Elbow elements

We assume that the undeformed pipe is of circular or nearly circular section. Let $\left(\varphi ,S\right)$ be the material coordinates of a point in the pipe wall middle surface, defined as follows (see Figure 1).

Figure 1. Elbow geometry.

In the undeformed configuration $\varphi$ measures the angular position of the point from the crown of the section such that $\varphi$ increases towards the extrados (on a straight segment the “crown” and “extrados” are arbitrarily defined as fixed positions around the pipe section) and S measures the distance along the pipe axis from some arbitrary origin. Let $\mathbf{x}\left(\varphi ,S\right)$ be the current and $\mathbf{X}\left(\varphi ,S\right)$ the reference positions of a point. Let $\overline{\mathbf{x}}\left(S\right)$ and $\overline{\mathbf{x}}\left(S\right)$ be the current and reference positions of a point on the pipe axis, and define $\mathbf{y}=\mathbf{x}-\overline{\mathbf{x}}$ as the offset of a midwall point from the pipe axis. Let $\mathbf{a}=[{\mathbf{a}}_1,{\mathbf{a}}_2,{\mathbf{a}}_3]$ be a right-handed director set of orthogonal, unit vectors, with ${\mathbf{a}}_3$ approximately tangent to the pipe axis. $\mathbf{A}=\left[{\mathbf{A}}_1,{\mathbf{A}}_2,{\mathbf{A}}_3\right]$ are the same directors in the reference configuration: we choose $\mathbf{A}$ so that the point $\varphi =0$ (the crown of an elbow section) lies on ${\mathbf{A}}_1$ and ${\mathbf{A}}_2$ points to the extrados $\left(\varphi =\pi /2\right).$ For simplicity we rewrite

where

and

Finally, we define a unit vector, $\mathbf{n}$, which will be made approximately outwardly normal to the pipe wall middle surface, as follows. Let

be a unit vector pointing around the pipe section $S=$constant, and let

Then write

where

with

This definition ensures that $\mathbf{n}\cdot \partial \mathbf{x}/\partial \varphi =0$: we will later introduce a penalty term to ensure that $\left|\mathbf{n}\cdot \partial \mathbf{x}/\partial S\right|$ is small as well.

The concept is that of a beam with deforming section, so that we first interpolate $\overline{\mathbf{x}}$ and $\mathbf{a}$ as functions of S and then interpolate $\mathbf{y}$ and $\mathbf{n}$ as functions of S and $\varphi$.

We assume the pipe axis position, $\overline{\mathbf{x}}$, is a polynomial in S:

where $H_n\left(S\right)$ are the usual polynomial interpolation functions of order $\left(N-1\right)$. The same functions are used to interpolate a rotation triplet,

which then gives

where $\mathbf{C}$ is the rigid rotation matrix.

Numerical experiments with standard shell models have shown that warping effects are important in the geometries and loadings of interest. We, therefore, introduce the Fourier/polynomial interpolation

where $H_m$ are polynomials of the same or lower order as the $H_n$ above

The number P gives the order of Fourier interpolation, and the terms “in-plane” and “out-of-plane” refer to symmetries that occur if the motion is symmetric about the plane of the pipe's initial curvature or antisymmetric about that plane.

To model ovalization we assume that

and

and for the direction $\mathbf{n}$

The $p=0$ and $p=1$ terms in $y^3$ and $u^t$ and the $p=0$ term in ${\gamma }^3$ are omitted since these relative quantities should not include rigid body motion. The implementation of the formulation has been based on choosing linear polynomials for the $H_n\left(S\right)$ and ${\overline{H}}_m\left(S\right)$ (referred to as element type ELBOW31) and quadratic polynomials for these functions (element type ELBOW32). Obviously element type ELBOW31 is the lowest order possible for such interpolation. However, we also provide an element in which the $H_n\left(S\right)$ are linear, while $M=1$ and ${\overline{H}}_1\left(S\right)=1$ ; that is, $\mathbf{y}$ and ${\gamma }^3$ are constant within an element and, thus, discontinuous from element to element. This is the level of approximation used in routine linear design formula analyses (Dodge and Moore, 1972) and that has been used previously in nonlinear cases (Hibbitt, 1974). We refer to this as element type ELBOW31B. For the linear elements (ELBOW31, ELBOW31B) one-point integration is used in each element with respect to S; the quadratic element uses a two-point Gauss rule.

The choice of P—the number of terms taken in the Fourier series—has been based on numerical experiments. Ohtsubo and Watanabe (1976) documented a detailed numerical investigation of this choice for their original element, and their results should be relevant to our formulation. For most practical thin-walled piping system applications, we find that six modes are sufficient to predict strains within a few percent.

The formulation is discrete Kirchhoff, using the Koiter-Sanders generalized section strains at each point of the pipe wall middle surface:

and

where

is the metric of the pipe wall middle surface and

is the curvature of the surface, where $\mathbf{n}$ is made normal to the surface and $G_{\alpha \beta }$ and $B_{\alpha \beta }$ are the same measures in the reference configuration—${\theta }^1=S$ and ${\theta }^2=\varphi$—and Greek indices span the range (1, 2).

The strain components at any point through the pipe wall are then defined by the Kirchhoff assumption as

where h is the material coordinate measuring position through the pipe wall: $-t/2\le h\le t/2$. In applications numerical integration is used through the wall. The integration rule varies from case to case but is usually a five, seven or nine point Simpson rule for inelastic material response involving creep and plasticity.

The discrete Kirchhoff constraint is imposed by associating a penalty with the strain

The penalty is obtained from the transverse shear stiffness of the pipe wall, suitably modified for thin walls in the manner of Hughes et al. (1977).

It seems reasonable to assume that the warping will be small, so that

Based on this, we write $\overline{\mathbf{n}}=\partial \mathbf{x}/\partial \theta \times {\mathbf{a}}_3+{\gamma }^3{\mathbf{a}}_3.$

Since $\mathbf{n}=\overline{\mathbf{n}}{\left(\overline{\mathbf{n}}\cdot \overline{\mathbf{n}}\right)}^{-\frac{1}{2}}$,

where $\mathbf{I}$ is the unit matrix.

We choose

and we impose a penalty to make $\left|\mathbf{n}\cdot \partial \mathbf{x}/\partial S\right|$ small. Then in the computation of the curvatures, $b_{\alpha \beta }$, we approximate $\mathbf{n}\cdot \partial \mathbf{x}/\partial S=0$, so that

This completes the basic definition of the element formulation. The formulation allows arbitrary rigid body motions to occur with no strain and uniform thermal expansion with constant strain. The detailed derivation of the strain variations, initial stress matrix, etc., follow directly from the definitions given above, although the forms are complicated. In the actual implementation the nonsymmetric terms in the initial stress matrix caused by our choice of definition of rotation—$\overline{\mathit{\omega }}$—as well as nonsymmetric terms in some load stiffnesses, are neglected. A major implementation approximation is in the modeling of inertia terms in the form of a mass matrix defined by

where $\rho$ is the mass density of the material.

This is a coarse approximation locally, but the intended applications concern analysis of pipelines, where you are mainly interested in inertia effects associated with locally low-mode response. Table 1 shows a comparison of free vibration frequencies for a single 90° bend restrained against rigid body motion on its axis at one end and prevented from warping (but allowed to ovalize) at both ends.

The agreement between the various models and the “standard shell element” is not very close, but we feel it is adequate for the intended use. More accurate modeling of inertia would incur a large penalty in computational cost.