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3.10. Consider the structural model shown. Determine the eigenvalue problem from which the critical load can be calculated. Use the direct method and the variational method to obtain the governing equations.
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P k u1 Rigid bar Frictionless hinges u2 Spring stiffness k Rigid bar kr = kL²
3.11. Establish the eigenproblem governing the stability of the system shown.
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P W Rigid bars, frictionless hinges 2k kr = kL² Spring stiffness k Spring stiffness k
3.12. The column structure in Exercise 3.11 is initially at rest under the constant force P (where P is below the critical load) when suddenly the force W is applied. Establish the governing equations of equilibrium. Assume the springs to be massless and that the bars have mass m per unit length.
3.13. Consider the analysis in Example 3.9. Assume \theta = \phi e^{-\lambda t} and \mathbf{Q} = \mathbf{0} and establish an eigenproblem corresponding to \lambda, \phi .
3.14. Consider the wall of three homogeneous slabs in Exercise 3.2. Formulate the heat transfer equations for a transient analysis in which the initial temperature distribution is given and suddenly \theta_{1} is changed to \theta_{1}^{\mathrm{new}} . Then assume \theta = \phi e^{-\lambda t} and \mathbf{Q} = \mathbf{0} , and establish an eigenproblem corresponding to \lambda, \phi . Assume that, for a unit cross-sectional area, each slab has a total heat capacity of c , and that for each slab the heat capacity can be lumped to the faces of the slab.
3.3 SOLUTION OF CONTINUOUS-SYSTEM MATHEMATICAL MODELS
The basic steps in the solution of a continuous-system mathematical model are quite similar to those employed in the solution of a lumped-parameter model (see Section 3.2). However, instead of dealing with discrete elements, we focus attention on typical differential elements with the objective of obtaining differential equations that express the element equilibrium requirements, constitutive relations, and element interconnectivity requirements. These differential equations must hold throughout the domain of the system, and before the solution can be calculated they must be supplemented by boundary conditions and, in dynamic analysis, also by initial conditions.
As in the solution of discrete models, two different approaches can be followed to generate the system-governing differential equations: the direct method and the variational method. We discuss both approaches in this section (see also R. Courant and D. Hilbert [A]) and illustrate the variational procedure in some detail because, as introduced in Section 3.3.4, this approach can be regarded as the basis of the finite element method.
3.3.1 Differential Formulation
In the differential formulation we establish the equilibrium and constitutive requirements of typical differential elements in terms of state variables. These considerations lead to a system of differential equations in the state variables, and it is possible that all compatibility requirements (i.e., the interconnectivity requirements of the differential elements) are already contained in these differential equations (e.g., by the mere fact that the solution is to be continuous). However, in general, the equations must be supplemented by additional differential equations that impose appropriate constraints on the state variables in order that all compatibility requirements be satisfied. Finally, to complete the formulation of the problem, all boundary conditions, and in a dynamic analysis the initial conditions, are stated.
For purposes of mathematical analysis it is expedient to classify problem-governing differential equations. Consider the second-order general partial differential equation in the domain x, y,
A (x, y) \frac {\partial^ {2} u}{\partial x ^ {2}} + 2 B (x, y) \frac {\partial^ {2} u}{\partial x \partial y} + C (x, y) \frac {\partial^ {2} u}{\partial y ^ {2}} = \phi \left(x, y, u, \frac {\partial u}{\partial x}, \frac {\partial u}{\partial y}\right) \tag {3.6}
where u is the unknown state variable. Depending on the coefficients in (3.6) the differential equation is elliptic, parabolic or hyperbolic:
B ^ {2} - A C \left\{ \begin{array}{l l} < 0 & \text { elliptic } \\ = 0 & \text { parabolic } \\ > 0 & \text { hyperbolic } \end{array} \right.
This classification is established when solving (3.6) using the method of characteristics because it is then observed that the character of the solutions is distinctly different for the three categories of equations. These differences are also apparent when the differential equations are identified with the different physical problems that they govern. In their simplest form the three types of equations can be identified with the Laplace equation, the heat conduction equation, and the wave equation, respectively. We demonstrate how these equations arise in the solution of physical problems by the following examples.
EXAMPLE 3.15: The idealized dam shown in Fig. E3.15 stands on permeable soil. Formulate the differential equation governing the steady-state seepage of water through the soil and give the corresponding boundary conditions.
For a typical element of widths dx and dy (and unit thickness), the total flow into the element must be equal to the total flow out of the element. Hence we have
\left(q \mid_ {y} - q \mid_ {y + d y}\right) d x + \left(q \mid_ {x} - q \mid_ {x + d x}\right) d y = 0
or -\frac{\partial q_y}{\partial y} dy dx - \frac{\partial q_x}{\partial x} dx dy = 0 (a)
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Impermeable dam h₁ h₂ L Flow of water Permeable soil x y x Impermeable rock
(a) Idealization of dam on soil and rock
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q|y+dy dx dy q|x q|x+dx q|y
(b) Differential element of soil
Figure E3.15 Two-dimensional seepage problem
Using Darcy's law, the flow is given in terms of the total potential \phi ,
q _ {x} = - k \frac {\partial \phi}{\partial x}; \quad q _ {y} = - k \frac {\partial \phi}{\partial y} \tag {b}
where we assume a uniform permeability k . Substituting from (b) into (a), we obtain the Laplace equation
k \left(\frac {\partial^ {2} \phi}{\partial x ^ {2}} + \frac {\partial^ {2} \phi}{\partial y ^ {2}}\right) = 0 \tag {c}
It may be noted that this same equation is also obtained in heat transfer analysis and in the solution of electrostatic potential and other field problems (see Chapter 7).
The boundary conditions are no-flow boundary conditions in the soil at x = -\infty and x = +\infty ,
\left. \frac {\partial \phi}{\partial x} \right| _ {x = - \infty} = 0; \quad \left. \frac {\partial \phi}{\partial x} \right| _ {x = + \infty} = 0 \tag {d}
at the rock-soil interface,
\left. \frac {\partial \phi}{\partial y} \right| _ {y = 0} = 0 \tag {e}
and at the dam-soil interface,
\frac {\partial \phi}{\partial y} (x, L) = 0 \quad \text { for } - \frac {h}{2} \leq x \leq + \frac {h}{2} \tag {f}
In addition, the total potential is prescribed at the water-soil interface,
\left. \phi (x, L) \right| _ {x < - (h / 2)} = h _ {1}; \quad \left. \phi (x, L) \right| _ {x > (h / 2)} = h _ {2} \tag {g}
The differential equation in (c) and the boundary conditions in (d) to (g) define the seepage flow steady-state response.
EXAMPLE 3.16: The very long slab shown in Fig. E3.16 is at a constant initial temperature \theta_{i} when the surface at x = 0 is suddenly subjected to a constant uniform heat flow input. The surface at x = L of the slab is kept at the temperature \theta_{i} , and the surfaces parallel to the x, z plane are insulated. Assuming one-dimensional heat flow conditions, show that the problem-governing differential equation is the heat conduction equation
k \frac {\partial^ {2} \theta}{\partial x ^ {2}} = \rho c \frac {\partial \theta}{\partial t}
where the parameters are defined in Fig. E3.16, and the temperature \theta is the state variable. State also the boundary and initial conditions.
We consider a typical differential element of the slab [see Fig. E3.16(b)]. The element equilibrium requirement is that the net heat flow input to the element must equal the rate of heat stored in the element. Thus
\left. q A \right| _ {x} - \left(q A \right| _ {x} + \left. A \frac {\partial q}{\partial x} \right| _ {x} d x) = \rho A c \frac {\partial \theta}{\partial t} \Bigg | _ {x} d x \tag {a}
The constitutive relation is given by Fourier's law of heat conduction
q = - k \frac {\partial \theta}{\partial x} \tag {b}
Figure E3.16 One-dimensional heat conduction problem
Substituting from (b) into (a) we obtain
k \frac {\partial^ {2} \theta}{\partial x ^ {2}} = \rho c \frac {\partial \theta}{\partial t} \tag {c}
In this case the element interconnectivity requirements are contained in the assumption that the temperature \theta be a continuous function of x and no additional compatibility conditions are applicable.
The boundary conditions are
\frac {\partial \theta}{\partial x} (0, t) = - \frac {q _ {0} (t)}{k} \quad ; \quad t > 0 \tag {d}
\theta (L, t) = \theta_ {i}
and the initial condition is \theta (x,0) = \theta_{i} (e)
The formulation of the problem is now complete, and the solution of (c) subject to the boundary and initial conditions in (d) and (e) yields the temperature response of the slab.
EXAMPLE 3.17: The rod shown in Fig. E3.17 is initially at rest when a load R(t) is suddenly applied at its free end. Show that the problem-governing differential equation is the wave equation
\frac {\partial^ {2} u}{\partial x ^ {2}} = \frac {1}{c ^ {2}} \frac {\partial^ {2} u}{\partial t ^ {2}}; \quad c = \sqrt {\frac {E}{\rho}}
where the variables are defined in Fig. E3.17 and the displacement of the rod, u, is the state variable. Also state the boundary and initial conditions.
The element force equilibrium requirements of a typical differential element are, using d'Alembert's principle,
\sigma A \left| _ {x} + A \frac {\partial \sigma}{\partial x} \right| _ {x} d x - \sigma A \left| _ {x} = \rho A \frac {\partial^ {2} u}{\partial t ^ {2}} \right| _ {x} d x \tag {a}
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u(x, t) x R₀
(a) Geometry of rod
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| t | R(t) |
|---|---|
| 0 | R₀ |
| 1 | R₀ |
Young's modulus E
Mass density ρ
Cross-sectional area A
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σA|ₓ σA|ₓ + dₓ dₓ
(b) Differential element
Figure E3.17 Rod subjected to step load
The constitutive relation is
\sigma = E \frac {\partial u}{\partial x} \tag {b}
Combining (a) and (b) we obtain
\frac {\partial^ {2} u}{\partial x ^ {2}} = \frac {1}{c ^ {2}} \frac {\partial^ {2} u}{\partial t ^ {2}} \tag {c}
The element interconnectivity requirements are satisfied because we assume that the displacement u is continuous, and no additional compatibility conditions are applicable.
The boundary conditions are
u (0, t) = 0
E A \frac {\partial u}{\partial x} (L, t) = R _ {0} \quad ; \quad t > 0 \tag {d}
and the initial conditions are
u (x, 0) = 0
(e)
\frac {\partial u}{\partial t} (x, 0) = 0
With the conditions in (d) and (e) the formulation of the problem is complete, and (c) can be solved for the displacement response of the rod.
Although we considered in these examples specific problems that are governed by elliptic, parabolic, and hyperbolic differential equations, the problem formulations illustrate in a quite general way some basic characteristics. In elliptic problems (see Exam-
ple 3.15) the values of the unknown state variables (or their normal derivatives) are given on the boundary. These problems are for this reason also called boundary value problems, where we should note that the solution at a general interior point depends on the data at every point of the boundary. A change in only one boundary value affects the complete solution; for instance, in Example 3.15 the complete solution for \phi depends on the actual value of h_1 . Elliptic differential equations generally govern the steady-state response of systems.
Comparing the governing differential equations given in Examples 3.15 to 3.17 it is noted that in contrast to the elliptic equation, the parabolic and hyperbolic equations (Examples 3.16 and 3.17, respectively) include time as an independent variable and thus define propagation problems. These problems are also called initial value problems because the solution depends on the initial conditions. We may note that analogous to the derivation of the dynamic equilibrium equations of lumped-parameter models, the governing differential equations of propagation problems are obtained from the steady-state equations by including the “resistance to change” (inertia) of the differential elements. Conversely, the parabolic and hyperbolic differential equations in Examples 3.16 and 3.17 would become elliptic equations if the time-dependent terms were neglected. In this way the initial value problems would be converted to boundary value problems with steady-state solutions.
We stated earlier that the solution of a boundary value problem depends on the data at all points of the boundary. Here lies a significant difference in the analysis of a propagation problem, namely, considering propagation problems the solution at an interior point may depend only on the boundary conditions of part of the boundary and the initial conditions over part of the interior domain.
3.3.2 Variational Formulations
The variational approach of establishing the governing equilibrium equations of a system was already introduced as an alternative to the direct approach when we discussed the analysis of discrete systems (see Section 3.2.1). As described, the essence of the approach is to calculate the total potential \Pi of the system and to invoke the stationarity of \Pi , i.e., \delta\Pi = 0 , with respect to the state variables. We pointed out that the variational technique can be effective in the analysis of discrete systems; however, we shall now observe that the variational approach provides a particularly powerful mechanism for the analysis of continuous systems. The main reason for this effectiveness lies in the way by which some boundary conditions (namely, the natural boundary conditions defined below) can be generated and taken into account when using the variational approach.
To demonstrate the variational formulation in the following examples, we assume that the total potential \Pi is given and defer the description of how an appropriate \Pi can be determined until after the presentation of the examples.
The total potential \Pi is also called the functional of the problem. Assume that in the functional the highest derivative of a state variable (with respect to a space coordinate) is of order m; i.e., the operator contains at most mth-order derivatives. We call such a problem a C^{m-1} variational problem. Considering the boundary conditions of the problem, we identify two classes of boundary conditions, called essential and natural boundary conditions.
The essential boundary conditions are also called geometric boundary conditions because in structural mechanics the essential boundary conditions correspond to prescribed
displacements and rotations. The order of the derivatives in the essential boundary conditions is, in a C^{m-1} problem, at most m - 1 .
The second class of boundary conditions, namely, the natural boundary conditions, are also called force boundary conditions because in structural mechanics the natural boundary conditions correspond to prescribed boundary forces and moments. The highest derivatives in these boundary conditions are of order m to 2m - 1.
We will see later that this classification of variational problems and associated boundary conditions is most useful in the design of numerical solutions.
In the variational formulations we will use the variational symbol \delta , already briefly employed in (3.1). Let us recall some important operational properties of this symbol; for more details, see, for example, R. Courant and D. Hilbert [A]. Assume that a function F for a given value of x depends on v (the state variable), dv/dx, \ldots, d^p v/dx^p , where p = 1, 2, \ldots . Then the first variation of F is defined as
\delta F = \frac {\partial F}{\partial v} \delta v + \frac {\partial F}{\partial (d v / d x)} \delta \left(\frac {d v}{d x}\right) + \dots + \frac {\partial F}{\partial \left(d ^ {p} v / d x ^ {p}\right)} \delta \left(\frac {d ^ {p} v}{d x ^ {p}}\right) \tag {3.7a}
This expression is explained as follows. We associate with v(x) a function \epsilon \eta(x) where \epsilon is a constant (independent of x) and \eta(x) is an arbitrary but sufficiently smooth function that is zero at and corresponding to the essential boundary conditions. We call \eta(x) a variation in v, that is \eta(x) = \delta v(x) [and of course \epsilon \eta(x) is then also a variation in v] and also have for the required derivatives
\frac {d ^ {n} \eta}{d x ^ {n}} = \frac {d ^ {n} \delta v}{d x ^ {n}} = \delta \left(\frac {d ^ {n} v}{d x ^ {n}}\right)
that is, the variation of a derivative of v is equal to the derivative of the variation in v. The expression (3.7a) then follows from evaluating
\delta F = \lim _ {\epsilon \rightarrow 0} \frac {F \left[ v + \epsilon \eta , \frac {d (v + \epsilon \eta)}{d x} , \dots , \frac {d ^ {p} (v + \epsilon \eta)}{d x ^ {p}} \right] - F \left(v , \frac {d v}{d x} , \dots , \frac {d ^ {p} v}{d x ^ {p}}\right)}{\epsilon} \tag {3.7b}
Considering (3.7a) we note that the expression for \delta F looks like the expression for the total differential dF; that is, the variational operator \delta acts like the differential operator with respect to the variables v, dv/dx, \ldots , d^{p}v/dx^{p} . These equations can be extended to multiple functions and state variables, and we find that the laws of variations of sums, products, and so on, are completely analogous to the corresponding laws of differentiation. For example, let F and Q be two functions possibly dependent on different state variables; then
\delta (F + Q) = \delta F + \delta Q; \quad \delta (F Q) = (\delta F) Q + F (\delta Q); \quad \delta (F) ^ {n} = n (F) ^ {n - 1} \delta F
In our applications the functions usually appear within an integral sign; and so, for example, we also use
\delta \int F (x) d x = \int \delta F (x) d x
We shall employ these rules extensively in the variational derivations and will use one important condition (which corresponds to the properties of \eta stated earlier), namely, that the variations of the state variables and of their (m - 1) st derivatives must be zero at and
corresponding to the essential boundary conditions, but otherwise the variations can be arbitrary.
Consider the following examples.
EXAMPLE 3.18: The functional governing the temperature distribution in the slab considered in Example 3.16 is
\Pi = \int_ {0} ^ {L} \frac {1}{2} k \left(\frac {\partial \theta}{\partial x}\right) ^ {2} d x - \int_ {0} ^ {L} \theta q ^ {B} d x - \theta_ {0} q _ {0} \tag {a}
and the essential boundary condition is
\theta_ {L} = \theta_ {i} \tag {b}
where \theta_0 = \theta(0, t) and \theta_L = \theta(L, t)
q^{B} is the heat generated per unit volume, and otherwise the same notation as in Example 3.16 is used. Invoke the stationarity condition on \Pi to derive the governing heat conduction equation and the natural boundary condition.
This is a C^{0} variational problem; i.e., the highest derivative in the functional in (a) is of order 1, or m = 1. An essential boundary condition, here given in (b), can therefore correspond only to a prescribed temperature, and a natural boundary condition must correspond to a prescribed temperature gradient or boundary heat flow input.
To invoke the stationarity condition \delta\Pi = 0 , we can directly use the fact that variations and differentiations are performed with the same rules. That is, using (3.7a) we obtain
\int_ {0} ^ {L} \left(k \frac {\partial \theta}{\partial x}\right) \left(\delta \frac {\partial \theta}{\partial x}\right) d x - \int_ {0} ^ {L} \delta \theta q ^ {B} d x - \delta \theta_ {0} q _ {0} = 0 \tag {c}
where also \delta (\partial \theta /\partial x) = \partial \delta \theta /\partial x . The same result is also obtained when using (3.7b), which gives here
\begin{array}{l} \delta \Pi = \lim _ {\epsilon \rightarrow 0} \left[ \frac {\left\{\int_ {0} ^ {L} \frac {1}{2} k \left(\frac {\partial \theta}{\partial x} + \epsilon \frac {\partial \eta}{\partial x}\right) ^ {2} d x - \int_ {0} ^ {L} (\theta + \epsilon \eta) q ^ {B} d x - (\theta_ {0} + \epsilon \eta | _ {x = 0}) q _ {0} \right\}}{\epsilon} \right. \\ - \frac {\left\{\int_ {0} ^ {L} \frac {1}{2} k \left(\frac {\partial \theta}{\partial x}\right) ^ {2} d x - \int_ {0} ^ {L} \theta q ^ {B} d x - \theta_ {0} q _ {0} \right\}}{\epsilon} \\ = \lim _ {\epsilon \rightarrow 0} \frac {\int_ {0} ^ {L} \left[ \epsilon k \frac {\partial \theta}{\partial x} \frac {\partial \eta}{\partial x} + \frac {1}{2} \epsilon^ {2} k \left(\frac {\partial \eta}{\partial x}\right) ^ {2} \right] d x - \int_ {0} ^ {L} \epsilon \eta q ^ {B} d x - \epsilon \eta | _ {x = 0} q _ {0}}{\epsilon} \\ = \int_ {0} ^ {L} k \frac {\partial \theta}{\partial x} \frac {\partial \eta}{\partial x} d x - \int_ {0} ^ {L} \eta q ^ {B} d x - \eta_ {0} q _ {0} \\ = 0 \\ \end{array}
where \eta_0 = \eta|_{x=0} and we would now substitute \delta\theta for \eta .
Now using integration by parts, ^{2} we obtain from (c) the following equation:
\underbrace {- \int_ {0} ^ {L} \left(k \frac {\partial^ {2} \theta}{\partial x ^ {2}} + q ^ {B}\right) \delta \theta d x} _ {①} + \underbrace {k \left. \frac {\partial \theta}{\partial x} \right| _ {x = L} \delta \theta_ {L}} _ {②} - \underbrace {\left[ k \left. \frac {\partial \theta}{\partial x} \right| _ {x = 0} + q _ {0} \right] \delta \theta_ {0}} _ {③} = 0 \tag {d}
To extract from (d) the governing differential equation and natural boundary condition, we use the argument that the variations on \theta are completely arbitrary, except that there can be no variations on the prescribed essential boundary conditions. Hence, because \theta_L is prescribed, we have \delta \theta_L = 0 and term ② in (d) vanishes.
Considering next terms ① and ③, assume that \delta\theta_0 = 0 but that \delta\theta is otherwise nonzero (except at x = 0 , where we have a sudden jump to a zero value). If (d) is to hold for any nonzero \delta\theta , we need to have ^3
k \frac {\partial^ {2} \theta}{\partial x ^ {2}} + q ^ {B} = 0 \tag {e}
Conversely, assume that \delta \theta is zero everywhere except at x = 0 ; i.e., we have \delta \theta_0 \neq 0 ; then (d) is valid only if
k \left. \frac {\partial \theta}{\partial x} \right| _ {x = 0} + q _ {0} = 0 \tag {f}
The expression in (f) represents the natural boundary condition.
The governing differential equation of the propagation problem is obtained from (e), specifying here that
q ^ {B} = - \rho c \frac {\partial \theta}{\partial t} \tag {g}
Hence (e) reduces to
k \frac {\partial^ {2} \theta}{\partial x ^ {2}} = \rho c \frac {\partial \theta}{\partial t}
We may note that until the heat capacity effect was introduced in the formulation in (g), the equations were derived as if a steady-state problem (and with q^{B} time-dependent a pseudo steady-state problem) was being considered. Hence, as noted previously, the formulation of the propagation problem can be obtained from the equation governing the steady-state response by simply taking into account the time-dependent “inertia term.”
EXAMPLE 3.19: The functional and essential boundary condition governing the wave propagation in the rod considered in Example 3.17 are
\Pi = \int_ {0} ^ {L} \frac {1}{2} E A \left(\frac {\partial u}{\partial x}\right) ^ {2} d x - \int_ {0} ^ {L} u f ^ {B} d x - u _ {L} R \tag {a}
and
u _ {0} = 0 \tag {b}
where the same notation as in Example 3.17 is used, u_{0} = u(0, t) , u_{L} = u(L, t) , and f^{B} is the body force per unit length of the rod. Show that by invoking the stationarity condition on \Pi the governing differential equation of the propagation problem and the natural boundary condition can be derived.
We proceed as in Example 3.18. The stationarity condition \delta \Pi = 0 gives
\int_ {0} ^ {L} \left(E A \frac {\partial u}{\partial x}\right) \left(\delta \frac {\partial u}{\partial x}\right) d x - \int_ {0} ^ {L} \delta u f ^ {B} d x - \delta u _ {L} R = 0
Writing \partial\delta u/\partial x for \delta(\partial u/\partial x) , recalling that EA is constant, and using integration by parts yields
- \int_ {0} ^ {L} \left(E A \frac {\partial^ {2} u}{\partial x ^ {2}} + f ^ {B}\right) \delta u d x + \left[ E A \frac {\partial u}{\partial x} \right| _ {x = L} - R \left[ \delta u _ {L} - E A \frac {\partial u}{\partial x} \right| _ {x = 0} \delta u _ {0} = 0