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Elements
\overline {{M}} _ {p r i m} ^ {m n} = \int_ {z} \rho z ^ {2} \mathrm{d} z \int_ {S} H ^ {m} (S) H ^ {n} (S) 2 \pi R \mathrm{d} S.
These primitive mass matrices are the same mass matrices that are used for the regular axisymmetric shell elements. We can also define the circumferential distribution matrices
f _ {1} ^ {p q} = \frac {1}{2 \pi} \int_ {0} ^ {2 \pi} R ^ {p} (\theta) R ^ {q} (\theta) \mathrm{d} \theta \quad \mathrm{and}
f _ {2} ^ {p q} = \frac {1}{2 \pi} \int_ {0} ^ {2 \pi} \sin p \theta \sin q \theta \mathrm{d} \theta .
The various components of the mass matrix then take the form
M _ {u _ {r}} ^ {m n p q} = M _ {p r i m} ^ {m n} f _ {1} ^ {p q}
M _ {u _ {z}} ^ {m n p q} = M _ {p r i m} ^ {m n} f _ {1} ^ {p q}
M _ {u _ {\theta}} ^ {m n p q} = M _ {p r i m} ^ {m n} f _ {2} ^ {p q}
M _ {\phi_ {r}} ^ {m n p q} = \overline {{M}} _ {p r i m} ^ {m n} f _ {2} ^ {p q}
M _ {\phi_ {z}} ^ {m n p q} = \overline {{M}} _ {p r i m} ^ {m n} f _ {2} ^ {p q}
M _ {\phi_ {\theta}} ^ {m n p q} = \overline {{M}} _ {p r i m} ^ {m n} f _ {1} ^ {p q}.
The circumferential distribution matrices can be evaluated for various values of the number of terms P in the Fourier series. After some calculations the following results are obtained:
P = 1:
f _ {1} ^ {p q} = \frac {1}{8} \left( \begin{array}{c c} 3 & 1 \\ 1 & 3 \end{array} \right), \qquad f _ {2} ^ {p q} = \frac {1}{2} (1).
P = 2:
f _ {1} ^ {p q} = \frac {1}{3 2} \left( \begin{array}{c c c} 7 & 2 & - 1 \\ 2 & 1 2 & 2 \\ - 1 & 2 & 7 \end{array} \right), \qquad f _ {2} ^ {p q} = \frac {1}{2} \left( \begin{array}{c c} 1 & 0 \\ 0 & 1 \end{array} \right).
P = 3:
f _ {1} ^ {p q} = \frac {1}{7 2} \left( \begin{array}{c c c c} 1 1 & 2 & - 2 & 1 \\ 2 & 2 0 & 4 & - 2 \\ - 2 & 4 & 2 0 & 2 \\ 1 & - 2 & 2 & 1 1 \end{array} \right), \qquad f _ {2} ^ {p q} = \frac {1}{2} \left( \begin{array}{c c c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right).
P = 4:
f _ {1} ^ {p q} = \frac {1}{1 2 8} \left( \begin{array}{c c c c c} 1 5 & 2 & - 2 & 2 & - 1 \\ 2 & 2 8 & 4 & - 4 & 2 \\ - 2 & 4 & 2 8 & 4 & - 2 \\ 2 & - 4 & 4 & 2 8 & 2 \\ - 1 & 2 & - 2 & 2 & 1 5 \end{array} \right), \qquad f _ {2} ^ {p q} = \frac {1}{2} \left( \begin{array}{c c c c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right).
3.6.8 Transverse shear stiffness in composite shells and offsets from the midsurface
Transverse shear stiffness
ABAQUS supports element types S3R, S3RS, S4R, S4RS, S4RSW, and S8R for the analysis of laminated composite shells. These elements are based on first-order transverse shear flexible theory in which the transverse shear strain is assumed to be constant through the thickness of the shell. This assumption necessitates the use of shear correction factors. The development of these factors also provides a basis for estimating interlaminar shear stresses in a composite section. This section describes the development of the transverse shear stiffness for element types S3R, S4R, and S8R.
Finite-strain shells
The transverse shear stiffness correction factors are easily shown to be 5 / 6 for isotropic plates. We want to establish the equivalent factors for laminated plates and sandwich constructions. For this purpose we calculate the distribution of transverse shear stress through the thickness of the shell, for the case of unidirectional bending and assuming linear elastic response. Then the shear strain energy, expressed in terms of section forces and strains, is equated to the strain energy of this distribution of transverse shear stresses. This method, outlined below, provides an approximate method for calculating interlaminar shear stresses and supplies reasonable estimates of transverse shear stiffnesses.
Consider a plate in the ( x , y ) plane. Assume only bending and shear in the x-direction, without gradients in the y-direction. Then the membrane forces in the shell are zero: N _ { x x } = N _ { y y } = N _ { x y } = 0 , and \partial / \partial y = 0 for all response variables. In this case equilibrium within the section in the x-direction is
Equation 3.6.8-1
\frac {\partial \sigma_ {x x}}{\partial x} + \frac {\partial \tau_ {x z}}{\partial z} = 0.
Moment equilibrium about the y-axis gives
Equation 3.6.8-2
V _ {x} + \frac {\partial M _ {x x}}{\partial x} = 0,
where V _ { x } is the transverse shear force per unit width in the plate and M _ { x x } is the bending moment per unit width for bending about the y-axis.
Elements
For the bending behavior we assume the strain varies linearly across the section:
\epsilon_ {\alpha \beta} = \overline {{\epsilon}} _ {\alpha \beta} - (z - z _ {0}) \overline {{\kappa}} _ {\alpha \beta},
where \overline { { \epsilon } } _ { \alpha \beta } is the membrane strain of the reference surface z = z _ { 0 } and \overline { { \kappa } } _ { \alpha \beta } is the curvature of that surface.
If the response of the shell is linear elastic, any in-plane component of stress at a point through the shell section is given by
Equation 3.6.8-3
\sigma_ {\alpha \beta} = D _ {\alpha \beta \gamma \delta} (\overline {{\epsilon}} _ {\gamma \delta} - (z - z _ {0}) \overline {{\kappa}} _ {\gamma \delta}),
where the plane stress elastic stiffness, D _ { \alpha \beta \gamma \delta } , is defined from the elasticity and orientation of the material at the particular layer of the shell. Greek subscripts take the range (1; 2).
Integrating through the thickness and inverting the resultant section stiffness provides the 6 \times 6 section flexibility matrix, [H ]:
\left\{ \begin{array}{l} \overline {{\epsilon}} _ {\alpha \beta} \\ \overline {{\kappa}} _ {\alpha \beta} \end{array} \right\} = [ H ] \left\{ \begin{array}{l} N _ {\alpha \beta} \\ M _ {\alpha \beta} \end{array} \right\}.
We have already assumed that N _ { x x } = N _ { y y } = N _ { x y } . We now also assume that M _ { y y } = M _ { x y } = 0 \mathrm { . } ; that is, that it is possible to have no bending in the y-direction without any restraining moments associated with the y-direction. This is clearly not the case for an unbalanced composite section, but we still use it as a simplifying assumption to obtain the shear correction factors. Thus,
Equation 3.6.8-4
\left\{ \begin{array}{l} \overline {{\epsilon}} _ {\alpha \beta} \\ \overline {{\kappa}} _ {\alpha \beta} \end{array} \right\} = \left\{H _ {i 4} \right\} M _ {x x},
where \{ H _ { i 4 } \} is the fourth column of [H]. Combining this result with the elastic stiffness at a point through the shell thickness provides the in-plane stress components in terms of M _ { x x } as
Equation 3.6.8-5
\sigma_ {x x} = \left(B _ {x 1} - (z - z _ {0}) B _ {x 2}\right) M _ {x x},
where
B _ {x 1} = D _ {x x x x} H _ {1 4} + D _ {x x y y} H _ {2 4} + D _ {x x x y} H _ {3 4}
and
B _ {x 2} = D _ {x x x x} H _ {4 4} + D _ {x x y y} H _ {5 4} + D _ {x x x y} H _ {6 4}.
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Combining the gradient of this equation in the x-direction with the equilibrium equations Equation 3.6.8-1 and Equation 3.6.8-2 yields a description of the variation of the transverse shear stress through the thickness of the plate:
Equation 3.6.8-6
\frac {\partial \tau_ {x z}}{\partial z} = \left(B _ {x 1} - (z - z _ {0}) B _ {x 2}\right) V _ {x}.
In calculating \partial \sigma _ { x x } / \partial x we have assumed that the elasticity and thickness of the composite section do not vary (or vary slowly) with position along the shell.
A laminated composite shell section consists of N layers 1; 2; 3; : : : with different values of ( B _ { x 1 } ^ { 1 } , B _ { x 2 } ^ { 1 } ) at layer 1, ( B _ { x 1 } ^ { 2 } , B _ { x 2 } ^ { 2 } ) at layer 2 , . . . ( B _ { x 1 } ^ { N } , B _ { x 2 } ^ { N } ) at layer N . Layer i extends from z _ { i } \mathrm { t } 0 \ z _ { i + 1 } and its thickness is t _ { i } = z _ { i + 1 } - z _ { i } . Integrating Equation 3.6.8-6 through the shell, using the boundary conditions \tau _ { x z } = 0 \mathrm { a t } z = 0 , \tau _ { x z } ^ { i } = \tau _ { x z } ^ { i + 1 } \mathrm { a t } z = z _ { i + 1 } and \tau _ { x z } = 0 \mathrm { a t } z = z _ { N + 1 } , gives the transverse shear stress in layer i as
Equation 3.6.8-7
\tau_ {x z} ^ {i} = \left[ B _ {x 1} ^ {i} (z - z _ {i}) - \left(\frac {1}{2} \left(z ^ {2} - z _ {i} ^ {2}\right) - z _ {x 0} (z - z _ {i})\right) B _ {x 2} ^ {i} + B _ {x 0} ^ {i} \right] V _ {x},
where
B _ {x 0} ^ {i} = \sum_ {j = 1} ^ {i - 1} t _ {j} \left[ B _ {x 1} ^ {j} - \left(\frac {1}{2} (z _ {j + 1} + z _ {j}) - z _ {x 0}\right) B _ {x 2} ^ {j} \right]
and
z _ {x 0} = \frac {\sum_ {i = 1} ^ {N} t _ {i} \left(\frac {1}{2} (z _ {i + 1} + z _ {i}) B _ {x 2} ^ {i} - B _ {x 1} ^ {i}\right)}{\sum_ {i = 1} ^ {N} t _ {i} B _ {x 2} ^ {i}}.
The subscript z _ { x 0 } is used instead of z _ { 0 } in this case because the result is associated with pure bending in the x-direction.
The variation of \tau _ { y z } through the shell thickness is obtained using a similar procedure, based on pure bending in the y-direction.
These results provide the estimates of interlaminar shear stresses.
We define the shear flexibility of the section by matching the shear strain energy obtained by integrating the elastic strain energy density associated with transverse shear stress distribution obtained above:
\frac {1}{2} \left\lfloor V _ {x} \right. \left. V _ {y} \right\rfloor \left[ F ^ {s} \right] \left\{V _ {x} \atop V _ {y} \right\} = \frac {1}{2} \sum_ {i = 1} ^ {N} \int_ {z _ {i}} ^ {z _ {i + 1}} \left\lfloor \tau_ {x z} \right. \left. \tau_ {y z} \right\rfloor \left[ F ^ {i} \right] \left\{ \begin{array}{l} \tau_ {x z} \\ \tau_ {y z} \end{array} \right\} d z,
Elements
where \left[ F ^ { s } \right] is the shear flexibility of the section and \left[ F ^ { i } \right] is the continuum transverse shear flexibility within layer i. Here we introduce the assumption that the transverse shear flexibility within a layer is not coupled to the in-plane flexibility. This is usually the case for shell constructions.
Substituting the relations for \tau _ { x z } ^ { i } and \tau _ { y z } ^ { i } into the above equation and integrating defines the shear flexibility of the section as
\begin{array}{l} F _ {x x} ^ {s} = \sum_ {i = 1} ^ {N} F _ {x x} ^ {i} t _ {i} \bigg [ (B _ {x 0} ^ {i}) ^ {2} + t _ {i} B _ {x 0} ^ {i} \left(B _ {x 1} ^ {i} - (z _ {i} - z _ {x 0}) B _ {x 2} ^ {i}\right) + \frac {1}{3} t _ {i} ^ {2} \left(B _ {x 1} ^ {i} - (z _ {i} - z _ {x 0}) B _ {x 2} ^ {i}\right) ^ {2} - \\ - \left. \frac {1}{4} t _ {i} ^ {3} B _ {x 2} ^ {i} \left(B _ {x 1} - (z _ {i} - z _ {x 0}) B _ {x 2} ^ {i}\right) ^ {2} + \frac {1}{2 0} t _ {i} ^ {4} (B _ {x 2} ^ {i}) ^ {2} \right] \\ \end{array}
F _ {y y} ^ {s} = \sum_ {i = 1} ^ {N} F _ {y y} ^ {i} t _ {i} \bigg [ (B _ {y 0} ^ {i}) ^ {2} + t _ {i} B _ {y 0} ^ {i} \left(B _ {y 1} ^ {i} - (z _ {i} - z _ {y 0}) B _ {y 2} ^ {i}\right) + \frac {1}{3} t _ {i} ^ {2} \left(B _ {y 1} ^ {i} - (z _ {i} - z _ {y 0}) B _ {y 2} ^ {i}\right) ^ {2} -
- \left. \frac {1}{4} t _ {i} ^ {3} B _ {y 2} ^ {i} (B _ {y 1} - (z _ {i} - z _ {y 0}) B _ {y 2} ^ {i}) ^ {2} + \frac {1}{2 0} t _ {i} ^ {4} (B _ {y 2} ^ {i}) ^ {2} \right]
\begin{array}{l} F _ {x y} ^ {s} = \sum_ {i = 1} ^ {N} F _ {x y} ^ {i} t _ {i} \bigg [ B _ {x 0} ^ {i} B _ {y 0} ^ {i} + \frac {1}{2} t _ {i} \left[ B _ {x 0} ^ {i} \left(B _ {y 1} ^ {i} - (z _ {i} - z _ {y 0}) B _ {y 2} ^ {i}\right) + B _ {y 0} ^ {i} (B _ {x 1} - (z _ {i} - z _ {x 0}) B _ {x 2} ^ {i} \right] + \\ + \frac {1}{3} t _ {i} ^ {2} (B _ {x 1} ^ {i} - (z _ {i} - z _ {x 0}) B _ {x 2} ^ {i}) (B _ {y 1} ^ {i} - (z _ {i} - z _ {y 0}) B _ {y 2} ^ {i}) - \\ - \frac {1}{8} t _ {i} ^ {3} \left[ B _ {x 2} ^ {i} (B _ {y 1} ^ {i} - (z _ {0} - z _ {y 0}) B _ {y 2} ^ {i}) + B _ {y 2} ^ {i} (B _ {x 1} ^ {i} - (z _ {0} - z _ {x 0}) B _ {x 2} ^ {i}) \right] + \\ \left. \right. + \left. \frac {1}{2 0} t _ {i} ^ {4} B _ {x 2} ^ {i} B _ {y 2} ^ {i} \right]. \\ \end{array}
The transverse shear stiffness of the section is then available as \left[ F _ { \alpha \beta } ^ { s } \right] ^ { - 1 } . Notice that F _ { 1 2 } ^ { s } will be nonzero if any layer is anisotropic or orthotropic in a local system (since then F _ { x y } ^ { i } will be nonzero).
Small-strain shells
When the shell resultant forces at each increment are computed for pre-integrated sections, the transverse shear forces for small-strain shell elements S3RS, S4RS, and S4RSW are computed using the transverse shear stiffness derived for finite-strain shells. For numerically integrated sections the transverse shear behavior is based on a simplified stiffness for improved computational performance. For single or multilayer isotropic sections and single layer orthotropic sections, the transverse shear force converges to the proper thin and thick shell transverse shear solution and the transverse shear stress is assumed to have a constant distribution. The transverse shear stiffness is approximate for multilayer orthotropic sections, where the transverse shear stress distribution is assumed piecewise constant. Convergence to the proper transverse shear behavior for this case may not be obtained as
shells become thick and principal material directions deviate from the principal section directions.
Offset: reference surface versus midsurface
In ABAQUS the geometry of the shell is defined by kinematic variables that exist at the nodes on the shell reference surface. The kinematics of the shell theory consist of measuring membrane strain on the reference surface and bending strain from the derivatives of the unit normal vector on the reference surface. The default reference surface is the shell midsurface. However, many situations arise in which it is more convenient to define the reference surface as offset from the midsurface. In this case we assume that the in-plane strain at any material point varies linearly across the section:
\epsilon_ {\alpha \beta} = \overline {{\epsilon}} _ {\alpha \beta} - (z - z _ {0}) \overline {{\kappa}} _ {\alpha \beta},
where \alpha and \beta represent the two orthogonal axes on the reference surface, \overline { { \epsilon } } _ { \alpha \beta } is the membrane strain of the reference surface, z _ { 0 } is the distance to the reference surface measured from the midsurface, and \overline { { \kappa } } _ { \alpha \beta } is the curvature of that surface. The positive values of z _ { 0 } are in the positive normal direction.
When z _ { 0 } = t / 2 _ { \mathrm { ; } } the top surface of the shell is the reference surface, where t is the shell thickness. The bottom surface of the shell becomes the reference surface when z _ { 0 } = - t / 2 . When z _ { 0 } = 0 , the midsurface represents the reference surface.
If the response of the shell is linear elastic, any in-plane component of stress at a point through the shell section, \sigma _ { \alpha \beta } , is given by
D _ {\alpha \beta \gamma \delta} (\overline {{\epsilon}} _ {\gamma \delta} - (z - z _ {0}) \overline {{\kappa}} _ {\gamma \delta}),
where the plane stress elastic stiffness, D _ { \alpha \beta \gamma \delta } , is defined from the elasticity and orientation of the material at the particular layer of the shell. Greek subscripts take the range (1; 2).
The section force and moment resultants per unit length can then be defined as
N _ {\alpha \beta} = \int_ {- z _ {0} - t / 2} ^ {- z _ {0} + t / 2} \sigma_ {\alpha \beta} d z,
M _ {\alpha \beta} = \int_ {- z _ {0} - t / 2} ^ {- z _ {0} + t / 2} \sigma_ {\alpha \beta} z d z.
Integrating the above equations through the thickness leads to the resultant section stiffness, [A]:
\left\{ \begin{array}{l} N _ {\alpha \beta} \\ M _ {\alpha \beta} \end{array} \right\} = [ A ] \left\{ \begin{array}{l} \overline {{\epsilon}} _ {\alpha \beta} \\ \overline {{\kappa}} _ {\alpha \beta} \end{array} \right\}.
3.6.9 Rotary inertia for 5 degree of freedom shell elements
Some of the shell elements in ABAQUS/Standard (S4R5, S8R5, S9R5, and STRI65) use two variables at a node to define the change in the shell normal at the node, \mathbf { n } ^ { N } , during an increment. At some nodes in these elements and in other elements we use the three components of the rotation triplet as the
Elements
rotation degrees of freedom. To provide inertia for all of these nodal variables, at a node N with two "rotation" variables we define
\partial^ {p} \mathbf {n} ^ {N} = \partial^ {p} \phi_ {2} ^ {\overline {{N}}} \mathbf {e} _ {1} ^ {\overline {{N}}} - \partial^ {p} \phi_ {1} ^ {\overline {{N}}} \mathbf {e} _ {2} ^ {\overline {{N}}},
where \partial ^ { p } denotes any time derivative or variation of the following quantity and the \mathbf { e } _ { \alpha } ^ { N } are the basis vectors used to define the rotational variables at the node during this increment. Barred superscripts are not summed.
This expression neglects any rotation of the basis system that occurs during the increment. This is an approximation in large-displacement analysis: it is adopted for the sake of simplicity, based on the argument that we are not attempting to model the rotary inertia accurately.
At a node at which we use the three global rotation components we define
\partial^ {p} \mathbf {n} ^ {N} = \partial^ {p} \phi_ {i} ^ {\overline {{N}}} \mathbf {e} _ {i} ^ {\overline {{N}}},
where in this case the \mathbf { e } _ { i } are the global Cartesian basis vectors. This definition is artificial and serves simply to associate a reasonable measure of inertia with the rotational degrees of freedom.
The first-order elements in ABAQUS use a lumped mass matrix. In this case M ^ { N M } is diagonal, so that the rotary inertia contribution at node N is
\eta M ^ {\overline {{N N}}} \delta \phi_ {\alpha} ^ {\overline {{N}}} \ddot {\phi} _ {\alpha} ^ {\overline {{N}}},
where ® sums over the number of rotation components used at the node (2 or 3).
For a consistent mass element the rotary inertia contribution is
\sum_ {M = 1} ^ {\mathrm{nodes}} \sum_ {N = 1} ^ {\mathrm{nodes}} M ^ {N M} \delta \phi_ {\alpha} ^ {N} \ddot {\phi} _ {\beta} ^ {M} \mathbf {e} _ {\alpha} ^ {N} \cdot \mathbf {e} _ {\beta} ^ {M},
where ® and \beta sum over the number of rotation components used at each node (2 or 3).
The time integration algorithms require initial conditions for each increment. For implicit integration these are the velocities and accelerations of the variables at the start of the increment, \dot { u } | ^ { 0 } and { \ddot { u } } | ^ { 0 } .
At a node where three global rotation components are used, these initial values are directly available from the solution to the previous increment. At a node where only two variables define the rotation, we convert variables from the basis of one increment to that of the next through the approximation
\partial^ {p} \phi_ {\alpha} | ^ {+} = \partial^ {p} \phi_ {\beta} | ^ {-} \mathbf {e} _ {\alpha} | ^ {+} \cdot \mathbf {e} _ {\beta} | ^ {-},
where j ¡ indicates a variable associated with the previous increment and \mid ^ { + } indicates a variable associated with the current increment. The justifications for this approximation are that it is simple, the incremental rotation will not be large anyway, and we are not trying to model the rotary inertia effect
with high accuracy.
3.7 Rebar
3.7.1 Rebar modeling in two dimensions
Let gi; i = 1; 2 be the element's usual isoparametric coordinates. Let r be an isoparametric coordinate along the line where the face of the element intersects the plane of reinforcement, with - 1 \leq r \leq 1 in an element (see Figure 3.7.1-1).
Figure 3.7.1-1 Rebar in a solid, two-dimensional element.
text_image
t r g₂ g₁ 2 3 1 r
The plane of reinforcement is always perpendicular to the element face.
The rebar will be integrated at one or two points, depending on the order of interpolation in underlying elements. The volume of integration (¢V ), position, rebar strain ("), and first and second variations of rebar strain (±" and d±") at each point are calculated as
\Delta V = \frac {A _ {r}}{S _ {r}} \left(\frac {\partial \mathbf {x}}{\partial r} \cdot \frac {\partial \mathbf {x}}{\partial r}\right) ^ {\frac {1}{2}} t _ {0} W _ {N},
where
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t _ { 0 }
is the original thickness for plane elements and 2 \pi x _ { 1 } for axisymmetric elements;
A _ { r }
is the rebar cross-sectional area;
S _ { r }
is the spacing of rebar (for axisymmetric elements S _ { r } = ( x _ { 1 } / x _ { 1 } ^ { 0 } ) S _ { r } ^ { 0 } , where x _ { 1 } ^ { 0 } is the radius where the spacing S _ { r } ^ { 0 } is given);
W _ { N }
is the Gauss weight associated with the integration point along the (r) line;
\mathbf {x} = \mathbf {x} (g _ {i})
is position; and
{\frac {\partial \mathbf {x}}{\partial r}} = {\frac {\partial \mathbf {x}}{\partial g _ {i}}} {\frac {\partial g _ {i}}{\partial r}}.
Strain is
\varepsilon = \frac {1}{2} \ln \left(\frac {d l ^ {2}}{d l _ {o} ^ {2}}\right),
where dl and d l _ { o } measure length along the rebar in the current and initial configurations, respectively.
For the deformations allowed in these elements,
\left(\frac {d l}{d l _ {o}}\right) ^ {2} = \cos^ {2} \alpha \lambda_ {r} ^ {2} + \sin^ {2} \alpha \lambda_ {t} ^ {2},
where ® is the orientation of the rebar from the plane of the model,
\lambda_ {r} ^ {2} = \frac {\partial \mathbf {x}}{\partial r} \cdot \frac {\partial \mathbf {x}}{\partial r} / \frac {\partial \mathbf {x} _ {o}}{\partial r} \cdot \frac {\partial \mathbf {x} _ {o}}{\partial r}
is the squared stretch ratio in the r-direction, and \lambda _ { t } is the stretch ratio in the thickness direction:
\lambda_ {t} = 1
for plane stress or plane strain;
\lambda_ {t} = t / t _ {o}
for generalized plane strain, where t is given in ``Generalized plane strain elements,'' Section 3.2.7; and
\lambda_ {t} = x _ {1} / x _ {1 _ {o}}
for axisymmetric elements.
From these results the first variation of strain is
\delta \varepsilon = \left(\frac {d l _ {o}}{d l}\right) ^ {2} (\cos^ {2} \alpha \frac {\partial \mathbf {x}}{\partial r} \cdot \frac {\partial \delta \mathbf {x}}{\partial r} / \frac {\partial \mathbf {x} _ {o}}{\partial r} \cdot \frac {\partial \mathbf {x} _ {o}}{\partial r} + \delta p _ {t}),
where
\delta p _ {t} = 0
for plane stress and plane strain,
\delta p _ {t} = \sin^ {2} \alpha t \delta t / t _ {o} ^ {2}
for generalized plane strain, and
\delta p _ {t} = \sin^ {2} \alpha x _ {1} \delta x _ {1} / x _ {1 _ {o}} ^ {2}
for axisymmetric cases.
The second variation of strain is then
d \delta \varepsilon = - 2 \left(\frac {d l _ {o}}{d l}\right) ^ {2} \left(\cos^ {2} \alpha \frac {\partial \mathbf {x}}{\partial r} \cdot \frac {\partial \delta \mathbf {x}}{\partial r} / \frac {\partial \mathbf {x} _ {o}}{\partial r} \cdot \frac {\partial \mathbf {x} _ {o}}{\partial r} + \delta p _ {t}\right)
\cdot \left(\cos^ {2} \alpha \frac {\partial \mathbf {x}}{\partial r} \cdot \frac {\partial d \mathbf {x}}{\partial r} / \frac {\partial \mathbf {x} _ {o}}{\partial r} \cdot \frac {\partial \mathbf {x} _ {o}}{\partial r} + d p _ {t}\right)
+ \left(\frac {d l _ {o}}{d l}\right) ^ {2} \left(\cos^ {2} \alpha \frac {\partial d \mathbf {x}}{\partial r} \cdot \frac {\partial \delta \mathbf {x}}{\partial r} / \frac {\partial \mathbf {x} _ {o}}{\partial r} \cdot \frac {\partial \mathbf {x} _ {o}}{\partial r} + d \delta p _ {t}\right).
3.7.2 Rebar modeling in three dimensions
Let g _ { i } , i = 1 , 2 , 3 ; be the isoparametric coordinates of the basic finite element in which the rebar are placed. Let r _ { \alpha } , \alpha = 1 , 2 , be isoparametric coordinates on the surface of reinforcement, with - 1 \leq r _ { \alpha } \leq 1 . Let t be a material coordinate along the rebar direction. See Figure 3.7.2-1.
Figure 3.7.2-1 Rebar in a solid, three-dimensional element.