Ribo Rodriguez 2019a

Introduction

The finite element method permits to solve the governing equation of a physical structure by means of numerical methods and one discretization. When the structure have one dimension much smaller than the other two, it is possible to simplify the problem as a shell finite element problem. The result of this analysis are the displacements and rotations in all the nodes of the mesh and the strengths in some interior element positions.

In order to take valuable information of the analysis, a good understanding and interpretation of the strengths is necessary. This document will explain what the strengths in shells are and how can they be used for dimensioning or evaluating a physical structure.

We are going to define mathematically the strengths in shells as they are used in the standard Reissner-Mindlin theory. In the next paragraph we are going to establish some relationships between the Shell strengths and the stresses tensor in some selected points in the Shell height.

Image 1. Shell is represented by its mid plane

Definitions

One structure can be considered a Shell if one of its dimensions is small compared to the other two.

Some definitions:

• Shell: is a plain or curved surface in the 3D space that geometrically is positioned in the mid plane of the structure (based on the thickness direction). This Shell and the thickness in every point over the Shell represents completely the geometry of the structure
• Thickness: for every point in the structure, the thickness is the small dimension , compared to the other two
• Local axes: for every point in the structure it is necessary to define an attached local axes system X’, Y’,Z’ where axe Z’ will always be perpendicular to the Shell. X’, Y’ can be any axes but must be defined for every point in the Shell prior to the calculation
• Section: is the part of the structure which is the intersection between the structure and a plane orthogonal to the Shell

and then, the following hypotheses can be considered as true:

• One planar section, after deformation will continue to be planar. The normal to the mid-surface remains straight but not necessarily perpendicular to the mid-surface
• The normal stress through the thickness is ignored (${\textstyle \sigma _{z'}=0}$)
• There is a linear variation of displacement across the plate thickness but the plate thickness does not change during deformation

Definition of the strengths in the shell

In any point of the structure there exists the stresses tensor. As we have simplified the shell structure by the mid plane, it is necessary to define some integrations of the stress tensor through the shell thickness so as they represent perfectly the stresses in the mid plane.

Image 2. Strengths in the shell

For any point in the Shell midplane, a set of strengths must define completely the stress tensor in any point in the thickness.

The strengths are three axial forces in the plane, three bending moments and two shears

Table 1. Definition of strengths

Axial force in X’	Axial force in Y’	Axial force in XY’
${\textstyle N_{x}=\int _{-t/2}^{t/2}{\sigma _{x}dz}}$	${\textstyle N_{y}=\int _{-t/2}^{t/2}{\sigma _{y}dz}}$	${\textstyle N_{xy}=\int _{-t/2}^{t/2}{\tau _{xy}dz}}$
Bending momment X’	Bending moment Y’	Twisting moment XY’
${\textstyle M_{x}=\int _{-t/2}^{t/2}{\sigma _{x}zdz}}$	${\textstyle M_{y}=\int _{-t/2}^{t/2}{\sigma _{y}zdz}}$	${\textstyle M_{xy}=\int _{-t/2}^{t/2}{\tau _{xy}zdz}}$
Shear force in X’	Shear force in Y’
${\textstyle Q_{x}=\int _{-t/2}^{t/2}{\tau _{xz}dz}}$	${\textstyle Q_{y}=\int _{-t/2}^{t/2}{\tau _{yz}dz}}$

All the integrals are in Z’ axe from -t/2 to t/2, being t the thickness of the shell in that point. Z’ is the local axis defined equal to the normal to the surface at the point.

We need to consider that the real ${\textstyle Q_{x}}$ and ${\textstyle Q_{y}}$ have a parabolic shape through the shell thickness with null values in the top and bottom layers. However, we accept the simplification of considering the shear as constant through all the thickness.

Image 3. Real parabolic shear stress is simplified as constant through the thickness

Finite element analysis

Finite element analysis is a method to solve numerically the differential equation that governs the structural behavior of the model. To do so, it is necessary to discretize the model into a mesh, which is a collection of elements connected between them by nodes. The variables to solve in our problem are the displacements and rotations of every node of the mesh. From them, it is possible to interpolate the value of displacements for every point over the structure and it is possible to calculate the strengths as a derived result of the displacements.

Image 4. The discretization is the shell mesh

Once discretized, the finite element problem becomes:

${\textstyle Ka=f}$ (1)

where ${\textstyle K}$ is the stifness matrix, ${\textstyle f}$ is the external forces vector and ${\textstyle a}$ is the displacements vector. As every node has 6 degrees of freedom (${\textstyle d_{x},d_{y},d_{z},\theta _{x},\theta _{y},\theta _{z}}$), the dimension of ${\textstyle a}$ is 6 x number of nodes.

If the problem is linear, ${\textstyle K}$ and ${\textstyle f}$ do not depend on ${\textstyle a}$ and the problem can be solved by simply solving the linear system once.

If the problem is non linear, ${\textstyle K}$ or ${\textstyle F}$ depend on ${\textstyle a}$ and then it is necessary to solve the problem by iterating through a non-linear solver.

If the problem is dynamic, the equation to solve becomes:

${\textstyle M{\ddot {a}}+D{\dot {a}}+Ka=f}$ (2)

Results of the finite element analysis

The finite element analysis is a resolution of a system of equations where the variables are the displacements and rotations defined in every node of the mesh. As these variables are defined in the nodes commons to neighbouring elements, the displacements and rotations fields is continuum throgh the elements by the own definition of the problem.

As described before, the strengths are the integration of the stress tensor through the thickness of the shell and the stresses are the derivatives of the displacements. Both the stresses and the strength are calculated inside every element. Specifically, they are calculated in the intagration points used to calculate the numerical integration inside the element during the analysis. These points are called Gauss points and are used due to the fact that the result is more precise on that points.

As the strengths are calculated inside every element, these leads to two effects that is important to consider:

1. The strengths fields is discontinuous between elements
2. When displaying the strengths in a postprocessing program, the software extrapolates the values calculated internally to new values in every boundary element. For relativelly small increments of strength for every Gauss point, it is possible that the calculated value for the nodes has a too great or too small magnitude.

Smoothing

As the strengths fields is discontinuous between elements, in some cases it might be convenient to calculate another field, similar to the previous one, that is continuous in the node. This is only possible between elements that share equal local axes definition. The smoothed field is calculating by comparing the values in every node from each of the neighbouring elements. Some kind of mean value is obtained and used as the value in the node.

This correction is generally good, as it provides a value that is closer in most cases to the real value. However, in extreme cases and specially near the shell boundary or in points over external elements or beams, it can lead to incorrect values. This effect can be maximized by adding up the smoothing correction to the extrapolation to nodes in boundary elements.

It is important to visualize and compare the non-smoothed results and the smoothed ones in order to understand better the strengths field.

Momentus in coarse mesh. Gauss points results

Momentus in coarse mesh. Smoothed results

Momentus in fine mesh. Gauss points results

Momentus in fine mesh. Smoothed results

We can summarize some of the results in table 2. We can see that the error, specially for the coarse mesh, is quite high.

Table 2. Some results for the different meshes

Mesh	Gauss points/Smooth	${\textstyle M_{x}}$ border	${\textstyle M_{x,max}}$	${\textstyle M_{x,min}}$
coarse	GP	-2.3	20	-77
coarse	SM	-2.3	13	-75
fine	GP	-0.17	5.8	-74
fine	SM	-0.17	4.6	-73

Main strengths and main stresses

For any tensor defined in a local axes system, there exists another local axes system in that point that maximizes and minimizes the diagonal values and converts to zero the non-diagonal values.

This system is calculated finding the eigenvalues and eigenvectors of the matrix.

The local axes system for main moments, main axial strengths and main stresses need not to be equivalent and typically will be different between them.

Note that the following matrices are symmetrical and ${\textstyle N_{xy}=N_{yx}}$, ${\textstyle M_{xy}=M_{yx}}$, ${\textstyle \tau _{xy}=\tau _{yx}}$, etc.

Strengths in local axes ${\textstyle L_{0}}$	Main strengths
${\textstyle {\begin{bmatrix}N_{x}&N_{xy}\\N_{yx}&N_{y}\end{bmatrix}}}$	${\textstyle {\begin{bmatrix}N_{11}&0\\0&N_{22}\end{bmatrix}}}$	${\textstyle LA_{N}}$
${\textstyle {\begin{bmatrix}M_{x}&M_{xy}\\M_{yx}&M_{y}\end{bmatrix}}}$	${\textstyle {\begin{bmatrix}M_{11}&0\\0&M_{22}\end{bmatrix}}}$	${\textstyle LA_{M}}$
${\textstyle {\begin{bmatrix}\sigma _{x}&\tau _{xy}&\tau _{xz}\\\tau _{yx}&\sigma _{y}&\tau _{yz}\\\tau _{zx}&\tau _{zy}&\sigma _{z}\end{bmatrix}}}$	${\textstyle {\begin{bmatrix}\sigma _{11}&0&0\\0&\sigma _{22}&0\\0&0&\sigma _{33}\end{bmatrix}}}$	${\textstyle LA_{S}}$

Where ${\textstyle LA_{0},LA_{N},LA_{M}}$ and ${\textstyle LA_{S}}$ will represent typically different local axes systems with a common Z’ axe.

Image 5. Main strengths in ${\textstyle LA_{M}}$ and ${\textstyle LA_{M}}$

Stresses in top and bottom layers

In our Shell model, we assume that stresses vary linearly through the thickness of the Shell. As a consequence, both the maximum and the minimum of any stress component will be located in either the top or the bottom layer for any Shell point.

Note: shear stresses in y’ and z’ have a real parabolic shape through the Shell thickness, with a maximum in the middle plane of the shell. However, in our simplified model, we assume a constant value for the shear stresses in the thickness. So, maximum and minimum are also in the external layers.

Top and bottom stresses are calculated in the following way:

${\textstyle W_{b}={\frac {I_{z}}{z_{max}}}={\frac {bh^{2}}{6}}}$

${\textstyle A_{b}=bh}$

then:

${\textstyle \sigma _{x}={\frac {N_{x}}{A_{b}}}\pm {\frac {M_{x}}{W_{b}}}}$	${\textstyle \sigma _{y}={\frac {N_{y}}{A_{b}}}\pm {\frac {M_{y}}{W_{b}}}}$	${\textstyle \sigma _{z}=0}$
${\textstyle \tau _{xy}={\frac {N_{xy}}{A_{b}}}\pm {\frac {M_{xy}}{W_{b}}}}$	${\textstyle \tau _{xz}={\frac {Q_{x}}{A_{b}}}}$	${\textstyle \tau _{yz}={\frac {Q_{y}}{A_{b}}}}$

Dimensioning metallic materials

Based on the stresses tensor calculated before, it is possible to calculate the Von Mises stress for the top and bottom layers following the formula:

$$\sigma _{vm}={\sqrt {\frac {(\sigma _{I}-\sigma _{II})^{2}+(\sigma _{II}-\sigma _{III})^{2}+(\sigma _{III}-\sigma _{I})^{2}}{2}}}$$ (3)

where ${\textstyle \sigma _{I},\sigma _{II},\sigma _{III}}$ are the main stresses associated to the stress tensor ${\textstyle \sigma _{x},\sigma _{y},\sigma _{z},\tau _{xy},\tau _{xz},\tau _{yz}}$.

The Von Mises stress as defined in (3) can be used as a yield criteria for the material. This is commonly used as yield criteria for metallic materials.

Dimensioning structural concrete

Structural concrete is a composite material made of concrete and steel bars. Several normatives, like Eurocode and ACI and other national regulations, offer guidelines to dimension and check the concrete cross section and the amount of steel.

For a model where the main axial forces axes ${\textstyle LA_{N}}$ and the main momentum axes ${\textstyle LA_{M}}$ are coincident with the element local axes ${\textstyle L_{0}}$, the strengths in the section will be ${\textstyle N_{x'}}$, ${\textstyle M_{x'}}$ and ${\textstyle Q_{x'}}$ in the ${\textstyle x'}$ direction and ${\textstyle N_{y'}}$, ${\textstyle M_{y'}}$ and ${\textstyle Q_{y'}}$ in the ${\textstyle y'}$ direction. With these strengths, it is possible to dimension by using the standard regulations.

In the general case, there is an additional twisting moment ${\textstyle M_{xy}}$ that is different the zero. It is not in the security side to dimension the section without considering this twisting moment in the verification of the main steel bars in both directions.

It is necessary to use some method, that satisfies the equilibrium and geometric compatibility in the section and that has some bearing strengths greater that the external actions.

One commonly used method is the Wood and Armer. In it, the bearing ${\textstyle m_{ux}}$ and ${\textstyle m_{uy}}$ are increased based on the value of ${\textstyle m_{xy}}$. These values are used for dimensioning.

The CEB-FIP Model Code proposes and alternative method where the thickness is divided into three layers and the strengths are converted into pairs of forces and shear forces for dimensioning.

Wood and Armer method

The Wood and Armer method is based on the normal moment yield criterion (Johansen’s yield criterion) and proposes the dimensioning of the maximum bear moments for some given directions ${\textstyle x}$ and ${\textstyle y}$ can be calculated as follows:

${\textstyle m_{ux}=m_{x}\pm \left|m_{xy}\right|}$ (4)

${\textstyle m_{uy}=m_{y}\pm \left|m_{xy}\right|}$ (5)

if ${\textstyle m_{x}}$ and ${\textstyle m_{y}}$ have different sign, and ${\textstyle m_{x}}$ is bigger then:

${\textstyle m_{ux}=m_{x}\pm \left|{\frac {m_{xy}^{2}}{m_{y}}}\right|}$ (6)

${\textstyle m_{uy}=0}$ (7)

Theoretical justification of the Wood and Armer method

Given external moments of ${\textstyle m_{x},m_{y}}$ and ${\textstyle m_{xy}}$, the moments in direction ${\textstyle n}$ where the angle between ${\textstyle x}$ and ${\textstyle n}$ is ${\textstyle \theta }$ are the following:

${\textstyle m_{n}=m_{x}\cos ^{2}\theta +m_{y}\sin ^{2}\theta +2m_{xy}\cos \theta \sin \theta }$ (8)

${\textstyle m_{nt}=(m_{y}-m_{x})\cos \theta \sin \theta +m_{xy}(\cos ^{2}\theta -sin^{2}\theta )}$ (9)

On the other side, if ${\textstyle m_{ux}}$ and ${\textstyle m_{uy}}$ are the moment capacities in axes x,y then, the bearing capacity of the section in direction ${\textstyle n}$ where the angle between ${\textstyle x}$ and ${\textstyle n}$ is ${\textstyle \theta }$ is the following:

${\textstyle m_{un}=m_{ux}\cos ^{2}\theta +m_{uy}\sin ^{2}\theta }$ (10)

Imposing that ${\textstyle m_{un}}$ in (10) must be bigger that ${\textstyle m_{n}}$ in (8) any direction, it is possible to justify that the Wood and Armer proposal is an upper bound of the capacity for any direction.

The justifications and details of the method can be described in [@r2](#r2) and [@r:3](#r:3).

CEB-FIP Model Code

The slab model is decomposed into three layers, each with 1/3 of the slab thickness. Moments are decomposed into pairs of forces and applied to the layers as normal forces and shear forces that satisfy equilibrium with the moments.

Dimensioning composite laminate material

A composite laminate material is built adding several layers of fiber that get glued together by the use of a resin. In order to dimension or checking its bearing capacity, it is typically necessary to apply a failure criterion to every one of its constitutive layers.

Every single layer can be defined by its components of fiber and resin and are normally orthotropic. This anisotropic composite materials will have different strengths in tension and compression.

It is common to use the Tsai–Wu failure criterion in order to check the capacity of each single layer.

Tsai–Wu failure criterion

The criterion is defined as follows:

${\textstyle \sum _{i=1}^{6}F_{i}\sigma _{i}+\sum _{i=1}^{6}\sum _{j=1}^{6}F_{ij}\sigma _{i}\sigma _{j}\leq 1}$ (11)

where ${\textstyle \sigma _{i}}$ is one of ${\textstyle (\sigma _{xx},\sigma _{yy},\sigma _{zz},\sigma _{yz},\sigma _{xz},\sigma _{xy})}$ and ${\textstyle F_{i}}$ and ${\textstyle F_{ij}}$ are some coefficients calculated from laboratory results.

For plane stress, where ${\textstyle \sigma _{z}=\sigma _{yz}=\sigma _{xz}=0}$, the criteria is reduced to:

${\textstyle F_{1}\sigma _{xx}+F_{2}\sigma _{yy}+F_{11}\sigma _{xx}^{2}+F_{22}\sigma _{yy}^{2}+F_{66}\tau _{xy}^{2}+2F_{12}\sigma _{xx}\sigma _{yy}\leq 1}$ (12)

where:

${\textstyle F_{1}={\frac {1}{\sigma _{1t}}}-{\frac {1}{\sigma _{1c}}}}$	${\textstyle F_{2}={\frac {1}{\sigma _{2t}}}-{\frac {1}{\sigma _{2c}}}}$
${\textstyle F_{11}={\frac {1}{\sigma _{1c}\sigma _{1t}}}}$	${\textstyle F_{22}={\frac {1}{\sigma _{2c}\sigma _{2t}}}}$
${\textstyle F_{66}={\frac {1}{\tau _{12}^{2}}}}$

References

[1] Oñate E. (1991) Cálculo de estructuras por el método de los elementos finitos. ISBN 8487867006

[2] Shin M., Bommer A., Deaton, J.B. and Alemdar, B.N. (2009) Twisting Moments in Two-Way Slabs. Concrete international. Available: https://www.academia.edu/15396812

[3] OBrien E., O’Dwyer D. (2015). A review of the Wood and Armer equations for torsion in bridge slabs. 93. 40-43

[4] Tsai S.W. and Wu E.M. (1971) A general theory of strength for anisotropic materials. J Compos Mater, 5. pp. 58-80