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# 数学代写|有限元方法代写finite differences method代考|Virtual Work

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## 数学代写|有限元代写Finite Element Method代考|Virtual Work

The term configuration means the simultaneous positions of all material points of a body. A body with specific geometric constraints takes different configurations under different loads. The set of configurations that satisfy the geometric constraints (e.g., geometric boundary conditions) of the system is called the set of admissible configurations (i.e., every configuration in the set corresponds to the solution of the problem for a particular set of loads on the system). Of all admissible configurations only one of them corresponds to the equilibrium configuration under a set of applied loads, and it is this configuration that also satisfies Newton’s second law. The admissible configurations for a fixed set of loads can be obtained from infinitesimal variations of the true configuration (i.e., infinitesimal movement of the material points). During such variations, the geometric constraints of the system are not violated, and all applied forces are fixed at their actual equilibrium values. When a mechanical system experiences such variations in its equilibrium configuration, it is said to undergo virtual displacements. These displacements need not have any relationship with the actual displacements. The displacements are called virtual because they are imagined to take place (i.e., hypothetical) with the actual loads acting at their fixed values.

For example, consider a beam fixed at $x=0$ and subjected to any arbitrary loading (e.g., distributed as well as point loads), as shown in Fig. 2.3.6. The possible geometric configurations the beam can take under the loads may be expressed in terms of the transverse deflection $w(x)$ and axial displacement $u(x)$. The support conditions require that
$$w(0)=0, \quad\left(-\frac{d w}{d x}\right)_{x=0}=0, \quad u(0)=0$$
These are called the geometric or displacement boundary conditions. Boundary conditions that involve specifying the forces applied on the beam are called force boundary conditions.

The set of all functions $w(x)$ and $u(x)$ that satisfy the geometric boundary conditions is the set of admissible configurations for this case. This set consists of pairs of elements $\left{\left(u_i, w_i\right)\right}$ of the form
$$\begin{gathered} u_1(x)=a_1 x, \quad w_1(x)=b_1 x^2 \ u_2(x)=a_1 x+a_2 x^2, \quad w_2(x)=b_1 x^2+b_2 x^3 \end{gathered}$$
where $a_i$ and $b_i$ are arbitrary constants. The pair $(u, w)$ that also satisfies, in addition to the geometric boundary conditions, the equilibrium equations and force boundary conditions (which require the precise nature of the applied loads) of the problem is the equilibrium solution. The virtual displacements, $\delta u(x)$ and $\delta w(x)$, must be necessarily of the form
$$\delta u_1=a_1 x, \delta w_1=b_1 x^2 ; \quad \delta u_2=a_1 x+a_2 x^2, \delta w_2=b_1 x^2+b_2 x^3$$
and so on, which satisfy the homogeneous form of the specified geometric boundary conditions:
$$\delta w(0)=0, \quad\left(\frac{d \delta w}{d x}\right)_{x=0}=0, \quad \delta u(0)=0$$

## 数学代写|有限元代写Finite Element Method代考|The Principle of Virtual Displacements

Consider the system of linear algebraic equations
\begin{aligned} & b_1=a_{11} x_1+a_{12} x_2+a_{13} x_3 \ & b_2=a_{21} x_1+a_{22} x_2+a_{23} x_3 \ & b_3=a_{31} x_1+a_{32} x_2+a_{33} x_3 \end{aligned}
We see that there are nine coefficients $a_{i j}, i, j=1,2,3$ relating the three coefficients $\left(b_1, b_2, b_3\right)$ to $\left(x_1, x_2, x_3\right)$. The form of these linear equations suggests writing down the coefficients $a_{i j}$ (jth components in the ith equation) in the rectangular array
$$\mathbf{A}=\left[\begin{array}{lll} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{array}\right]$$
This rectangular array $\mathbf{A}$ of numbers $a_{i j}$ is called a matrix, and the quantities $a_{i j}$ are called the elements of matrix $\mathbf{A}$.

If a matrix has $m$ rows and $n$ columns, we will say that is $m$ by $n(m \times n)$, the number of rows always being listed first. The element in the $i$ th row and $j$ th column of a matrix $\mathbf{A}$ is generally denoted by $a_{i j}$, and we will sometimes designate a matrix by $\mathbf{A}=[A]=\left[a_{i j}\right]$. A square matrix is one that has the same number of rows as columns. An $n \times n$ matrix is said to be of order $n$. The elements of a square matrix for which the row number and the column number are the same (that is, $a_{i i}$ for any fixed $i$ ) are called diagonal elements. A square matrix is said to be a diagonal matrix if all of the off-diagonal elements are zero. An identity matrix or its unit matrix, denoted by $\mathbf{I}=[I]$, is a diagonal matrix whose elements are all 1’s. Examples of diagonal and identity matrices are:
$$\left[\begin{array}{rrrr} 6 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 3 & 0 \ 0 & 0 & 0 & -2 \end{array}\right], \quad \mathbf{I}=\left[\begin{array}{llll} 1 & 0 & 0 & 0 \ 0 & 1 & 0 & 0 \ 0 & 0 & 1 & 0 \ 0 & 0 & 0 & 1 \end{array}\right]$$
If the matrix has only one row or one column, we will normally use only a single subscript to designate its elements. For example,
$$\mathbf{X}=\left{\begin{array}{l} x_1 \ x_2 \ x_3 \end{array}\right}, \quad \mathbf{Y}=\left{\begin{array}{lll} y_1 & y_2 & y_3 \end{array}\right}$$
denote a column matrix and a row matrix, respectively. Row and column matrices can be used to denote the components of a vector.

## 数学代写|有限元代写Finite Element Method代考|Virtual Work

$$w(0)=0, \quad\left(-\frac{d w}{d x}\right)_{x=0}=0, \quad u(0)=0$$

$$\begin{gathered} u_1(x)=a_1 x, \quad w_1(x)=b_1 x^2 \ u_2(x)=a_1 x+a_2 x^2, \quad w_2(x)=b_1 x^2+b_2 x^3 \end{gathered}$$

$$\delta u_1=a_1 x, \delta w_1=b_1 x^2 ; \quad \delta u_2=a_1 x+a_2 x^2, \delta w_2=b_1 x^2+b_2 x^3$$

$$\delta w(0)=0, \quad\left(\frac{d \delta w}{d x}\right)_{x=0}=0, \quad \delta u(0)=0$$

## 数学代写|有限元代写Finite Element Method代考|The Principle of Virtual Displacements



& b_1=a_{11} x_1+a_{12} x_2+a_{13} x_3 \
& b_2=a_{21} x_1+a_{22} x_2+a_{23} x_3 \
& b_3=a_{31} x_1+a_{32} x_2+a_{33} x_3




\ mathbf{一}=左[开始{数组}{微光}
A_ {11} & A_ {12} & A_ {13} \
A_ {21} & A_ {22} & A_ {23} \
A_ {31} & A_ {32} & A_ {33}





6 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 3 & 0 \
0 & 0 & 0 & -2
1 & 0 & 0 & 0 \
0 & 1 & 0 & 0 \
0 & 0 & 1 & 0

## MATLAB代写

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