# Simply Supported Beam

A simple linear elastic beam subjected to a point force is presented in this tutorial. The aim here is to show the various analysis capabilities of COSSAN using a toy problem. To start with, the effect of the structural uncertainties on the mid-span displacement will be assessed using simulation analysis. Next, the failure probability of exceeding a certain threshold value for this mid-span displacement will be estimated within the context of reliability analysis.

## Description of the Problem

### Geometry

The geometry of the simply supported (all translational displacements are constrained on the left end, while right end is free to move in x direction) beam is shown in the figure below. The beam has a rectangular cross section and a point force is applied at the point $\alpha$=25 mm (L = 100mm).

### Input parameters

Four parameters are considered in the problem:

• Point force (P)
• Young modulus of the material (E)
• Width of (the cross section of) the beam (b)
• Height of (the cross section of) the beam (h)

### Analytical solution

The displacement of the mid-span of the beam can be evaluated analytically as: $u_y = -11 \cdot P \cdot 100^3/(768 EI)$ where uy  denotes the vertical displacement, I is the inertia moment of the cross section given as $I = \frac{bh^{3}}{12}$

## Preparation of the Input

### Summary of the probabilistic model

The following table summaries the random variables that have been used for simulation analysis:

 Please improve this article by expanding it. check the consistency of the units between the table and the figures
 Summary of the Probabilistic Model Structural Property Distribution Mean Value Std Dev Young's Modulus (E) Normal 210 GPa 10 GPa Point Force (f) Lognormal 10 N 1.4 N Height of the beam (h) Uniform 5 mm 0.577 mm

### Creation of the Random Variables

The three Random variables described in previous section are created. Random variables are the inputs of analysis considering uncertainties (reliability, simulation, etc).

Creation of the Random variable related to the Young modulus.
Creation of the Random variable related to the force.
Creation of the Random variable related to the eight of the beam.

### Creation of the Random Variable Set

A Random Variable Set containing all the existing Random Variables is created.

Creation of the Random variable Set.

No correlation is considered in this example. Thus the data in the tab Correlation were not changed.

### Creation of a Parameter

The width of the beam (b=8.1mm) is modelled using a Parameter. Parameters are the deterministic input for simulation analysis. These can be used for values that may need to be changed easily within COSSAN-X.

Creation of the Random variable related to the width of the beam.

### Creation of a Function

 Please improve this article by expanding it. The functions for the Inertia22 Inertia_torsion Area and G are missing

The inertia moment will be expressed using a Function, which depends on the height and width of the beam

Creation of the Function computing the inertia moment of the beam

## Evaluators

Evaluators are used to evalaute the response/output of a defined model. In order to represent various types of evaluators, a Matlab script will be used to calculate the analytical solution, while different FE solvers will be utilizied (although not necessary for such a simple problem) in order show the interaction with the 3rd party software.

### Matlab script

In order to evaluate the analytical solution, a Matlab script, where the output (mid-span displacement) is called u_y.Here structures are used for the inputs and outputs.

Creation of the script, metadata tab.

The syntax of the (predefined loop of the) script is the following:

 %% 1. retrieving inputs
% applied strength
f = Tinput(i).f;

% Young modulus
E =  Tinput(i).E

% Inertia moment
Inertia = Tinput(i).Inertia;

%% 2. computation of the vertical displacement
VerticalDisplacement = -11*f*100^3  / (768 * E * Inertia);

%% 3. preparation of outputs
Toutput(i).u_y = VerticalDisplacement;


During the execution of the script, the following operations are performed:

• The inputs are retrieved and put inside a temporary variable. This operation is performed for the readability of the script.
• The vertical displacement at the mid-point of the beam is computed using the analytical formula. The result are assigned to a temporary variable for readability.
• The vertical displacement is assigned to the output of the script. The name assigned in the output of the script (u_y in this case) must be the same as the one defined in the metadata tab.
Creation of the script, script tab.

### Interaction with the 3rd party FE-Solvers

Any analysis requiring the interaction with  3rd party software requires to perform the following operations:

• Providing the input files
• Definition of a connector
• Preparation of the injector
• Preparation of the extractor

In the following, the problem at hand will be analysed using various 3rd party FE solvers, i.e. Nastran, Abaqus and Ansys.

#### Connector interacting with ABAQUS

The finite element models consists of 41 nodes and 40 beam elements. The original input file can be found at:

The nodal displacement are printed in the beam.dat file using the following commands:

 *NODE PRINT
U,
*EL PRINT
S,
*EL FILE
S,
*NODE FILE
U


The connector to Abaqus is created. The flag interactive ask_delete=off is required. Two files are used in file management: the original input file (beam.inp) and one output file (beam.dat). The connector needs to be configured so that all the results from the finite element analysis are named beam.dat. The input file will be used in order to inject values from COSSAN-X (realizations of the Random variables,...). The output file will be used to recover results from the simulations.

Definition of the connector to Abaqus

Identifiers are set in order to inject the quantities of interest at the right position in the input file. In this example, identifiers are defined for the cross section, the inertia moment of the beam, the material properties and the point force.

Creation of the injector
Creation of the injector (continued)

The properties of the identifiers needs to be set so that the values are properly replaced (same number of characters as in the original file, etc). The name of each identifier is identical to the name of the object to inject.

Properties of the identifier

The extractor is used to retrieve the quantity of interest. In this example this is the vertical displacement at the mid-point of the beam, that is to say the displacement of node 21 in the y-direction

First an anchor is defined at the header of the table of the nodal displacements. Then an identifier is set at the position of the quantity of interest in the output file. The response properties have to be set: only one value is extracted, it will be given the name u_y.

Properties of the response

The response has to be linked to the anchor by dragging it.

Response and anchor in the extractor

#### Connector interacting with ANSYS

The input file for the ANSYS model can be found at:

A script calling Ansys has been created, it can be found at:

The nodal displacement are printed in the beam.out.

The connector to Ansys is created. Two files are used in file management: the original input file (beam) and one output file (beam.out). The connector needs to be configured so that all the results from the finite element analysis are named beam.out. The input file will be used in order to inject values from COSSAN-X (realizations of the Random variables,...). The output file will be used to recover results from the simulations.

Definition of the connector to Nastran

Identifiers are set in order to inject the quantities of interest at the right position in the input file. In this example, identifiers are defined for the cross section, the inertia moment of the beam, the material properties and the point force.

Creation of the injector

The properties of the identifiers needs to be set so that the values are properly replaced (same number of characters as in the original file, etc). The name of each identifier is identical to the name of the object to inject.

Properties of the identifier

The extractor is used to retrieve the quantity of interest. In this example this is the vertical displacement at the mid-point of the beam, that is to say the displacement of node 21 in the y-direction

First an anchor is defined at the header of the table of the nodal displacements. Then an identifier is set at the position of the quantity of interest in the output file. The response properties have to be set: only one value is extracted, it will be given the name u_y.

Properties of the response

The response has to be linked to the anchor by dragging it.

Response and anchor in the extractor

#### Connector interacting with NASTRAN

The input file for NASTRAN can be found at:

The connector to Nastran is created. The flag scr=yes news=no bat=no old=no is required at the end of the execution command. Two files are used in file management: the original input file (beam.bdf) and one output file (beam.f06). The input file will be used in order to inject values from COSSAN-X (realizations of the Random variables,...). The output file will be used to recover results from the simulations.

Definition of the connector to Nastran

Identifiers are set in order to inject the quantities of interest at the right position in the input file. In this example, identifiers are defined for the cross section, the inertia moment of the beam, the material properties and the point force.

Definition of the injector
Definition of the injector (continued)
Definition of the injector (continued)

The name of each identifier is identical to the name of the object to inject to inject. The format is that of Nastran.

Properties of the identifier

The extractor is used to retrieve the quantity of interest. In this example this is the vertical displacement at the mid-point of the beam, that is to say the displacement of node 21 in the y-direction

First an anchor is defined at the header of the table of the nodal displacements. Then an identifier is set at the position of the quantity of interest in the output file. The response properties have to be set: only one value is extracted, it will be given the name u_y.

Properties of the response

The response has to be linked to the anchor by dragging it.

Response and anchor in the extractor

## Model Definition

Here simulation analysis will be performed with the Simply Supported Beam model.

### Definition of the physical model

A physical model has to be defined to link the evaluator with the inputs.

Creation of the physical model.

### Monte Carlo simulation

The model previously defined is used to preform a Monte Carlo simulation. A Simulation analysis is started.

Wizard of the analysis.

The Monte Carlo simulation is performed using 100 samples.

Wizard of the simulation analysis. Start of a Monte Carlo simulation with 100 samples

## Reliability Analysis

### Definition of performance function

A performance function requires to be defined before performing reliability analysis. In this example failure is defined as a displacement of the mid-point of the beam below a critical value. Here failure occurs when the displacement is below -0.015 mm. A parameter whose value is the critical displacement is created.

Then a performance function is created. The capacity is the displacement of the mid-point of the beam, the demand is the critical displacement. Failure occurs when the value of the performance function is less then zero.

Creation of the performance function

### Definition of probabilistic model

A probabilistic model needs to be defined. It is based on the physical model used to perform simulation analysis.

Creation of the performance function
 Physical model The parameter defining the critical displacement needs to be added to the physical model on which the probabilistic model is based

### Estimation of the failure probability

The failure probability objects can be displayed. They provide relevant information about the analysis (failure probability, coefficient of variation of the analysis, etc)

Result from reliability analysis