrix is the inverse of the stiffness matrix: which upon comparison with Eq. (10-5) demonstrates that the flexibility matrix is the inverse of the stiffness matrix:
这与式 (10-5) 比较后表明，柔度矩阵是刚度矩阵的逆： 这与式 (10-5) 比较后表明，柔度矩阵是刚度矩阵的逆：
$$ $$
\mathbf{k}^{-1}=\widetilde{\mathbf{f}} \mathbf{k}^{-1}=\widetilde{\mathbf{f}}\tag{10-10}
$$ $$
In practice, the evaluation of stiffness coefficients by direct application of the definition, as implied in Fig. 10-2, may be a tedious computational problem. In many cases, the most convenient procedure for obtaining the stiffness matrix is direct evaluation of the flexibility coefficients and inversion of the flexibility matrix. In practice, the evaluation of stiffness coefficients by direct application of the definition, as implied in Fig. 10-2, may be a tedious computational problem. In many cases, the most convenient procedure for obtaining the stiffness matrix is direct evaluation of the flexibility coefficients and inversion of the flexibility matrix.
@ -128,7 +128,7 @@ A property which is very important in structural-dynamics analysis can be derive
Case 1: Case 1:
$$ $$
\begin{array}{c}{Loads\ a:\ {W_{a a}=\frac{1}{2}\,\sum p_{i a}{v_{i a}}=\frac{1}{2}\,{\bf p}_{a}{}^{T}{\bf v}_{a}}}\\ Loads\ b:\ {{{ W}_{b b}+W_{a b}=\frac{1}{2}\,{\bf p}_{b}{}^{T}{\bf v}_{b}+{\bf p}_{a}{}^{T}{\bf v}_{b}}}\\ Total:\ {{W_{1}=W_{a a}+W_{b b}+W_{a b}=\frac{1}{2}\,{\bf p}_{a}{}^{T}{\bf v}_{a}+\frac{1}{2}\,{\bf p}_{b}{}^{T}{\bf v}_{b}+{\bf p}_{a}{}^{T}{\bf v}_{b}}}\end{array} \begin{array}{c}{Loads\ a:\ {W_{a a}=\frac{1}{2}\,\sum p_{i a}{v_{i a}}=\frac{1}{2}\,{\bf p}_{a}{}^{T}{\bf v}_{a}}}\\ Loads\ b:\ {{{ W}_{b b}+W_{a b}=\frac{1}{2}\,{\bf p}_{b}{}^{T}{\bf v}_{b}+{\bf p}_{a}{}^{T}{\bf v}_{b}}}\\ Total:\ {{W_{1}=W_{a a}+W_{b b}+W_{a b}=\frac{1}{2}\,{\bf p}_{a}{}^{T}{\bf v}_{a}+\frac{1}{2}\,{\bf p}_{b}{}^{T}{\bf v}_{b}+{\bf p}_{a}{}^{T}{\bf v}_{b}}}\end{array}\tag{10-11}
$$ $$
Note that the work done by loads $a$ during the application of loads $b$ is not multiplied by $\frac{1}{2}$ ; they act at their full value during the entire displacement $\mathbf{v}_{b}$ . Now if the loads are applied in reverse sequence, the work done is: Note that the work done by loads $a$ during the application of loads $b$ is not multiplied by $\frac{1}{2}$ ; they act at their full value during the entire displacement $\mathbf{v}_{b}$ . Now if the loads are applied in reverse sequence, the work done is:
@ -149,7 +149,7 @@ FIGURE 10-3 Two independent load systems and resulting deflections.
Case 2 : Case 2 :
$$ $$
\begin{array}{c}Loads\ b:\ {{W_{b b}=\frac{1}{2}\,{\bf p}_{b}^{\ T}{\bf v}_{b}}}\\ Loads\ a:\ {{W_{a a}+W_{b a}=\frac{1}{2}\,{\bf p}_{a}^{\ T}{\bf v}_{a}+{\bf p}_{b}^{\ T}{\bf v}_{a}}}\\ Total:\ {{W_{2}=W_{b b}+W_{a a}+W_{b a}=\frac{1}{2}\,{\bf p}_{b}^{\ T}{\bf v}_{b}+\frac{1}{2}\,{\bf p}_{a}^{\ T}{\bf v}_{a}+{\bf p}_{b}^{\ T}{\bf v}_{a}}}\end{array} \begin{array}{c}Loads\ b:\ {{W_{b b}=\frac{1}{2}\,{\bf p}_{b}^{\ T}{\bf v}_{b}}}\\ Loads\ a:\ {{W_{a a}+W_{b a}=\frac{1}{2}\,{\bf p}_{a}^{\ T}{\bf v}_{a}+{\bf p}_{b}^{\ T}{\bf v}_{a}}}\\ Total:\ {{W_{2}=W_{b b}+W_{a a}+W_{b a}=\frac{1}{2}\,{\bf p}_{b}^{\ T}{\bf v}_{b}+\frac{1}{2}\,{\bf p}_{a}^{\ T}{\bf v}_{a}+{\bf p}_{b}^{\ T}{\bf v}_{a}}}\end{array}\tag{10-12}
$$ $$
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 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
@ -686,21 +686,43 @@ In the preceding discussion, two different levels of approximation have been con
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 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 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
在前面的讨论中，已经考虑了两种不同程度的近似方法来评估质量、弹性刚度、几何刚度和外部载荷特性：(1) 一种基本方法，仅考虑结构的平移自由度；(2) 一种“一致”方法，该方法同时考虑旋转位移和平移位移。基本方法易于应用得多；不仅单元特性定义更简单，而且对于给定的结构组合，分析中需要考虑的坐标数量也少得多。原则上，一致方法应该会带来更高的结果精度，但实际上，改进通常很小。显然，在分析中，旋转自由度远不如平移项重要。一致方法的主要优点是，对结构响应的所有能量贡献都以一致的方式进行评估，这使得可以就振动频率的界限得出某些结论；然而，这种优势很少能抵消所需的额外工作。
当弹性刚度矩阵通过有限元方法或任何其他在矩阵中包含旋转自由度的程序制定时，基本的集中质量方法会带来一个特殊问题。如果所有其他特性的评估都排除了这些自由度，那么在运动方程可以写出之前，也有必要将它们从刚度矩阵中排除。
从刚度矩阵中消除这些不必要的自由度的过程称为静力凝聚。为了本次讨论的目的，假设旋转自由度和平移自由度已被分离，以便方程 (9-5) 可以写成划分形式。
$$ $$
\left[\mathbf{k}_{t t}\quad\mathbf{k}_{t\theta}\right]\;\left\{\mathbf{v}_{t}\atop\mathbf{v}_{\theta}\right\}=\left\{\mathbf{f}_{\mathrm{St}}\atop\mathbf{f}_{S\theta}\right\}=\left\{\begin{array}{l l}{\mathbf{f}_{\mathrm{St}}}\\ {\mathbf{f}_{\mathrm{S}}}\end{array}\right\}=\left\{\begin{array}{l l}{\mathbf{f}_{\mathrm{St}}}\\ {\mathbf{0}}\end{array}\right\} \begin{bmatrix}
\mathbf{k}_{tt} & \mathbf{k}_{t\theta} \\
\mathbf{k}_{\theta t} & \mathbf{k}_{\theta\theta}
\end{bmatrix}
\begin{Bmatrix}
\mathbf{v}_t \\
\mathbf{v}_{\theta}
\end{Bmatrix}
=
\begin{Bmatrix}
\mathbf{f}_{St} \\
\mathbf{f}_{S\theta}
\end{Bmatrix}
=
\begin{Bmatrix}
\mathbf{f}_{St} \\
0
\end{Bmatrix}
\tag{10-44}
$$ $$
where $\mathbf{v}_{t}$ represents the translations and $\mathbf{v}_{\theta}$ the rotations, with corresponding subscripts to identify the submatrices of stiffness coefficients. Now, if none of the other force vectors acting in the structure include any rotational components, it is evident that the elastic rotational forces also must vanish, that is, $\mathbf{f}_{S\theta}=\mathbf{0}$ . When this static constraint is introduced into Eq. (10-44), it is possible to express the rotational displacements in terms of the translations by means of the second submatrix equation, with the result where $\mathbf{v}_{t}$ represents the translations and $\mathbf{v}_{\theta}$ the rotations, with corresponding subscripts to identify the submatrices of stiffness coefficients. Now, if none of the other force vectors acting in the structure include any rotational components, it is evident that the elastic rotational forces also must vanish, that is, $\mathbf{f}_{S\theta}=\mathbf{0}$ . When this static constraint is introduced into Eq. (10-44), it is possible to express the rotational displacements in terms of the translations by means of the second submatrix equation, with the result
其中 $\mathbf{v}_{t}$ 代表平移，而 $\mathbf{v}_{\theta}$ 代表转动，相应的下标用于识别刚度系数的子矩阵。此时，如果作用在结构中的其他力矢量都不包含任何转动分量，那么很明显弹性转动力也必须消失，即 $\mathbf{f}_{S\theta}=\mathbf{0}$。当将此静力约束引入到方程 (10-44) 中时，可以通过第二个子矩阵方程以平移的形式表示转动位移，结果是
$$ $$
\mathbf{v}_{\theta}=-\mathbf{k}_{\theta\theta}^{\phantom{\theta}-1}\,\mathbf{k}_{\theta t}\,\mathbf{v}_{t} \mathbf{v}_{\theta}=-\mathbf{k}_{\theta\theta}^{\phantom{\theta}-1}\,\mathbf{k}_{\theta t}\,\mathbf{v}_{t}
$$ $$
Substituting this into the first of the submatrix equations of Eq. (10-44) leads to Substituting this into the first of the submatrix equations of Eq. (10-44) leads to
将此代入(10-44)式中的第一个子矩阵方程，得到
$$ $$
\left(\mathbf{k}_{t t}-\mathbf{k}_{t\theta}\mathbf{\Deltak}_{\theta\theta}^{\prime}^{-1}\mathbf{\Deltak}_{\theta t}\right)\mathbf{\Deltav}_{t}=\mathbf{f}_{\mathrm{St}} (\mathbf{k}_{tt} - \mathbf{k}_{t\theta} \mathbf{k}_{\theta\theta}^{-1} \mathbf{k}_{\theta t}) \mathbf{v}_t = \mathbf{f}_{St}
$$ $$
or or
@ -712,15 +734,18 @@ $$
where where
$$ $$
\mathbf{k}_{t}=\mathbf{k}_{t t}-\mathbf{k}_{t\theta}\mathbf{\Deltak}_{\theta\theta}^{\phantom{}}-1\mathbf{\Deltak}_{\theta t} \mathbf{k}_t = \mathbf{k}_{tt} - \mathbf{k}_{t\theta} \mathbf{k}_{\theta\theta}^{-1} \mathbf{k}_{\theta t} \tag{10-47}
$$ $$
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. 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.
是平动弹性刚度。该刚度矩阵适用于与其他基本属性表达式一起使用；换句话说，它是图10-2中所示的刚度矩阵类型。
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. 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 The stiffness submatrix associated with the rotational degrees of freedom of Example E10-1 is
示例 E10-3。为了演示静力凝聚过程的应用，将从示例 E10-1 中评估的刚度矩阵中消除两个旋转自由度。得到的凝聚刚度矩阵将只保留框架的平移自由度，因此将与示例 E10-2 中推导的集中质量矩阵兼容。
示例 E10-1 中与旋转自由度相关的刚度子矩阵是
$$ $$
\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] \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]
$$ $$
@ -732,13 +757,13 @@ $$
$$ $$
When this is used in Eq. (10-45), the rotational degrees of freedom can be expressed in terms of the translation: When this is used in Eq. (10-45), the rotational degrees of freedom can be expressed in terms of the translation:
当将其代入方程 (10-45) 时，旋转自由度可以表示为平动：
$$ $$
\left\{{\begin{array}{c}{{v_{2}}}\\ {{v_{3}}}\end{array}}\right\}=-{\frac{L}{32E I}}\,\left[{\begin{array}{r r}{{3}}&{{-1}}\\ {{}}&{{}}\\ {{-1}}&{{3}}\end{array}}\right]\,{\frac{2E I}{L^{3}}}\,\left\{{\begin{array}{c}{{3L}}\\ {{3L}}\end{array}}\right\}\,v_{1}=-{\frac{3}{8L}}\,\left\{{\begin{array}{c}{{1}}\\ {{1}}\end{array}}\right\}\,v_{1} \left\{{\begin{array}{c}{{v_{2}}}\\ {{v_{3}}}\end{array}}\right\}=-{\frac{L}{32E I}}\,\left[{\begin{array}{r r}{{3}}&{{-1}}\\ {{}}&{{}}\\ {{-1}}&{{3}}\end{array}}\right]\,{\frac{2E I}{L^{3}}}\,\left\{{\begin{array}{c}{{3L}}\\ {{3L}}\end{array}}\right\}\,v_{1}=-{\frac{3}{8L}}\,\left\{{\begin{array}{c}{{1}}\\ {{1}}\end{array}}\right\}\,v_{1}
$$ $$
The condensed stiffness given by Eq. (10-47) then is The condensed stiffness given by Eq. (10-47) then is
由式 (10-47) 给出的凝聚刚度则为
$$ $$
\mathbf{k}_{t}=\frac{2E I}{L^{3}}\left(12-<3L\quad3L>\left\{\begin{array}{l}{\frac{3}{8L}}\\ {\frac{3}{8L}}\end{array}\right\}\right)=\frac{2E I}{L^{3}}\:\frac{39}{4} \mathbf{k}_{t}=\frac{2E I}{L^{3}}\left(12-<3L\quad3L>\left\{\begin{array}{l}{\frac{3}{8L}}\\ {\frac{3}{8L}}\end{array}\right\}\right)=\frac{2E I}{L^{3}}\:\frac{39}{4}
$$ $$