add raw source
This commit is contained in:
김경종
+481
@@ -0,0 +1,481 @@
|
||||
<!-- source-page: 31 -->
|
||||
|
||||
from Eq. 1.2-9 that $k_{11} = F_{i}$ , $k_{21} = M_{i}$ , $k_{31} = F_{j}$ , and $k_{41} = M_{j}$ . These quantities are shown in Fig. 1.2-2c, all directed in the assumed positive sense. Let E = elastic modulus and I = moment of inertia of the cross-sectional area. We regard Fig. 1.2-2c as a beam cantilevered from its right end and loaded at its left end, and apply equations of beam theory and statics. Thus
|
||||
|
||||
$$
|
||||
w = 1 \text { at node } i \quad 1 = \frac {k _ {1 1} L ^ {3}}{3 E I} - \frac {k _ {2 1} L ^ {2}}{2 E I} \tag {1.2-10}
|
||||
$$
|
||||
|
||||
$$
|
||||
\theta = 0 \text { at node } i \quad 0 = \frac {k _ {1 1} L ^ {2}}{2 E I} - \frac {k _ {2 1} L}{E I} \tag {1.2-11}
|
||||
$$
|
||||
|
||||
$$
|
||||
\Sigma (\text { forces }) = 0 \quad 0 = k _ {1 1} + k _ {3 1} \tag {1.2-12}
|
||||
$$
|
||||
|
||||
$$
|
||||
\Sigma (\text { moments }) = 0 \quad 0 = k _ {2 1} + k _ {4 1} - k _ {1 1} L \tag {1.2-13}
|
||||
$$
|
||||
|
||||
Solution of these equations yields
|
||||
|
||||
$$
|
||||
k _ {1 1} = - k _ {3 1} = \frac {1 2 E I}{L ^ {3}} / \quad \text { and } \quad k _ {2 1} = k _ {4 1} = \frac {6 E I}{L ^ {2}} \tag {1.2-14}
|
||||
$$
|
||||
|
||||
Stiffness coefficients in columns 2, 3, and 4 of [k] are determined by applying similar arguments to Figs. 1.2-2d, 1.2-2e, and 1.2-2f in turn. The resulting stiffness matrix is exact, not approximate (provided that transverse shear deformation is ignored and deflections are small, as is commonly the case).
|
||||
|
||||
Having defined [k] in terms of E, I, and L, one is prepared to solve many problems of plane beams, such as that in Fig. 1.2-3. Generating the finite element model involves converting the distributed load into concentrated nodal loads. The conversion recipe for the loading shown in Fig. 1.2-3 is discussed in Section 4.3.
|
||||
|
||||
One can combine the stiffness matrices of Eqs. 1.2-3 and 1.2-9 to produce a 6 by 6 matrix [k] for an element that has two translational and one rotational d.o.f. at each end. Such elements can be used to analyze a plane frame. These elements are discussed in Sections 4.2 and 7.5.
|
||||
|
||||
# 1.3 ELEMENT ASSEMBLY AND SOLUTION FOR UNKNOWNS
|
||||
|
||||
$$
|
||||
c P _ {- 1} = 1 + 1 \frac {1}{2}
|
||||
$$
|
||||
|
||||
We consider a very simple example, which illustrates briefly how elements are put together to form a finite element structure and how a solution for displacements and stresses is obtained. These matters are discussed in detail in Chapter 2. In
|
||||
|
||||
|
||||
|
||||
<details>
|
||||
<summary>text_image</summary>
|
||||
|
||||
L1
|
||||
L2
|
||||
</details>
|
||||
|
||||
(a)
|
||||
|
||||
|
||||
|
||||
<details>
|
||||
<summary>text_image</summary>
|
||||
|
||||
qL₂/2
|
||||
qL₂²/12
|
||||
qL₂²/12
|
||||
qL₂/2
|
||||
①
|
||||
②
|
||||
L₁
|
||||
L₂
|
||||
</details>
|
||||
|
||||
(b)
|
||||
Figure 1.2-3. (a) Cantilever beam carrying a uniformly distributed load q (force per unit length). (b) A two-element model, showing nodal loads produced by q.
|
||||
|
||||
<!-- source-page: 32 -->
|
||||
|
||||
nonstructural problems the matrices have other names, but manipulations are the same.
|
||||
|
||||
Assembly. An axially loaded bar structure is shown in Fig. 1.3-1a; a two-element model of it is shown in Fig. 1.3-1b. The stiffness matrix of a typical element is given by Eq. 1.2-3. Stiffness coefficients associated with elements 1 and 2 in Fig. 1.3-1 are abbreviated as
|
||||
|
||||
$$
|
||||
k _ {1} = \frac {A _ {1} E _ {1}}{L _ {1}} \quad \text { and } \quad k _ {2} = \frac {A _ {2} E _ {2}}{L _ {2}} \tag {1.3-1}
|
||||
$$
|
||||
|
||||
The d.o.f. are axial displacements $u_{1}$ , $u_{2}$ , and $u_{3}$ , where 1, 2, and 3 are arbitrary labels assigned to identify the structure nodes. $^{2}$ Now imagine two hypothetical states: in the first, only element 1 is present; in the second, only element 2 is present. Thus the respective structure stiffness matrices would be
|
||||
|
||||
$$
|
||||
\left[ \begin{array}{c c c} u _ {1} & u _ {2} & u _ {3} \\ k _ {1} & - k _ {1} & 0 \\ - k _ {1} & k _ {1} & 0 \\ 0 & 0 & 0 \end{array} \right] \quad \text { and } \quad \left[ \begin{array}{c c c} u _ {1} & u _ {2} & u _ {3} \\ 0 & 0 & 0 \\ 0 & k _ {2} & - k _ {2} \\ 0 & - k _ {2} & k _ {2} \end{array} \right] \tag {1.3-2}
|
||||
$$
|
||||
|
||||
element 1 only
|
||||
|
||||
element 2 only
|
||||
|
||||
where column headings indicate the d.o.f. associated with the matrix coefficients. As elements are put together to form a finite element model, element matrices are put together to form the structure matrix. By direct addition of the preceding matrices, the structure stiffness matrix is
|
||||
|
||||
$$
|
||||
\left[ \begin{array}{l} F _ {0 1} \quad \text { Set } + i \cdot c \\ S + t u c t + v c \end{array} \right] \quad \left[ \begin{array}{l} \left[ \mathbf {K} \right] = \left[ \begin{array}{c c c} u _ {1} & u _ {2} & u _ {3} \\ k _ {1} & - k _ {1} & 0 \\ - k _ {1} & k _ {1} + k _ {2} & - k _ {2} \\ 0 & - k _ {2} & k _ {2} \\ \frac {0}{o} & \frac {1}{o} & \frac {1}{o} \end{array} \right] \end{array} \right. \tag {1.3-3}
|
||||
$$
|
||||
|
||||
One can easily check that each column of [K] represents an equilibrium set of nodal forces associated with activation of the corresponding d.o.f. This should be no surprise, as the method of activating each d.o.f. in turn can be used to generate either an element matrix [k] or a structure matrix [K].
|
||||
|
||||
|
||||
|
||||
<details>
|
||||
<summary>text_image</summary>
|
||||
|
||||
P ← A₁, E₁
|
||||
L₁ ← L₂
|
||||
2
|
||||
A₂, E₂
|
||||
3
|
||||
</details>
|
||||
|
||||
(a)
|
||||
|
||||
|
||||
|
||||
<details>
|
||||
<summary>text_image</summary>
|
||||
|
||||
u₁
|
||||
u₂
|
||||
u₃
|
||||
1
|
||||
2
|
||||
3
|
||||
①
|
||||
②
|
||||
L₁
|
||||
L₂
|
||||
</details>
|
||||
|
||||
(b)
|
||||
Figure 1.3-1. (a) An axially loaded bar. (b) Finite elements used to model the bar.
|
||||
|
||||
$^{2}$ Node labels i and j in Figs. 1.2-1 and 1.2-2 are called element or local node labels. Numbers would serve as well. They are used only temporarily, when element properties are defined. When elements are assembled, local labels are replaced by structure or global node labels (such as 1, 2, and 3 in Fig. 1.3-1). These matters are discussed in Section 2.7.
|
||||
|
||||
<!-- source-page: 33 -->
|
||||
|
||||
Matrix [K] of Eq. 1.3-3 is singular. Physically, this means that the structure in Fig. 1.3-1b is unsupported and can undergo a rigid-body translation. For solving a particular problem, at least one of the d.o.f. must be prescribed.
|
||||
|
||||
Solution for Unknowns. To impose the displacement boundary condition appropriate to Fig. 1.3-1, we must enforce the constraint $u_{3} = 0$ . This can be done by discarding row 3 and column 3 from [K] of Eq. 1.3-3. What remains is a 2 by 2 system to be solved for $u_{1}$ and $u_{2}$ . (This system can also be obtained by activating $u_{1}$ and $u_{2}$ . in turn, while $u_{3}$ is kept at zero, and calculating the loads that must be applied to d.o.f. $u_{1}$ and $u_{2}$ .) Thus the problem of Fig. 1.3-1a is described by the matrix equation
|
||||
|
||||
$$
|
||||
\left[ \begin{array}{c c} k _ {1} & - k _ {1} \\ - k _ {1} & k _ {1} + k _ {2} \end{array} \right] \left\{ \begin{array}{l} u _ {1} \\ u _ {2} \end{array} \right\} = \left\{ \begin{array}{c} - P \\ 0 \end{array} \right\} \tag {1.3-4}
|
||||
$$
|
||||
|
||||
The right-hand side indicates that node 1 carries a load P in the negative direction and that node 2 carries no externally applied load. The stiffness matrix in Eq. 1.3-4 is nonsingular. Therefore, Eqs. 1.3-4 can be solved for $u_{1}$ and $u_{2}$ . These results are
|
||||
|
||||
$$
|
||||
u _ {1} = - \frac {P}{k _ {1}} - \frac {P}{k _ {2}} \quad \text { and } \quad u _ {2} = - \frac {P}{k _ {2}} \tag {1.3-5}
|
||||
$$
|
||||
|
||||
Finally we convert displacements to strains and strains to stresses as follows. In elements 1 and 2, respectively, with $u_{3} = 0$ ,
|
||||
|
||||
$$
|
||||
\begin{array}{l} \text {Up to Now} \\ \text {App. of Direct,} \\ \text {Sail. Mesh,} \\ \text {(No diff.)} \end{array} \left\{ \begin{array}{l} \checkmark \sigma_ {1} = E _ {1} \epsilon_ {1} = E _ {1} \frac {u _ {2} - u _ {1}}{L _ {1}} = \frac {E _ {1}}{L _ {1}} \frac {P}{A _ {1} E _ {1} / L _ {1}} = \frac {P}{A _ {1}} \\ \checkmark \sigma_ {2} = E _ {2} \epsilon_ {2} = E _ {2} \frac {u _ {3} - u _ {2}}{L _ {2}} = \frac {E _ {2}}{L _ {2}} \frac {P}{A _ {2} E _ {2} / L _ {2}} = \frac {P}{A _ {2}} \end{array} \right. \tag {1.3-6a}
|
||||
$$
|
||||
|
||||
The preceding displacements and stresses are exact for this particular problem.
|
||||
|
||||
Notation. Our symbols for element and structure stiffness equations are, respectively,
|
||||
|
||||
$$
|
||||
[ \mathbf {k} ] \{\mathbf {d} \} = \{\overline {{{\mathbf {r}}}} \} \quad \text { and } \quad [ \mathbf {K} ] \{\mathbf {D} \} = \{\mathbf {R} \} \tag {1.3-7}
|
||||
$$
|
||||
|
||||
where [k] and [K] are stiffness matrices, {d} and {D} are vectors of nodal d.o.f., and {r} and {R} are vectors of nodal loads. Subsequently (in Section 2.6) it will be worthwhile to distinguish between loads applied to an element and loads applied by an element. Displacements {d} produce loads {r} consistent with static equilibrium and applied to an element. If we define {r} as loads equal and opposite to {r}, that is,
|
||||
|
||||
$$
|
||||
\{\mathbf {r} \} = - \{\vec {\mathbf {r}} \} \tag {1.3-8}
|
||||
$$
|
||||
|
||||
then $\{r\}$ represents loads applied to nodes by deformed elements. Similarly, in the assembled structure, $\{R\}$ represents loads applied to nodes, now by external sources rather than by deformed elements.
|
||||
|
||||
<!-- source-page: 34 -->
|
||||
|
||||
# 1.4 SUMMARY OF FINITE ELEMENT HISTORY
|
||||
|
||||
①Beginning in 1906, researchers suggested a “lattice analogy” for stress analysis [1.2–1.4]. The continuum was replaced by a regular pattern of elastic bars. Properties of the bars were chosen in a way that caused displacements of the joints to approximate displacements of points in the continuum. The method sought to capitalize on well-known methods of structural analysis.
|
||||
|
||||
Courant appears to have been the first to propose the finite element method as we know it today. In a 1941 mathematics lecture, published in 1943, he used the principle of stationary potential energy and piecewise polynomial interpolation over triangular subregions to study the Saint-Venant torsion problem [1.5]. Courant's work was ignored until engineers had independently developed it.
|
||||
|
||||
None of the foregoing work was of much practical value at the time because there were no computers available to generate and solve large sets of simultaneous algebraic equations. It is no accident that the development of finite elements coincided with major advances in digital computers and programming languages.
|
||||
|
||||
By 1953 engineers had written stiffness equations in matrix format and solved the equations with digital computers [1.6]. Most of this work took place in the aerospace industry. At the time, a large problem was one with 100 d.o.f. In 1953, at the Boeing Airplane Company, Turner suggested that triangular plane stress elements be used to model the skin of a delta wing [1.7]. This work, published almost simultaneously with similar work done in England [1.8,1.9], marks the beginning of widespread use of finite elements. Much of this early work went unrecognized because of company policies against publication [1.10].
|
||||
|
||||
The name “finite element method” was coined by Clough in 1960. The practical value of the method was soon obvious. New elements for stress analysis applications were developed, largely by intuition and physical argument. In 1963 the finite element method gained respectability when it was recognized as having a sound mathematical foundation: it can be regarded as the solution of a variational problem by minimization of a functional. Thus the method was seen as applicable to all field problems that can be cast in a variational form. Papers about the application of finite elements to problems of heat conduction and seepage flow appeared in 1965.
|
||||
|
||||
Large general-purpose finite element computer programs emerged during the late 1960s and early 1970s. Examples include ANSYS, ASKA, and NASTRAN. Each of these programs included several kinds of elements and could perform static, dynamic, and heat transfer analysis. Additional capabilities were soon added. Also added were preprocessors (for data input) and postprocessors (for results evaluation). These processors rely on graphics and make it easier, faster, and cheaper to do finite element analysis. Graphics development became intensive in the early 1980s as hardware and software for interactive graphics became available and affordable.
|
||||
|
||||
A general-purpose finite element program typically contains over 100,000 lines of code and usually resides on a mainframe or a superminicomputer. However, in the mid-1980s, adaptations of general-purpose programs began to appear on personal computers. Hundreds of analysis and analysis-related programs are now available, large and small, general and narrow, cheap and expensive, for lease or for purchase.
|
||||
|
||||
Ten papers about finite elements were published in 1961, 134 in 1966, and 844 in 1971. By 1976, two decades after engineering applications began, the cumulative
|
||||
|
||||
<!-- source-page: 35 -->
|
||||
|
||||
total of publications about finite elements exceeded 7000. By 1986, the total was about 20,000.
|
||||
|
||||
# 1.5 STRAIN-DISPLACEMENT RELATIONS
|
||||
|
||||
The relation between strain and displacement is a key ingredient in the formulation of finite elements for stress analysis problems. In the present section we consider general strain-displacement relations in Cartesian coordinates. Alternative special forms of the relations, such as forms used for solids of revolution and for plate bending, are stated where they are used.
|
||||
|
||||
In Fig. 1.5-1, perpendicular lines 01 and 02 are drawn on a plane sheet of material before the sheet is loaded. As a result of loading the lines become $0^{\prime}1^{\prime}$ and $0^{\prime}2^{\prime}$ . Displacements u and v are functions of the coordinates: $u = u(x, y)$ and $v = v(x, y)$ . We assume that displacement increments $^{3}$ such as $\overline{u} \cup_{a} dx$ are small in comparison with u and v. By definition, normal strain is the ratio of change in length to original length. Therefore,
|
||||
|
||||
$$
|
||||
\epsilon_ {x} = \frac {L _ {0 ^ {\prime} 2 ^ {\prime}} - L _ {0 2}}{L _ {0 2}} = \frac {[ d x + (u + \frac {d y}{d x} d x) - u ] - d x}{d x} = u _ {x} = \frac {d y}{d x} \tag {1.5-1}
|
||||
$$
|
||||
|
||||
A similar analysis yields the y-direction normal strain as
|
||||
|
||||
$$
|
||||
\epsilon_ {y} = v _ {, y} \text { 与 } \frac {2 0}{2 0} \tag {1.5-2}
|
||||
$$
|
||||
|
||||
Shear strain in the “engineering definition” is defined as the amount of change in a right angle. Because displacement increments are small, $\beta_{1} \approx \tan \beta_{1}$ and $\beta_{2} \approx \tan \beta_{2}$ . Therefore, the engineering shear strain is
|
||||
|
||||
$$
|
||||
\gamma_ {x y} = \underline {{{\beta_ {1}}}} + \underline {{{\beta_ {2}}}} = \frac {(u + u , y d y) - u}{d y} + \frac {(v + v , x d x) - v}{d x} = u, y + v, x \tag {1.5-3}
|
||||
$$
|
||||
|
||||
|
||||
|
||||
<details>
|
||||
<summary>text_image</summary>
|
||||
|
||||
y,v
|
||||
u + u,y dy
|
||||
v + v,y dy
|
||||
1'
|
||||
1
|
||||
β₁
|
||||
β₂
|
||||
2'
|
||||
0'
|
||||
v
|
||||
0
|
||||
2
|
||||
v + v₁,x dx
|
||||
x,u
|
||||
u
|
||||
dx
|
||||
u + u₁,x dx
|
||||
</details>
|
||||
|
||||
Figure 1.5-1. Displacement and distortion of differential lengths dx and dy.
|
||||
|
||||
$^{3}$ The notation $u_{,x}$ means $\partial u/\partial x$ . In general, a comma followed by a literal subscript indicates partial differentiation with respect to that subscript. Thus, for example, $\phi_{,xy} = \partial^{2}\phi/\partial x \partial y$ .
|
||||
|
||||
<!-- source-page: 36 -->
|
||||
|
||||
Collecting results, we have the two-dimensional strain-displacement relations:
|
||||
|
||||
$$
|
||||
\epsilon_ {x} = u _ {, x} \checkmark \epsilon_ {y} = v _ {, y} \checkmark \gamma_ {x y} = u _ {, y} + v _ {, x} \checkmark \tag {1.5-4}
|
||||
$$
|
||||
|
||||
In three dimensions, with w the z-direction displacement, a similar analysis yields the strains of Eqs. 1.5-4 and also
|
||||
|
||||
$$
|
||||
\epsilon_ {z} = w _ {, z} \checkmark \quad \gamma_ {y z} = v _ {, z} + w _ {, y} \checkmark \quad \gamma_ {z x} = u _ {, z} + w _ {, x} \checkmark \tag {1.5-5}
|
||||
$$
|
||||
|
||||
where now u, v, and w are all functions of x, y, and z. The foregoing relations can be stated in matrix operator form as follows. In two and three dimensions, respectively,
|
||||
|
||||
$$
|
||||
\left\{ \begin{array}{l} \dot {\epsilon_ {x}} \\ \epsilon_ {y} \\ \gamma_ {x y} \end{array} \right\} = \left[ \begin{array}{c c} \frac {\partial}{\partial x} & 0 \\ 0 & \frac {\partial}{\partial y} \\ \frac {\partial}{\partial y} & \frac {\partial}{\partial x} \end{array} \right] \left\{ \begin{array}{l} u \\ v \end{array} \right\} \quad \text { and } \quad \left\{ \begin{array}{l} \epsilon_ {x} \\ \epsilon_ {y} \\ \epsilon_ {z} \\ \gamma_ {x y} \\ \gamma_ {y z} \\ \gamma_ {z x} \end{array} \right\} = \left[ \begin{array}{c c c} \frac {\partial}{\partial x} & 0 & 0 \\ 0 & \frac {\partial}{\partial y} & 0 \\ 0 & 0 & \frac {\partial}{\partial z} \\ \frac {\partial}{\partial y} & \frac {\partial}{\partial x} & 0 \\ 0 & \frac {\partial}{\partial z} & \frac {\partial}{\partial y} \\ \frac {\partial}{\partial z} & 0 & \frac {\partial}{\partial x} \end{array} \right] \left\{ \begin{array}{l} u \\ v \\ w \end{array} \right\} \tag {1.5-6}
|
||||
$$
|
||||
|
||||
Plane Beams. We adopt standard beam theory in which transverse shear deformation is ignored and all deformations and strains are expressed in terms of the lateral displacement w. In Fig. 1.5-2, rotation $w_{,x}$ is assumed to be small, and plane cross sections are assumed to remain plane and normal to the deformed
|
||||
|
||||
|
||||
|
||||
<details>
|
||||
<summary>text_image</summary>
|
||||
|
||||
z,w
|
||||
dx
|
||||
P
|
||||
z
|
||||
C
|
||||
t/2
|
||||
x,u
|
||||
t/2
|
||||
</details>
|
||||
|
||||
(a)
|
||||
|
||||
|
||||
|
||||
<details>
|
||||
<summary>text_image</summary>
|
||||
|
||||
u = -zw_x
|
||||
w_x
|
||||
P
|
||||
z
|
||||
C
|
||||
w
|
||||
w_x
|
||||
dx
|
||||
</details>
|
||||
|
||||
(b)
|
||||
Figure 1.5-2. (a) Differential slice of a beam that lies along the x axis, before loading. (b) The same slice after loading. Transverse shear deformation is assumed to be negligible.
|
||||
|
||||
<!-- source-page: 37 -->
|
||||
|
||||
axis of the beam after it is bent. Therefore, axial displacement u is seen to be $u = -zw_{,x}$ . The negative sign means that with z and $w_{,x}$ both positive, displacement u is in the negative x direction. Accordingly, since $\epsilon_{x} = u_{,x}$ ,
|
||||
|
||||
$$
|
||||
\epsilon_ {x} = - z w _ {, x x} \tag {1.5-7}
|
||||
$$
|
||||
|
||||
where $w_{xx}$ is called the curvature of the beam.
|
||||
|
||||
Notation. A compact notation serves to symbolize any strain-displacement relation. Let $\{\epsilon\}$ represent the strains and $\{\mathbf{u}\}$ represent the displacements. Then the equation
|
||||
|
||||
$$
|
||||
\{\epsilon \} = [ \partial ] \{\mathbf {u} \} \tag {1.5-8}
|
||||
$$
|
||||
|
||||
represents either of Eqs. 1.5-6. It is only necessary to infer the proper size and proper contents of the three matrices $\{\epsilon\}$ , $[\partial]$ , and $\{\mathbf{u}\}$ . For beam bending, the matrices are all 1 by 1 (scalars):
|
||||
|
||||
$$
|
||||
\text { In beams: } \quad \epsilon_ {x} = [ \partial ] w, \quad \text { where } \quad [ \partial ] = - z \frac {d ^ {2}}{d x ^ {2}} \tag {1.5-9}
|
||||
$$
|
||||
|
||||
# 1.6 THEORY OF STRESS AND DEFORMATION
|
||||
|
||||
Fundamental concepts, definitions, and equations used in the analysis of stress and deformation are discussed in the discipline called theory of elasticity. These fundamentals are used in solving problems by both classical and finite element methods. Elasticity theory states conditions that must be met by an exact solution, and therefore helps us in judging the shortcomings or range of applicability of approximate solutions. For simplicity, the following summary is presented in Cartesian coordinates only, and the two-dimensional case is emphasized. More general arguments may be found in texts on theory of elasticity.
|
||||
|
||||
Equilibrium. Figure 1.6-1a shows a plane differential element (not a finite element!). We will develop equations stating that the differential element is in equilibrium under forces applied to it. Forces come from stresses on the edges and from body forces.
|
||||
|
||||
Body forces $F_{x}$ and $F_{y}$ are applied to all material points and have dimensions of force per unit volume. They can arise from gravity, acceleration, a magnetic field, and so on. They are considered positive when acting in positive coordinate directions. On each differential element of volume (dV = t dx dy, where t = thickness), $F_{x}$ and $F_{y}$ produce differential forces $F_{x}$ dV and $F_{y}$ dV.
|
||||
|
||||
In general, stresses and body forces are functions of the coordinates. Thus, for example, $\sigma_{x,x}$ is the rate of change of $\sigma_x$ with respect to $x$ , and $\sigma_{x,x} dx$ is the amount of change of $\sigma_x$ over distance $dx$ . For constant thickness $t$ , static equilibrium of forces in the $x$ direction, $\Sigma f_x = 0$ , requires that
|
||||
|
||||
$$
|
||||
\begin{array}{l} - \sigma_ {x} t d y - \tau_ {x y} t d x + (\sigma_ {x} + \sigma_ {x, x} d x) t d y \\ + \left(\tau_ {x y} + \tau_ {x y, y} d y\right) t d x + F _ {x} t d x d y = 0 \tag {1.6-1} \\ \end{array}
|
||||
$$
|
||||
|
||||
<!-- source-page: 38 -->
|
||||
|
||||
|
||||
|
||||
<details>
|
||||
<summary>text_image</summary>
|
||||
|
||||
σy + σy,y dy
|
||||
τxy + τxy,y dy
|
||||
Fy
|
||||
dy
|
||||
Fx
|
||||
τxy + τxy,x dx
|
||||
σx + σx,x dx
|
||||
dx
|
||||
τxy
|
||||
σy
|
||||
</details>
|
||||
|
||||
(a)
|
||||
|
||||
|
||||
|
||||
<details>
|
||||
<summary>text_image</summary>
|
||||
|
||||
y
|
||||
Normal
|
||||
Φy
|
||||
dy
|
||||
ds
|
||||
Φx
|
||||
σx
|
||||
τxy
|
||||
dx
|
||||
τxy
|
||||
σy
|
||||
x
|
||||
Structure boundary
|
||||
</details>
|
||||
|
||||
(b)
|
||||
Figure 1.6-1. (a) Stresses and body forces that act on a plane differential element of constant thickness. (b) Surface tractions $\Phi_{x}$ and $\Phi_{y}$ on an arbitrarily oriented edge in the xy plane.
|
||||
|
||||
There is a corresponding $y$ -direction equilibrium equation, $\Sigma f_y = 0$ . After simplification, the two equilibrium equations are
|
||||
|
||||
$$
|
||||
\sqrt {x} \text { direction: } \sigma_ {x, x} + \tau_ {x y, y} + F _ {x} = 0 \tag {1.6-2a}
|
||||
$$
|
||||
|
||||
$$
|
||||
\text { y direction: } \quad \tau_ {x y, x} + \sigma_ {y, y} + F _ {y} = 0 \tag {1.6-2b}
|
||||
$$
|
||||
|
||||
In many problems the effects of body forces are far less important than the effects of loads applied to the surface of the structure. Then $F_{x}$ and $F_{y}$ are set to zero. Note that Eqs. 1.6-2 are derived from equilibrium considerations alone: as no material properties are invoked, Eqs. 1.6-2 are applicable whether or not the body is linearly elastic.
|
||||
|
||||
Although derived for the case of static equilibrium, Eqs. 1.6-2 are also valid if acceleration is present. The right-hand sides of the Newton's law equations $\Sigma f_{x} = ma_{x}$ and $\Sigma f_{y} = ma_{y}$ become negligible in comparison with $\Sigma f_{x}$ and $\Sigma f_{y}$ as the element size shrinks to zero.
|
||||
|
||||
Compatibility. When a body is deformed without breaking, no cracks appear in stretching, no kinks appear in bending, and no part overlaps another. Stated more elegantly, this is the compatibility condition: the displacement field is continuous and single valued.
|
||||
|
||||
The compatibility equation $\epsilon_{x,yy} + \epsilon_{y,xx} = \gamma_{xy,xy}$ states the relation that exists among the strains if a plane displacement field is compatible. Most finite element methods are based on displacements rather than stresses. Thus, each element invokes a displacement field that is continuous and single valued. Therefore the compatibility equation is automatically satisfied, as one may see by substituting Eqs. 1.5-4 into it. (However, the displacement field may not satisfy the equilibrium equations. This point is discussed further in the following.)
|
||||
|
||||
Boundary Conditions. Boundary conditions consist of prescriptions of displacement and of stress. For example, in Fig. 1.1-2a the left edge does not move, so the displacement boundary condition along x = 0 is u = v = 0. Along the top edge, the stress boundary condition is $\sigma_{y} = \tau_{xy} = 0$ to the left of the zone that carries pressure p, and $\sigma_{y} = -p$ and $\tau_{xy} = 0$ beneath pressure p. On the right
|
||||
|
||||
<!-- source-page: 39 -->
|
||||
|
||||
edge, $\sigma_{x} = \tau_{xy} = 0$ . Along the curved lower edge, surface tractions $\Phi_x$ and $\Phi_y$ are zero. Surface tractions are related to stresses in the manner now described.
|
||||
|
||||
Surface tractions $\Phi_x$ and $\Phi_y$ in Fig. 1.6-1b have units of stress. They are applied on a boundary (in contrast to body forces, which act throughout a volume). Tractions are $x$ - and $y$ -direction force increments divided by the boundary area increment $dA$ on which they act. In Fig. 1.6-1b, $dA = t ds$ , where $t =$ thickness. Note that $dA$ is not either of the projected areas $t dx$ or $t dy$ . Equilibrium of $x$ - and $y$ -direction forces in Fig. 1.6-1b requires that $\Phi_x t ds = \sigma_x t dy + \tau_{xy} t dx$ and $\Phi_y t ds = \tau_{xy} t dy + \sigma_y t dx$ . But $dy / ds = \ell$ and $dx / ds = m$ , where $\ell$ and $m$ are direction cosines of the outward normal to the boundary. Thus
|
||||
|
||||
$$
|
||||
x \text { direction: } \Phi_ {x} = \ell \sigma_ {x} + m \tau_ {x y} \tag {1.6-3a}
|
||||
$$
|
||||
|
||||
$$
|
||||
y \text { direction: } \Phi_ {y} = \ell \tau_ {x y} + m \sigma_ {y} \tag {1.6-3b}
|
||||
$$
|
||||
|
||||
Equations 1.6-3 define a relation among stresses at an arbitrarily curved edge when tractions $\Phi_{x}$ and $\Phi_{y}$ are prescribed along that edge. Like Eqs. 1.6-2, Eqs. 1.6-3 do not require that the body be linearly elastic. One can check that Eqs. 1.6-3 yield the correct stress boundary conditions along the top and right edges of Fig. 1.1-2a.
|
||||
|
||||
Three Dimensions. In solids, arguments analogous to those preceding lead to analogous results. The equilibrium and surface traction equations are, respectively,
|
||||
|
||||
$$
|
||||
\sigma_ {x, x} + \tau_ {x y, y} + \tau_ {z x, z} + F _ {x} = 0 \quad \Phi_ {x} = \ell \sigma_ {x} + m \tau_ {x y} + n \tau_ {z x}
|
||||
$$
|
||||
|
||||
$$
|
||||
\tau_ {x y, x} + \sigma_ {y, y} + \tau_ {y z, z} + F _ {y} = 0 \quad \text { and } \quad \Phi_ {y} = \ell \tau_ {x y} + m \sigma_ {y} + n \tau_ {y z} \tag {1.6-4}
|
||||
$$
|
||||
|
||||
$$
|
||||
\tau_ {z x, x} + \tau_ {y z, y} + \sigma_ {z, z} + F _ {z} = 0 \quad \Phi_ {z} = \ell \tau_ {z x} + m \tau_ {y z} + n \sigma_ {z}
|
||||
$$
|
||||
|
||||
where $\ell$ , m, and n are direction cosines of an outward normal to the surface. There are six compatibility equations that relate the six strains used in solids. They are not presented here.
|
||||
|
||||
We observe that if stresses $\{\sigma\}$ are arrayed in the order used in Section 1.7, then the notation of Eq. 1.5-8 permits us to write the equilibrium equations in either two or three dimensions as
|
||||
|
||||
$$
|
||||
[ \partial ] ^ {T} \{\boldsymbol {\sigma} \} + \{\mathbf {F} \} = \{\mathbf {0} \} \tag {1.6-5}
|
||||
$$
|
||||
|
||||
where $[\partial]$ is given by Eqs. 1.5-6 and $\{\mathbf{F}\}$ is $\lfloor F_x\quad F_y\rfloor^T$ or $\lfloor F_x\quad F_y\quad F_z\rfloor^T$ .
|
||||
|
||||
Remarks. Consider a linearly elastic body. If one finds a stress field or a displacement field that simultaneously satisfies equilibrium, compatibility, and boundary conditions, then one has found a solution to the problem posed. The solution is both unique and exact within the assumptions made (such as linearity and homogeneity).
|
||||
|
||||
How do these observations relate to finite element analysis? If elements are based on polynomial displacement fields, then compatibility prevails within elements. Suitably chosen displacement fields also provide compatibility between elements and satisfy displacement boundary conditions. The differential equations
|
||||
|
||||
<!-- source-page: 40 -->
|
||||
|
||||
of equilibrium and boundary conditions on stress are satisfied only approximately. The approximation improves as more elements are used and, barring computational difficulties, the exact solution is achieved in the limit of an infinitely refined mesh.
|
||||
|
||||
Some finite elements are incompatible: adjacent elements can overlap or gap apart between nodes. Compatibility is then satisfied only within elements and at node points. Thus there is only approximate satisfaction of equilibrium, stress boundary conditions, and compatibility. If interelement compatibility tends to be restored as more elements are used, it is still possible to converge toward the exact solution as the mesh is refined. (Incompatible elements are subsequently treated in more detail.)
|
||||
|
||||
Other finite element methods are based on fields other than displacement. A stress field model may satisfy equilibrium equations a priori. Mesh refinement would then yield a better approximation of compatibility conditions.
|
||||
|
||||
# 1.7 STRESS-STRAIN-TEMPERATURE RELATIONS
|
||||
|
||||
The finite element method deals easily with rather general material properties and with both thermal and mechanical loading. We therefore devote this entire section to elastic properties and thermal relations.
|
||||
|
||||
General. The stress vector $\{\sigma\}$ and the strain vector $\{\epsilon\}$ are, respectively,
|
||||
|
||||
$$
|
||||
\{\boldsymbol {\sigma} \} = \left\lfloor \sigma_ {x} \quad \sigma_ {y} \quad \sigma_ {z} \quad \tau_ {x y} \quad \tau_ {y z} \quad \tau_ {z x} \right] ^ {T} \tag {1.7-1a}
|
||||
$$
|
||||
|
||||
and
|
||||
|
||||
$$
|
||||
\{\epsilon \} = \left\lfloor \epsilon_ {x} \quad \epsilon_ {y} \quad \epsilon_ {z} \quad \gamma_ {x y} \quad \gamma_ {y z} \quad \gamma_ {z x} \right\rfloor^ {T} \tag {1.7-1b}
|
||||
$$
|
||||
|
||||
Ignoring the effect of temperature change, which will be treated subsequently, we symbolize the isothermal stress–strain relation as
|
||||
|
||||
$$
|
||||
\{\epsilon \} = [ \mathrm{C} ] \{\sigma \} \quad \text { or as } \quad \{\sigma \} = [ \mathrm{E} ] \{\epsilon \} \tag {1.7-2}
|
||||
$$
|
||||
|
||||
where [C] is a symmetric matrix of material compliances, [E] is a symmetric matrix of material stiffnesses, and $[\mathbf{E}] = [\mathbf{C}]^{-1}$ (examples follow). In the most general case of anisotropy, [C] and [E] each contain 21 independent coefficients.
|
||||
|
||||
An orthotropic material is an anisotropic material that displays extreme values of stiffness in mutually perpendicular directions. These directions are called principal directions of the material. An example is wood cut from a log, which is stiffest in the axial direction, least stiff in the circumferential direction, and of intermediate stiffness in the radial direction. If x, y, and z are principal directions, then normal stresses $\sigma_{x}$ , $\sigma_{y}$ , and $\sigma_{z}$ are independent of shear strains $\gamma_{xy}$ , $\gamma_{yz}$ , and $\gamma_{zy}$ . [E] for an orthotropic material contains only nine independent coefficients.
|
||||
|
||||
Equations 1.7-2 express Hooke's law: stress is directly proportional to strain. This rule is an approximation limited to small strains and certain materials.
|
||||
Reference in New Issue
Block a user