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Elements
u (g) = \frac {1}{2} (1 - g) u _ {1} + \frac {1}{2} (1 + g) u _ {2},
and for the 3-node element,
u (g) = \frac {1}{2} g (g - 1) u _ {1} + (1 - g ^ {2}) u _ {2} + \frac {1}{2} g (g + 1) u _ {3},
where u _ { 1 } , u _ { 2 } , and u _ { 3 } are the values of a variable at the nodes and u ( g ) is the interpolated value of this variable.
Strain measure
These are one-dimensional elements, and the only strain considered is that along the axis of the element. The stretch ratio along the axis is
\lambda = \frac {d l}{d L},
where l measures length along the truss axis in the current configuration:
d l = \sqrt {\frac {d \mathbf {x}}{d g} \cdot \frac {d \mathbf {x}}{d g}} d g,
and d L measures length along the axis in the original configuration.
For geometrically nonlinear analysis we use a logarithmic strain measure:
\varepsilon = \ln \left(\frac {d l}{d L}\right).
First variation of strain
The first variation of strain is
\delta \varepsilon = \frac {d g}{d l} \mathbf {t} \cdot \frac {d \delta \mathbf {x}}{d g},
where
\mathbf {t} = \frac {d g}{d l} \frac {d \mathbf {x}}{d g}
is a unit tangent along the truss axis.
Second variation of strain
The second variation of strain is
d \delta \varepsilon = \left(\frac {d g}{d l}\right) ^ {2} \frac {d \delta \mathbf {x}}{d g} \cdot [ \mathbf {I} - 2 \mathbf {t} \mathbf {t} ] \cdot \frac {d d \mathbf {x}}{d g}.
Integration
Stiffness
The linear truss is a constant strain element and so is integrated exactly. The quadratic truss is integrated numerically using two Gauss points.
Mass and consistent loads
A linear truss has two Gauss points. A quadratic truss has three Gauss points.
Virtual work contribution
The virtual work contribution from the stress in a truss element is
\delta W = \int_ {l} a \sigma \delta \varepsilon d l,
where a is the current cross-sectional area of the truss, ¾ is the "true" (Cauchy) stress along the truss, " is the logarithmic strain, and l is the length of the element.
Since we assume the truss is incompressible, a dl = A dL, where A is the original area and L the original length of the truss. So,
\delta W = \int_ {L} A \sigma \delta \varepsilon d L.
This is the form in which the internal virtual work contribution is used for truss elements.
Mixed (hybrid) forms
"Hybrid" truss elements are also available in ABAQUS/Standard. In those elements the axial force at the integration points is taken as an additional variable, with the compatibility condition introduced to define these variables. The formulation is identical to that used for the hybrid beam elements ( ``Hybrid beam elements,'' Section 3.5.4), without the bending terms.
3.4.3 Axisymmetric membranes
ABAQUS includes two libraries of axisymmetric membrane elements, MAX and MGAX, whose geometry is axisymmetric (bodies of revolution) and that can be subjected to axially symmetric loading conditions. In addition, MGAX elements support torsion loading and general material anisotropy.
Therefore, MGAX elements will be referred to as generalized axisymmetric membrane elements, and
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MAX elements will be referred to as regular axisymmetric membrane elements. In both cases the body of revolution is generated by revolving a line that represents the membrane surface (a membrane has negligible thickness) about an axis (the symmetry axis) and is described readily in cylindrical coordinates r , z , and \theta . The radial and axial coordinates of a point on this cross-section are denoted by r and z, respectively. At \theta = 0 , the radial and axial coordinates coincide with the global Cartesian X \cdot - and Y -coordinates.
If the loading consists of radial and axial components that are independent of µ and the material is either isotropic or orthotropic, with µ being a principal material direction, the displacement at any point will have only radial ( u _ { r } ) and axial ( u _ { z } ) components. The only nonzero stress components are \sigma _ { s s } and \sigma _ { \theta \theta } , where s denotes a length measuring coordinate along the line representing the membrane surface on any _ { r - z } plane. The deformation of any _ { r - z } plane (or, more precisely, any r { \mathord { \left/ { \vphantom { \sum } } \right. \kern - delimiterspace } } \mathrm { i n e } ) completely defines the state of stress and strain in the body. Consequently, the geometric model is described by discretizing the reference cross-section at \theta = 0 .
If one allows for a circumferential component of loading (which is independent of \theta ) and general material anisotropy, displacements and stress fields become three-dimensional. However, the problem remains axisymmetric in the sense that the solution does not vary as a function of \theta , and the deformation of the reference _ { r - z } cross-section characterizes the deformation in the entire body. The motion at any point will have--in addition to the aforementioned radial and axial displacements--a twist \phi (in radians) about the z-axis, which is independent of µ. There will also be a nonzero in-plane shear stress, \sigma _ { s \theta } , as a result of the deformation.
This section describes the formulation of the generalized axisymmetric membrane elements. The formulation of the regular axisymmetric membrane elements is a subset of this formulation.
Kinematic description
The coordinate system used with both families of elements is the cylindrical system ( r , z , \theta ) , where r measures the distance of a point from the axis of the cylindrical system, z measures its position along this axis, and \theta measures the angle between the plane containing the point and the axis of the coordinate system and some fixed reference plane that contains the coordinate system axis. The order in which the coordinates and displacements are taken in these elements is based on the convention that z is the second coordinate. This order is not the same as that used in three-dimensional elements in ABAQUS, in which z is the third coordinate; nor is it the order ( r , \theta , z ) that is usually taken in cylindrical systems.
Let \mathbf { e } _ { R } , \mathbf { e } _ { Z } , and \mathbf { e } _ { \Theta } be unit vectors in the radial, axial, and circumferential directions at a point in the undeformed state, as shown in Figure 3.4.3-1.
Figure 3.4.3-1 Cylindrical coordinate system and definition of position vectors.
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text_image
Y e₀ eᵣ X e₀ eᵣ X θ φ θ e_z, e_z X
The reference position X of the point can be represented in terms of the original radius, R, and the axial position, Z :
\mathbf {X} = R \mathbf {e} _ {R} + Z \mathbf {e} _ {Z},
Likewise, let \mathbf { e } _ { r } , \mathbf { e } _ { z } , , and \mathbf { e } _ { \theta } be unit vectors in the radial, axial, and circumferential directions at a point in the deformed state. As shown in Figure 3.4.3-1, the radial and circumferential base vectors depend on the µ coordinate: { \mathbf e } _ { r } = { \mathbf e } _ { r } ( \theta ) and \mathbf { e } _ { \theta } = \mathbf { e } _ { \theta } ( \theta ) .
The current position, x, of the point can be represented in terms of the current radius, r, and the current axial position, z, as
\mathbf {x} = r \mathbf {e} _ {r} + z \mathbf {e} _ {z}.
The general axisymmetric motion at a point on the membrane surface can be described by
Equation 3.4.3-1
r (R, Z) = R + u _ {r} (R, Z),
z (R, Z) = Z + u _ {z} (R, Z),
\theta (R, Z, \Theta) = \Theta + \phi (R, Z).
As this description implies, the degrees of freedom u _ { r } , u _ { z } . , and \phi are independent of £. Moreover, the reference cross-section of interest is at \Theta = 0 ; however, for the benefit of the mathematical analysis to follow, it is important that £ be considered an independent variable in the above expression for µ.
Parametric interpolation and integration
The following isoparametric interpolation scheme is used for the motion:
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u _ {r} = N ^ {N} (g) \bar {u _ {r}} ^ {N},
u _ {z} = N ^ {N} (g) \bar {u _ {z}} ^ {N},
\phi = N ^ {N} (g) \bar {\phi} ^ {N},
where g is the isoparametric coordinate in the reference r-z cross-section at \Theta = 0 ; and \bar { u _ { r } } ^ { N } , \bar { u _ { z } } ^ { N } , \bar { \phi } ^ { N } are the nodal degrees of freedom. The interpolation functions are identical to those used for truss elements (see ``Truss elements,'' Section 3.4.2). All elements use reduced integration.
Deformation gradient
For a material point the deformation gradient F is defined as the gradient of the current position, x, with respect to the original position, X, and is given by
Equation 3.4.3-2
\mathbf {F} = \frac {\partial \mathbf {x}}{\partial \mathbf {X}}.
The components of the deformation gradient require that two sets of orthonormal basis vectors be defined. In the undeformed configuration the basis vectors are defined by
\mathbf {E} _ {i} = \frac {\partial \mathbf {X}}{\partial S _ {i}},
where the S _ { i } denote length measuring coordinates in the reference configuration along the element length and the hoop direction, respectively. Thus,
\mathbf {E} _ {1} = \frac {\partial \mathbf {X}}{\partial S}, \mathbf {E} _ {2} = \frac {\partial \mathbf {X}}{R \partial \Theta}.
In the current configuration ABAQUS formulates the equations in terms of a fixed spatial basis with respect to the axisymmetric twist degree of freedom. The basis vectors convect with the material. However, because of the axisymmetry of the model in the deformed configuration, these vectors can be defined at £ = 0 as
\mathbf {e} _ {i} = \frac {\partial \mathbf {x}}{\partial s _ {i}},
where the s _ { i } denote length measuring coordinates in the current configuration along the element length and the hoop direction, respectively. Thus, the basis vectors in the reference and current configurations can be written as
Equation 3.4.3-3
\mathbf {E} _ {1} = \frac {\partial \mathbf {X}}{\partial S}, \mathbf {E} _ {2} = \frac {\partial \mathbf {X}}{R \partial \Theta}, \mathbf {e} _ {1} = \frac {\partial \mathbf {x}}{\partial s}, \mathbf {e} _ {2} = \frac {\partial \mathbf {x}}{r \partial \theta},
where S and s are length measuring coordinates along the element length in the reference and current
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configurations, respectively. The components of the deformation gradient in the two sets of basis vectors may be computed as
F _ {\alpha \beta} = \mathbf {e} _ {\alpha} \cdot \mathbf {F} \cdot \mathbf {E} _ {\beta}.
Using the definitions of the basis vectors in Equation 3.4.3-3, the components of the deformation gradient tensor are
F _ {1 1} = \frac {\partial r}{\partial s} \frac {\partial r}{\partial S} + \frac {\partial z}{\partial s} \frac {\partial z}{\partial S},
F _ {2 1} = r \frac {\partial \phi}{\partial S},
F _ {1 2} = 0,
F _ {2 2} = \frac {r}{R}.
Virtual work
As discussed in ``Equilibrium and virtual work,'' Section 1.5.1, the formulation of equilibrium (virtual work) requires the virtual velocity gradient, which takes the form
\delta \mathbf {L} = \delta \mathbf {F} \cdot \mathbf {F} ^ {- 1},
where ±F represents the first variation of the deformation gradient tensor. Alternatively, the virtual velocity gradient can be written as
\delta \mathbf {L} = \frac {\partial \delta \mathbf {x}}{\partial \mathbf {x}}.
Recall that ABAQUS formulates the finite element equations in terms of a fixed spatial basis with respect to the axisymmetric twist degree of freedom. Therefore, the desired result for ±F does not simply follow from the linearization of Equation 3.4.3-2. Namely, the contributions from the variations
\delta \mathbf {e} _ {r} = \delta \phi \mathbf {e} _ {\theta} \quad \text { and } \quad \delta \mathbf {e} _ {\theta} = - \delta \phi \mathbf {e} _ {r}
arising from the spin of the coordinate system must be canceled out. To this end, F can be modified according to
\tilde {\mathbf {F}} = \mathbf {R} \cdot \mathbf {F},
where R = I instantaneously, but its variation is given by
\delta \mathbf {R} = \widehat {\delta \boldsymbol {\phi}} \cdot \mathbf {R},
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where \widehat { \delta \phi } is skew-symmetric with components
\left[ \widehat {\delta \phi} \right] = \left[ \begin{array}{c c c} 0 & \delta \phi & 0 \\ - \delta \phi & 0 & 0 \\ 0 & 0 & 0 \end{array} \right],
with respect to the basis \mathbf { e } _ { r } , \mathbf { e } _ { z } , and \mathbf { e } _ { \theta } at \theta = 0
With this modification, the corotational virtual deformation gradient is given by
\delta \tilde {\mathbf {F}} = \delta \mathbf {R} \cdot \mathbf {F} + \mathbf {R} \cdot \delta \mathbf {F} = \widehat {\delta \pmb {\phi}} \cdot \mathbf {F} + \delta \mathbf {F},
and the corotational virtual velocity gradient by
\delta \tilde {\mathbf {L}} = \delta \tilde {\mathbf {F}} \cdot \tilde {\mathbf {F}} ^ {- 1} = \widehat {\delta \phi} + \delta \mathbf {L}.
The individual components of \delta \tilde { \mathbf { L } } are given by
\begin{array}{l} \delta \tilde {L} _ {1 1} = \frac {\partial r}{\partial s} \frac {\partial \delta r}{\partial s} + \frac {\partial z}{\partial s} \frac {\partial \delta z}{\partial s}, \\ \delta \tilde {L} _ {2 1} = r \frac {\partial \delta \phi}{\partial s}, \\ \delta \tilde {L} _ {3 1} = \frac {\partial r}{\partial s} \frac {\partial \delta z}{\partial s} - \frac {\partial z}{\partial s} \frac {\partial \delta r}{\partial s}, \\ \delta \tilde {L} _ {1 2} = 0, \\ \delta \tilde {L} _ {2 2} = \frac {\delta r}{r}, \\ \delta \tilde {L} _ {3 2} = 0. \\ \end{array}
The components \delta \tilde { L } _ { i 3 } are not determined by the kinematics.
Stiffness in the current state
The second variation has the usual contribution:
d \delta \tilde {D} _ {i j} = s y m (\delta \tilde {L} _ {k i} d \tilde {L} _ {k j} - 2 \delta \tilde {D} _ {i k} d \tilde {D} _ {k j}).
Moreover, there are additional contributions from d \delta \tilde { L } _ { i j } , which are given by
\begin{array}{l} d \delta \tilde {L} _ {1 1} = 0, \\ d \delta \tilde {L} _ {2 1} = \delta r \frac {\partial d \phi}{\partial s} + d r \frac {\partial \delta \phi}{\partial s}, \\ d \delta \tilde {L} _ {1 2} = 0, \\ d \delta \tilde {L} _ {2 2} = 0. \\ \end{array}
The remaining terms do not contribute since \sigma _ { i 3 } = 0 .
3.5 Beam elements
3.5.1 Beam element overview
The element library in ABAQUS contains several types of beam elements. A "beam" in this context is an element in which assumptions are made so that the problem is reduced to one dimension mathematically: the primary solution variables are functions of position along the beam axis only. For such assumptions to be reasonable, it is intuitively clear that a beam must be a continuum in which we can define an axis such that the shortest distance from the axis to any point in the continuum is small compared to typical lengths along the axis. This idea is made more precise in the detailed derivations in ``Beam element formulation,'' Section 3.5.2. There are several levels of complexity in the assumptions upon which the reduction to a one-dimensional problem can be made, and different beam elements in ABAQUS use different assumptions.
The simplest approach to beam theory is the classical Euler-Bernoulli assumption, that plane cross-sections initially normal to the beam's axis remain plane, normal to the beam axis, and undistorted. The beam elements in ABAQUS that use cubic interpolation (element types B23, B33, etc.) all use this assumption, implemented in the context of arbitrarily large rotations but small strains. The Euler-Bernoulli beam elements are described in ``Euler-Bernoulli beam elements,'' Section 3.5.3. This approximation can also be used to formulate beams for large axial strains as well as large rotations. The beam elements in ABAQUS that use linear and quadratic interpolation ( B21, B22, B31, B32, etc.) are based on such a formulation, with the addition that these elements also allow "transverse shear strain"; that is, the cross-section may not necessarily remain normal to the beam axis. This extension leads to Timoshenko beam theory (Timoshenko, 1956) and is generally considered useful for thicker beams, whose shear flexibility may be important. (These elements in ABAQUS are formulated so that they are efficient for thin beams--where Euler-Bernoulli theory is accurate--as well as for thick beams: because of this they are the most effective beam elements in ABAQUS.) The large-strain formulation in these elements allows axial strains of arbitrary magnitude, but quadratic terms in the nominal torsional strain are neglected compared to unity, and the axial strain is assumed to be small in the calculation of the torsional shear strain: thus, while the axial strain may be arbitrarily large, only "moderately large" torsional strain is modeled correctly, and then only when the axial strain is not large. We assume that, throughout the motion, the radius of curvature of the beam is large compared to distances in the cross-section: the beam cannot fold into a tight hinge. A further assumption is that the strain in the beam's cross-section is the same in any direction in the cross-section and throughout the section. Some additional assumptions are made in the derivation of these elements: these are introduced in the detailed derivation in ``Beam element formulation,'' Section 3.5.2.
For certain important designs the beam is constructed from thin segments made up into an open section. The response of such open sections is strongly effected by warping, when material particles move out of the plane of the section along lines parallel to the beam axis so as to minimize the shearing between lines along the wall of the section and along the beam axis. The beam element formulation (``Beam element formulation,'' Section 3.5.2) includes provision for such effects. Beam elements that allow for warping of open sections (B31OS, B32OS etc.) are also derived. The particular
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approach used for modeling open section warping in ABAQUS is based on the assumption that the warping amplitude is never large anywhere along the beam axis because the warping will be constrained at some points along the beam--perhaps because one or both ends of the beam are built into a stiff structure or because some form of transverse stiffeners are added.
The regular beam elements can be used for slender and moderately thick beams. For extremely slender beams, for which the length to thickness ratio is O(103) or more and geometrically nonlinear analysis is required (such as pipelines), convergence may become very poor. For such cases use of the hybrid elements, in which the axial (and transverse) forces are treated as independent degrees of freedom, can be beneficial. The hybrid beam formulation is described in ``Hybrid beam elements,'' Section 3.5.4. Distributed pressure loads applied to beams (for example, due to wind or current) will rotate with the beam, leading to follower force effects. The derivation of the load stiffness that accounts for this effect is presented in ``Pressure load stiffness for beam elements,'' Section 3.5.5.
In some piping applications thin-walled, circular, relatively straight pipes are subjected to relatively large magnitudes of internal pressure. This has the effect of creating high levels of hoop stress around the wall of the pipe section so that, if the section yields plastically, the axial yield stress will be different in tension and compression because of the interaction with this hoop stress. The PIPE elements allow for this effect by providing uniform radial expansion of the cross-section caused by internal pressure.
In other piping cases thin-walled straight pipes might be subjected to large amounts of bending so that the section collapses ("Brazier collapse"); or a section of pipe may already be curved in its initial configuration--it might be an "elbow." In such cases the ovalization and, possibly, warping, of the cross-section may be important: these effects can reduce the bending stiffness of the member by a factor of five or more in common piping designs. For materially linear analysis these effects can be incorporated by making suitable adjustments to the section's bending stiffness (by multiplying the bending stiffness calculated from beam theory by suitable flexibility factors); but when nonlinear material response is a part of the problem it is necessary to model this ovalization and warping explicitly. Elbow elements are provided for that purpose; they are described in ``Elbow elements,'' Section 3.9.1. Elbow elements look like beam elements to the user, but they incorporate displacement variables that allow ovalization and warping and so are much more complex in their formulation. In particular, ovalization of the section implies a strong gradient of strain with respect to position through the wall of the pipe: this requires numerical integration through the pipe wall, on top of that used around the pipe section, to capture the material response. This makes the elbow elements computationally more expensive than beams.
Since consideration of planar deformation only provides considerable simplification in formulating beam elements, for each beam element type in ABAQUS a corresponding beam element is provided that only moves in the (X; Y ) plane. However, the open section beams are provided only in three dimensions for reasons that are obvious.
3.5.2 Beam element formulation
At a given stage in the deformation history of the beam, the position of a material point in the cross-section is given by the expression
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\hat {\mathbf {x}} (S, S ^ {\alpha}) = \mathbf {x} (S) + f (S) S ^ {\alpha} \mathbf {n} _ {\alpha} (S) + w (S) \psi (S ^ {\alpha}) \mathbf {t} (S).
In this expression \mathbf { x } ( S ) is the position of a point on the centerline, { \mathbf { n } } _ { \alpha } ( S ) are unit orthogonal direction vectors in the plane of the beam section, t(S) is the unit vector orthogonal to \mathbf { n } _ { 1 } and { \mathbf { n } } _ { 2 } , \psi ( S ^ { \alpha } ) is the warping function of the section, w ( S ) is the warping amplitude, and f ( S ) is a cross-sectional scaling factor depending on the stretch of the beam.
These quantities are functions of the beam axis coordinate S and the cross-sectional coordinates S ^ { \alpha } , which are assumed to be distances measured in the original (reference) configuration of the beam. The warping function is chosen such that the value at the origin of the section vanishes: \psi ( 0 ) = 0 .
It is assumed that at the integration points along the beam, the beam section directions are approximately orthogonal to the beam axis tangent s given by
\mathbf {s} = \lambda^ {- 1} \frac {d \mathbf {x}}{d S},
where ¸ is the axial stretch given by
\lambda = \left| \frac {d \mathbf {x}}{d S} \right|.
The normality condition is enforced numerically by penalizing the transverse shear strains
\gamma_ {\alpha} = \mathbf {s} \cdot \mathbf {n} _ {\alpha}.
This condition is assumed to be satisfied exactly in the original configuration.
In what follows, \epsilon _ { \alpha } ^ { \beta } is the alternator
\epsilon_ {1} ^ {2} = - \epsilon_ {2} ^ {1} = 1, \epsilon_ {1} ^ {1} = \epsilon_ {2} ^ {2} = 0.
The curvature of the beam is defined by
b _ {\alpha} = \epsilon_ {\alpha} ^ {\beta} \mathbf {t} \cdot \frac {d \mathbf {n} _ {\beta}}{d S},
and the twist of the beam follows from
b = \mathbf {n} _ {2} \cdot \frac {d \mathbf {n _ {1}}}{d S} = - \mathbf {n} _ {1} \cdot \frac {d \mathbf {n _ {2}}}{d S}.
The "bicurvature" of the beam is defined by
\chi = \frac {d w}{d S}.