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3D surface plot with grid lines and a small inset grid, no text or symbols present
Fig. 6.1(c) The 9-node Lagrangian quadrilateral element (continued).
For each application, a computer code is developed which allows the solution of practical problems. The computation times of elasto-plastic problems are relatively high with solution costs being typically ten times those of the corresponding linear elastic analysis. Of course a direct comparison would depend on the extent of plastic yielding and how close to the ultimate load carrying capacity a solution is sought. In view of these relatively high computer costs it is essential that the codes developed should be as efficient as possible and that any numerical techniques which reduce the computational requirements be employed. Since the main aim of this text is to fulfil a teaching role some compromise must however be inevitably made between program clarity and efficiency. The applicability of the programs presented is demonstrated by the solution of practical examples. Detailed user instructions for all of the computer programs presented in Part II of this text are provided in Appendix II.
In Section 6.2 the basic expressions for the linear elastic finite element analysis of two-dimensional continua and plate bending problems are presented. Section 6.3 outlines the principles of isoparametric element formulation with particular attention being given to the role of numerical integration. Standard subroutines pertaining to linear elastic finite element analysis are reviewed in Section 6.4 and some subroutines common to the three nonlinear applications considered in Chapters 7, 8 and 9 are presented in Section 6.5.
6.2 Virtual work expressions for various solid mechanics applications
6.2.1 Introduction
In this section we briefly describe various two-dimensional solid mechanics finite element applications in the elastic range only. Later in Chapters 7–9 we demonstrate how elasto-plastic or elasto-viscoplastic behaviour may be included in these applications using finite elements.
In Part I we presented some very simple finite element representations. By contrast, in Part II we include numerically integrated isoparametric quadrilateral elements.
6.2.2 Virtual work expression
If a body is subjected to a set of body forces b then by the Virtual Work Principle we can write
\int_ {\Omega} [ \delta \epsilon ] ^ {T} \sigma d \Omega - \int_ {\Omega} [ \delta \boldsymbol {u} ] ^ {T} \boldsymbol {b} d \Omega - \int_ {\Gamma_ {t}} [ \delta \boldsymbol {u} ] ^ {T} \boldsymbol {t} d \Gamma = 0, \tag {6.1}
where \sigma is the vector of stresses, t is the vector of boundary tractions, \delta u is the vector of virtual displacements, \delta \epsilon is the vector of associated virtual strains, \Omega is the domain of interest, \Gamma_{t} is that part of the boundary on which boundary tractions are prescribed and \Gamma_{u} is that part of the boundary on which displacements are prescribed.
6.2.3 Plane stress
Consider some typical plane stress problems shown in Fig. 6.2. Typically a thin plate is subjected to loads applied in the xy plane, that is the plane of the structure. ^{(2)} The thickness of the plate is assumed to be small compared with the plan dimensions in the xy plane. Stresses are assumed to be constant through the thickness of the plate and \sigma_{z} , \tau_{zx} and \tau_{zy} are ignored. Thus the displacements may now be expressed as
\boldsymbol {u} = [ u, v ] ^ {T}, \tag {6.2}
where u and v are the in-plane displacements in the x and y directions respectively.
The strain components may be listed in the vector
\epsilon = [ \epsilon_ {x}, \epsilon_ {y}, \gamma_ {x y} ] ^ {T}, \tag {6.3}
where for small displacements the normal strains are given as
\epsilon_ {x} = \frac {\partial u}{\partial x}, \quad \epsilon_ {y} = \frac {\partial v}{\partial y},
Fig. 6.2 Typical plane stress problems.
and the shear strain is given as
\gamma_ {x y} = \frac {\dot {c} u}{\dot {c} y} + \frac {\dot {c} v}{\dot {c} x}.
Note that virtual displacements are listed in the vector
\delta \boldsymbol {u} = [ \delta u, \delta v ] ^ {T}, \tag {6.4}
and the associated virtual strains are
\delta \epsilon = \left[ \frac {\dot {c} (\delta u)}{\dot {c} x}, \frac {\dot {c} (\delta v)}{\dot {c} y}, \frac {\dot {c} (\delta u)}{\dot {c} y} + \frac {\dot {c} (\delta v)}{\dot {c} x} \right] ^ {T}. \tag {6.5}
The relevant stress-strain relationships may be written as
\sigma = D \epsilon , \tag {6.6}
where
\sigma = \left[ \sigma_ {x}, \sigma_ {y}, \tau_ {x y} \right] ^ {T},
in which \sigma_{x} and \sigma_{y} are the normal stresses and \tau_{xy} is the shear stress.
For linear elastic situations the stress-strain or constitutive matrix is given as
\boldsymbol {D} = \frac {E}{(1 - \nu^ {2})} \left[ \begin{array}{c c c} 1 & \nu & 0 \\ \nu & 1 & 0 \\ 0 & 0 & \frac {(1 - \nu)}{2} \end{array} \right], \tag {6.7}
in which E and \nu are the elastic modulus and Poisson's ratio respectively.
The body forces b are written as
\boldsymbol {b} = [ b _ {x}, b _ {y} ] ^ {T}, \tag {6.8}
in which b_{x} and b_{y} are the body forces per unit volume in the x and y directions respectively.
Boundary tractions t may be expressed as
\boldsymbol {t} = [ t _ {x}, t _ {y} ] ^ {T}, \tag {6.9}
in which t_{x} and t_{y} are the boundary tractions per unit length.
An element of volume d\Omega is given as
d \Omega = t d x d y, \tag {6.10}
where t is the plate thickness.
6.2.4 Plane strain
For plane strain problems the thickness dimension normal to a certain plane (say the xy plane) is large compared with the typical dimensions in the xy plane and the body is subjected to loads in the xy plane only. For plane strain problems ^{(2)} it may be assumed that the displacements in the z direction are negligible and that the in-plane displacements u and v are independent of z. Figure 6.3 illustrates some typical plane strain problems.
The displacements are then listed in the vector
\boldsymbol {u} = [ u, v ] ^ {T}, \tag {6.11}
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Pure geometric diagram of a trapezoidal structure with horizontal and vertical lines, no text or symbols present
(a)
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Concentric circles with radial arrows, no text or symbols present
(b)
Fig. 6.3 Typical plane strain problems.
in which u and v are the in-plane displacements in the x and y directions respectively.
The in-plane strain components may be expressed as
\epsilon = [ \epsilon_ {x}, \epsilon_ {y}, \gamma_ {x y} ] ^ {T}, \tag {6.12}
where \epsilon_{x} , \epsilon_{y} and \gamma_{xy} have the same meaning as the strain components in plane stress applications.
Again the virtual displacements and associated virtual strains are respectively given as
\delta \boldsymbol {u} = [ \delta u, \delta v ] ^ {T}, \tag {6.13}
and
\delta \epsilon = \left[ \frac {\partial (\delta u)}{\partial x}, \frac {\partial (\delta v)}{\partial y}, \frac {\partial (\delta u)}{\partial y} + \frac {\partial (\delta v)}{\partial x} \right] ^ {T}. \tag {6.14}
The stress–strain relationships may be written in the form
\sigma = D \epsilon , \tag {6.15}
where the stresses \sigma = [\sigma_x, \sigma_y, \tau_{xy}]^T have the same meaning as the stresses in plane stress applications.
For linear elastic materials the stress-strain or constitutive matrix D is given as
\boldsymbol {D} = \frac {E}{(1 + \nu) (1 - 2 \nu)} \left[ \begin{array}{c c c} (1 - \nu) & \nu & 0 \\ \nu & (1 - \nu) & 0 \\ 0 & 0 & \frac {(1 - 2 \nu)}{2} \end{array} \right]. \tag {6.16}
Note that the stress normal to the xy plane is nonzero and may be evaluated as
\sigma_ {z} = \nu (\sigma_ {x} + \sigma_ {y}). \tag {6.17}
The body forces b and surface tractions t have the same meaning as those adopted for plane stress problems.
A typical element of volume is given as
d \Omega = d x d y. \tag {6.18}
under the assumption that a unit slice of the problem is being analysed.
6.2.5 Axisymmetric solids
For a three-dimensional solid which is symmetrical about its centreline axis (which coincides with the z axis) and which is subjected to loads and boundary conditions that are symmetrical about this axis, then the behaviour ^{(2)} is independent of the circumferential coordinate \theta . Figure 6.4 shows a typical axisymmetric solid.
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Axisymmetric loading Axisymmetric loading r, u z, w
Fig. 6.4 A typical axisymmetric solid.
The displacements may here be expressed as
\boldsymbol {u} = [ u, w ] ^ {T}, \tag {6.19}
where u and w are the displacements in the r and z directions respectively.
The nonzero strains are given as
\epsilon = [ \epsilon_ {r}, \epsilon_ {\theta}, \epsilon_ {z}, \gamma_ {r z} ] ^ {T}, \tag {6.20}
where for small displacements, the normal strains are given as
\epsilon_ {r} = \frac {\partial u}{\partial r}, \quad \epsilon_ {\theta} = \frac {u}{r} \quad \text { and } \quad \epsilon_ {z} = \frac {\partial w}{\partial z},
and the shear strain is
\gamma_ {r z} = \frac {\partial u}{\partial z} + \frac {\partial w}{\partial r}.
Virtual displacements and associated virtual strains are respectively given as
\delta \boldsymbol {u} = [ \delta \boldsymbol {u}, \delta \boldsymbol {w} ] ^ {T}, \tag {6.21}
and
\delta \epsilon = \left[ \frac {\dot {c} (\delta u)}{\partial r}, \frac {\delta u}{r}, \frac {\dot {c} (\delta w)}{\dot {c} z}, \frac {\dot {c} (\delta u)}{\dot {c} z} + \frac {\dot {c} (\delta w)}{\dot {c} r} \right] ^ {T}. \tag {6.22}
The stress-strain relationships are given as
\sigma = D \epsilon , \tag {6.23}
where \sigma = [\sigma_{r}, \sigma_{\theta}, \sigma_{z}, \tau_{rz}]^{T} , in which \sigma_{r}, \sigma_{\theta} and \sigma_{z} are the normal stresses in the r, \theta and z directions respectively and \tau_{rz} is the shear stress in the rz plane.
For linear elastic materials, the stress–strain matrix is given as
D = \frac {E}{(1 + \nu) (1 - 2 \nu)} \left[ \begin{array}{c c c c} (1 - \nu) & \nu & 0 & 0 \\ \nu & (1 - \nu) & \nu & 0 \\ 0 & \nu & (1 - \nu) & 0 \\ 0 & 0 & 0 & \frac {(1 - 2 \nu)}{2} \end{array} \right] \tag {6.24}
The body forces are given as
\boldsymbol {b} = [ b _ {r}, b _ {z} ] ^ {T}, \tag {6.25}
where b_{r} and b_{z} are the body forces/unit volume in the r and z direction respectively.
The boundary tractions may be expressed as
\boldsymbol {t} = [ t _ {r}, t _ {z} ] ^ {T}, \tag {6.26}
where t_{r} and t_{z} are the boundary tractions/unit surface in the r and z directions.
An elemental volume is given as
d \Omega = 2 \pi r d r d z. \tag {6.27}
6.2.6 Mindlin plates
In Mindlin plate theory it is possible to allow for transverse shear deformation. It thus offers an alternative to classical Kirchhoff thin plate theory. The main assumptions are that:
(a) displacements are small compared with the plate thickness,
(b) the stress normal to the midsurface of the plate is negligible,
(c) normals to the midsurface before deformation remain straight but not necessarily normal to the midsurface after deformation.
A typical Mindlin plate is shown in Fig. 6.5. Note that Mindlin plate theory is the two-dimensional equivalent of Timoshenko beam theory which was discussed in Chapter 5.
The main displacement parameters can be expressed
\boldsymbol {u} = [ w, \theta_ {x}, \theta_ {y} ] ^ {T}, \tag {6.28}
in which w is the lateral plate displacement normal to the xy plane and variables \theta_{x} and \theta_{y} are the normal rotations in the xz and yz planes. Here it should be noted that
\theta_ {x} = \frac {\dot {c} w}{\partial x} - \phi_ {x} \quad \text { and } \quad \theta_ {y} = \frac {\dot {c} w}{\dot {c} y} - \phi_ {y}, \tag {6.29}
where \theta_{x} and \theta_{y} are the rotations of the normal in the xz and yz planes
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z, w Qx Mxy Qy Myx My Myx Myx Myx Mxy Mxy Qy Qx θx x θy Fig. 6.5. A typical Mindlin plate
Fig. 6.5 A typical Mindlin plate.
respectively and are integrated measures of the transverse shear strain. In thin plate theory it is assumed that shear rotations \phi_{x} and \phi_{y} , defined below, are equal to zero.
The strains, or more exactly the strain resultants, may be expressed as
\epsilon = [ r _ {x}, r _ {y}, r _ {x y}, \phi_ {x}, \phi_ {y} ] ^ {T}, \tag {6.30}
where the curvatures are given as
r _ {x} = - \frac {\partial \theta_ {x}}{\partial x} \quad \text { and } \quad r _ {y} = - \frac {\partial \theta_ {y}}{\partial y},
and the twisting curvature is
r _ {x y} = - \left(\frac {\partial \theta_ {y}}{\partial x} + \frac {\partial \theta_ {x}}{\partial y}\right).
The shear strains are expressed as
\phi_ {x} = \left(\frac {\partial w}{\partial x} - \theta_ {x}\right) \quad \text { and } \quad \phi_ {y} = \left(\frac {\partial w}{\partial y} - \theta_ {y}\right). \tag {6.31}
Virtual displacements and rotations and associated virtual curvatures and shear strains are respectively given as
\delta \boldsymbol {u} = [ \delta w, \delta \theta_ {x}, \delta \theta_ {y} ] ^ {T}, \tag {6.32}
and
\delta \epsilon = \left[ - \frac {\partial (\delta \theta_ {x})}{\partial x}, - \frac {\partial (\delta \theta_ {y})}{\partial y}, - \frac {\partial (\delta \theta_ {x})}{\partial y} - \frac {\partial (\delta \theta_ {y})}{\partial x}, \right.
\left. \frac {\partial (\delta w)}{\partial x} - \delta \theta_ {x}, \frac {\partial (\delta w)}{\partial y} - \delta \theta_ {y} \right] ^ {T}. \tag {6.33}
The constitutive relationships are given in the form
\sigma = D \epsilon , \tag {6.34}
where
\sigma = [ M _ {x}, M _ {y}, M _ {x y}, Q _ {x}, Q _ {y} ] ^ {T},
in which M_{x} and M_{y} are the direct bending moments and M_{xy} is the twisting moment. The quantities Q_{x} and Q_{y} are the shear forces in the xz and yz planes.
For an isotropic elastic material
\boldsymbol {D} = \left[ \begin{array}{c c c c c} D & \nu D & 0 & 0 & 0 \\ \nu D & D & 0 & 0 & 0 \\ 0 & 0 & \frac {(1 - \nu)}{2} D & 0 & 0 \\ 0 & 0 & 0 & S & 0 \\ 0 & 0 & 0 & 0 & S \end{array} \right], \tag {6.35}
in which for a plate of thickness t
D = \frac {E t ^ {3}}{1 2 (1 - \nu^ {2})} \quad \text { and } \quad S = \frac {G t}{1 . 2},
where G is the shear modulus and the factor 1.2 is a shear correction term.
Here we will not consider surface tractions. For a more complete discussion of this and other aspects of Mindlin plate theory the reader is directed to the work of Hughes and his coworkers. ^{(3)} We will only consider body forces of the form
\boldsymbol {b} = [ q, 0, 0 ] ^ {T}, \tag {6.36}
where q is the lateral distributed loading per unit area.
An elemental plate area is given as
d \Omega = d x d y. \tag {6.37}
6.3 Isoparametric finite element representation
6.3.1 Governing equations
In this section we present the discretised governing equations for the solid mechanics applications described in Sections 6.2.3–6.2.6. In a finite element representation, the displacements and strains and their virtual counterparts may be expressed by the relationships
\boldsymbol {u} = \sum_ {i = 1} ^ {n} N _ {i} \boldsymbol {d} _ {i}, \quad \delta \boldsymbol {u} = \sum_ {i = 1} ^ {n} N _ {i} \delta \boldsymbol {d} _ {i}, \tag {6.38}
\epsilon = \sum_ {i = 1} ^ {n} B _ {i} d _ {i}, \quad \delta \epsilon = \sum_ {i = 1} ^ {n} B _ {i} \delta d _ {i}, \tag {6.39}
where, for node i , d_{i} is the vector of nodal variables, ^{*} \delta d_{i} is the vector of virtual nodal variables, N_{i} = I N_{i} is the matrix of global shape functions \dagger and B_{i} is the global strain-displacement matrix. The total number of nodes in the whole mesh is n .
If (6.38) and 6.39) are substituted into the virtual work expression (6.1) then we obtain
\sum_ {i = 1} ^ {n} \left[ \delta \boldsymbol {d} _ {i} \right] ^ {T} \left\{\int_ {\Omega} \left[ \boldsymbol {B} _ {i} \right] ^ {T} \sigma d \Omega - \int_ {\Omega} \left[ N _ {i} \right] ^ {T} \boldsymbol {b} d \Omega - \int_ {\Gamma_ {t}} \left[ N _ {i} \right] ^ {T} \boldsymbol {t} d \Gamma \right\} = 0, \tag {6.40}
and since (6.40) must be true for an arbitrary set of virtual displacements \delta d_{i} then we have for each node i an equation of the form
\int_ {\Omega} \left[ \boldsymbol {B} _ {i} \right] ^ {T} \sigma d \Omega - \int_ {\Omega} \left[ N _ {i} \right] ^ {T} \boldsymbol {b} d \Omega - \int_ {\Gamma_ {t}} \left[ N _ {t} \right] ^ {T} \boldsymbol {t} d \Gamma = 0. \tag {6.41}
If we use C(0) isoparametric finite element representations we can evaluate contributions to (6.41) separately from each element.
The displacements can be expressed in the usual way as
\boldsymbol {u} ^ {(e)} = \sum_ {i = 1} ^ {r} N _ {i} ^ {(e)} \boldsymbol {d} _ {i} ^ {(e)}, \tag {6.42}
where, for local node i of element e, N^{(e)} = I N^{(e)} is the matrix of shape functions and the vector of variables is d_{i}^{(e)} . There are r local nodes in each element e.
Typical 4-, 8- and 9-node isoparametric element shape functions are shown and listed in Figs. 6.1(a), (b) and (c) respectively.
Note that in an isoparametric representation we may use the following representation for the x and y coordinates within an element