LESM - Linear Elements Structure Model

This is the main driver file of LESM. This is a MATLAB program for linear-elastic, displacement-based, static analysis of two-dimensional and three-dimensional linear elements structure models, using the direct stiffness method. For each structural analysis, the program assembles a system of equilibrium equations, solves the system and displays the analysis results.

The program handles five different types of analysis models of reticulated structures: 2D (plane) Truss, 2D (plane) Frame, Grillage, 3D (spatial) Truss, and 3D (spatial) Frame.

In addition, two types of flexural behavior for linear elements are considered: Euler-Bernoulli (Navier theory) elements and Timoshenko elements.

The Object-Oriented Programming (OOP) paradigm is adopted in the implementation of the analysis process. The use of OOP is justified by the clarity, organization and generality that this programming paradigm provides to the code.

The program may be used in a non-graphical version or in a GUI (Graphical User Interface) version. The non-graphical version reads a structural model from a neutral format file and prints model information and analysis results in the the default output (MATLAB command window). In the GUI version, a user may create a structural model with attributes through the program graphical interface. The program can save and read a structural model data stored in a neutral format file. The standalone executable of the GUI version can be downloaded from the LESM website: https://web.tecgraf.puc-rio.br/lesm/

Authors
History
Linear Elements Models
Coordinate Systems
Analysis Model Types
Structural Element Types
Local Axes of an Element
Load Types
Components of Concentrated Nodal Loads
Components of Distributed Loads on Elements
Components of Thermal Loads on Elements
Materials
Cross-Sections
Object Oriented Classes
Auxiliary Functions and Files
Non-graphical version

Authors

Luiz Fernando Martha (lfm@tecgraf.puc-rio.br)

Rafael Lopez Rangel (rafaelrangel@tecgraf.puc-rio.br)

Pontifical Catholic University of Rio de Janeiro - PUC-Rio

Department of Civil and Environmental Engineering and Tecgraf Institute of Technical-Scientific Software Development of PUC-Rio (Tecgraf/PUC-Rio)

History

@version 1.1

Initial version: August 2017

Initially prepared for the course ENG 1240 - Analise de Estruturas III, 2016, second term, Department of Civil Engineering, PUC-Rio. Developed in the undergraduate thesis "Development of a Graphic Program for Structural Analysis of Linear Element Models", by Rafael Lopez Rangel, advised by Professor Luiz Fernando Martha, Department of Civil Engineering, PUC-Rio, December, 2016. Extended in the undergraduate thesis "Extensao de Programa Grafico para Analise de Trelicas e Porticos Espaciais via MATLAB e GUI", by Murilo Felix Filho, advised by Professor Luiz Fernando Martha, Department of Civil Engineering, PUC-Rio, July, 2017.

This version of the program is an accompanying software for the book "Analise Matricial de Estruturas com Orientacao a Objetos", Elsevier, 2018, by Luiz Fernando Martha.

Linear Elements Models

Linear element models can be trusses, frames or grillages, which are made up of uniaxial members with one dimension much larger than the others, such as bars (members with axial behavior) or beams (members with flexural behavior). In the present context, members of any type are generically referred to as elements.

The analysis process of this type of model is based on a nodal dicretization, where a node is a joint between two or more elements, or any element end.

The final analysis result is composed by a the results from a global analysis and a local analysis. The result from the global analysis is computed with nodal displacements and rotations, and the result of a local analysis is computed with the effect of distributed loads over elements. These analyzes use analytical expressions of linear element behavior. Therefore, the sum of the two analysis provides the analytical solution of the problem.

Coordinate Systems

Two types of coordinate systems are used to locate points or define directions in space. Both of these systems are cartesian, orthogonal and right-handed.

Global system

The global system XYZ is an absolute reference. Its origin is fixed and it gives unique coordinates to each point in space. In the present context, the global system is referenced by uppercase letters.

Local system

The local system xyz is a relative reference. Its origin is located at the beginning of each element (initial node) and the local x-axis is always in the same direction of the element longitudinal axis. Local y-axis and z-axis are in the principal moment of inertia directions of the element cross-section. The local system is referenced by lowercase letters.

Analysis Model Types

The LESM program considers only static linear-elastic structural analysis of 2D (plane) linear elements models and 3D (spatial) linear elements models, which can be of the following types:

Truss analysis

A truss model is a common form of analysis model, with the following assumptions:

Truss elements are bars connected at their ends only, and they are connected by friction-less pins. Therefore, a truss element does not present any secondary bending moment or torsion moment induced by rotation continuity at joints.
A truss model is loaded only at nodes. Any load action along an element, such as self weight, is statically transferred as concentrated forces to the element end nodes.
Local bending of elements due to internal loads is neglected, when compared to the effect of global load acting on the truss.
Therefore, there is only one type of internal force in a truss element: axial internal force, which may be tension or compression.
A 2D truss model is considered to be laid in the XY-plane, with only in-plane behavior, that is, there is no displacement transversal to the truss plane.
Each node of a 2D truss model has two d.o.f.'s (degrees of freedom): a horizontal displacement in X direction and a vertical displacement in Y direction.
Each node of a 3D truss model has three d.o.f.'s (degrees of freedom): displacements in X, Y and Z directions.

Frame analysis

These are the basic assumptions of a frame model:

Frame elements are usually rigidly connected at joints. However, a frame element might have a hinge (rotation liberation) at an end or hinges at both ends.
It is assumed that a hinge in a 2D frame element releases continuity of rotation in the out-of-plane direction, while a hinge in a 3D frame element releases continuity of rotation in all directions.
A 2D frame model is considered to be laid in the XY-plane, with only in-plane behavior, that is, there is no displacement transversal to the frame plane.
Internal forces at any cross-section of a 2D frame element are: axial force (local x direction), shear force (local y direction), and bending moment (about local z direction).
Internal forces at any cross-section of a 3D frame element are: axial force, shear forces (local y direction and local z direction), bending moments (about local y direction and local z direction), and torsion moment (about x direction).
Each node of a 2D frame model has three d.o.f.'s: a horizontal displacement in X direction, a vertical displacement in Y direction, and a rotation about the Z-axis.
Each node of a 3D frame model has six d.o.f.'s: displacements in X, Y and Z directions, and rotations about X, Y and Z directions.

Grillage analysis

A grillage model is a common form of analysis model for building stories and bridge decks. Its key features are:

It is a 2D model, which in LESM is considered in the XY-plane.
Beam elements are laid out in a grid pattern in a single plane, rigidly connected at nodes. However, a grillage element might have a hinge (rotation liberation) at an end or hinges at both ends.
It is assumed that a hinge in a grillage element releases continuity of both bending and torsion rotations.
By assumption, there is only out-of-plane behavior, which includes displacement transversal to the grillage plane, and rotations about in-plane axes.
Internal forces at any cross-section of a grillage element are: shear force (local z direction), bending moment (about local y direction), and torsion moment (about local x direction). By assumption, there is no axial force in a grillage element.
Each node of a grillage model has three d.o.f.'s: a transversal displacement in Z direction, and rotations about the X and Y directions.

Structural Element Types

For frame or grillage models, whose elements have bending effects, two types of beam elements are considered:

Navier (Euler-Bernoulli) beam element.
Timoshenko beam element

In Euler-Bernoulli flexural behavior, it is assumed that there is no shear deformation. As a consequence, bending of a linear structure element is such that its cross-section remains plane and normal to the element longitudinal axis.

In Timoshenko flexural behavior, shear deformation is considered in an approximated manner. Bending of a linear structure element is such that its cross-section remains plane but it is not normal to the element longitudinal axis.

In truss models, the two types of elements may be used indistinguishably, since there is no bending behavior of a truss element, and Euler-Bernoulli elements and Timoshenko elements are equivalent for the axial behavior.

Local Axes of an Element

In 2D models of LESM, the local axes of an element are defined uniquely in the following manner:

The local z-axis of an element is always in the direction of the global Z-axis, which is perpendicular to the model plane and its positive direction points out of the screen.
The local x-axis of an element is its longitudinal axis, from its initial node to its final node.
The local y-axis of an element lays on the global XY-plane and is perpendicular to the element x-axis in such a way that the cross-product (x-axis * y-axis) results in a vector in the global Z direction.

In 3D models of LESM, the local y-axis and z-axis are defined by an auxiliary vector vz = (vzx,vzy,vzz), which is an element property and should be specified as an input data of each element:

The local x-axis of an element is its longitudinal axis, from its initial node to its final node.
The auxiliary vector lays in the local xz-plane of an element, and the cross-product (vz * x-axis) defines the the local y-axis vector.
The direction of the local z-axis is then calculated with the cross-product (x-axis * y-axis).

In 2D models, the auxiliary vector vz is automatically set to (0,0,1). In 3D models, it is important that the auxiliary vector is not parallel to the local x-axis; otherwise, the cross-product (vz * x-axis) is zero and local y-axis and z-axis will not be defined.

Load Types

There are four types of loads considered in the LESM program:

Concentrated nodal force and moment in global axes directions.
Uniformely distributed force on elements, spanning its entire length, in local or in global axes directions.
Linearly distributed force on elements, spanning its entire length, in local or in global axes directions.
Temperature variation along elements local axes (temperature gradient).

In addition, nodal prescribed displacements and rotations may be specified.

It is assumed that there is a single load case.

Components of Concentrated Nodal Loads

In 2D truss models, concentrated nodal loads are force components in global X and global Y directions.
In 3D truss models, concentrated nodal loads are force components in global X, global Y and global Z directions.
In 2D frame models, concentrated nodal loads are force components in global X and global Y directions, and a moment component about global Z direction.
In 3D frame models, concentrated nodal loads are force and moment components in global X, global Y and global Z directions.
In grillage models, concentrated nodal loads are a force component in global Z direction, and moment components about global X and global Y directions.

Components of Distributed Loads on Elements

In 2D truss or 2D frame models, uniformely or linearly distributed forces have two components, which are in global X and Y directions, or in local x and y directions.
In 3D truss or 3D frame models, uniformely or linearly distributed forces have three components, which are in global X, Y and Z directions, or in local x, y and z directions.
In grillage models, uniformely or linearly distributed forces have only one component, which is in global Z direction.

Components of Thermal Loads on Elements

In 2D truss and 2D frame models, thermal loads are specified by a temperature gradient relative to element local y-axis, and the temperature variation on its center of gravity.
In 3D truss and 3D frame models, thermal loads are specified by the temperature gradients relative to element local y and z axes, and the temperature variation on its center of gravity.
In grillage models, thermal loads are specified by a temperature gradient relative to element local z-axis, and the temperature variation on its center of gravity.

The temperature gradient relative to an element local axis is the difference between the temperature variation on its bottom face (face on the negative side of local axis) and its upper face (face on the positive side of local axis).

Materials

All materials in LESM are considered to have linear elastic bahavior. In adition, homogeneous and isotropic properties are also considered, that is, all materials have the same properties at every point and in all directions.

Cross-Sections

All cross-sections in LESM are considered to be of a generic type, which means that particular shapes are not specified, only their geometric properties are provided, such as area, moment of inertia and height.

Object Oriented Classes

This program adopts an Object Oriented Programming (OOP) paradigm, in which the following OOP classes are implemented:

Auxiliary Functions and Files

Non-graphical version

clc;
clear;
close all;

% Specify file name (edit only this line to change file name for target model)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
fileName = 'Models/Frame2D/Frame2D_1.lsm';
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Initialize analysis driver object
drv = Drv();

% Open input file with model information
fid = fopen(fileName,'rt');

% Check for valid input file
if fid > 0
    % Read model information
    fprintf(1,'Pre-processing...\n');
    [vs,print] = readFile(fid,drv);

    % Check input file version compatibility
    if vs == 1
        % Process provided data according to the direct stiffness method
        drv.fictRotConstraint(1); % Create ficticious rotation constraints
        status = drv.process();   % Process data
        drv.fictRotConstraint(0); % Remove ficticious rotation constraints

        % Print analysis results
        if status == 1
            print.results();
        end
    else
        fprintf(1,'This file version is not compatible with the program version!\n');
    end
else
    fprintf(1,'Error opening input file!\n');
end