# 数学代写|有限元方法代写Finite Element Method代考|Imposition of Boundary Conditions and the Condensed Equations

#### Doug I. Jones

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## 数学代写|有限元方法代写Finite Element Method代考|Imposition of Boundary Conditions and the Condensed Equations

At this step of the analysis, we must impose the particular boundary conditions of the problem being analyzed. The type of essential (also known as geometric) boundary conditions for a specific beam problem depends on
the nature of the geometric support. Table 5.2.1 contains a list of commonly used geometric supports for beams. The natural (also called force) boundary conditions involve the specification of generalized forces when the corresponding primary variables are not constrained. One must bear in mind that one and only one element of each of the following pairs must be specified at every node of the finite element mesh of a problem:
$$\left[w \text { or } V=-\frac{d}{d x}\left(E I \frac{d^2 w}{d x^2}\right)\right] \text { and }\left[\theta_x \equiv-\frac{d w}{d x} \text { or } M=-E I \frac{d^2 w}{d x^2}\right]$$
At an interior node, we impose the continuity of generalized displacements and balance of generalized forces as discussed in Eqs. (5.2.29) and (5.2.30).
There are two alternative ways to include the effect of a linear elastic spring (extensional as well as torsional). (1) Include it through the boundary condition for the appropriate degree of freedom (see Table 5.2.1). (2) Include the spring as another finite element, whose element equations are given by Eq. (3.3.2). In the former case, after assembly of the element equations, the secondary variable in the direction of the spring action is replaced by the negative of the spring constant times the associated primary variable. Let $V_I$ and $M_I$ denote the secondary variables (transverse force and bending moment, respectively) associated with the transverse and rotational degrees of freedom and $Q_0$ and $M_0$ be their specified values at global node $I$. Then, we have Table 5.2.1 Types of commonly used support conditions for beams and frames.

## 数学代写|有限元方法代写Finite Element Method代考|Finite Element Model

In this section we are interested in solving Eq. (4.3.1) to determine the velocity $u(y)$ due to an applied pressure gradient, $-d P / d x$. Equation (4.3.1) is a special case of the model equation (3.4.1), with the following correspondence:
$$f=-\frac{d P}{d x}=\text { constant } \equiv f_0, \quad a=\mu=\text { constant, } \quad c=0, \quad x=y$$
Therefore, the finite element equations in (3.4.32b) and (3.4.33) are valid for this problem:
$$\begin{gathered} \mathbf{K}^e \mathbf{u}^e=\mathbf{f}^e+\mathbf{Q}^e \ K_{i j}^e=\int_{y_a}^{y_b} \mu \frac{d \psi_i^e}{d y} \frac{d \psi_j^e}{d y} d y, \quad f_i^e=\int_{y_a}^{y_b}\left(-\frac{d P}{d x}\right) \psi_i^e d y \ Q_1^e=-\left.\left(\mu \frac{d u}{d y}\right)\right|_a, \quad Q_2^e=\left.\left(\mu \frac{d u}{d y}\right)\right|_b \end{gathered}$$

\begin{aligned} & V_I+k_{\mathrm{s}} w=Q_0 \text { or } V_I=-k_s w+Q_0 \text { for vertical spring } \ & M_I+\mu_s \theta_x=M_0 \text { or } M_I=-\mu_s \theta_x+M_0 \text { for torsional spring } \end{aligned}
For example, consider the case of a beam of length $L$ and constant bending stiffness $E I$, clamped at the left end and supported vertically by a linear elastic spring at the right end, and loaded by uniformly distributed force of intensity $q_0$, as shown in Fig. 5.2.10(a). Using one-element model of the beam, we obtain
$$\frac{2 E I}{L^3}\left[\begin{array}{rrrr} 6 & -3 L & -6 & -3 L \ -3 L & 2 L^2 & 3 L & L^2 \ -6 & 3 L & 6 & 3 L \ -3 L & L^2 & 3 L & 2 L^2 \end{array}\right]\left{\begin{array}{l} U_1 \ U_2 \ U_3 \ U_4 \end{array}\right}=\frac{q_0 L}{12}\left{\begin{array}{r} 6 \ -L \ 6 \ L \end{array}\right}+\left{\begin{array}{l} Q_1^1 \ Q_2^1 \ Q_3^1 \ Q_4^1 \end{array}\right}$$
The boundary conditions on the primary variables $\left(w, \theta_x\right)$ are
$$w(0)=\theta_x(0)=0 \rightarrow U_1=U_2=0$$
The boundary conditions on the secondary variables $(V, M)$ are
$$V(L)=-k_s w(L), M(L)=0 \rightarrow Q_3^1=-k_s U_3, Q_4^1=0$$

# 有限元方法代考

## 数学代写|有限元方法代写Finite Element Method代考|Imposition of Boundary Conditions and the Condensed Equations

$$\left[w \text { or } V=-\frac{d}{d x}\left(E I \frac{d^2 w}{d x^2}\right)\right] \text { and }\left[\theta_x \equiv-\frac{d w}{d x} \text { or } M=-E I \frac{d^2 w}{d x^2}\right]$$

## 数学代写|有限元方法代写Finite Element Method代考|Finite Element Model

$$f=-\frac{d P}{d x}=\text { constant } \equiv f_0, \quad a=\mu=\text { constant, } \quad c=0, \quad x=y$$

$$\begin{gathered} \mathbf{K}^e \mathbf{u}^e=\mathbf{f}^e+\mathbf{Q}^e \ K_{i j}^e=\int_{y_a}^{y_b} \mu \frac{d \psi_i^e}{d y} \frac{d \psi_j^e}{d y} d y, \quad f_i^e=\int_{y_a}^{y_b}\left(-\frac{d P}{d x}\right) \psi_i^e d y \ Q_1^e=-\left.\left(\mu \frac{d u}{d y}\right)\right|_a, \quad Q_2^e=\left.\left(\mu \frac{d u}{d y}\right)\right|_b \end{gathered}$$

\begin{aligned} & V_I+k_{\mathrm{s}} w=Q_0 \text { or } V_I=-k_s w+Q_0 \text { for vertical spring } \ & M_I+\mu_s \theta_x=M_0 \text { or } M_I=-\mu_s \theta_x+M_0 \text { for torsional spring } \end{aligned}

$$\frac{2 E I}{L^3}\left[\begin{array}{rrrr} 6 & -3 L & -6 & -3 L \ -3 L & 2 L^2 & 3 L & L^2 \ -6 & 3 L & 6 & 3 L \ -3 L & L^2 & 3 L & 2 L^2 \end{array}\right]\left{\begin{array}{l} U_1 \ U_2 \ U_3 \ U_4 \end{array}\right}=\frac{q_0 L}{12}\left{\begin{array}{r} 6 \ -L \ 6 \ L \end{array}\right}+\left{\begin{array}{l} Q_1^1 \ Q_2^1 \ Q_3^1 \ Q_4^1 \end{array}\right}$$

$$w(0)=\theta_x(0)=0 \rightarrow U_1=U_2=0$$

$$V(L)=-k_s w(L), M(L)=0 \rightarrow Q_3^1=-k_s U_3, Q_4^1=0$$

## 有限元方法代写

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