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Sf sf, fB fV, fY i RCy Z, W z, w y, v x, u Y, V Wj j Uj Vi Nodal point j Finite element m X, U Su
Figure 4.1 General three-dimensional body with an 8-node three-dimensional element
Problem Statement
Consider the equilibrium of a general three-dimensional body such as that shown in Fig. 4.1. The body is located in the fixed (stationary) coordinate system X, Y, Z. Considering the body surface area, the body is supported on the area S_{u} with prescribed displacements U^{S_{u}} and is subjected to surface tractions f^{S_{f}} (forces per unit surface area) on the surface area S_{f} .¹
^{1} We may assume here, for simplicity, that all displacement components on S_{u} are prescribed, in which case S_{u} \cup S_{f} = S and S_{u} \cap S_{f} = 0 . However, in practice, it may well be that at a surface point the displacement(s) corresponding to some direction(s) is (are) imposed, while corresponding to the remaining direction(s) the force component(s) is (are) prescribed. For example, a roller boundary condition on a three-dimensional body would correspond to an imposed zero displacement only in the direction normal to the body surface, while tractions are applied (which are frequently zero) in the remaining directions tangential to the body surface. In such cases, the surface point would belong to S_{u} and S_{f} . However, later, in our finite element formulation, we shall first remove all displacement constraints (support conditions) and assume that the reactions are known, and thus consider S_{f} = S and S_{u} = 0 , and then, only after the derivation of the governing finite element equations, impose the displacement constraints. Hence, the assumption that all displacement components on S_{u} are prescribed may be used here for ease of exposition and does not in any way restrict our formulation.
In addition, the body is subjected to externally applied body forces f^{B} (forces per unit volume) and concentrated loads R_{C}^{i} (where i denotes the point of load application). We introduce the forces R_{C}^{i} as separate quantities, although each such force could also be considered surface tractions f^{S_{f}} over a very small area (which would usually model the actual physical situation more accurately). In general, the externally applied forces have three components corresponding to the X, Y, Z coordinate axes:
\mathbf {f} ^ {B} = \left[ \begin{array}{l} f _ {X} ^ {B} \\ f _ {Y} ^ {B} \\ f _ {Z} ^ {B} \end{array} \right]; \quad \mathbf {f} ^ {S _ {f}} = \left[ \begin{array}{l} f _ {X} ^ {S _ {f}} \\ f _ {Y} ^ {S _ {f}} \\ f _ {Z} ^ {S _ {f}} \end{array} \right]; \quad \mathbf {R} _ {C} ^ {i} = \left[ \begin{array}{l} R _ {C X} ^ {i} \\ R _ {C Y} ^ {i} \\ R _ {C Z} ^ {i} \end{array} \right] \tag {4.1}
where we note that the components of \mathbf{f}^B and \mathbf{f}^{S_f} vary as a function of X, Y, Z (and for \mathbf{f}^{S_f} the specific X, Y, Z coordinates of S_f are considered).
The displacements of the body from the unloaded configuration are measured in the coordinate system X, Y, Z and are denoted by U, where
\mathbf {U} (X, Y, Z) = \left[ \begin{array}{l} U \\ V \\ W \end{array} \right] \tag {4.2}
and \mathbf{U} = \mathbf{U}^{S_u} on the surface area S_u . The strains corresponding to \mathbf{U} are
\boldsymbol {\epsilon} ^ {T} = \left[ \begin{array}{l l l l l l} \epsilon_ {X X} & \epsilon_ {Y Y} & \epsilon_ {Z Z} & \gamma_ {X Y} & \gamma_ {Y Z} & \gamma_ {Z X} \end{array} \right] \tag {4.3}
where \epsilon_{XX} = \frac{\partial U}{\partial X};\quad \epsilon_{YY} = \frac{\partial V}{\partial Y};\quad \epsilon_{ZZ} = \frac{\partial W}{\partial Z} (4.4)
\gamma_ {X Y} = \frac {\partial U}{\partial Y} + \frac {\partial V}{\partial X}; \quad \gamma_ {Y Z} = \frac {\partial V}{\partial Z} + \frac {\partial W}{\partial Y}; \quad \gamma_ {Z X} = \frac {\partial W}{\partial X} + \frac {\partial U}{\partial Z}
The stresses corresponding to € are
\boldsymbol {\tau} ^ {T} = \left[ \begin{array}{l l l l l l} \tau_ {X X} & \tau_ {Y Y} & \tau_ {Z Z} & \tau_ {X Y} & \tau_ {Y Z} & \tau_ {Z X} \end{array} \right] \tag {4.5}
where \tau = \mathbf{C}\epsilon +\tau^{\prime} (4.6)
In (4.6), C is the stress-strain material matrix and the vector \tau' denotes given initial stresses [with components ordered as in (4.5)].
The analysis problem is now the following.
Given
the geometry of the body, the applied loads \mathbf{f}^{S_f} , \mathbf{f}^B , \mathbf{R}_C^i , i = 1, 2, \ldots , the support conditions on S_u , the material stress-strain law, and the initial stresses in the body.
Calculate
the displacements U of the body and the corresponding strains \epsilon and stresses \tau .
In the problem solution considered here, we assume linear analysis conditions, which require that
The displacements be infinitesimally small so that (4.4) is valid and the equilibrium of the body can be established (and is solved for) with respect to its unloaded configuration.
The stress-strain material matrix can vary as a function of X, Y, Z but is constant otherwise (e.g., C does not depend on the stress state).
We consider nonlinear analysis conditions in which one or more of these assumptions are not satisfied in Chapters 6 and 7.
To calculate the response of the body, we could establish the governing differential equations of equilibrium, which then would have to be solved subject to the boundary conditions (see Section 3.3). However, closed-form analytical solutions are possible only when relatively simple geometries are considered.
The Principle of Virtual Displacements
The basis of the displacement-based finite element solution is the principle of virtual displacements (which we also call the principle of virtual work). This principle states that the equilibrium of the body in Fig. 4.1 requires that for any compatible small ^{2} virtual displacements (which are zero at and corresponding to the prescribed displacements) ^{3} imposed on the body in its state of equilibrium, the total internal virtual work is equal to the total external virtual work:
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Internal virtual work External virtual work ℜ ∫v ε^T τ dV = ∫v U^T f^B dV + ∫s_f U^{s_f^T} f^{s_f} dS + ∑i U^{iT} R^i_C Stresses in equilibrium with applied loads Virtual strains corresponding to virtual displacements U̅ (4.7)
where the \overline{U} are the virtual displacements and the \overline{\epsilon} are the corresponding virtual strains (the overbar denoting virtual quantities).
The adjective “virtual” denotes that the virtual displacements (and corresponding virtual strains) are not “real” displacements which the body actually undergoes as a consequence of the loading on the body. Instead, the virtual displacements are totally independent
^{2} We stipulate here that the virtual displacements be “small” because the virtual strains corresponding to these displacements are calculated using the small strain measure (see Example 4.2). Actually, provided this small strain measure is used, the virtual displacements can be of any magnitude and indeed we later on choose convenient magnitudes for solution.
^{3} We use the wording “at and corresponding to the prescribed displacements” to mean “at the points and surfaces and corresponding to the components of displacements that are prescribed at those points and surfaces.”
from the actual displacements and are used by the analyst in a thought experiment to establish the integral equilibrium equation in (4.7).
Let us emphasize that in (4.7),
The stresses \tau are assumed to be known quantities and are the unique stresses ^{4} that exactly balance the applied loads.
The virtual strains \bar{\epsilon} are calculated by the differentiations given in (4.4) from the assumed virtual displacements \overline{U} .
The virtual displacements \overline{U} must represent a continuous virtual displacement field (to be able to evaluate \overline{\epsilon} ), with \overline{U} equal to zero at and corresponding to the prescribed displacements on S_{u} ; also, the components in \overline{U}^{S_{f}} are simply the virtual displacements \overline{U} evaluated on the surface S_{f} .
All integrations are performed over the original volume and surface area of the body, unaffected by the imposed virtual displacements.
To exemplify the use of the principle of virtual displacements, assume that we believe (but are not sure) to have been given the exact solution displacement field of the body. This given displacement field is continuous and satisfies the displacement boundary conditions on S_{u} . Then we can calculate \epsilon and \tau (corresponding to this displacement field). The vector \tau lists the correct stresses if and only if the equation (4.7) holds for any arbitrary virtual displacements \overline{\mathbf{U}} that are continuous and zero at and corresponding to the prescribed displacements on S_{u} . In other words, if we can pick one virtual displacement field \overline{\mathbf{U}} for which the relation in (4.7) is not satisfied, then this is proof that \tau is not the correct stress vector (and hence the given displacement field is not the exact solution displacement field).
We derive and demonstrate the principle of virtual displacements in the following examples.
EXAMPLE 4.2: Derive the principle of virtual displacements for the general three-dimensional body in Fig. 4.1.
To simplify the presentation we use indicial notation with the summation convention (see Section 2.4), with x_{i} denoting the ith coordinate axis ( x_{1} \equiv X, x_{2} \equiv Y, x_{3} \equiv Z ), u_{i} denoting the ith displacement component ( u_{1} \equiv U, u_{2} \equiv V, u_{3} \equiv W ), and a comma denoting differentiation.
The given displacement boundary conditions are u_{i}^{S_{u}} on S_{u} , and let us assume that we have no concentrated surface loads, that is, all surface loads are contained in the components f_{i}^{S_{f}} .
The solution to the problem must satisfy the following differential equations (see, for example, S. Timoshenko and J. N. Goodier [A]):
\tau_ {i j, j} + f _ {i} ^ {B} = 0 \quad \text { throughout the body } \tag {a}
with the natural (force) boundary conditions
\tau_ {i j} n _ {j} = f _ {i} ^ {S _ {f}} \quad \text { on } S _ {f} \tag {b}
and the essential (displacement) boundary conditions
u _ {i} = u _ {i} ^ {S _ {u}} \quad \text { on } S _ {u} \tag {c}
where S = S_{u} \cup S_{f}, S_{u} \cap S_{f} = 0 , and the n_{j} are the components of the unit normal vector to the surface S of the body.
Consider now any arbitrarily chosen continuous displacements \overline{u}_{i} satisfying
\overline {{{u}}} _ {i} = 0 \quad \text { on } S _ {u} \tag {d}
Then (\tau_{ij,j} + f_i^B)\overline{u}_i = 0
and therefore, \int_{V}(\tau_{ij,j} + f_i^B)\overline{u}_i dV = 0 (e)
We call the \overline{u}_i virtual displacements. Note that since the \overline{u}_i are arbitrary, (e) can be satisfied if (and only if) the quantity in the parentheses vanishes. Hence (e) is equivalent to (a).
Using the mathematical identity (\tau_{ij}\overline{u}_i)_{,j} = \tau_{ij,j}\overline{u}_i + \tau_{ij}\overline{u}_{i,j} , we obtain from (e),
\int_ {V} \left[ \left(\tau_ {i j} \overline {{{u}}} _ {i}\right) _ {, j} - \tau_ {i j} \overline {{{u}}} _ {i, j} + f _ {i} ^ {B} \overline {{{u}}} _ {i} \right] d V = 0
Next, using the identity \int_{V} (\tau_{ij} \overline{u}_i)_{,j} dV = \int_{S} (\tau_{ij} \overline{u}_i) n_j dS , which follows from the divergence theorem ^5 (see, for example, G. B. Thomas and R. L. Finney [A]), we have
\int_ {V} \left(- \tau_ {i j} \overline {{u}} _ {i, j} + f _ {i} ^ {B} \overline {{u}} _ {i}\right) d V + \int_ {S} \left(\tau_ {i j} \overline {{u}} _ {i}\right) n _ {j} d S = 0 \tag {f}
In light of (b) and (d), we obtain
\int_ {V} \left(- \tau_ {i j} \overline {{u}} _ {i, j} + f _ {i} ^ {B} \overline {{u}} _ {i}\right) d V + \int_ {S _ {f}} f _ {i} ^ {S _ {f}} \overline {{u}} _ {i} ^ {S _ {f}} d S = 0 \tag {g}
Also, because of the symmetry of the stress tensor (\tau_{ij} = \tau_{ji}) , we have
\tau_ {i j} \overline {{{u}}} _ {i, j} = \tau_ {i j} \left[ \frac {1}{2} \left(\overline {{{u}}} _ {i, j} + \overline {{{u}}} _ {j, i}\right) \right] = \tau_ {i j} \overline {{{\epsilon}}} _ {i j}
and hence we obtain from (g) the required result, (4.7),
\int_ {V} \tau_ {i j} \overline {{{\epsilon}}} _ {i j} d V = \int_ {V} f _ {i} ^ {B} \overline {{{u}}} _ {i} d V + \int_ {S _ {f}} f _ {i} ^ {S _ {f}} \overline {{{u}}} _ {i} ^ {S _ {f}} d S \tag {h}
Note that in (h) we use the tensor notation for the strains; hence, the engineering shear strains used in (4.7) are obtained by adding the appropriate tensor shear strain components, e.g., \bar{\gamma}_{XY} = \bar{\epsilon}_{12} + \bar{\epsilon}_{21} . Also note that by using (b) [and (d)] in (f), we explicitly introduced the natural boundary conditions into the principle of virtual displacements (h).
EXAMPLE 4.3: Consider the bar shown in Figure E4.3.
(a) Specialize the equation of the principle of virtual displacements (4.7) to this problem.
(b) Solve for the exact response of the mechanical model.
(c) Show that for the exact displacement response the principle of virtual displacements is satisfied with the displacement patterns (i) \overline{u} = ax and (ii) \overline{u} = ax^2 , where a is a constant.
(d) Assume that the stress solution is
\tau_ {x x} = \frac {F}{\frac {3}{2} A _ {0}}
^{5} The divergence theorem states: Let F be a vector field in volume V; then
\int_ {V} F _ {i, i} d V = \int_ {S} \mathbf {F} \cdot \mathbf {n} d S
where n is the unit outward normal on the surface S of V.
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A = A₀(2 - x/L) x, u F L Young's modulus E Force F
Figure E4.3 Bar subjected to concentrated load F
i.e., that \tau_{xx} is the force F divided by the average cross-sectional area, and investigate whether the principle of virtual displacements is satisfied for the displacement patterns given in (c).
The principle of virtual displacements (4.7) specialized to this bar problem gives
\int_ {0} ^ {L} \frac {d \overline {{u}}}{d x} E A \frac {d u}{d x} d x = \overline {{u}} \left| F _ {x = L} \right. \tag {a}
The governing differential equations are obtained using integration by parts (see Example 3.19):
\overline {{{u}}} E A \frac {d u}{d x} \Bigg | _ {0} ^ {L} - \int_ {0} ^ {L} \overline {{{u}}} \frac {d}{d x} \left(E A \frac {d u}{d x}\right) d x = \overline {{{u}}} \Bigg | _ {x = L} F \tag {b}
Since \overline{u}\big|_{x = 0} = 0 and \overline{u} is arbitrary otherwise, we obtain from (b) (see Example 3.18 for the arguments used),
\frac {d}{d x} \left(E A \frac {d u}{d x}\right) = 0 \quad \text { differential equation of equilibrium } \tag {c}
E A \left. \frac {d u}{d x} \right| _ {x = L} = F \quad \text { force or natural boundary condition } \tag {d}
Of course, in addition we have the displacement boundary condition u|_{x=0} = 0 . Integrating (c) and using the boundary conditions, we obtain as the exact solution of the mathematical model,
u = \frac {F L}{E A _ {0}} \ln \left(\frac {2}{2 - x / L}\right) \tag {e}
Next, using (e) and \overline{u} = ax and \overline{u} = ax^2 in equation (a), we obtain
\int_ {0} ^ {L} a \frac {F}{A _ {0} (2 - x / L)} A _ {0} \left(2 - \frac {x}{L}\right) d x = a L F \tag {f}
and
\int_ {0} ^ {L} 2 a x \frac {F}{A _ {0} (2 - x / L)} A _ {0} \left(2 - \frac {x}{L}\right) d x = a L ^ {2} F \tag {g}
Equations (f) and (g) show that for the exact displacement /stress response the principle of virtual displacements is satisfied with the assumed virtual displacements.
Now let us employ the principle of virtual displacements with \tau_{xx} = \frac{2}{3}(F/A_{0}) and use first \overline{u} = ax and then \overline{u} = ax^{2} . We obtain with \overline{u} = ax ,
\int_ {0} ^ {L} a \frac {2}{3} \frac {F}{A _ {0}} A _ {0} \left(2 - \frac {x}{L}\right) d x = a L F
which shows that the principle of virtual displacements is satisfied with this virtual displacement field. For \overline{u} = ax^2 , we obtain
\int_ {0} ^ {L} 2 a x \frac {2}{3} \frac {F}{A _ {0}} A _ {0} \left(2 - \frac {x}{L}\right) d x \neq a L ^ {2} F
and this equation shows that \tau_{xx} = \frac{2}{3}(F/A_{0}) is not the correct stress solution.
The principle of virtual displacements can be directly related to the principle that the total potential \Pi of the system must be stationary (see Sections 3.3.2 and 3.3.4). We study this relationship in the following example.
EXAMPLE 4.4: Show how for a linear elastic continuum the principle of virtual displacements relates to the principle of stationarity of the total potential.
Assuming a linear elastic continuum with zero initial stresses, the total potential of the body in Fig. 4.1 is
\Pi = \frac {1}{2} \int_ {V} \epsilon^ {T} \mathbf {C} \epsilon d V - \int_ {V} \mathbf {U} ^ {T} \mathbf {f} ^ {B} d V - \int_ {S _ {f}} \mathbf {U} ^ {S _ {f} ^ {T}} \mathbf {f} ^ {S _ {f}} d S - \sum_ {i} \mathbf {U} ^ {i T} \mathbf {R} _ {C} ^ {i} \tag {a}
where the notation was defined earlier, and we have
\tau = \mathbf {C} \epsilon
with C the stress-strain matrix of the material.
Invoking the stationarity of \Pi , i.e., evaluating \delta\Pi = 0 with respect to the displacements (which now appear in the strains) and using the fact that C is symmetric, we obtain
\int_ {V} \boldsymbol {\delta} \boldsymbol {\epsilon} ^ {T} \mathbf {C} \boldsymbol {\epsilon} d V = \int_ {V} \boldsymbol {\delta} \mathbf {U} ^ {T} \mathbf {f} ^ {B} d V + \int_ {S _ {f}} \delta \mathbf {U} ^ {S _ {f} ^ {T}} \mathbf {f} ^ {S _ {f}} d S + \sum_ {i} \delta \mathbf {U} ^ {i ^ {T}} \mathbf {R} _ {C} ^ {i} \tag {b}
However, to evaluate \Pi in (a) the displacements must satisfy the displacement boundary conditions. Hence in (b) we consider any variations on the displacements but with zero values at and corresponding to the displacement boundary conditions, and the corresponding variations in strains. It follows that invoking the stationarity of \Pi is equivalent to using the principle of virtual displacements, and indeed we may write
\delta \epsilon \equiv \overline {{{\epsilon}}}; \quad \delta \mathbf {U} \equiv \overline {{{\mathbf {U}}}}; \quad \delta \mathbf {U} ^ {S _ {f}} \equiv \overline {{{\mathbf {U}}}} ^ {S _ {f}}; \quad \delta \mathbf {U} ^ {i} \equiv \overline {{{\mathbf {U}}}} ^ {i}
so that (b) reduces to (4.7).
It is important to realize that when the principle of virtual displacements (4.7) is satisfied for all admissible virtual displacements with the stresses \tau “properly obtained” from a continuous displacement field U that satisfies the displacement boundary conditions on S_{u} , all three fundamental requirements of mechanics are fulfilled:
- Equilibrium holds because the principle of virtual displacements is an expression of equilibrium as shown in Example 4.2.
- Compatibility holds because the displacement field
\mathbf{U}is continuous and satisfies the displacement boundary conditions. - The stress-strain law holds because the stresses
\tauhave been calculated using the constitutive relationships from the strains\epsilon(which have been evaluated from the displacements\mathbf{U}).
So far we have assumed that the body being considered is properly supported, i.e., that there are sufficient support conditions for a unique displacement solution. However, the principle of virtual displacements also holds when all displacement supports are removed and the correct reactions (then assumed known) are applied instead. In this case the surface area S_{f} on which known tractions are applied is equal to the complete surface area S of the body (and S_{u} is zero) ^{6} . We use this basic observation in developing the governing finite element equations. That is, it is conceptually expedient to first not consider any displacement boundary conditions, develop the governing finite element equations accordingly, and then prior to solving these equations impose all displacement boundary conditions.
Finite Element Equations
Let us now derive the governing finite element equations. We first consider the response of the general three-dimensional body shown in Fig. 4.1 and later specialize this general formulation to specific problems (see Section 4.2.3).
In the finite element analysis we approximate the body in Fig. 4.1 as an assemblage of discrete finite elements interconnected at nodal points on the element boundaries. The displacements measured in a local coordinate system x, y, z (to be chosen conveniently) within each element are assumed to be a function of the displacements at the N finite element nodal points. Therefore, for element m we have
\mathbf {u} ^ {(m)} (x, y, z) = \mathbf {H} ^ {(m)} (x, y, z) \hat {\mathbf {U}} \tag {4.8}
where \mathbf{H}^{(m)} is the displacement interpolation matrix, the superscript m denotes element m , and \hat{\mathbf{U}} is a vector of the three global displacement components U_i , V_i , and W_i at all nodal points, including those at the supports of the element assemblage; i.e., \hat{\mathbf{U}} is a vector of dimension 3N ,
\hat {\mathbf {U}} ^ {T} = \left[ U _ {1} V _ {1} W _ {1} \quad U _ {2} V _ {2} W _ {2} \quad \dots \quad U _ {N} V _ {N} W _ {N} \right] \tag {4.9}
We may note here that more generally, we write
\hat {\mathbf {U}} ^ {T} = \left[ \begin{array}{l l l l l} U _ {1} & U _ {2} & U _ {3} & \dots & U _ {n} \end{array} \right] \tag {4.10}
where it is understood that U_{i} may correspond to a displacement in any direction X, Y, or Z, or even in a direction not aligned with these coordinate axes (but aligned with the axes of another local coordinate system), and may also signify a rotation when we consider beams, plates, or shells (see Section 4.2.3). Since \hat{U} includes the displacements (and rota-
tions) at the supports of the element assemblage, we need to impose, at a later time, the known values of \hat{U} prior to solving for the unknown nodal point displacements.
Figure 4.1 shows a typical finite element of the assemblage. This element has eight nodal points, one at each of its corners, and can be thought of as a “brick” element. We should imagine that the complete body is represented as an assemblage of such brick elements put together so as to not leave any gaps between the element domains. We show this element here merely as an example; in practice, elements of different geometries and nodal points on faces and in the element interiors may be used.
The choice of element and the construction of the corresponding entries in \mathbf{H}^{(m)} (which depend on the element geometry, the number of element nodes/degrees of freedom, and convergence requirements) constitute the basic steps of a finite element solution and are discussed in detail later.
Although all nodal point displacements are listed in \hat{U} , it should be realized that for a given element only the displacements at the nodes of the element affect the displacement and strain distributions within the element.
With the assumption on the displacements in (4.8) we can now evaluate the corresponding element strains,
\boldsymbol {\epsilon} ^ {(m)} (x, y, z) = \mathbf {B} ^ {(m)} (x, y, z) \hat {\mathbf {U}} \tag {4.11}
where \mathbf{B}^{(m)} is the strain-displacement matrix; the rows of \mathbf{B}^{(m)} are obtained by appropriately differentiating and combining rows of the matrix \mathbf{H}^{(m)} .
The purpose of defining the element displacements and strains in terms of the complete array of finite element assemblage nodal point displacements may not be obvious now. However, we will see that by proceeding in this way, the use of (4.8) and (4.11) in the principle of virtual displacements will automatically lead to an effective assemblage process of all element matrices into the governing structure matrices. This assemblage process is referred to as the direct stiffness method.
The stresses in a finite element are related to the element strains and the element initial stresses using
\boldsymbol {\tau} ^ {(m)} = \mathbf {C} ^ {(m)} \boldsymbol {\epsilon} ^ {(m)} + \boldsymbol {\tau} ^ {I (m)} \tag {4.12}
where \mathbf{C}^{(m)} is the elasticity matrix of element m and \tau^{I(m)} are the given element initial stresses. The material law specified in \mathbf{C}^{(m)} for each element can be that for an isotropic or an anisotropic material and can vary from element to element.
Using the assumption on the displacements within each finite element, as expressed in (4.8), we can now derive equilibrium equations that correspond to the nodal point displacements of the assemblage of finite elements. First, we rewrite (4.7) as a sum of integrations over the volume and areas of all finite elements:
\sum_ {m} \int_ {V ^ {(m)}} \overline {{{\boldsymbol {\epsilon}}}} ^ {(m) T} \boldsymbol {\tau} ^ {(m)} d V ^ {(m)} = \sum_ {m} \int_ {V ^ {(m)}} \overline {{{\mathbf {u}}}} ^ {(m) T} \mathbf {f} ^ {B (m)} d V ^ {(m)}
+ \sum_ {m} \int_ {S _ {1} ^ {(m)}, \dots , S _ {q} ^ {(m)}} \overline {{{\mathbf {u}}}} ^ {S (m) ^ {T}} \mathbf {f} ^ {S (m)} d S ^ {(m)} + \sum_ {i} \overline {{{\mathbf {u}}}} ^ {i ^ {T}} \mathbf {R} _ {C} ^ {i} \tag {4.13}
where m = 1, 2, \ldots, k , where k = number of elements, and S_{1}^{(m)}, \ldots, S_{q}^{(m)} denotes the element surfaces that are part of the body surface S. For elements totally surrounded by other elements no such surfaces exist, whereas for elements on the surface of the body one or more such element surfaces are included in the surface force integral. Note that we assume in (4.13) that nodal points have been placed at the points where concentrated loads are applied, although a concentrated load can of course also be included in the surface force integrals.
It is important to note that since the integrations in (4.13) are performed over the element volumes and surfaces, for efficiency we may use a different and any convenient coordinate system for each element in the calculations. After all, for a given virtual displacement field, the internal virtual work is a number, as is the external virtual work, and this number can be evaluated by integrations in any coordinate system. Of course, it is assumed that for each integral in (4.13) only a single coordinate system for all variables is employed; e.g., \overline{\mathbf{u}}^{(m)} is defined in the same coordinate system as \mathbf{f}^{B(m)} . The use of different coordinate systems is in essence the reason why each of the integrals can be evaluated very effectively in general element assemblages.
The relations in (4.8) and (4.11) have been given for the unknown (real) element displacements and strains. In our use of the principle of virtual displacements we employ the same assumptions for the virtual displacements and strains
\boxed {\overline {{{\mathbf {u}}}} ^ {(m)} (x, y, z) = \mathbf {H} ^ {(m)} (x, y, z) \overline {{{\mathbf {U}}}}} \tag {4.14}
\overline {{{\boldsymbol {\epsilon}}}} ^ {(m)} (x, y, z) = \mathbf {B} ^ {(m)} (x, y, z) \overline {{{\hat {\mathbf {U}}}}} \tag {4.15}
In this way the element stiffness (and mass) matrices will be symmetric matrices.
If we now substitute into (4.13), we obtain
\begin{array}{l} \bar {\mathbf {U}} ^ {T} \left[ \sum_ {m} \int_ {V ^ {(m)}} \mathbf {B} ^ {(m) T} \mathbf {C} ^ {(m)} \mathbf {B} ^ {(m)} d V ^ {(m)} \right] \hat {\mathbf {U}} = \bar {\mathbf {U}} ^ {T} \left[ \left\{\sum_ {m} \int_ {V ^ {(m)}} \mathbf {H} ^ {(m) T} \mathbf {f} ^ {B (m)} d V ^ {(m)} \right\} \right. \\ + \left\{\sum_ {m} \int_ {S _ {1} ^ {(m)}, \dots , S _ {q} ^ {(m)}} \mathbf {H} ^ {S (m) T} \mathbf {f} ^ {S (m)} d S ^ {(m)} \right\} \tag {4.16} \\ \left. - \left\{\sum_ {m} \int_ {V ^ {(m)}} \mathbf {B} ^ {(m) T} \boldsymbol {\tau} ^ {I (m)} d V ^ {(m)} \right\} + \mathbf {R} _ {C} \right] \\ \end{array}
where the surface displacement interpolation matrices \mathbf{H}^{S(m)} are obtained from the displacement interpolation matrices \mathbf{H}^{(m)} in (4.8) by substituting the appropriate element surface coordinates (see Examples 4.7 and 5.12) and R_{C} is a vector of concentrated loads applied to the nodes of the element assemblage.
We should note that the ith component in R_{c} is the concentrated nodal force that corresponds to the ith displacement component in \hat{U} . In (4.16) the nodal point displacement vectors \hat{U} and \hat{U} of the element assemblage are independent of element m and are therefore taken out of the summation signs.
To obtain from (4.16) the equations for the unknown nodal point displacements, we apply the principle of virtual displacements n times by imposing unit virtual displacements