김경종
27 KiB
(a) Sinusoidal distributed lateral load q = q_0 \sin(\pi x / L) over 0 < x < L .
(b) Uniformly distributed lateral load q = q_0 over 0 < x < L .
Section 15.4
15.9 A cable carries constant axial tension T and contacts an elastic foundation of modulus B (force per unit length per unit of deflection w). The left and right ends of the cable are loaded by the respective forces F_{L} and F_{R} , which act perpendicular to the elastic foundation. The sketch shows the cable in its deflected position.
(a) Show that the governing differential equation is Tw_{xx} - Bw = 0 .
(b) Use the Galerkin method to establish formulas for element matrices analogous to those in Eq. 15.4-9.
text_image
F_L w x F_R T T
Elastic foundation
Problem 15.9
15.10 The equation of motion of a string is Tw_{,xx} - \rho_L\ddot{w} = 0 , where T = constant axial tension, w = small lateral displacement, x = axial coordinate, \rho_L = mass per unit length, and \ddot{w} = d^2 w / dt^2 .
(a) Formulate finite element matrices by the Galerkin method. Let the element have two d.o.f.
(b) Consider a simply supported uniform string of length 2L . Model it by two elements, each of length L , and of the type formulated in part (a). Solve for the fundamental frequency of vibration. (The exact answer is \omega^2 = \pi^2 T / 4\rho_L L^2 .)
(c) Can the problem of part (b) be solved using a single element? Explain how or why not.
15.11 Let F = constant axial force, positive in tension, and B = elastic foundation modulus (force per unit length per unit of deflection w ). With F and B taken into account, the differential equation of a beam becomes EIw_{,xxx} - q - Fw_{,xx} + Bw = 0 . Formulate expressions for the element matrices associated with F and B in a form analogous to Eq. 15.4-20.
15.12 For the beam problem, Eqs. 15.4-13 to 15.4-20, demonstrate the treatment of interelement moments and shear forces, in the fashion of Eq. 15.4-12.
15.13 Let the end cross sections of element 1-2 in Fig. 15.4-3 have areas A_{1} and A_{2} , respectively. If A between ends is a linear function of x , what equation replaces Eq. 15.4-24? Assume that k is constant.
15.14 Starting with Eq. 15.4-23, demonstrate the assembly of element thermal "load" vectors. Use Eq. 15.4-12 as a guide.
15.15 In the differential equations cited in (a) and (b) below, g = g(x) and u = u(x) . Let the approximating field for an element of length L be \bar{u} = \lfloor \mathbf{N} \rfloor \{\mathbf{d}\} , as usual. In each case, what is a formula for [k] in the equation [\mathbf{k}]\{\mathbf{d}\} = \{\mathbf{r}\} in terms of g and \lfloor \mathbf{N} \rfloor ? (You may ignore boundary conditions in this exercise.) (a) gu_{,x} = 0 . (b) gu_{,xx} = 0 .
Section 15.6
15.16 Show that Eq. 15.6-7 can also be obtained from Eq. 15.6-3 and the stationary condition of the functional
\Pi = \frac {1}{2} \int \int \left(k _ {x} \phi_ {, x} ^ {2} + k _ {y} \phi_ {, y} ^ {2} - 2 Q \phi\right) d x d y - \int q _ {B} \phi d S
15.17 The Helmholz equation, p_{,xx} + p_{,yy} + p_{,zz} + (\omega / c)^2 p = 0 , governs acoustic modes of vibration in a cavity with rigid walls. Here p = p(x, y, z) represents the amplitude of sinusoidally varying pressure, \omega is the circular frequency, and c is the speed of sound in the medium. The boundary condition is p_{,n} = 0 , where n is a direction normal to the wall. Derive formulas for finite element matrices, using the assumed pressure amplitude field p = |\mathbf{N}| \{\mathbf{p}_e\} .
15.18 In cylindrical coordinates and with k_{x} = k_{y} = k = constant, Eq. 15.6-1 becomes
\frac {1}{r} \frac {\partial}{\partial r} \left(r \frac {\partial \phi}{\partial r}\right) + \frac {1}{r ^ {2}} \frac {\partial^ {2} \phi}{\partial \theta^ {2}} + \frac {\partial^ {2} \phi}{\partial z ^ {2}} + \frac {Q}{k} = 0
For a solid of revolution, the nonessential boundary condition is k(\ell \phi_r + n\phi_{rz}) - q_B = 0 , where \ell and n are direction cosines of a normal to the surface of the solid. For this problem, formulate equations analogous to Eqs. 15.6-7.
15.19 The differential equation for wind-driven circulation in a shallow lake is
\psi_ {, x x} + \psi_ {, y y} + A \psi_ {, x} + B \psi_ {, y} + C = 0
where \psi is the stream function and A, B , and C are functions of x and y . With h = depth, depthwise average velocities are u = \psi_y / h and v = -\psi_x / h . Coordinates x and y are tangent to the lake surface. The nonessential boundary condition is \psi_n = 0 on the shoreline, where n is a direction normal to the shoreline. Derive a finite element formulation by the Galerkin method [15.8]. A symbolic result is desired, analogous to Eq. 15.6-7, not details of a particular element.
15.20 Complete the development outlined in Eqs. 15.6-8 to 15.6-13; that is, verify that Eqs. 4.1-5 and 4.1-6 are produced.
15.21 An isotropic, flat disk has unit thickness and inner and outer radii r_{i} and r_{0} . The disk is set spinning about its center at constant angular velocity \omega . The differential equation of equilibrium is
\frac {1}{r} \frac {d}{d r} \left(r \sigma_ {r}\right) - \frac {\sigma_ {\theta}}{r} + \rho \omega^ {2} r = 0
where \sigma_{r} = radial stress, \sigma_{\theta} = circumferential stress, and \rho = mass density. Generate formulas for element matrices. Express your results in terms of the shape function matrix and its derivatives, as in Eq. 15.6-7.
15.22 Consider an elastic, axially symmetric solid under axially symmetric loads. Use arguments analogous to those of Eqs. 15.6-8 to 15.6-13 to formulate finite element matrices. For simplicity, omit body forces and initial stresses and strains. Express your results in terms of the shape function matrix and its derivatives, as in Eq. 15.6-7.
HEAT CONDUCTION AND SELECTED FLUID PROBLEMS
Heat conduction equations are reviewed and used to generate finite element formulations. Thermal transients are discussed. Certain problems of acoustics and flow, primarily those whose differential equation is of the same form as the heat conduction equation, are also treated.
16.1 INTRODUCTION TO HEAT CONDUCTION PROBLEMS
Heat conduction analysis may be performed to determine material temperatures and rates of heat flow. The temperature distribution may also be needed in order to perform an analysis for thermally induced stress. Fortunately, it is possible to use a single mesh layout for both problems: a computer program can read a single data file, compute temperatures at nodes, then use these temperatures in stress analysis. Remarks about thermally induced stress, and when a temperature gradient may not produce stress, appear in Sections 1.7 and 4.7. Thermally induced nodal loads are accounted for by terms in Eq. 4.1-6.
A finite element formulation of steady-state heat conduction produces equations of the form [K]\{T\} = \{R\} , where \{T\} contains nodal temperatures of the structure. Matrices [K] and \{R\} can be generated by the method of making a suitable functional stationary or by a weighted residual method.
Quantities used in our discussion are as follows. The unit of heat or energy is J = 1 joule = 1 \, \text{N} \cdot \text{m} .
c = specific heat (J/kg·°C) h = heat transfer coefficient, or film coefficient (J/m²·s·°C) k = thermal conductivity (J/m·s·°C) Q = rate of internal heat generation per unit volume (J/m³·s) q = heat flux per unit area (J/m²·s) q_B = prescribed flux normal to a surface (J/m²·s) ρ = mass density (kg/m³) T = temperature (°C) T_f = fluid temperature (°C) [used with h] t = time (s) T = ∂T/∂t (°C/s)
In the foregoing units, actual fluids and solids display numerical values in the approximate ranges 10^{2} < c < 10^{4} , 10 < h < 10^{5} , and 10^{-2} < k < 500 .
Heat transfer into a solid across a fluid boundary layer is given by q =
h(T_{f} - T) , where T is the surface temperature of the solid and T_{f} is the fluid temperature on the other side of the boundary layer. The heat transfer coefficient h depends on the nature of the fluid, the geometry of the surface, and the dynamics of fluid motion past the surface.
Sources of O include resistance to electric current and chemical reactions.
In this chapter, the foregoing material properties are taken as independent of temperature unless specifically stated otherwise. Realistically, k is a function of T. Also, \partial k/\partial T may be positive or negative, depending upon the material and sometimes the temperature as well. Similarly, h may depend on temperature. Temperature dependence of k and/or h makes the heat conduction equations nonlinear. This problem is briefly considered in Section 16.5. Radiation is another important source of nonlinearity.
The starting point for heat conduction analysis is the Fourier heat conduction equation, which is
q = - k \frac {\partial T}{\partial x} \tag {16.1-1}
This equation states that heat flux q in direction x is proportional to the gradient of temperature in direction x. The negative sign indicates that heat flow is opposite to the direction of temperature increase.
16.2 A ONE-DIMENSIONAL EXAMPLE
Consider a straight but tapered bar, Fig. 16.2-1. Heat flows across end surfaces. Due to convection, heat also flows across lateral surfaces with the flux rate h(T_{f} - T) , where T_{f} is the temperature of surrounding fluid. This flux is directed into the bar if T_{f} > T . Heat is also assumed to be generated internally at rate Q per unit volume. We assume that temperature T in the bar varies only with x, and ask for a finite element formulation that will yield T = T(x) in the steady-state condition.
In the steady-state condition, the net rate of heat flow into any differential element is zero. With volume element dV = A dx and surface area increment dS = p dx, where p = p(x) is the perimeter of the cross section, Fig. 16.2-1b yields
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Cross-sectional area A = A(x) Perimeter p = p(x) T = T₀ (prescribed) Section A-A q = q_R (prescribed) dx Aq Aq + d(Aq) QA dx + h(T_f - T) p dx (a) (b)
Figure 16.2-1. (a) A bar of varying circular cross section. A typical element 1–2 is shaded. (b) Contributions to heat flow through a differential element.
A q - [ A q + d (A q) ] + Q A d x + h (T _ {f} - T) p d x = 0 \tag {16.2-1}
Combining Eqs. 16.1-1 and 16.2-1 and dividing by dx, we obtain the governing differential equation
\frac {d}{d x} (A k T _ {, x}) + Q A + h (T _ {f} - T) p = 0 \tag {16.2-2}
Boundary conditions at the ends are
T = T _ {0} \text { at } x = 0 \quad \text { and } \quad T _ {, x} = - q _ {R} / k \text { at } x = L _ {T} \tag {16.2-3}
These boundary conditions are respectively essential and nonessential.
A finite element formulation can be developed from the following functional:
\Pi = \int \left[ \frac {1}{2} A k T _ {, x} ^ {2} + \frac {1}{2} h p T ^ {2} - (Q A + h p T _ {f}) T \right] d x \tag {16.2-4}
The standard manipulations of calculus of variations show that Eq. 16.2-2 and the nonessential boundary condition are produced by the stationary condition d\Pi = 0 (see Eq. 3.7-6). We interpolate temperature T and temperature gradient T_{,x} along an element from nodal temperatures \{T_{e}\} :
T = \lfloor \mathbf {N} \rfloor \{\mathbf {T} _ {e} \} \quad \text { and } \quad T _ {, x} = \lfloor \mathbf {N}, _ {x} \rfloor \{\mathbf {T} _ {e} \} \tag {16.2-5}
For the two-node element shown in Fig. 16.2-1, \{\mathbf{T}_e\} = \lfloor T_1 - T_2 \rfloor^T and N_1 = (L - x) / L , N_2 = x / L . Next, because T = T^T and T^2 = T^T T , Eqs. 16.2-4 and 16.2-5 yield, for a single element,
\begin{array}{l} \Pi_ {e} = \frac {1}{2} \left\{\mathbf {T} _ {e} \right\} ^ {T} \underbrace {\int_ {0} ^ {L} \left\lfloor \mathbf {N} , _ {x} \right] ^ {T} A k \left\lfloor \mathbf {N} , _ {x} \right\rfloor d x} _ {[ \mathbf {k} ]} \left\{\mathbf {T} _ {e} \right\} + \frac {1}{2} \left\{\mathbf {T} _ {e} \right\} ^ {T} \underbrace {\int_ {0} ^ {L} \left\lfloor \mathbf {N} \right] ^ {T} h p \left\lfloor \mathbf {N} \right\rfloor d x} _ {[ \mathbf {h} _ {\mathrm{ls}} ]} \left\{\mathbf {T} _ {e} \right\} \\ - \{\mathbf {T} _ {e} \} ^ {T} \underbrace {\int_ {0} ^ {L} \lfloor \mathbf {N} \rfloor^ {T} Q A d x} _ {\{\mathbf {r} _ {Q} \}} - \{\mathbf {T} _ {e} \} ^ {T} \underbrace {\int_ {0} ^ {L} \lfloor \mathbf {N} \rfloor^ {T} h p T _ {f} d x} _ {\{\mathbf {r} _ {1 s} \}} \tag {16.2-6} \\ \end{array}
The subscript is stands for “lateral surface.” Quantities such as A, p, and Q are in general functions of x, and may be interpolated from nodal values if so desired.
For the entire finite element structure, \Pi is given by the summation \Pi = \Sigma \Pi_e , which implies the expansion of element arrays to “structure size,” with the global nodal temperature array \{\mathbf{T}\} replacing the several element arrays \{\mathbf{T}_e\} , and with [\mathbf{K}] = \Sigma [\mathbf{k}] , and so on, following the same matrix assembly procedures described in Section 2.7. Next, \Pi is made stationary with respect to temperatures \{\mathbf{T}\} . Writing the stationary condition for an element rather than for the structure, we have
\left\{\frac {\partial \Pi}{\partial \mathbf {T} _ {e}} \right\} = \{\mathbf {0} \} \quad \text { yields } \quad ([ \mathbf {k} ] + [ \mathbf {h} _ {\mathrm{ls}} ]) \{\mathbf {T} _ {e} \} = \{\mathbf {r} _ {Q} \} + \{\mathbf {r} _ {\mathrm{ls}} \} \tag {16.2-7}
After assembly of elements, the corresponding global equations can be written in the form
([ \mathbf {K} ] + [ \mathbf {H} _ {\mathrm{ls}} ]) \{\mathbf {T} \} = \{\mathbf {R} \} \quad \text { where } \quad \{\mathbf {R} \} = \{\mathbf {P} \} + \sum \{\mathbf {r} _ {e} \} \tag {16.2-8}
where, recalling structural terminology, [K] + [H_{ls}] is analogous to a stiffness matrix, \{T\} is analogous to nodal displacements, \{r_{e}\} = \{r_{Q}\} + \{r_{ls}\} is analogous to nodal loads from elements (caused, e.g., by self-weight), and \{P\} is analogous to externally applied loads. For the bar in Fig. 16.2-1, \{P\} contains a single nonzero term—namely -q_{R}A_{R} —which represents the prescribed heat flux at the right end (out of the bar, in this case). Our sign convention for boundary values of q is that they are considered positive when heat flows into the body or element. The boundary condition at the left end is imposed by assigning T = T_{0} at the leftmost node, which is analogous to assigning a known nonzero value to a displacement d.o.f. in a structural problem.
The foregoing formulation is also easy to obtain by the Galerkin method, as described in Section 15.4. Equation 15.4-24 states [k] for a two-node element.
16.3 HEAT CONDUCTION IN A PLANE
In this section we consider the mathematical formulation of heat flow in a plane of unit thickness. Symbols defined in Section 16.1 are used. Radiation heat transfer is not included.
Governing Equation. For a thermally orthotropic material, Fig. 16.3-1, Eq. 16.1-1 yields the heat fluxes
q _ {r} = - k _ {r} T _ {, r} \quad \text { and } \quad q _ {s} = - k _ {s} T _ {, s} \tag {16.3-1}
where k_{r} and k_{s} are principal thermal conductivities in the principal material directions r and s. Temperature gradients in the various coordinate directions are given by chain rule differentiation, that is, by
\left\{ \begin{array}{l} T _ {, r} \\ T _ {, s} \end{array} \right\} = [ \Lambda ] \left\{ \begin{array}{l} T _ {, x} \\ T _ {, y} \end{array} \right\} \quad \text { where } \quad [ \Lambda ] ^ {\cdot} = \left[ \begin{array}{l l} x _ {, r} & y _ {, r} \\ x _ {, s} & y _ {, s} \end{array} \right] = \left[ \begin{array}{l l} \cos \beta & \sin \beta \\ - \sin \beta & \cos \beta \end{array} \right] \tag {16.3-2}
text_image
s β y r β x
Figure 16.3-1. A layered material with principal directions r and s.
text_image
q_y + q_y, y dy q_x dy q_x + q_x, x dx dx q_y
Figure 16.3-2. Heat flux through sides of a differential element.
text_image
y x v α S (boundary)
Figure 16.3-3. Plane region with outward normal vector \nu on its boundary S.
Heat flux q is a vector and transforms like displacement, that is, \lfloor q_x \quad q_y \rfloor^T = [\Lambda]^T \lfloor q_r \quad q_s \rfloor^T . Combining this with Eqs. 16.3-1 and 16.3-2, we obtain
\left\{ \begin{array}{l} q _ {x} \\ q _ {y} \end{array} \right\} = - [ \kappa ] \left\{ \begin{array}{l} T _ {, x} \\ T _ {, y} \end{array} \right\} \quad \text { where } \quad [ \kappa ] = \left[ \begin{array}{l l} k _ {x} & k _ {x y} \\ k _ {x y} & k _ {y} \end{array} \right] = [ \Lambda ] ^ {T} \left[ \begin{array}{l l} k _ {r} & 0 \\ 0 & k _ {s} \end{array} \right] [ \Lambda ] \tag {16.3-3}
For a body of unit thickness, the rate of heat generation in a differential element dx dy is Q dx dy. If lateral surfaces of the body are insulated, heat flux across the boundary of a differential element is as depicted in Fig. 16.3-2. The net inward flow of heat is
Q d x d y - \left(q _ {x, x} d x\right) d y - \left(q _ {y, y} d y\right) d x = \left(Q - q _ {x, x} - q _ {y, y}\right) d x d y \tag {16.3-4}
In general, heat flow produces a time rate of change of stored energy—namely, c\rho \, dx \, dy \, \dot{T} . Therefore, Q - q_{x,x} - q_{y,y} = c\rho \dot{T} . Combining this with Eq. 16.3-3, we obtain
\frac {\partial}{\partial x} \left(k _ {x} T _ {, x} + k _ {x y} T _ {, y}\right) + \frac {\partial}{\partial y} \left(k _ {x y} T _ {, x} + k _ {y} T _ {, y}\right) + Q = c \rho \dot {T} \tag {16.3-5}
If a lateral surface z = constant is not insulated, so that there is a convective transfer of heat across a lateral surface of the plane body, the heat flux q = h(T_f - T) flows into the body across the lateral surface. If this transfer occurs on both lateral surfaces of the body, and if h and T_f are the same on both surfaces, then the term 2h(T_f - T) must be added to the left-hand side of Eq. 16.3-5. A portion of a lateral surface may be neither insulated nor subject to convection; instead, an inward flux q_1 may be prescribed. Then q_1 must be added to the left-hand side of Eq. 16.3-5 (or add q_1 / \tau for a body of thickness \tau ).
If the medium is homogeneous and isotropic, then k_{xy} = 0 and k_x = k_y = k , where k is a constant. Thus Eq. 16.3-5 becomes
k (T _ {, x x} + T _ {, y y}) + Q = c \rho \dot {T} \tag {16.3-6}
If, in addition, Q = 0 and a steady state prevails (\dot{T} = 0) , then we obtain Laplace's equation, T_{,xx} + T_{,yy} = 0 .
Boundary Conditions. One may prescribe temperature on part (or all) of the boundary and heat flux on another part (or all). On any one part, temperature or flux is prescribed, not both. In general, the prescribed quantities may be functions of time. The following boundary conditions apply on boundary S, Fig. 16.3-3, not on the lateral surfaces cited below Eq. 16.3-5.
The essential boundary condition is a prescription of temperature on part or all of S. Alternative names are first or Dirichlet boundary condition.
The nonessential boundary condition is a prescription of heat flux (possibly zero) on part or all of S. Alternative names are second or Neumann boundary condition. For a thermally isotropic material, flux in direction \nu of Fig. 16.3-3 is q_{\nu} = -kT_{,\nu} . In addition, by the chain rule,
T _ {, \nu} = T _ {, x} x _ {, \nu} + T _ {, y} y _ {, \nu} = T _ {, x} \ell_ {B} + T _ {, y} m _ {B} \tag {16.3-7}
where \ell_{B} and m_{B} are direction cosines of \nu . Adopting the convention that prescribed boundary heat flux q_{B} is positive when directed into the body, the non-essential boundary condition is therefore q_{B} = -q_{\nu} , or
q _ {B} = k \left(T, _ {x} \ell_ {B} + T, _ {y} m _ {B}\right) \tag {16.3-8}
If the body is thermally orthotropic, we write q_{\nu} = q_{x}\cos \alpha + q_{y}\sin \alpha = q_{x}\ell_{B} + q_{y}m_{B} and obtain q_{x} and q_{y} from Eq. 16.3-3. Therefore, with q_{B} = -q_{\nu} ,
q _ {B} = \left(k _ {x} T _ {, x} + k _ {x y} T _ {, y}\right) \ell_ {B} + \left(k _ {x y} T _ {, x} + k _ {y} T _ {, y}\right) m _ {B} \tag {16.3-9}
If there is convection heat transfer across all or part of boundary S, then q_{B} is replaced by h(T_{f} - T) for this part of S. Convection heat transfer across a lateral surface is not considered part of the boundary condition as it does not appear on S.
Functional. A functional for plane heat conduction is
\begin{array}{l} \Pi = \iint \left(\frac {1}{2} \left\{ \begin{array}{l} T _ {, x} \\ T _ {, y} \end{array} \right\} ^ {T} [ \kappa ] \left\{ \begin{array}{l} T _ {, x} \\ T _ {, y} \end{array} \right\} - Q T + \rho c \dot {T} T\right) d x d y \\ - \int h \left(T _ {f} T - \frac {1}{2} T ^ {2}\right) d S - \int q _ {B} T d S \tag {16.3-10} \\ \end{array}
in which the surface integrals are each evaluated on the portion of S subject to convection or prescribed flux. If there is also convection across a lateral surface, the term h(T_{f}T - T^{2}/2) must be subtracted from the integrand of the double integral, once for each lateral surface involved. With T, T_{,x} , and T_{,y} subject to variation, one can show by calculus of variations that the condition \delta\Pi = 0 produces Eqs. 16.3-5 and 16.3-9.
16.4 GENERAL SOLIDS AND SOLIDS OF REVOLUTION
The equations of Section 16.3 can be written in a form that also serves for general solids and solids of revolution. The governing equation, Eq. 16.3-5, is
\{\partial \} ^ {T} ([ \kappa ] \{\mathbf {T} _ {\partial} \}) + Q = c \rho \dot {T} \tag {16.4-1}
where \{\partial\}^{T} is a differential operator and \{T_{\partial}\} contains temperature gradients (examples follow). Allowing for either prescribed flux or convection on S, the non-essential boundary condition, Eq. 16.3-9, is
q _ {B} = \{\boldsymbol {\mu} \} ^ {T} [ \boldsymbol {\kappa} ] \{\mathbf {T} _ {\partial} \} \quad \text { or } \quad h (T _ {f} - T) = \{\boldsymbol {\mu} \} ^ {T} [ \boldsymbol {\kappa} ] \{\mathbf {T} _ {\partial} \} \tag {16.4-2}
where \{\mu\} contains direction cosines of a normal to boundary S. The functional, Eq. 16.3-10, is
\Pi = \int_ {V} \left(\frac {1}{2} \left\{\mathbf {T} _ {\partial} \right\} ^ {T} [ \boldsymbol {\kappa} ] \left\{\mathbf {T} _ {\partial} \right\} - Q T + \rho c \dot {T} T\right) d V - \int_ {S} \left(q _ {B} T + h T _ {f} T - \frac {1}{2} h T ^ {2}\right) d S \tag {16.4-3}
where, for the plane problem with unit thickness discussed in Section 16.3, dV = (1)dx dy = dx dy , and
\{\partial \} = \left\{ \begin{array}{l} \partial / \partial x \\ \partial / \partial y \end{array} \right\} \quad \left\{\mathbf {T} _ {\partial} \right\} = \left\{ \begin{array}{l} T _ {, x} \\ T _ {, y} \end{array} \right\} \quad \left\{\boldsymbol {\mu} \right\} = \left\{ \begin{array}{l} \ell_ {B} \\ m _ {B} \end{array} \right\} \tag {16.4-4}
and [\kappa] is given by Eq. 16.3-3. In structural terms, \{\mathbf{T}_{\partial}\} is analogous to strains \{\epsilon\} and [\kappa] is analogous to material property matrix [E].
General Solids. Extension from two dimensions to three requires that we include T_{xz} in \{T_{\partial}\} , include k_{z} , k_{yz} , and k_{zx} in [\kappa] , and do volume integration over dV = dx dy dz. Equations 16.4-1, 16.4-2, and 16.4-3 apply, with
\{\partial \} = \left\{ \begin{array}{l} \partial / \partial x \\ \partial / \partial y \\ \partial / \partial z \end{array} \right\} \quad \left\{\mathbf {T} _ {\partial} \right\} = \left\{ \begin{array}{l} T _ {, x} \\ T _ {, y} \\ T _ {, z} \end{array} \right\} \quad \left\{\boldsymbol {\mu} \right\} = \left\{ \begin{array}{l} \ell_ {B} \\ m _ {B} \\ n _ {B} \end{array} \right\} \tag {16.4-5}
If principal thermal conductivities are written as the diagonal matrix [k_r \quad k_s \quad k_t] , then [\kappa] is [\kappa] = [\Lambda]^T [k_r \quad k_s \quad k_t][\Lambda] , where [\Lambda] is given by Eq. 7.2-1.
Solids of Revolution. The coordinates are r (radial), \theta (circumferential), and z (axial). As compared with the general solid, [\kappa] for a solid of revolution is computed in the same way, dr and r \, d\theta replace dx and dy , T_{,r} and T_{,\theta}/r replace T_{,x} and T_{,y} , dV = r \, dr \, d\theta \, dz , and dS = r \, d\theta \, db , where db is an increment of boundary length in an rz plane. Direction cosines in [\Lambda] pertain to angles between principal material axes and coordinate directions r , \theta , and z . In Eqs. 16.4-1, 16.4-2, and 16.4-3, we use
\{\partial \} = \left\{ \begin{array}{c} (1 / r) + \partial / \partial r \\ (1 / r) \partial / \partial \theta \\ \partial / \partial z \end{array} \right\} \quad \left\{\mathbf {T} _ {\partial} \right\} = \left\{ \begin{array}{c} T _ {, r} \\ T _ {, \theta} / r \\ T _ {, z} \end{array} \right\} \quad \left\{\boldsymbol {\mu} \right\} = \left\{ \begin{array}{c} \ell_ {B} \\ 0 \\ n _ {B} \end{array} \right\} \tag {16.4-6}
A boundary-normal \pmb{\nu} has no \theta component, so \ell_B = \cos (\nu, x) , m_B = 0 , and n_B = \cos (\nu, z) .
If the temperature field and the material properties are axially symmetric, then all derivatives with respect to \theta vanish, and the problem is mathematically two-dimensional.
A nonsymmetric temperature field can be treated by Fourier series, as explained for stress analysis in Section 10.5. Thus a three-dimensional problem is replaced by a series of two-dimensional problems. If, for example, \theta is a principal material direction and the temperature field is symmetric with respect to the \theta = 0 plane, one can use T = \sum \overline{T}_n \cos n\theta , where \overline{T}_n is a function of n, r , and z but is independent of \theta . Boundary conditions are written in terms of their Fourier series components, and \overline{T} is determined for n = 0 , n = 1 , n = 2 , and so on. Superposition of these solutions yields the resultant temperature field.
16.5 FINITE ELEMENT FORMULATION
The reader may wish to review Section 3.10, in which element matrices are formulated for a special case of plane heat conduction. The same procedures