A primer on shell structures

A primer on shell structures

Through this post, I attempt to provide a starter pack for shell structures. This includes trying to understand their structural behaviour, a very brief overview of shell-kinematics and the basic methodology for shell analysis through FEM. The theory and implementation follows the widely popular Kirchhoff-Love theory described for thin shells (thickness/length < 20)

Realistically, all structures designed and constructed to carry and/or transfer mechanical loads, exists in the 3-D space and trying to understand their behaviour constitutes what is broadly known as structural analysis. But a complete analysis, considering all the intricacies and nuances of the material, geometry and loading is ideally not possible, owing to the extreme complexity, resources and time required. Hence we resort to idealized mathematical models of the structural components, also called structural elements, which almost accurately represent/mimics the real-world response of the structure and are fairly easier to understand and handle because of their reduced dimensions. To name a few; a truss, beam, column, plate, shell etc.. are some of the common generalizations of these structural elements. The nomenclature is defined by the manner in which the structure (element) carries the load and this further corresponds to the mathematics (read GOVERNING DIFFERENTIAL EQUATION) of the structure. For instance, if a horizontal member supports the load by developing axial force, we call it a bar element.

Different load carrying modes

A shell is one such structural element. It is the idealization of a curved 3-D solid structure and predominantly resists the applied loads through bending action and membrane action. In addition to having properties and response similar to that of similarly shaped plate structures, shell analysis requires knowledge of curvature and differential geometry, which makes them the most complicated of all structural elements. But shell structures possess large strength density and the ability to span large areas, which makes them very efficient and appealing.

Shell Action

To understand how does a ‘structural shell’ works, it is necessary to understand two other structures; namely plates and membranes. A plate is geometrically understood as a solid with its thickness dimension very small compared to its in-plane dimensions and having straight surfaces. In terms of Mechanics, plates are structures that resist transverse loads predominantly through bending action. They can be considered as a 2-D equivalent of a ‘beam structure model’. A membrane, on the other hand, carries these transverse loads by developing in-plane stresses. This is as shown below.

Membrane Action

The geometry of a shell structure is described similar to that of plates, except that an additional ‘curvature‘ creeps in. In other words, shells are curved plates or plates with curvature. Another way of putting it is to understand plates as shell structures with a radius of curvature = 0. To sustain the applied loads, shell structures develop both bending and membrane actions. So in terms of mechanics, shell = plate + membrane

A plate element develops 5 force components $(M_{xx}, M_{yy}, M_{xy}=M_{yx}, V_x, V_y)$ whereas a membrane develops 3 force components $(N_{xx}, N_{yy}, N_{xy}=N_{yx})$ upon transverse loading and a shell develops all the above 8 components.

In terms of FEA, a typical plate finite element has 3 DOF per node (in-plane rotations $\theta_x, \theta_y$ + out-of-plane rotation $u_z$). A membrane element also has 3 DOF per node (in-plane translations $u_x, u_y$ + out-of-plane rotation $\theta_z)$. A typical shell element in FEA combines all these and have 6 DOF per node, i.e all the 3 translations and 3 rotations.

Shell Kinematics

As mentioned before, shells are curved geometries. An outcome of this is that the position vector of every point in the structure is uniquely defined. If we suppose a curvilinear coordinate system $(\xi_1, \xi_2, \zeta)$ to parametrize the shell geometry, a position vector to any point in the volume of the shell may be given as $\mathcal{R}\left(\xi^{1}, \xi^{2}, \zeta{}\right)=\boldsymbol{r}\left(\xi^{1}, \xi^{2}\right)+\zeta \boldsymbol{n}\left(\xi^{1}, \xi^{2}\right)$ where $\boldsymbol{n}$ is a unit vector and $\zeta \in [-0.5t, 0.5t], t$ being the ‘uniform’ thickness of the shell. Thus, recognition of the proper differential geometry is fundamental in analyzing shell structures. Here, I will limit to a basic overview

Differential Geometry of a shell surface

The first principle of shell-analysis is dimension reduction. Without much proof or deliberation, we can assume that the entire mechanics & behavior of a solid-shell-structure may be represented by its middle surface itself. Thus we can represent covariant base vectors and normal vector to the middle surface such that

$$\boldsymbol{a}_{1}=\frac{\partial \boldsymbol{x}}{\partial \xi^{1}}, \quad \boldsymbol{a}_{2}=\frac{\partial \boldsymbol{x}}{\partial \xi^{2}}, \quad \boldsymbol{n}=\frac{\boldsymbol{a}_{1} \times \boldsymbol{a}_{2}}{\left\|\boldsymbol{a}_{1} \times \boldsymbol{a}_{2}\right\|}$$

We use the Kirchhoff-Love theory here to specify thin shells. This is analogous to the Euler-Bernoulli beam theory but extended to plates and shells such that transverse shear deformations are completely neglected. This leads to the assumptions that the total strain measure in a shell varies linearly through the thickness and can be written as a sum of (a) in-plane membrane strain, and (b) out-of-plane bending strain such that

$$\epsilon_{\alpha \beta}=e_{\alpha \beta}+\zeta \kappa_{\alpha \beta} \ \ \ where \ \alpha, \beta = \xi^1, \xi^2$$

$$\begin{aligned}e_{\alpha \beta} &=\frac{1}{2}\left(\boldsymbol{u}_{, \alpha} \cdot \boldsymbol{a}_{\beta}+\boldsymbol{u}_{, \beta} \cdot \boldsymbol{a}_{\alpha}\right) \\\kappa_{\alpha \beta} &=-\boldsymbol{u}_{, \alpha \beta} \cdot \boldsymbol{n} \\&+\frac{1}{\left\|\boldsymbol{a}_{1} \times \boldsymbol{a}_{2}\right\|}\left[\boldsymbol{u}_{, \xi} \cdot\left(\boldsymbol{a}_{\alpha, \beta} \times \boldsymbol{a}_{2}\right)+\boldsymbol{u}, \boldsymbol{\eta} \cdot\left(\boldsymbol{a}_{1} \times \boldsymbol{a}_{\alpha, \beta}\right)\right] \\&+\frac{\boldsymbol{n} \cdot \boldsymbol{a}_{\alpha, \beta}}{\left\|\boldsymbol{a}_{1} \times \boldsymbol{a}_{2}\right\|}\left[\boldsymbol{u}_{, \xi} \cdot\left(\boldsymbol{a}_{2} \times \boldsymbol{n}\right)+\boldsymbol{u}_{, \eta} \cdot\left(\boldsymbol{n} \times \boldsymbol{a}_{1}\right)\right]\end{aligned}$$

Sketches and Diagrams: Abhinav Gupta

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