Let r be the undeformed shell midsurface position, ξi be element local (possibly nonorthogonal) coordinates with origin in the shell midsurface, and n be the normal to the undeformed midsurface. The thickness of the shell is d, which can vary over the element. The local coordinates ξ1 and ξ2 follow the midsurface, and ξ3 is the coordinate in the normal direction. The normal coordinate has a value of −d/2 on the bottom side of the element, and +d/2 on the top side.

The position of the deformed midsurface is r + u, and the normal after deformation is n + a. To keep the normal a unit vector requires that

In a geometrically linear analysis Equation 5-1 is replaced by the simpler linearized form

The vectors r, u, n, and a are interpolated by the nth-order Lagrange basis functions. The basic assumption is that the position of a point within the shell after deformation has a linear dependence of the thickness coordinate, and thus is

The indices α and β range from 1 to 2. The transverse shear strains in local covariant components are

The constitutive relation for the shell elements is a plane stress assumption, as is customary in shell theory. The strain component in the normal direction ε33 is thus irrelevant. The different parts of the strain tensors above can be written out as

In a geometrically linear analysis, the nonlinear terms (products between u, a, and their derivatives) disappear. In all study types, the contributions from the parts καβ and ωα are ignored. They are small unless the element has an extremely high ratio between thickness and radius of curvature, in which case the errors from using shell theory are large anyway.

where rR is the position of the meshed reference surface and ζo is the offset distance.

where t is the tangent to the edge. For an internal edge, it is possible that there is a discontinuity in thickness or offset. In such a case, the line scale factor will be an average. Edge conditions in are not well defined in such situations because the position of the midsurface can be discontinuous. In practice, errors caused by such effects are small.

The MITC formulation (Ref. 1) does not take the strain components directly from the shape functions of the element. Instead, meticulously selected interpolation functions are selected for the individual strain components. The values of the interpolated strains are then at selected points in the element tied to the value that would be computed from the shape functions. The interpolation functions and tying points are specific to each element shape and order.

Each contribution to the virtual work of the element is numerically integrated over the reference surface while the integration in the thickness direction is performed analytically. The computation of the strain energy from transverse shear deformations uses a correction factor of 5/6 to compensate for the difference between the assumed constant average shear strain and the true parabolic distribution.

In regions where the discretized surface is smooth (which is always the case for plates), the normal of the shell surface is uniquely defined. When two or more shell elements meet at an angle, each element must however keep its own normal direction. It is thus not possible to have the same set of degrees of freedom for the displacement of the normal in such a point. This is automatically handled by the program. The automatic search for these fold lines compares the normals of all boundaries sharing an edge. If the angle between the normals is larger than a certain angle (with a default of 3 degrees) it is considered as a fold line. For a fold line, the assumption is that the angle between the shell normals remains constant. This gives

where the values or j and k range over the number of shells elements with different normals. The third term in Equation 5-2 is relevant only in a large deformation analysis because it is nonlinear. A special case occurs when two adjacent boundaries are parallel but their normal vectors have opposite directions. In this case the special constraint

All functions of ζ are assumed to be of the form ζn. Odd powers will integrate to zero, so typically the cross thickness integration will give factors d (for the case n = 0) and d3/12 (for the case n = 2). The thickness d can be a function of the position.

Because the normal vector displacements are quantities which are less intuitive than the more customary nodal rotations, it is possible to specify the prescribed values in terms of nodal rotations as well as in terms of the normal vector displacement. The representation by normal vector direction is insensitive to whether the analysis is geometrically nonlinear or not. The direction of the shell normal is prescribed in the deformed state, n0. The prescribed values for the actual degrees of freedom, a0, are internally computed as

where Ω0 is the vector of prescribed nodal rotations. This relation is fully defined only when all three components of Ω0 are given. It is also possible to prescribe only one or two of the components of Ω0, while leaving the remaining components free. Because it has no relevance to prescribe the rotation about the normal direction of the shell, it is only possible prescribe individual rotations in a shell local system. In this case, the result becomes one or two constraint relations between the components of a0. The directions are the edge local coordinate system where t1 is the tangent to the edge and t2 is perpendicular and inward from the edge, in the plane of the shell. These constraints are formulated as

Here Ω0i is the prescribed rotation around the axis ti.

where R is a standard rotation matrix, representing the finite rotation about the given rotation vector.

Initial velocities are always given using an angular velocity vector ω as

for the antisymmetry case. Here t1 and t2 are two perpendicular directions in the plane of the shell.

When applied to an edge, there is a local coordinate system where t1 is the tangent to the edge, and t2 is perpendicular and in the plane of the shell. The assumption is then that t2 is the normal to the symmetry or antisymmetry plane. The constraints are

When symmetry or antisymmetry conditions are specified in a general coordinate system with axis directions ei, i = 1, 2, 3 with e1 as the normal to the symmetry/antisymmetry plane the constraints are

In axisymmetric models, the only possible symmetry plane is the one having the Z-axis as normal. In this case, you can use the Symmetry Plane boundary condition. The imposed constraint is

where the forces (F) and moments (M) can be distributed over a boundary or an edge or concentrated in a point. The contribution from the normal vector displacement a is only included in a geometrically nonlinear analysis. Loads are always referred to the midsurface of the element. In the special case of a follower load, defined by its pressure p, the force intensity is

Each part of the covariant strain (γαβ, χαβ, ζα) is transformed separately. They correspond to membrane, bending, and shear action, respectively, and it is thus possible to separate the stresses from each of these actions. The membrane stress is defined as

where D is the plane stress constitutive matrix, Ni are the initial membrane forces, and γi the initial membrane strains. The influence of thermal strains is included through the midsurface temperature Tm, and the hygroscopic swelling through the midsurface moisture concentration, cm.The membrane stress can be considered as the stress at the midsurface, or as the average through the thickness.

where χi is the initial value of the bending part of the strain tensor (actually: the curvature), and Mi are the initial bending and twisting moments. ΔT is the temperature difference between the top and bottom surface of the shell, and Δcmo is the difference in moisture concentration between the top and bottom. The bending stress is the stress at the top surface of the shell if no membrane stress is present.

where G represent the transverse shear moduli, ζi is the initial average shear strain, and Qi are the initial transverse shear forces. The correction factor 5/6 ensures that the stresses are averaged so that they correspond to the ratio between shear force and thickness. The corresponding strains ζ and ζi are averaged in an energy sense.

where z is a parameter ranging from −1 (bottom surface) to +1 (top surface). The computation of the shear stress at a certain level in the element uses the following analytical parabolic stress distribution:

Many quantities for a shell can best be interpreted in a local coordinate system aligned to the shell surface. Material data, initial stresses and stress results are always represented in this local coordinate system. You specify the orientation of the local directions in a Shell Local System node under the Linear Elastic Material.

If a Boundary System is selected, then the orientation of the shell local system is fully defined by the boundary system. When using a boundary system, it also possible to control the orientation of the shell normal by selecting the Reverse normal direction check box.

1. The local z direction ezl is the positive geometry normal of the shell surface.

2. The local x direction exl is the projection of the first direction in the material coordinate system (ex1) on the shell surface

This projection cannot be performed if ex1 is normal to the shell. In that case, the second axis ex2 of the material system instead defines exl using the same procedure. Thus, if

Many features, such as an edge load, allow input in an edge local coordinate system. The orthogonal local edge coordinate system directions xl, yl, and zl are defined so that:

When an edge is shared between two or more boundaries, the directions may not always be unique. It is then possible to use the control Face Defining the Local Orientations to select from which boundary the normal direction zl should be picked. The default is Use face with lowest number.

If the geometry selection contains several edges, the only available option is Use face with lowest number, since the list of adjacent boundaries would then be different for each edge. For each edge in the selection, the face with the lowest number attached to that edge is then used for the definition of the normal orientation.

For visualization and results evaluation, predefined variables include all nonzero stress and strain tensor components, principal stresses and principal strains, in-plane and out-of-plane forces, moments, and von Mises and Tresca equivalent stresses. It is possible to evaluate the stress and strain tensor components and equivalent stresses at an arbitrary distance from the midsurface. The parameter zshell (variable name shell.z) is found in, for example, the Parameters table of the Settings window of a surface plot. Its can be set to a value from −1 (downside) to +1 (upside). A value of 0 means the midsurface of the shell. The default value is given in the Default through-thickness result location section of the Shell interface.

Stresses and strains are available both in the global coordinate system and in the shell local system as described in Local Coordinate Systems.

The Shell data set is tailored for display of results computed in the Shell interface. The purpose is to simultaneously show results on the top and bottom surfaces of the shell with a separation which matches the physics thickness. A Shell data set with appropriate settings is generated together with the default plots, and some of the defaults plots make use of it.

When you evaluate stress and strain results, it is usually better to use the results at Gauss points than at the element nodes. This is particularly important when you use the MITC formulation, and when there are inelastic strain contributions like thermal expansion. You achieve do this by applying the gpeval() operator to the selected result quantity.

The default stress plots generated from the Shell interface show the von Mises stress at Gauss points, using an expression like gpeval(4,shell.mises).

The inertial properties mass (m) and moment of inertia tensor (I) of a rigid shell take the finite thickness into account. They are computed as:

where E3 and XM are the identity matrix and the center of mass of the rigid domain, respectively. The last term in I is accounting for the finite thickness, and there the fact that nT ⋅ n = 1 has been taken into account.

This section describes the theory and assumptions behind the multiphysics couplings Solid-Thin Structure Connection and Solid-Beam Connection. Only the shell version of the connection is described in detail, since the beam version is a direct specialization to 2D.

Figure 5-2 illustrates the first two cases. The shell edge can be part of the definition of the solid but there is no assumption about that.

where the subscripts sld and sh represent “solid” and “shell”, respectively, and a′ is the displacement of the shell normal, represented in the local directions.

This connection, however, enforces a ‘plane strain condition’ in the solid, which is not consistent with shell theory and which causes local unphysical stresses if Poisson’s ratio is nonzero. This effect disappears within a few elements from the connection, and the approximation can, in many situations, be acceptable. This constraint is enforced if Method is set to Rigid in the multiphysics coupling.

The transverse displacement is then, after integration with respect to z′:

Thus, in order to obtain a stress-free transverse displacement, this deformation must be allowed. Note that the last term makes the solid thinner on one side of the midsurface and thicker on the other. The constant term — that is, the displacement at the midsurface of the solid — is known from the shell midsurface displacement. The other two terms depend on the stress state and thus on derivatives of the displacement, which are not readily available. The K1 term is caused by the membrane action and the K2 term by the bending action.

Note that as the K1 term is related to membrane action, it cannot contribute to the transverse shear stress. Since the derivative in the x′ direction cannot be controlled by a condition on the boundary, it is necessary to make an assumption about u′(z′). An integration with respect to z′ gives

This shows that a third power of z′ is required in order to be able to represent the correct shear strain contribution.

Here qu, qw1, and qw2 are the new dependent variables defined on the shell edge. They are dimensionless, due to the multiplication with the shell thickness, d. The constants ku = 3/5 and kw2 = 1/5 are explained below. The variable qw1 is proportional to the membrane axial strain, the variable qw2 is proportional to the bending strain, and the variable qu is proportional to the transverse shear strain. Since these variable are directly related to strains, the shape function order used is one order lower than for the displacements.

If no extra equations defining qu, qw1, and qw2 are introduced, these variables try to adapt to proper values through the reaction forces on the solid. The reaction force for u′ is the traction σx′ and the reaction force for w′ is the traction τx′z′. When taking the variation of the new dependent variables, these enforce the following constraints:

The equation with qw1 is trivially fulfilled because the shear stress is an even function of z′. Inserting the known stress distributions gives equations that can be solved for ku and kw2.

The fact that az′ = 0 has been used when formulating Equation 5-3.

The only coordinate value needed is actually z′. For the other two coordinates only the direction is important. The coordinate in the normal direction can be computed as

This definition of z′ assumes that the thickness of the solid does not change significantly. Under geometric nonlinearity, the computation should be based on the current geometry.

Let Φ be the matrix that transforms displacements from the global system to the local system:

The only transformation actually needed is thus the projection of the q vector. For a linear case, the transformation can be written as

where N is the undeformed shell normal (shell.an) and tl is the shell edge tangent (shell.tle). The coefficient s is either 1 or −1, and is selected so that the x′ direction coincides with the outward normal of the solid.

This is the analogous case in 2D. Beam theory assumes that the stress in the out-of-plane direction is zero. It is thus only physically sound to connect to a model where the plane stress assumption is used. The derivation above still remains valid with the following exceptions: the displacements in the local y’ direction are zero and the tangent direction tl is replaced by the out-of-plane direction.

The case where the shell is parallel to the boundary of the solid exists in two versions — Shared and Parallel.

where ζ is a distance from the solid to the shell. The right hand side is mapped from the shell to the solid using an extrusion operator. The default is that ζ is half the shell thickness, but you can also use the geometrical distance between the boundaries, or a user-defined distance.

You can create the appropriate couplings by adding a Solid-Thin Structure Connection multiphysics coupling. The theory of this connection is outlined below.

where n is the normal to the shell. The rotation vector in the beam can be expressed in the shell degrees of freedom as

The constraints are actually formed on the shell edge, and the degrees of freedom are taken from the beam using a General Extrusion operator which maps values from the closest point on the beam to the shell.

The definitions of n, t1, and t2 may however be discontinuous over a shell edge. For this reason, the constraint is formed using values from one boundary only if several boundaries share the edge. Another complications arises when the edge is a fold line, that is when the boundaries that meet do not have a common normal direction. On a fold line all three rotational degrees of freedom do exist in the shell and should then be connected to the corresponding degrees of freedom in the shell. In this case, also a third rotational constraint is formed.

This coupling is intended for the case when the beam is not in the same plane as the boundary modeled by shell theory. This case is called Shell boundaries to beam points in the Shell-Beam Connection node. Physically, this can be seen as a beam with one end welded to a plate. In order to get a correct stiffness representation of such a connection, it is necessary that the beam is connected to an area of the shell which is similar to the actual physical width of the beam. The connected area on the shell does not have to fit a boundary in the geometry. It is however necessary that the mesh size is such that there are at least three nodes within the connected area.

This case is identical to the previous case, with the only exception that the selection in the Shell interface is an edge. The beam can have any orientation relative to the shell edge. This case is called Shell edges to beam points in the Shell-Beam Connection node.