title: Understanding the strengths in shells
author: Ramon Ribo
email: ramsan@compassis.com
affiliation: http://www.compassis.com
Keywords: Shells, Finite elements, Strengths, Stresses
date: 2019-09-03
HTML header: <script src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.2/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>

css: css/markdown.css

Understanding the strengths in shells

Introduction

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.

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:

Thickness: for every point in the structure, the thickness is the small dimension , compared to the other two
Shell: is a plain or curve 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
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 stress through the thickness is ignored
There is a linear variation of displacement across the plate thickness but that 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.

File:Shell strengths.svg

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

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'
${\textstyle Q_{x}=\int _{-t/2}^{t/2}{\tau _{x}dz}}$	${\textstyle Q_{y}=\int _{-t/2}^{t/2}{\tau _{y}dz}}$

[Definition of strengths]

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 that point.

We need to consider that the real ${\textstyle Q_{x}}$ and ${\textstyle Q_{y}}$ has 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.

File:Shear strengths.svg

Real parabolic shear stress is simplified as constant through the thickness

Finite element analysis

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:

The strengths fields is discontinuous between elements
When displaying the strengths in a postprocessing program, the software extrapolates the valus 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.

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:

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

Based on the stresses tensor calculated before, it is possible to calculate Von Mises invariant 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}}}

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

Dimensioning structural concrete

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

References

[1]: 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
[2] ANALYSIS OF SLAB AND YIELD LINE THEORY. Available: http://shodhganga.inflibnet.ac.in/bitstream/10603/108563/11/11_chapter%202.pdf
[3] Oñate E. (1991) Cálculo de estructuras por el método de los elementos finitos. ISBN 8487867006

Back to Top