A landing gear assembly consists of various components viz. Lower side stay, Upperside stay, Locking actuators, Extension actuators, Tyres, and Locking pins to name a few. Each unit having a specific operation to deal with, in this project the main unit being studied is the lower brace. The primary objective is to analyse stresses in the element of the lower brace unit using strength of materials or RDM method and Finite Element Method (FEM) and compare both. Using the obtained data a suitable material is proposed for the component. The approach used here is to study the overall behaviour of the element by taking up each aspect, finally summing up the total effect of all the aspects in the functioning of the element.
Isoparametric bilinear quadrilateral element _ ppt presentationFilipe Giesteira
The theoretical formulation of an isoparametric element, from the Lagrange family. In addition, the MATLAB code of the FEM from the bilinear element with four nodes was also implemented.
Assignment developed in the scope of the finite element method course, lectured at FEUP (Faculdade de Engenharia da Universidade do Porto).
Response of dynamic systems to harmonic excitation is discussed. Single degree of freedom systems are considered. For general damped multi degree of freedom systems, see my book Structural Dynamic Analysis with Generalized Damping Models: Analysis (e.g., in Amazon http://buff.ly/NqwHEE)
Isoparametric bilinear quadrilateral element _ ppt presentationFilipe Giesteira
The theoretical formulation of an isoparametric element, from the Lagrange family. In addition, the MATLAB code of the FEM from the bilinear element with four nodes was also implemented.
Assignment developed in the scope of the finite element method course, lectured at FEUP (Faculdade de Engenharia da Universidade do Porto).
Response of dynamic systems to harmonic excitation is discussed. Single degree of freedom systems are considered. For general damped multi degree of freedom systems, see my book Structural Dynamic Analysis with Generalized Damping Models: Analysis (e.g., in Amazon http://buff.ly/NqwHEE)
Mumbai University.
Mechanical Engineering
SEM V
Dynamics of Machinery
Governors and Gyroscopes:
1.1 Governors: Introduction to Centrifugal and Inertia governors, Force analysis of governors-
Porter and Hartnell governors, Performance characteristics of governors, Governors effort
and power
1.2 Gyroscope: Introduction, Gyroscopic couple and its effect on spinning bodies, naval ships
during steering, pitching, rolling and their stabilization. Effect of gyroscopic and centrifugal
couples, permissible speeds on curve paths, gyroscopic effect due to lateral misalignment of
rigid disc mounted on shaft.
09
2
2.1 Static and Dynamic force analysis, in slider crank mechanism (neglecting mass of
connecting rod and crank), Engine force analysis, Turning moment on crank shaft
2.2 Dynamically equivalent systems, to convert rigid body in to two mass with and without
correction couple
06
3
3.1 Basic Concepts of Vibration:
Vibration and oscillation, causes and effects of vibrations, Vibration parameters - springs,
mass, damper, damper models, Motion- periodic, non-periodic, degree of freedom, static
equilibrium position, vibration classification, steps involved in vibration analysis
3.2 Free Undamped Single Degree of Freedom Vibration System:
Longitudinal, transverse, torsional, vibration system, methods for formulation of differential
equations by Newton, Energy, Lagrangian and Rayleigh's method
08
4
4.1 Free Damped Single Degree of Freedom Vibration System:
Viscous damped system - under damped, critically damped, over damped; Logarithmic
decrement; Coulomb's damping
4.2 Equivalent Single Degree of Freedom Vibration System:
Conversion of multi-springs, multi masses, multi-dampers into a single spring and damper
with linear or rotational co-ordinate system, Introduction to free multi-degree of freedom
vibration systems
07
5
5.1 Forced Single Degree of Freedom Vibratory System:
Analysis of linear and torsional systems subjected to harmonic force excitation and harmonic
motion excitation (excluding elastic damper)
5.2 Vibration Isolation and Transmissibility:
Force Transmissibility, motion transmissibility, typical isolators & mounts.
5.3 Vibration Measuring instruments: Principle of seismic instruments, vibrometer,
accelerometer - undamped and damped, Introduction to conditioning monitoring and fault
diagnosis
6
6.1 Rotor Dynamics:
Critical speed of single rotor, undamped and damped
6.2 Balancing: Static and Dynamic balancing of multi rotor system, balancing of reciprocating
masses in In-line engines, V-engines (excluding other radial engines)
CONSISTENT AND LUMPED MASS MATRICES IN DYNAMICS AND THEIR IMPACT ON FINITE EL...IAEME Publication
There are two strategies in the finite element analysis of dynamic problems related to natural frequency determination viz. the consistent / coupled mass matrix and the lumped mass matrix. Correct determination of natural frequencies is extremely important and forms the basis of any further NVH (Noise vibration and harshness) calculations and Impact or crash analysis. It has been thought by the finite element community that the consistent mass matrix should not be used as it leads to a higher computational cost and this opinion has been prevalent since 1970. We are of the opinion that in today’s age where computers have become so fast we can use the consistent mass matrix on relatively coarse meshes with an advantage for better accuracy rather than going for finer meshes and lumped mass matrix
This session covers the basics that are required to analyse indeterminate trusses to maximum of two degree indeterminacy which includes,
strain energy stored due to axial loads and bending stresses,
maxwell's reciprocal deflection theorem,
Betti's law,
castigliano's theorems,
problems based on castigliano's theorems on beams and frames,
unit load method,
problems on trusses with unit load method,
lack of fit in trusses,
temperature effect on truss members.
Unit 5- balancing of reciprocating masses, Dynamics of machines of VTU Syllabus prepared by Hareesha N Gowda, Asst. Prof, Dayananda Sagar College of Engg, Blore. Please write to hareeshang@gmail.com for suggestions and criticisms.
Mumbai University.
Mechanical Engineering
SEM V
Dynamics of Machinery
Governors and Gyroscopes:
1.1 Governors: Introduction to Centrifugal and Inertia governors, Force analysis of governors-
Porter and Hartnell governors, Performance characteristics of governors, Governors effort
and power
1.2 Gyroscope: Introduction, Gyroscopic couple and its effect on spinning bodies, naval ships
during steering, pitching, rolling and their stabilization. Effect of gyroscopic and centrifugal
couples, permissible speeds on curve paths, gyroscopic effect due to lateral misalignment of
rigid disc mounted on shaft.
09
2
2.1 Static and Dynamic force analysis, in slider crank mechanism (neglecting mass of
connecting rod and crank), Engine force analysis, Turning moment on crank shaft
2.2 Dynamically equivalent systems, to convert rigid body in to two mass with and without
correction couple
06
3
3.1 Basic Concepts of Vibration:
Vibration and oscillation, causes and effects of vibrations, Vibration parameters - springs,
mass, damper, damper models, Motion- periodic, non-periodic, degree of freedom, static
equilibrium position, vibration classification, steps involved in vibration analysis
3.2 Free Undamped Single Degree of Freedom Vibration System:
Longitudinal, transverse, torsional, vibration system, methods for formulation of differential
equations by Newton, Energy, Lagrangian and Rayleigh's method
08
4
4.1 Free Damped Single Degree of Freedom Vibration System:
Viscous damped system - under damped, critically damped, over damped; Logarithmic
decrement; Coulomb's damping
4.2 Equivalent Single Degree of Freedom Vibration System:
Conversion of multi-springs, multi masses, multi-dampers into a single spring and damper
with linear or rotational co-ordinate system, Introduction to free multi-degree of freedom
vibration systems
07
5
5.1 Forced Single Degree of Freedom Vibratory System:
Analysis of linear and torsional systems subjected to harmonic force excitation and harmonic
motion excitation (excluding elastic damper)
5.2 Vibration Isolation and Transmissibility:
Force Transmissibility, motion transmissibility, typical isolators & mounts.
5.3 Vibration Measuring instruments: Principle of seismic instruments, vibrometer,
accelerometer - undamped and damped, Introduction to conditioning monitoring and fault
diagnosis
6
6.1 Rotor Dynamics:
Critical speed of single rotor, undamped and damped
6.2 Balancing: Static and Dynamic balancing of multi rotor system, balancing of reciprocating
masses in In-line engines, V-engines (excluding other radial engines)
CONSISTENT AND LUMPED MASS MATRICES IN DYNAMICS AND THEIR IMPACT ON FINITE EL...IAEME Publication
There are two strategies in the finite element analysis of dynamic problems related to natural frequency determination viz. the consistent / coupled mass matrix and the lumped mass matrix. Correct determination of natural frequencies is extremely important and forms the basis of any further NVH (Noise vibration and harshness) calculations and Impact or crash analysis. It has been thought by the finite element community that the consistent mass matrix should not be used as it leads to a higher computational cost and this opinion has been prevalent since 1970. We are of the opinion that in today’s age where computers have become so fast we can use the consistent mass matrix on relatively coarse meshes with an advantage for better accuracy rather than going for finer meshes and lumped mass matrix
This session covers the basics that are required to analyse indeterminate trusses to maximum of two degree indeterminacy which includes,
strain energy stored due to axial loads and bending stresses,
maxwell's reciprocal deflection theorem,
Betti's law,
castigliano's theorems,
problems based on castigliano's theorems on beams and frames,
unit load method,
problems on trusses with unit load method,
lack of fit in trusses,
temperature effect on truss members.
Unit 5- balancing of reciprocating masses, Dynamics of machines of VTU Syllabus prepared by Hareesha N Gowda, Asst. Prof, Dayananda Sagar College of Engg, Blore. Please write to hareeshang@gmail.com for suggestions and criticisms.
Advanced mathematical analysis of chassis integrated platform designed for un...Dr.Vikas Deulgaonkar
The present work deals with advanced mathematical stress analysis of a platform integrated structure mounted on vehicle chassis designed for unconventional type of loading pattern. The perceptible loading cases in the present analysis comprise static load and its effect on the platform/structure by usage of simple shear force & bending moment diagrams. Deflection analysis using conventional Macaulay’s method invokes the structures suitability for the transportation. Present analysis accentuates on the design stage aspects of the platform as this research is a step in proposed doctoral study. A different type of combination of longitudinal and cross members in platform/frame design is formulated. Present design is anticipated after analysis of all possible combinations& orientations of longitudinal and cross members. Determination of section properties of longitudinal and cross members of the platform & deduction of elementary stress based on the unconventional load pattern are the fundamental steps in design and analysis of structure. Peculiarity of this analysis is the usage of combined section modulus of three members for computation of stress. Present research provides a tool that can be used prior to computer aided design and finite element analysis.
OPTIMIZATION OF AERODYNAMIC AND STRUCTURAL PERFORMANCES OF A WIND TURBINE BLA...IAEME Publication
The purpose of this study is to optimize the energy efficiency of a wind turbine
blade and reduce its cost. In this paper, we define several optimization targets such as
maximizing Cl / Cd ratio and minimizing the deformation and mass of the blade. To
solve this multi-objectives optimization problem, we used the ant colony heuristic
optimization method on a blade model computed by the BEM and the FEM methods.
The optimization results are compared with the results obtained by the BEM method.
Structural analysis "slope-deflection moethod and robot modeling comparison" ...Omar Daher
structural analysis using both manual calculations"slope deflection" and software"Robot" for a frame.Term protect solution.See
https://www.slideshare.net/OmarDaher6/cvle-312-term-project.
Fluid-Structure Interaction Over an Aircraft WingIJERDJOURNAL
ABSTRACT:- Aircraft is a brilliant man-made structure which helps us to fly over the world. At the same time, aircraft is a complex structure to be checked and maintained for the aero elasticity due to aerodynamic properties. In this paper, the fluid-structure interaction problem in super critical NASA SC(2)-0412 airfoil is discussed. The main aim of this project is to find the best performance and deformation limit of the wing on different Mach numbers. This project is completely done by numerical methods of designing the wing using CATIA and flow properties in Computational Fluid Dynamics (CFD) method. Finally, the structural analysis for deformation is analysed in ANSYS. The analytical approach of fluid-structure interaction over an Aircraft wing is complex.
OPTIMIZATION AND FATIGUE ANALYSISOF A CRANE HOOK USING FINITE ELEMENT METHODijmech
Stress analysis plays an important role in the design of structures like crane hook under loading conditions.
Crane hook is a reliable lifting component being used in industries. Structure failure of crane hook occurs
because of the stress induced due to repetitive loading and unloading conditions. In this study, solid
modeling of crane hook having trapezoidal cross-section referring to one of its existing design is done
using SOLIDWORKS. Further, analyses are carried out in ANSYS Workbench and nCode DesignLife. The
lengths of two parallel sides of the cross-section of crane hook are varied and different candidates are
obtained for loading capacity of 30 ton on the basis of Mass, total Displacement and Von-Mises stress.This
is done to reduce weight and balance economy. Further, out of these candidates, best candidates are
considered and fatigue analysis is performed on these candidates.
EShape Optimization Of A Suspension Bellcrank Using 3d Finite Element MethodsIJERA Editor
The paper represents an application of an optimization procedure for a mass and stress optimization of the bellcrank of a double wishbone suspension system of SRM University’s FormulaSAE vehicle. The used optimization procedure, so-called Fully Stressed Design, is based on an indirect approach utilizing optimum criteria. The aim of the optimization was to achieve the lowest possible mass of the construction taking into consideration the allowed resistance and also to investigate and analyze the structural stress distribution of bellcrank at the real time condition during damping process and the spring actuation.
Modeling and Structural Analysis of a Wing [FSI ANSYS&MATLAB] BahaaIbrahim10
In our study, analyzing aircraft’s wing with the old assumptions will not give an exact solution but
this solution (total deformation) changes according to the geometry of the cross-section of the beam, so
the total deformation of the beam may be greater or lower than the exact solution. In these two cases,
the solution is not acceptable as in the first case which the deformation is greater than the exact solution
will make more weight and cost, and Engineers design aircraft at minimum weight and less cost. But in
the second case which will make lower deformation than exact solution will be much risky as the aircraft
could fail at any time, and this case much dangerous because it threatens the life of people.
Similar to Analysis of CANADAIR CL-215 retractable landing gear. (20)
FORTRAN is used as a numerical and scientific computing language. The main objective of the lab work is to understand FORTRAN language using which we solve simple numerical problems and compare different methodologies. Through this project we make use of various functions of FORTRAN and solve a FDM simple heat equation problem applying different conditions viz. Dirichlet and Von Neumann. The given problems are solved analytically then built and compiled using a free integrated development environment called CODE::BLOCKS [1] which is an open source platform for FORTRAN and C.
Thermochemical study of Energy associated with reactions using python.Nagesh NARASIMHA PRASAD
Thermochemistry deals with the study of the energy and heat associated with chemical reactions or physical transformations. With numerous products and reactants it’s a very complex phenomenon to analyse and sometimes even impossible to obtain accurate results analytically. Using a programming language to solve or compute such complex problems gives reliable results. CANTERA is an open source tool based on python programming language which can be used to solve chemical, thermodynamics processes. We make use of CANTERA in this project to obtain chemical composition of products and compare it with analytical solutions and also to calculate flame temperature, enthalpy etc.
vFORTRAN is used as a numerical and scientific computing language. The main objective of the lab work is to understand FORTRAN language using which we solve simple numerical problems and compare different methodologies. Through this project we make use of various functions of FORTRAN and solve a simple projectile problem and also LAPACK library to solve a tridiagonal matrix problem. We use DGESV and DGTSV functions to make it possible. The given problems are built and compiled using a free integrated development environment called CODE::BLOCKS [1] which is an open source platform for FORTRAN and C.
Design and Testing of Magneto Rheological Damper for Vehicle SuspensionNagesh NARASIMHA PRASAD
Conventional suspension systems have been ruling the automobile industry since ages but with the advancement in material technology it has led us to a new kind of suspension systems known as Active Suspension systems which involves smart materials like Magneto Rheological Fluids (MR), Electro Rheological Fluids (ER) etc. In present day conventional suspension system, we have a coiled spring setup with a simple damper filled with oil in motorcycles which are passive in nature. But off late in 21st century the development of smart materials bought in new dimensions of research in the suspension technology which has led to the era of active suspension systems. Now we can see usage of Semi Active Suspension/Active suspension system in cars and heavy vehicles which enhances the life of automobile, ride comfort and drive control. Though these systems have been developed for high end cars and military vehicles, the implementation of this technology to motorcycles has been a topic of research interest to various motorcycle industries.
We hence designed and machined an Magneto-Rheological Damper to Motorcycles, this system senses the irregularities of the road with various sensors and tries to stiffen and soften the suspension system in motorcycles in turn providing safe and snuggly ride with increased control during rough terrain. Thereby with this suggested concept an alternative design consideration of suspension systems has been offered.
The presentation was prepared for an Technical Paper Presentation competition. It contains basic conceptual explanations pertaining to the BWB concept.
A Presentation on Reusable Launcher Technology , with reference and basis of SpaceX Technologies Falcon 9 reusable rocket. With basic slides explaining the overview of the technology presented. ( No analytical or numerical issues addressed)
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
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Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
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Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Pile Foundation by Venkatesh Taduvai (Sub Geotechnical Engineering II)-conver...
Analysis of CANADAIR CL-215 retractable landing gear.
1. Structures - Materials Project Work
By
NARASIMHA PRASAD Nagesh & PENKULINTI Sai Sreenivas
Submitted to
MAHAJAN Prasad Baliram
2. ABSTRACT
Landing gear is one of the critical subsystems of an aircraft. The need to design landing gear with
minimum weight, minimum volume, high performance, improved life, and reduced life cycle cost have
posed many challenges to landing gear designers and practitioners. Further, it is essential to reduce the
landing gear design and development cycle time while meeting all the regulatory and safety requirements.
Many technologies have been developed over the years to meet these challenges in design and
development of landing gear. This project presents the study of lower brace of a CANADAIR CL-215
landing gear.
CONTENTS:
List of figures. & tables.
1. Introduction 1
2. Aim and Objective 1
3. Study and Analysis 3
3.1 Kinematic load estimation 3
3.2 Stress estimation by RDM method 6
3.2.1 Cut method, assumptions and equations 7
3.2.2 Global coordinate system to local coordinate system 8
3.2.3 Forces in beam 10
3.2.4 Equivalent Von Mises Stress 10
3.2.5 Code and Results 12
3.3 Stress estimation by ABAQUS (FEM) 13
4. Comparison between RDM & FEM 15
CONCLUSION 16
REFERENCE 17
APPENDIX 18
3. LIST OF FIGURES & TABLES.
Fig. 1: Canadair CL-215.
Fig. 2: Canadair CL-215 Landing gear Linking graph (a) and physical representation (b).
Fig. 3: Landing Gear lower brace element representation (a) and links configurations (b).
Table 1: Basic link systems.
Table 2: Link configuration and tensors.
Table 3: Element force and moment balance.
Fig. 4: 2D diagram to obtain elemental length.
Fig. 5: MATLAB code for solving simplified system of equation.
Fig. 6: Top and side views of element BF cross section.
Fig. 7: Front view of element BF.
Fig. 8: Cut method to obtain moment & loads.
Fig. 9: Transformation in z axis 𝜃1 =
𝜋
2
.
Fig. 10: Transformation in y axis 𝜃2 = 37.058°
.
Fig. 11: Transformation in z axis 𝜃3 = 𝜋.
Fig. 12: Stress-Strain curve of ductile material.
Fig. 13: MATLAB code for Von Mises equation solving.
Fig. 14: Graph plotting x coordinate with stress.
Fig. 15: Constrained couple applied on surfaces.
Fig. 16: Load and Boundary condition application.
Fig. 17: Meshing of part (Within limits of student version).
Fig. 18: Final Von Mises contour representation.
4. Structures-Materials Project
Narasimha Prasad Nagesh & Penkulinti Sai Sreenivas Page | 1
I. Analysis of CANADAIR CL-215 retractable landing gear.
1. INTRODUCTION
Of the many components that must be defined in an aircraft, the Landing gear is one of the most crucial.
It must be placed in the right position and proper structural orientation in order to ensure smooth and
effective realization of its purpose. The landing gear supports the craft when it is not flying, allowing it to
take off, land, and taxi without damage. Faster aircraft have retractable undercarriages, which fold away
during flight to reduce drag.
Fig. 1: Canadair CL-215.
The Canadair CL-215 is a twin-engine, high-wing general-purpose amphibious aircraft. It features an
atypically spacious fuselage for an amphibian, which is designed to accommodate for the operational
needs of various roles that the aircraft was developed to perform. The CL-215 can be used as an airborne
firefighting platform, in which capacity it is used as a water bomber; it has been claimed to be the first
aircraft designed to withstand the severe aerodynamic and hydrodynamic loads imposed by such usage.
Beyond the water bomber role, the CL-215 was designed for use in other capacities, such as a search and
rescue platform, passenger transport, and freighter; for this purpose, the cabin can be configured in various
different ways, including a flexible combi configuration. Under typical operations, these applications
would harness the aircraft's ability to land and take-off from the water, the hull having been designed to
enable its use upon the open seas.
2. AIM AND OBJECTIVE
A landing gear assembly consists of various components viz. Lower side stay, Upper side stay, Locking
actuators, Extension actuators, Tyres and Locking pins to name a few. Each unit having a specific
operation to deal with, in this project the main unit being studied is the lower brace.
5. Structures-Materials Project
Narasimha Prasad Nagesh & Penkulinti Sai Sreenivas Page | 2
Fig. 2: Canadair CL-215 Landing gear Linking graph (a) and physical representation (b).
The aircraft’s built up mass is 17100Kgs which should be supported by the landing gear during operation
which may vary according to operational conditions. More specifically the lower brace is reconstructed as
in the Fig. 3(a). Few assumptions are considered to make the problem simple which are,
1. All the joints are perfect with no friction.
2. No distributed loads across individual elements.
3. Joints are permanent.
4. Whole unit follows linear elastic behaviour.
Fig. 3: Landing Gear lower brace element representation (a) and links configurations (b).
The primary objective is to analyse stresses in any one element of the lower brace unit using strength of
materials or RDM method and Finite Element Method (FEM) and compare the both. Using the obtained
data a suitable material is proposed for the component. The approach used here is to study the overall
behaviour of the element by taking up each aspect, finally summing up the total effect of all the aspects in
the functioning of the element.
The step wise approach is as below,
a. Evaluation of loads
b. Evaluation of loads using Strength of materials concept.
c. Evaluation of loads using Finite Element Methods.
d. Evaluation of Material Selection.
6. Structures-Materials Project
Narasimha Prasad Nagesh & Penkulinti Sai Sreenivas Page | 3
3. STUDY AND ANALYSIS
3.1 KINEMATIC LOAD ESTIMATION.
To make a load calculation for any mechanical component one should also obtain the Degree of static
indeterminacy. Number of unknowns are equal to number of equations hence the system is externally
determinate.
By the Fig. 2(a,b) the tensor for each link can be tabulated. The basic three types of links are as follows,
Link Torseur Link diagram
Pivot [T] = [
𝑿 𝟎
𝒀 𝑴
𝒁 𝑵
]
Rotule [T] = [
𝑿 𝟎
𝒀 𝟎
𝒁 𝟎
]
Spherical Cylinder [T] = [
𝟎 𝟎
𝒀 𝟎
𝒁 𝟎
]
Table 1: Basic link systems.
Inspecting the lower brace of the structural system one can make comparison with one of the three basic
links. This is done in order to ease the problem solving technique where all loads and moments should be
defined with respect to their direction.
Sl No. Link. Configuration Tensor.
1 B(4-1) Pivot
[
𝑋41 0
𝑌41 𝑀41
𝑍41 𝑁41
]
2 C(4-2) Spherical Cylinder
[
0 0
𝑌42 0
𝑍42 0
]
3 C(4-3) Rotule
[
𝑋43 0
𝑌43 0
𝑍43 0
]
4 D(0-3) Rotule
[
𝑋03 0
𝑌03 0
𝑍03 0
]
5 E(0-2) Pivot
[
𝑋02 0
𝑌02 𝑀02
𝑍02 𝑁02
]
7. Structures-Materials Project
Narasimha Prasad Nagesh & Penkulinti Sai Sreenivas Page | 4
6 F(0-1) Pivot
[
𝑋01 0
𝑌01 𝑀01
𝑍01 𝑁01
]
Table 2: Link configuration and Tensors.
Being defined all the tensors for each links, considering these a system of equation can be formulated to
solve for the unknown forces and momentum values. In order to do so we need to consider the ground
force tensor as below,
TA = [
4𝑚𝑔 0
0 0
3𝑚𝑔 0
] where, m = 8550Kgs. And g = 10 m/s2
.
With all the data available consider each element to give the Force and Moment balance which can be
achieved by the summation of the Tensor values and equating those to the external force or moment. In
turn forming the system of equations required to obtain unknowns,
Force balance by,
F1 + F2 = 0
Moment balance by,
M + Lx(F) = 0
Cross products of length and forces are solved utilizing an online tool (link provided in reference).
Element Force Balance Moment Balance (Moment acting)
BF (1)
X41 + X01 = 0
Y41 + Y01 = 0
Z41 + Z01 = 0
BFy×Z01 – BFz×Y01 = 0
M41 + M01 + BFz×X01 = 0
N41 + N01 - BFy×X01 = 0
CE (2)
X02 = 0
Y42 + Y02 = 0
Z42 + Z02 = 0
CEy×Z01 – CEz×Y01 = 0
M02 + CEz×X01 = 0
N02 - CEy×X01 = 0
CD (3)
X43 + X03 = 0
Y43 + Y03 = 0
Z43 + Z03 = 0
CDy×Z01 – CDz×Y01 = 0
CEz×X01 = 0
- CDy×X01 = 0
A BC (4)
4mg - X43 – X41 = 0
-Y41 – Y43 - Y42 = 0
3mg - Z41 + Z43 - Z42 = 0
ABy×Z41 -ABz×Y41 +ACy×Z42 -ACz×Y42
+ACy×Z43 - ACz×Y43 = 0
M41 + ABz×X41 + ACz×X43 = 0
N41 - ABy×X41 - ACy×X43 = 0
Table 3: Element force and moment balance.
By solving the set of equations in Table 2 the 23 unknown force and moment values can be obtained. The
respective lengths for the given elemental distance are obtained from the provided drawing named
“Maquette Canadair CL215 – 3” the same is provided in the Fig. 4.
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Fig. 4: 2D diagram to obtain elemental length.
The basic equations simplification is done by hand and been attached in appendix. The remaining
unknowns are solved using a MATLAB program, the same has been saved in the name as given below,
PROGRAM NAME: matfin.m (present in ZIP file)
Initially syms command is used to declare variables
Gravity constant g and mass m are stated
Linear equations which are to be solved are declared under eqns
Syntax [A,b] = equationsToMatrix(eqns) is used to convert the equations to matrix form
Next, vars=symvars(eqns) syntax is used that automatically detects the variables in the equations
and returns the coefficient matrix which follows the order determined by symvar
x = Ab is used the find the unknown variables.
Fig. 5: MATLAB code for simplified solving system of equation.
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𝐴 =
[
0 1 0 0 1.323 0 0 0 0
0 0 0 0 1 0 1 0 0
0 0 1 0 0 1.732 0 0 0
0 0 0 0 0 1 0 1 0
1 0 0 1 0 0 0 0 0
1.864 0 0 0 0 0 0 0 1
0 −1 −1 −1 0 0 0 0 0
0 0 0 0 0 0 −1 −1 −1
0 −0.816 −1.338 −1.338 0 0 0.504 0.588 0.588]
𝑏 =
(
0
0
0
0
0
0
0
−256500
0 )
, 𝑥 =
(
𝑌03
𝑌41
𝑌42
𝑌43
𝑍01
𝑍02
𝑍41
𝑍42
𝑍43)
Results, in KN 𝑥 =
(
−245
−329
84
245
248.6
−48.4
−248.6
48.47
456 )
The final torseurs are,
[T01] = [
−342 0
329 531
248.6 −506
] [T41] = [
342 0
−329 −279
−248.6 172
] [T02] = [
0 0
−84 0
−48.4 0
]
[T42] = [
0 0
84 0
48.47 0
] [T03] = [
0 0
−245 0
−456 0
] [T43] = [
0 0
245 0
456 0
]
Hence the obtained values of forces and moments which are concerning our scope of work (Element BF
or 1, end torseurs) are as given below,
[T41]B = [
𝟑𝟒𝟐 𝟎
−𝟑𝟐𝟗 −𝟐𝟕𝟗
−𝟐𝟒𝟗 𝟏𝟕𝟐
] [T01]F = [
−𝟑𝟒𝟐 𝟎
𝟑𝟐𝟗 𝟓𝟑𝟏
𝟐𝟒𝟗 −𝟓𝟎𝟔
]
Note: Forces are in KN and Moments are in K-Nm.
Now the coordinates must be rectified and transformed into local system in order to ease the solving
methodology.
3.2 STRESS ESTIMATION BY RDM METHOD.
RDM Method is the analytical method used to determine the stresses and deformations of the structure
and is used to compare it with other solutions and also validate the model. The link or component BF
can be considered as beam and stresses are estimated and some of the assumptions made are i.e., the
material is
i. Elastic.
ii. Linear.
iii. Homogeneous.
iv. Isotropic.
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Fig. 6: Top and Side views of element BF cross section.
Fig. 7: Front view of element BF.
3.2.1 Cut Method, Assumptions And Equations.
Résistance des Matériaux (RDM) method or Strength of Materials (SOM) is used to calculate the
stresses and deformations in the given structure. The objective is to design the structure according to
criteria of resistance, admissible deformation and acceptable financial cost.
Fig. 8: Cut section method to obtain moment & loads.
Assumptions,
Elastic – The material returns to its initial shape after a loading & unloading cycle.
Linear – The deformations are proportional to the stresses.
Homogeneous – The material is of the same nature throughout its mass.
B G
L-x x
F
Arbitrary section G
N
L
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Isotropic – The properties of the material are identical in all directions.
Static or Cohesion torseur is used to describe the Mechanical actions in a member,
𝑇⃗ = [
𝑁 𝑀𝑡
𝑇𝑦 𝑀 𝑦
𝑇𝑧 𝑀𝑧
]
Where, N is Normal Force, Ty shear force in y direction and Tz shear force in z direction. Mt is the
moment of torsion and its vector has direction x, My is the bending moment along y and Mz bending
moment along z.
Note:
The Neutral axis of the beam lies along the x-axis. It is the mean fibre where centre of gravity
lies.
The principal axes are represented by Gy and Gz and they cross through the CG.
The y-axis lies along the width of the beam and the z-axis lies along the depth of the beam.
𝑀𝑧 = 𝑀𝑧 + 𝑇𝑦 × (𝐿 − 𝑥)
Stresses in beam are given by,(in MPa.)
𝝈 𝒙𝒙 =
𝑵
𝒃 × 𝑯(𝒙)
+
[𝑴 𝒛 + 𝑻 𝒚 × (𝑳 − 𝒙)] × 𝒚
𝑰 𝒛
𝝈 𝒙𝒚 =
𝑻 𝒚
𝒃 × 𝑯(𝒙) × (𝟏 + 𝜼)
+
𝑴𝒕 × 𝒛
𝑱
𝝈 𝒙𝒛 =
𝑴𝒕 × 𝒚
𝑱
+
𝑻 𝒛
𝒃 × 𝑯(𝒙) × (𝟏 + 𝜼)
Where, y is
𝐻(𝑥)
2
and z is
𝑧
2
.
3.2.2 Global Coordinate System To Local Coordinate System.
In 3-dimensional analysis it is necessary to convert section properties, forces and deflections between
coordinate systems defined by the individual structural members (local coordinates) and the common
coordinate system defining the entire structure (global coordinates).
Rz (𝜃1 =
𝜋
2
)
Ry (𝜃2 = 37.058°)
Ry (𝜃3 = 𝜋)
Rotational matrix along z axis,
Rz = [
cos 𝜃 sin 𝜃 0
−sin 𝜃 cos 𝜃 0
0 0 1
]
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Rotational matrix along y axis,
Ry= [
cos 𝜃 0 −sin 𝜃
0 1 0
sin 𝜃 0 cos 𝜃
]
Fig. 9: Transformation in z axis 𝜽 𝟏 =
𝝅
𝟐
.
Fig. 10: Transformation in y axis 𝜽 𝟐 = 𝟑𝟕. 𝟎𝟓𝟖°
.
Fig. 11: Transformation in z axis 𝜽 𝟑 = 𝝅.
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Matrix multiplication,
= [
−1 0 0
0 1 0
0 0 −1
] × [
0.797 0 −0.602
0 1 0
0.602 0 0.797
] × [
0 1 0
−1 0 0
0 0 1
]
∴ Final Rotational Matrix is,
[
𝟎 −𝟎. 𝟕𝟗𝟕 𝟎. 𝟔𝟎𝟐
−𝟏 𝟎 𝟎
𝟎 −𝟎. 𝟔𝟎𝟐 −𝟎. 𝟕𝟗𝟕
]
Transformation of a torseurs is done to ease the visualisation of forces acting on the element in a simple
standard coordinate system. Rather than transforming the magnitude of the force accordingly at each
point, it is prefered to rotate the element to local (standard) axis system.
3.2.3 Forces In Beam.
Torseurs T01 and T41 after the transformation,
𝑻⃗⃗ 𝟎𝟏 = [
−𝟏𝟏𝟑. 𝟏𝟗𝟑 −𝟕𝟐𝟗. 𝟒𝟓𝟗
𝟑𝟒𝟐 𝟎
−𝟑𝟗𝟕. 𝟔𝟗𝟐 𝟖𝟒. 𝟑𝟗𝟑
]
𝑻⃗⃗ 𝟒𝟏 = [
𝟏𝟏𝟑. 𝟏𝟗𝟑 𝟑𝟐𝟔. 𝟑𝟓𝟖
−𝟑𝟒𝟐 𝟎
𝟑𝟗𝟕. 𝟔𝟗𝟐 𝟑𝟎. 𝟗𝟖𝟏
]
Note: Forces are in KN and Moments are in K-Nm.
3.2.4 Equivalent Von Mises Stress.
Von Mises stress is a value used to determine if a given material will yield or fracture. It is mostly used
for ductile materials such as metals. The Von Mises criterion states that if the von Mises stress of material
under load is equal or greater than the yield limit of the same material under simple tension, which is easy
to determine experimentally, then the material will yield.
Note: Complex loads/stress tensors that act in a member : 𝜎̿(𝑛) = (
𝜎𝑥𝑥 𝜎𝑥𝑦 𝜎𝑥𝑧
𝜎 𝑦𝑥 𝜎 𝑦𝑦 𝜎 𝑦𝑧
𝜎𝑧𝑥 𝜎𝑧𝑦 𝜎𝑧𝑧
)
Fig. 12 represents the stress-strain curve for the ductile material, 𝜎 𝑦𝑖𝑒𝑙𝑑 is the yield stress and if the 𝜎 𝑎𝑥𝑖𝑎𝑙
(axial stress) is less than the yield stress then it is in elastic region and if 𝜎 𝑦𝑖𝑒𝑙𝑑 = 𝜎 𝑎𝑥𝑖𝑎𝑙 then it represents
the start of plasticity.
Ry (𝜃3 = 𝜋) Ry (𝜃2 = 37.058°) Rz (𝜃1 =
𝜋
2
)
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Fig. 12: Stress-Strain curve of ductile material.
The general form of von mises criterion is:
𝝈 𝒆𝒒 =
𝟏
√𝟐
[(𝝈 𝒙𝒙 − 𝝈 𝒚𝒚)
𝟐
+ (𝝈 𝒚𝒚 − 𝝈 𝒛𝒛)
𝟐
+ (𝝈 𝒛𝒛 − 𝝈 𝒙𝒙) 𝟐
+ 𝟔(𝝈 𝒙𝒚
𝟐
+ 𝝈 𝒙𝒛
𝟐
+ 𝝈 𝒚𝒛
𝟐
)]
𝟏
𝟐
Here, 𝜎𝑥𝑥, 𝜎 𝑦𝑦 and 𝜎𝑧𝑧 are the Normal stresses. 𝜎𝑥𝑦, 𝜎𝑥𝑧 and 𝜎 𝑦𝑧 are the Shear stresses. Few tensors are
to be neglected for which the definition and subsequent explanation in the following paragraph provides
clarification.
It is a tensor that consists of nine components 𝜎𝑖𝑗that completely define the state of stress at a point
inside a material in the deformed state or configuration. It is of Rank 2 or it has 2 basis vectors per
component, for example 𝜎𝑖𝑗 here i represents direction the area is perpendicular to and j represents the
direction of the force.
In our case we consider x plane and direction of the force also in x axis, hence the stress and shear
components that contemplates are𝜎𝑥𝑥, 𝜎𝑥𝑦, 𝜎𝑥𝑧, 𝜎 𝑦𝑥 𝑎𝑛𝑑 𝜎𝑧𝑥. From the Cauchy’s second law of motion
𝜎𝑖𝑗 = 𝜎𝑗𝑖which leads to the conclusion that stress tensor is symmetric.
If equivalent von mises stress 𝜎𝑒𝑞 is equal to the applied stress then plasticity starts.
𝝈̿(𝒏) =
[
𝑵
𝒃 × 𝒉(𝒙)
+
[𝑴 𝒛 + 𝑻 𝒚 × (𝑳 − 𝒙)] × 𝒚
𝑰 𝒛(𝒙)
𝑻 𝒚
𝒃 × 𝒉(𝒙) × (𝟏 + 𝜼)
+
𝑴𝒕 × 𝒛
𝑱(𝒙)
𝑴𝒕 × 𝒚
𝑱(𝒙)
+
𝑻 𝒛
𝒃 × 𝒉(𝒙) × (𝟏 + 𝜼)
𝑻 𝒚
𝒃 × 𝒉(𝒙) × (𝟏 + 𝜼)
+
𝑴𝒕 × 𝒛
𝑱(𝒙)
𝟎 𝟎
𝑴𝒕 × 𝒚
𝑱(𝒙)
+
𝑻 𝒛
𝒃 × 𝒉(𝒙) × (𝟏 + 𝜼)
𝟎 𝟎
]
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σxx =
−1048.084
ℎ(𝑥)
+
51.785 ×106
2 × ℎ(𝑥)2
−
38 × 103
2 × ℎ(𝑥)2
σxy =
−112.08 × 103
ℎ(𝑥)
σxz =
−12.370 × 106
ℎ(𝑥)2
−
2.83 ×103
ℎ(𝑥)
𝝈 𝒆𝒒 =
𝟏
√𝟐
× [(𝝈 𝒙𝒙
𝟐) + (−𝝈 𝒙𝒙) 𝟐
+ 𝟔 × (𝝈 𝒙𝒚
𝟐
+ 𝝈 𝒙𝒛
𝟐
)]
𝟎.𝟓
Note: The above Stresses are in MPa.
3.2.5 Code And Results.
Determining point where maximum von mises stress acts. Algorithm and code for the same is attached
below.
PROGRAM NAME: ma2.m (in zip file attached).
Initially syms syntax is used to declare the variables.
Area function (H) is declared.
Normal and shear stresses are declared namely, sxx, sxy and sxz.
Equivalent Von Mises stress equation is stated i.e. sigmaeq.
A simple syntax, fplot (f, xinterval) is used wherein it plots a curve for the specified interval
which helps in determining the point where maximum Von Mises stress is acting. Where,
f=sigmaeq and xinterval is [0,1116].
Fig. 13: MATLAB code for Von Mises equation solving.
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Fig. 14: Graph plotting x coordinate with stress.
The result or the Von Mises Stress value which we got is ~840 MPa that can be seen from the Fig. 14.
3.3 STRESS ESTIMATION BY ABAQUS (FEM).
Finite Element Method (FEM) is computational approach for solving problems governed by differential
equations. The concept is more useful for complicated geometries, loadings and material properties where
analytical solution is difficult or cannot be obtained. A large object is divided into many elements which
is called discretization. Each element will have a simple equation defining the geometry and other physical
characters, equations of all elements form together a global system to define the entire system.
Various FEM solving tools are used viz. ANSYS, Abaqus, DIANA and Nastran to name some. Here
Abaqus CAE tool is used, to model, define and solve the problem. The three main steps in any FEM
problem are,
i. Idealization.
ii. Discretization
iii. Solution.
Taking up each step, the process of idealization & discretization in this particular case is elaborated below,
1. The geometric modelling of the lower brace is accomplished using Abaqus CAE tool. Properties
of the model is defined as 200GPa and 0.3 which are Young’s modulus and Poisions ratios
respectively.
2. In order to apply the loads and boundary conditions appropriately, a constrained couple is created
on a reference point on the side phase of the model. This is done in order to bind the surface to a
reference point in turn assisting the application of moment load as represented in Fig. 15.
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Fig. 15: Constrained Couple applied on surfaces.
3. Now, the loads are applied on their respective application points i.e. as the load on the lower brace
effects by the holes through which the connecting rod passes and links the next element. As the
hole is not defined in this model due to many limitations, the loads are applied on the face edges
where the holes would be present. The moments are applied on the coupled reference points,
graphical representation of the same is given in Fig. 16.
Fig. 16: Load and boundary condition application.
4. The next step is discretization, using Abaqus student version the node numbering limit is 1000.
Hence using seed part instance the global element size is defined as 75, this leads in creating n
number of elements within the node limits.
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Fig. 17: Meshing of part (Within limits of Student Version).
After this the model is submitted to analysis in Abaqus solver, the principle aim is to compare the value
obtained in RDM and FEM methods. To do so the Von Mises stress value can be considered. Explaining
the concept,
1. The taper in the section was considered while defining the varying cross section of the element
for which the location on x coordinate for the maximum von mises stress is 0 i.e. which is almost
the starting point of the taper.
2. The value of the Von Mises obtained at x=0 in RDM method is ~840 MPa. (elaboration in previous
section)
3. In the same way the Von Mises value in FEM is around in the range of 715MPa – 915MPa. As
the whole region of taper starting is one element. The contour has varying values.
Fig. 18: Final Von Mises contour representation.
4. COMPARISION BETWEEN RDM & FEM.
To make a conclusion between both the methods used, it’s important to understand the basic difference
on how the stress calculations are obtained.
As explained in FEM section the object is discretized into many elements which are solved later
by combining into a global equation form. This might lead to high concentration of errors that
might sum up as the analysis is carried out.
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In RDM method the geometry is simplified to ease the calculations, thus reducing the accuracy.
But in FEM the structure is meshed which gives results at particular node and element but as the
mesh finess increases the results obtained may not be as desired since even for a small
perturbation, displacement or stress values are obtained at undesired region.
As the elements and nodes increases CPU cost and also the processing time increases but when it
comes to RDM method those above stated constraints doesn’t matter.
RDM method is used to solve simple structures but FEM is used to solve simple and complex
structures. In our case the Von mises stress result percentage error between RDM and
FEM(abaqus) is 4.54.
CONCLUSION.
The whole purpose of this project is to study the structural credibility of selected elements in CANADAIR
CL-215 landing gear system. To do so one should have a knowledge on geometry, mechanical properties,
loading, stress variation and material behaviour. Taking into consideration all the aspects mentioned
before, study of the structure is performed.
In order to analyse the stress variation and distribution, the first step is to define the joints and obtaining
the unknown forces and moments. Using these values and the geometric variation the principal stress
matrix is generated. Considering the Von Mises stress theory, which states that “If Von Mises stress is
equal to or more than yield stress plastic deformation starts making the design fail”. Therefore to provide
a stable and safe structure the Von Mises stress value should be respected. According to the same a suitable
material must be selected. Having a simple geometry in this case RDM method is incorporated but the
same cannot be done if a highly complex and non-homogenous system is considered, then Finite Element
Method is used.
Using FEM the model is created with proper mechanical properties and orientation, the Von Mises value
obtained is almost nearby to one calculated in RDM, showing the credibility of comparing both the
techniques. This also confirms the validity of FEM in structural problems which makes it easy to solve
complex geometries.
Summarizing all the results the main objective is to study and select the most suitable material for this
particular element for which a specific tool is required. Hence with the available possibilities it can be
suggested that the Titanium alloy of grade 6, Ti-6Al-2Sn-4Zr-6Mo can be used. Having yield strength of
1100MPa. It would be a safe select. But it should be noted that this is not the final material as cost
effectiveness is not considered in this process.
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REFERENCES
1. http://dsk.ippt.pan.pl/docs/abaqus/v6.13/books/usi/default.htm?startat=pt03ch19s07hlb02.html
2. http://www.dhondt.de/ccx_2.15.pdf
3. https://academy.3ds.com/en/software/abaqus-student-edition
4. https://abaqus-docs.mit.edu/2017/English/SIMACAEOUTRefMap/simaout-c-
output.htm#simaout-c-output-t-TheDataFile-sma-topic1
5. “Introduction to Finite Element Method (FEM) for Structural mechanics” reference book by
Marianne Beringhier.
6. https://en.wikipedia.org/wiki/Canadair_CL-215
7. https://www.symbolab.com/solver/vector-cross-product-calculator
8. https://abaqus-docs.mit.edu/2017/English/SIMACAECSTRefMap/simacst-c-coupling.htm
9. http://dsk.ippt.pan.pl/docs/abaqus/v6.13/books/usi/default.htm?startat=pt03ch16s09hlb02.html
10. https://www.simscale.com/docs/content/simwiki/fea/what-is-von-mises-stress.html
11. ps://bertoldi.seas.harvard.edu/files/bertoldi/files/abaqusinputfilemanualv1.pdf?m=1444417191
12. https://www.simscale.com/docs/content/simwiki/fea/what-is-von-mises-stress.html
13. https://en.wikipedia.org/wiki/Cauchy_stress_tensor