E1. Generalities

The different post-treated quantities may change significantly according to the objective of the finite element analysis.

Usually the studied quantities in the pre-dimensioning phase (a sketch or preliminary draft) are different from those searched in the pre-design and design phases. Other quantities of interest are needed when analyzing existing structures to quantify structural risks.

It is tempting to consider:

Pre-dimensioning phases (sketch, preliminary draft, …): quantities such as displacements, deflection relative to the span, …
Dimensioning phase: density of reinforcement for slabs, reinforcement area for beams and columns, crack opening, …
Structural analysis: stress state (principal stresses in concrete: directions, signs, tri-axiality and its values, comparison with criteria such as Ottosen, Rankine, Drucker-Prager if the analysis is elastic), principal strains: directions, signs and its values, damage, stresses in the rebars, plastic strains, …

It is important to keep in mind that not all quantities of interest are accessible for every type of element and constitutive models. The type of element determines the nature of the degrees of freedom: the nodes and moments on solid elements are not accessible. Analogously, a model simulating plasticity cannot provide damage results. These details may seem trivial, but they are often forgotten by the users, so it is important to mention it.

Stresses and strains

As exposed in the “Generalities” chapter, the stress and strain fields are not obtained directly when solving a mechanical problem with a FE solver. They are obtained from the displacements by computing the derivative of the interpolated displacement fields. Using the same notation as in the “Generalities” chapter:

In linear-elasticity, the strain-stress relation can be written as (H being Hooke’s matrix):

In this kind of procedure, the strain and stress fields evolve according to the interpolation functions of degree inferior to one of the displacement functions.

For non-linear analyses, the stresses and strains cannot be obtained by simply multiplying Hooke’s matrix by the derivative of the nodal displacements. They are based on the values of the element at the Gauss points, and the shape functions are used to extrapolate the results to the rest of the element and nodes. This procedure allows the stresses to evolve according to the same degree as the interpolation functions and not their derivatives.

As an example, on an element with n nodes, n_pg Gauss points located at the coordinates ξn_pg with shape functions N_i, we compute the least-square minimization between the interpolated field evaluated from the nodal values searched and the known Gaussian values.

Building elementary nodal values starting from the values at the Gauss points in 1D

So, either it is a matter of minimizing the function:

Or for each node i among the n nodes of the element:

Which can be expressed in matrix form. The matrices are finally computed for all the isoparametric elements of reference:

And so, it is possible to directly obtain the nodal stresses:

Thus, it is possible to have several nodal values of strains or stresses for a given node, shared by multiple elements. The other nodal values can be deduced from those values.

Discontinuity of the elementary nodal values in 1D

Several methods can be used to recover the continuity of strains or stresses (if the material is the same in the 2 adjacent elements) to have a single nodal value of strain (or stress). Those methods consist of smoothening the stress and strain fields by using certain algorithms on the totality of the structure (by least-square minimization). It can also be done by computing the average of the values on a given node shared by multiple elements as shown in the figure below.

Example of the calculation of an averaged nodal value starting from elementary nodal values in 1D

For convenience one might prefer obtaining smooth elementary values. However, when we look for precision, it is convenient to give priority to the values obtained at the Gauss points.

Going from continuum mechanics to structural mechanics stresses

The automatic methods to compute reinforcement rely on algorithms that take into account combined axial forces and bending moments based on plate and shell torsors with 8 components, or deviated combined axial forces and bending moments based on beam torsors with 6 components.

Some structures may need, for various reasons, to be modeled using solid finite elements (dam, structures such as prestressed tanks: liquefied natural gas tanks, retaining structures). To avoid computations based on 8-component torsors, it is necessary to rebuild the Structural Mechanics internal forces starting from the stress fields along the segment as shown in the figure below.

Segment along which the stresses are used as a reference to rebuilding a shell type torsor

We consider the stress tensor expressed along the segment shown in the previous figure:

To be accurate, for thin plates satisfying the Kirchhoff-Love hypothesis, we consider the following torsor with 10 components that will be associated with the stresses according to the following equivalence principle:

This continuous integration is discrete along the segment and can typically be computed from the stress values expressed at the Gauss points or the nodes.

Multiple integration techniques can be used (Trapezoidal rule, Gauss, Lobatto, etc …)

A continuous expression will be kept for the sake of simplicity, even if it is not rigorous in the discrete context of the values obtained by the FE method.

Taking into consideration the Cauchy theorem equivalent to the shear stresses at the adjacent faces, this torsor is reduced to 8 components:

Considering that this plate is an isotropic slab implies that the contribution of membrane efforts is neglected.

Then, it becomes a 5 component torsor (considering that the notation M_x represents the bending moment activating the rebars in the direction e_x), which are the internal mechanical actions in a plate element:

(1) Shear efforts outside of the plane (z-direction) with x as the normal direction

(2) Shear efforts outside of the plane (z-direction) with y as the normal direction

(3) Bending moment activating the rebars in the x-direction

(4) Bending moment activating the rebars in the y-direction

(5) Torsion moment on the cross-section of the slab with x or y as the normal direction

Section method: elementary reduction elements (EF) to beam-type structural reduction elements

Depending on the stress state it might be convenient to build a model of the whole structure with plate elements including the zones where these are not appropriate such as posts and lintels.

In these zones, one must restore a beam-type torsor to take into consideration more accurately the local behavior, in particular, to estimate the required reinforcement.

A widely used technique to solve this problem consists of “cutting” the structure fictively and estimating the efforts at the cross-section.

This “cut” must be chosen wisely to respect the Euler-Bernoulli hypothesis stating that cross-sections must remain plane. It is usually the case for posts and lintels.

Example of cross-sections in a lintel or post passing or not through nodes

We then build in the local coordinate system of the cross-section the beam-type reduction elements substituting the shell-type reduction elements. The latter can be the efforts at the nodes or evaluated at any point of the element coinciding with the section line and analyzed utilizing elementary form functions.

For the example illustrated above, it can be written for the cutting line, where local and global coordinate systems coincide, that:

(1) Axial forces at the cross-section of the associated beam, i.e. the “cut plane”

(2) Shear forces on the plane

(3) Shear forces off the plane

(4) Bending moment on the plane

(5) Bending moment off the plane

(6) Torsion moment