- Strain gauges oriented in different directions and arranged in a rosette configuration are used to determine principal strains and stresses at a point.
- Several types of rosette configurations are commonly used, including rectangular, delta, and tee rosettes with varying numbers and orientations of strain gauges.
- Simultaneous equations relating strain readings from each gauge to the principal strains are solved to determine the principal strain values and orientations.
- Principal stresses are then calculated from the principal strains using stress-strain relationships accounting for Poisson's ratio.
This PPT discusses Fatigue and Fracture mechanism, some history and problems. It has included on research paper. You can refer the literature review for further study of the topic.
This is a ppt which will give u a better understanding of fracture toughness of a material in short time. It also has great exposure to testing method that we do in our laboratory class in undergraduate courses. So good luck with slide.
This PPT discusses Fatigue and Fracture mechanism, some history and problems. It has included on research paper. You can refer the literature review for further study of the topic.
This is a ppt which will give u a better understanding of fracture toughness of a material in short time. It also has great exposure to testing method that we do in our laboratory class in undergraduate courses. So good luck with slide.
Lecture slides on the calculation of the bending stress in case of unsymmetrical bending. The Mohr's circle is used to determine the principal second moments of area.
This presentation gives an information about: photoelasticity, covering syllabus of Unit-3, of Experimental stress analysis subject for BE course under Visvesvaraya Technological University (VTU), Belgaum.
Stress concentrations produced by discontinuities in structures such as holes, notches, and fillets will be introduced in this section. The stress concentration factor will be defined. The concept of fracture toughness will also be introduced.
Stresses and its components - Theory of Elasticity and PlasticityAshishVivekSukh
Stress at any section is internal resistance offered by metal against the deformation caused by applied load.
It is Internal resistance pre-unit area.
When a metal is subjected to a load, it is deformed, no matter how strong the metal.
If the load is small, the distortion will probably disappear when the load is removed.
If the distortion disappears and the metal returns to its original dimensions upon removal of the load, the strain is called elastic strain.
If the distortion disappears and the metal remains distorted, the strain type is called plastic strain
Lecture slides on the calculation of the bending stress in case of unsymmetrical bending. The Mohr's circle is used to determine the principal second moments of area.
This presentation gives an information about: photoelasticity, covering syllabus of Unit-3, of Experimental stress analysis subject for BE course under Visvesvaraya Technological University (VTU), Belgaum.
Stress concentrations produced by discontinuities in structures such as holes, notches, and fillets will be introduced in this section. The stress concentration factor will be defined. The concept of fracture toughness will also be introduced.
Stresses and its components - Theory of Elasticity and PlasticityAshishVivekSukh
Stress at any section is internal resistance offered by metal against the deformation caused by applied load.
It is Internal resistance pre-unit area.
When a metal is subjected to a load, it is deformed, no matter how strong the metal.
If the load is small, the distortion will probably disappear when the load is removed.
If the distortion disappears and the metal returns to its original dimensions upon removal of the load, the strain is called elastic strain.
If the distortion disappears and the metal remains distorted, the strain type is called plastic strain
Rotation in 3d Space: Euler Angles, Quaternions, Marix DescriptionsSolo Hermelin
Mathematics of rotation in 3d space, a lecture that I've prepared.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks!
Fore more presentations, please visit my website at
http://www.solohermelin.com/
Elasticity, Plasticity and elastic plastic analysisJAGARANCHAKMA2
It is actually the basis of structural engineering to study elasticity and plasticity analysis. So people who are also studying in various fields of structure and need to analyze finite element analysis also need to study this basis.
ملزمة الرياضيات للصف السادس الاحيائي الفصل الثاني القطوع المخروطية 2022 anasKhalaf4
طبعة جديدة ومنقحة
حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
PROBABILITY DISTRIBUTION OF SUM OF TWO CONTINUOUS VARIABLES AND CONVOLUTIONJournal For Research
All physical subjects, involving random phenomena, something depending upon chance, naturally find their own way to theory of Statistics. Hence there arise relations between the results derived for hose random phenomena in different physical subjects and the concepts of Statistics. Convolution theorem has a variety of applications in field of Fourier transforms and many other situations, but it bears beautiful applications in field of statistics also .Here in this paper authors want to discuss some notions of Electrical Engineering in terms of convolution of some probability distributions.
Basic Mathematics (non-calculus) for k-12 students in B.C. Canada. Intended as a guide for teaching basic math to young learners, and uploaded as a personal favor to my friend Oliver Cougur. This is a supplement teaching/learning material, and functions as a 'cheat sheet' for instructors and/or students.
This is not intended as curriculum material. I guarantee nothing. I claim no ownership or discovery of any of the material in this document, however I reserve my right of creative expression for materials contained. This document may not be sold, copied or altered in anyway by anyone.
Please report any errors to s.grantwilliam@ieee.org
Mathematical analysis of a non-uniform tetradecahedron having 2 congruent reg...Harish Chandra Rajpoot
All the important parameters of a non-uniform tetradecahedron, having 2 congruent regular hexagonal faces, 12 congruent trapezoidal faces & 18 vertices lying on a spherical surface with a certain radius, have been derived by the author by applying "HCR's Theory of Polygon" to calculate solid angle subtended by each regular hexagonal & trapezoidal face & their normal distances from the center of non-uniform tetradecahedron, inscribed radius, circumscribed radius, mean radius, surface area & volume. These formulas are very useful in the analysis, designing & modeling of various non-uniform polyhedra.
Differential Geometry for Machine LearningSEMINARGROOT
References:
Differential Geometry of Curves and Surfaces, Manfredo P. Do Carmo (2016)
Differential Geometry by Claudio Arezzo
Youtube: https://youtu.be/tKnBj7B2PSg
What is a Manifold?
Youtube: https://youtu.be/CEXSSz0gZI4
Shape analysis (MIT spring 2019) by Justin Solomon
Youtube: https://youtu.be/GEljqHZb30c
Tensor Calculus
Youtube: https://youtu.be/kGXr1SF3WmA
Manifolds: A Gentle Introduction,
Hyperbolic Geometry and Poincaré Embeddings by Brian Keng
Link: http://bjlkeng.github.io/posts/manifolds/,
http://bjlkeng.github.io/posts/hyperbolic-geometry-and-poincare-embeddings/
Statistical Learning models for Manifold-Valued measurements with application to computer vision and neuroimaging by Hyunwoo J.Kim
ملزمة الرياضيات للصف السادس التطبيقي الفصل الثاني القطوع المخروطية 2022anasKhalaf4
طبعة جديدة ومنقحة
حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
NUMERICAL SIMULATIONS OF HEAT AND MASS TRANSFER IN CONDENSING HEAT EXCHANGERS...ssuser7dcef0
Power plants release a large amount of water vapor into the
atmosphere through the stack. The flue gas can be a potential
source for obtaining much needed cooling water for a power
plant. If a power plant could recover and reuse a portion of this
moisture, it could reduce its total cooling water intake
requirement. One of the most practical way to recover water
from flue gas is to use a condensing heat exchanger. The power
plant could also recover latent heat due to condensation as well
as sensible heat due to lowering the flue gas exit temperature.
Additionally, harmful acids released from the stack can be
reduced in a condensing heat exchanger by acid condensation. reduced in a condensing heat exchanger by acid condensation.
Condensation of vapors in flue gas is a complicated
phenomenon since heat and mass transfer of water vapor and
various acids simultaneously occur in the presence of noncondensable
gases such as nitrogen and oxygen. Design of a
condenser depends on the knowledge and understanding of the
heat and mass transfer processes. A computer program for
numerical simulations of water (H2O) and sulfuric acid (H2SO4)
condensation in a flue gas condensing heat exchanger was
developed using MATLAB. Governing equations based on
mass and energy balances for the system were derived to
predict variables such as flue gas exit temperature, cooling
water outlet temperature, mole fraction and condensation rates
of water and sulfuric acid vapors. The equations were solved
using an iterative solution technique with calculations of heat
and mass transfer coefficients and physical properties.
KuberTENes Birthday Bash Guadalajara - K8sGPT first impressionsVictor Morales
K8sGPT is a tool that analyzes and diagnoses Kubernetes clusters. This presentation was used to share the requirements and dependencies to deploy K8sGPT in a local environment.
HEAP SORT ILLUSTRATED WITH HEAPIFY, BUILD HEAP FOR DYNAMIC ARRAYS.
Heap sort is a comparison-based sorting technique based on Binary Heap data structure. It is similar to the selection sort where we first find the minimum element and place the minimum element at the beginning. Repeat the same process for the remaining elements.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
Contact with Dawood Bhai Just call on +92322-6382012 and we'll help you. We'll solve all your problems within 12 to 24 hours and with 101% guarantee and with astrology systematic. If you want to take any personal or professional advice then also you can call us on +92322-6382012 , ONLINE LOVE PROBLEM & Other all types of Daily Life Problem's.Then CALL or WHATSAPP us on +92322-6382012 and Get all these problems solutions here by Amil Baba DAWOOD BANGALI
#vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore#blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #blackmagicforlove #blackmagicformarriage #aamilbaba #kalajadu #kalailam #taweez #wazifaexpert #jadumantar #vashikaranspecialist #astrologer #palmistry #amliyaat #taweez #manpasandshadi #horoscope #spiritual #lovelife #lovespell #marriagespell#aamilbabainpakistan #amilbabainkarachi #powerfullblackmagicspell #kalajadumantarspecialist #realamilbaba #AmilbabainPakistan #astrologerincanada #astrologerindubai #lovespellsmaster #kalajaduspecialist #lovespellsthatwork #aamilbabainlahore #Amilbabainuk #amilbabainspain #amilbabaindubai #Amilbabainnorway #amilbabainkrachi #amilbabainlahore #amilbabaingujranwalan #amilbabainislamabad
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
An Approach to Detecting Writing Styles Based on Clustering Techniquesambekarshweta25
An Approach to Detecting Writing Styles Based on Clustering Techniques
Authors:
-Devkinandan Jagtap
-Shweta Ambekar
-Harshit Singh
-Nakul Sharma (Assistant Professor)
Institution:
VIIT Pune, India
Abstract:
This paper proposes a system to differentiate between human-generated and AI-generated texts using stylometric analysis. The system analyzes text files and classifies writing styles by employing various clustering algorithms, such as k-means, k-means++, hierarchical, and DBSCAN. The effectiveness of these algorithms is measured using silhouette scores. The system successfully identifies distinct writing styles within documents, demonstrating its potential for plagiarism detection.
Introduction:
Stylometry, the study of linguistic and structural features in texts, is used for tasks like plagiarism detection, genre separation, and author verification. This paper leverages stylometric analysis to identify different writing styles and improve plagiarism detection methods.
Methodology:
The system includes data collection, preprocessing, feature extraction, dimensional reduction, machine learning models for clustering, and performance comparison using silhouette scores. Feature extraction focuses on lexical features, vocabulary richness, and readability scores. The study uses a small dataset of texts from various authors and employs algorithms like k-means, k-means++, hierarchical clustering, and DBSCAN for clustering.
Results:
Experiments show that the system effectively identifies writing styles, with silhouette scores indicating reasonable to strong clustering when k=2. As the number of clusters increases, the silhouette scores decrease, indicating a drop in accuracy. K-means and k-means++ perform similarly, while hierarchical clustering is less optimized.
Conclusion and Future Work:
The system works well for distinguishing writing styles with two clusters but becomes less accurate as the number of clusters increases. Future research could focus on adding more parameters and optimizing the methodology to improve accuracy with higher cluster values. This system can enhance existing plagiarism detection tools, especially in academic settings.
Online aptitude test management system project report.pdfKamal Acharya
The purpose of on-line aptitude test system is to take online test in an efficient manner and no time wasting for checking the paper. The main objective of on-line aptitude test system is to efficiently evaluate the candidate thoroughly through a fully automated system that not only saves lot of time but also gives fast results. For students they give papers according to their convenience and time and there is no need of using extra thing like paper, pen etc. This can be used in educational institutions as well as in corporate world. Can be used anywhere any time as it is a web based application (user Location doesn’t matter). No restriction that examiner has to be present when the candidate takes the test.
Every time when lecturers/professors need to conduct examinations they have to sit down think about the questions and then create a whole new set of questions for each and every exam. In some cases the professor may want to give an open book online exam that is the student can take the exam any time anywhere, but the student might have to answer the questions in a limited time period. The professor may want to change the sequence of questions for every student. The problem that a student has is whenever a date for the exam is declared the student has to take it and there is no way he can take it at some other time. This project will create an interface for the examiner to create and store questions in a repository. It will also create an interface for the student to take examinations at his convenience and the questions and/or exams may be timed. Thereby creating an application which can be used by examiners and examinee’s simultaneously.
Examination System is very useful for Teachers/Professors. As in the teaching profession, you are responsible for writing question papers. In the conventional method, you write the question paper on paper, keep question papers separate from answers and all this information you have to keep in a locker to avoid unauthorized access. Using the Examination System you can create a question paper and everything will be written to a single exam file in encrypted format. You can set the General and Administrator password to avoid unauthorized access to your question paper. Every time you start the examination, the program shuffles all the questions and selects them randomly from the database, which reduces the chances of memorizing the questions.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
Forklift Classes Overview by Intella PartsIntella Parts
Discover the different forklift classes and their specific applications. Learn how to choose the right forklift for your needs to ensure safety, efficiency, and compliance in your operations.
For more technical information, visit our website https://intellaparts.com
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdf
experimental stress analysis-Chapter 2
1. Experimental Stress Analysis
Department of Mechanical Engineering Page 1
Strain Analysis Method
Introduction:
For completely defining the strain or stress at a point on the surface of a component or structure,
generally it is necessary to measure strains along three different directions at that point. Multiple
element strain gauges or rosettes with strain gauges oriented along fixed directions are used for
this purpose.
When both the magnitude and directions of the principal strains at a point are unknown, a three
element strain gage is needed for the complete definition of strain at that point. Consider the case
where the three gauges in the rosette are placed at arbitrary angles related to the x-and y axis.
The strain along these directions A, B and C are related to strains ∈ ,∈ ,
∈ =∈ +∈ + sin cos
∈ =∈ +∈ + sin cos
∈ =∈ +∈ + sin cos
Where , and are the angles between the x-axis and the directions A, B and C respectively.
The magnitudes of strains ∈ , ∈ and ∈ are obtained through measurements on gauges oriented
along these directions. Hence ∈ ,∈ , can be found out by solving the simultaneous eq
The principal strains and principal directions are then determined through
∈ =
1
2
∈ +∈ +
1
2
∈ −∈ +
⁄
∈ =
1
2
∈ +∈ −
1
2
∈ −∈ +
⁄
tan2∅ = ∈ −∈⁄
2. Experimental Stress Analysis
Department of Mechanical Engineering Page 2
Here ∅ is the angle between the x axis and the principal axis corresponding to strain ∈ . From
the principal strains ∈ and∈ , the principal stress and can be determined from
= (∈ + ∈ ) (1 − )⁄
= (∈ + ∈ ) (1 − )⁄
Several multiple element rosettes with gauges oriented along specified directions are
commercially available. These rosettes are denoted by the angles along which the gauges are
oriented in them as the three element rectangular rosettes, delta russets, four element rectangular
rosettes and tee-delta rosette.
Two element rectangular rosettes:
This rosette is suitable only when the directions of principal strain are known. The gage a is
arranged along the maximum strain direction chosen along the x-axis so that = 0 and the gage
b is set along the minimum strain direction so that = 90
Fig: Two gage rosette
The strain along these directions A, B is
∴ ∈ =∈
∈ =∈
Hence, ∈ = ∈ , ∈ = ∈ , = (∈ − ∈ )
The principal stress and can be
= (∈ + ∈ ) ∗
(1 − )
= (∈ + ∈ ) ∗
(1 − )
3. Experimental Stress Analysis
Department of Mechanical Engineering Page 3
=
2(1 + )
(∈ − ∈ )
Three element rectangular rosettes:
In this rosette the three gage are laid out so that the axis of gauges B and C are at 45o
and 90 o
respectively to the axis of gage A. taking the OA axis to be coincident with the O x-axis, the
angles corresponding to the gauges A, B and C in the three- element rectangular rosette are
= 0 = 45 = 90
Than
∴ ∈ =∈ …………………………….. (1)
∈ =
1
2
∈ +∈ + … … … … … … … (2)
∈ =∈ … … … … … … … … … … … … … . (3)
We can rewrite these eq in terms of∈ , ∈ , are obtained as
∴ ∈ =∈
∈ =∈
= 2 ∈ − ( ∈ + ∈ )……………… (4)
The principal strains are given by
∈ =
1
2
∈ +∈ +
1
2
∈ −∈ +
⁄
∈ =
1
2
(∈ +∈ ) +
1
2
(∈ −∈ ) + 2 ∈ − ( ∈ + ∈ )
⁄
4. Experimental Stress Analysis
Department of Mechanical Engineering Page 4
∈ =
1
2
∈ +∈ −
1
2
∈ −∈ +
⁄
∈ = (∈ +∈ ) − (∈ −∈ ) + 2 ∈ − ( ∈ + ∈ )
⁄
………………….. (5)
Maximum shear strains
= ∈ −∈ +
⁄
= {(∈ −∈ ) + (2 ∈ −∈ −∈ ) } ⁄
……………… (6)
Principal strain directions are
tan2∅ = ∈ −∈⁄
tan 2∅ = [ 2 ∈ − ∈ − ∈ ] (∈ −∈ )⁄ ……. (7)
Substituting eq 5 value in the general eq of the principal stress and and we get
= (∈ + ∈ ) (1 − )⁄
=
2
∈ +∈
(1 − )
+
1
(1 + )
{(∈ −∈ ) + ( 2 ∈ − ∈ − ∈ ) } ⁄
= (∈ + ∈ ) (1 − )⁄
=
2
∈ +∈
(1 − )
−
1
(1 + )
{(∈ −∈ ) + ( 2 ∈ − ∈ − ∈ ) } ⁄
Maximum shear stress is given by
=
2(1 + )
=
2(1 + )
{(∈ −∈ ) + (2 ∈ −∈ −∈ ) } ⁄
Three element delta rosettes:
In a three element delta rosette three gauges are placed at angular disposition of 0o
,120o
, 240o
.
for a delta rosette,
10. Experimental Stress Analysis
Department of Mechanical Engineering Page 10
= ∈ −∈ +
⁄
= (∈ −∈ ) +
4
3
(∈ −∈ )
⁄
Maximum shear stress is given by
=
2(1 + )
=
2(1 + )
(∈ −∈ ) +
4
3
(∈ −∈ )
⁄
Transverse sensitivity error in rosettes:
In the analysis presented above the strain gage readings were assumed to be free from errors due
to transverse sensitivity effects. However, in actual practice the effect of transverse sensitivity
should be studied while measuring the stresses in a biaxial stress field with strain gauges. If it is
found that the error due to transverse sensitivity effect is significant, the strain gage readings
should be corrected for it.
Two element rectangular rosette:
Consider first a two-element rosette, with the gage axes aliened with two perpendicular axes x
and y on the test surface. It is assumed that the individual gage elements in the rosette have the
same transverse sensitivity.
Where∈ and ∈ are strains indicated by gauges along x and y directions respectively. ∈ and
∈ are the corresponding corrected strains.
∈ =
1 −
1 −
∈ − ∈
∈ =
1 −
1 −
∈ − ∈
The equations given above are for the gage elements oriented along any two orthogonal axes, x
and y. in actual practice the two element rectangular rosette is generally used with the axes of the
gage oriented along the principal axes. In such case x and y axes would denote the principal
axes.
∈ =
1 −
1 −
(∈ − ∈ )
11. Experimental Stress Analysis
Department of Mechanical Engineering Page 11
∈ =
1 −
1 −
(∈ − ∈ )
Three element rectangular rosette:
The individual strain readings,∈ , ∈ , ∈ , of a three element rectangular rosette can be corrected
for transverse sensitivity effects as follows. The corrected strains∈ , ∈ along the orthogonal
axes A and B respectively are
∈ =
1 −
1 −
(∈ − ∈ )
∈ =
1 −
1 −
(∈ − ∈ )
From the condition (∈ +∈ ) is an invariant, the strain along axis D orthogonal to axis B can be
estimated as
∈ =∈ +∈ −∈
Substituting the orthogonal strains∈ and ∈ , the corresponding strain ∈ is obtained.
∈ =
1 −
1 −
∈ − (∈ +∈ −∈ )
The corrected strains are used along with equations given to determine the principal strains,
stress and principal directions.
Three element delta rosette:
Expressions for correcting individual strain readings from a delta rosette for transverse
sensitivity effects can be easily derived for the case where all the three gage element have the
same transverse sensitivity k.
∈ =
1 −
1 −
1 +
3
∈ −
2
3
(∈ +∈ )
∈ =
1 −
1 −
1 +
3
∈ −
2
3
(∈ +∈ )
∈ =
1 −
1 −
1 +
3
∈ −
2
3
(∈ +∈ )
12. Experimental Stress Analysis
Department of Mechanical Engineering Page 12
Shear strain gauges:
Strain gauges do not respond to shear strains. However the relationship between shear and
normal strains can be utilized to obtain from a strain rosette an output directly proportional to the
shear strain in the surface.
The two strain gauges a and b oriented so that the x axis bisects the angle between the gage axes.
Strain along two gage axes is
∈ =
∈ +∈
2
+
∈ −∈
2
cos2 +
2
sin 2
∈ =
∈ +∈
2
+
∈ −∈
2
cos2 −
2
sin 2
From the above eq. The shear strain is
=
∈ −∈
sin 2
From above eq the difference in the normal strain sensed by any two arbitrarily oriented gauges
in a uniform strain field is directly proportional to the shear strain along an axis bisecting the
included angle between the strain gage axes. When the included angle is 900
, i.e. the rosette is a
two-element rectangular rosette, above eq can be reduced to
=∈ −∈
Hence by orienting a two-element rectangular rosette such that the x-axis bisects the 900
angle
between the gage elements and connecting the gage elements in the adjacent arm of a
Wheatstone bridge, an output from the rosette equal to the shear strain can be obtained
directly.
13. Experimental Stress Analysis
Department of Mechanical Engineering Page 13
Stress gauges:
In some application it may be desirable to have an output from a single strain gage directly
proportional to the axial stress in a particular direction. Such gauges are known as stress gauges.
For eg, if one wishes to determine the stress at say, five stations along specified directions under
dynamic loading, the use of stress gage in place of strain rosettes results in considerable saving
in equipment. The principle of operation of a stress gage is given below.
Stress gauge
Mohr’s circle of strain
14. Experimental Stress Analysis
Department of Mechanical Engineering Page 14
Fig 1 shows the sketch of a stress gage with its axis along the x-axis. The gage is oriented such
that the x-axis bisects the angle 2 between the grid elements A and B of this gage.
The strains along the grid elements A and B are given by
∈ = ( + ) + ( − ) 2( − )…………. (a)
∈ = ( + ) + ( − ) 2( + )………… (b)
The average of these will be
∈ +∈ = ( + ) + ( − )[ 2( + ) + 2( − )]…………. (c)
On expanding the cosine terms in above eq
∈ +∈ = ( + ) + ( − ) 2 2 …………. (d)
From Mohr’s strain circle
∈ +∈ =∈ +∈ ………..... (e)
∈ −∈ = (∈ −∈ )cos2 ………. (f)
Substituting the values in eq e and f in eq d and simplifying
1
2
∈ +∈ =
1
2
+ +
1
2
− 2
1
2
∈ +∈ =
1
2
+ +
1
2
− (2 − 1)
1
2
∈ +∈ =
1
2
+ + − −
1
2
−
1
2
∈ +∈ = − +
1
2
∈ +∈ = + (1 − )
1
2
∈ +∈ = +
∈ +∈ = + ……………….(g)
If is so chosen that it is equal to tan √ then
=
15. Experimental Stress Analysis
Department of Mechanical Engineering Page 15
=
=
w.k.t
1 = +
Substituting the value of in above eq
(1 + ) = 1
= ……………. (h)
Substituting these values in eq g yields
∈ +∈ = + …………….. (i)
However,
= ∈ + ∈ ………(j)
Substituting the values of ∈ + ∈ from eq I and j gives
= ∈ +∈ ………………. (1)
It may be noted that ∈ +∈ is the strain indicated by the stress gage, i.e.
( ⁄ )
. thus
the stress along the x-axis is obtained by multiplying the strain indicated by the stress gage
with /(1 − ) .
If the direction of the maximum principal stress is known, a single conventional strain gage
can be used as shown in fig 3 to directly measure the principal stress . In this case as is zero
from eq a and b,
∈ =∈ =∈ ……… (k)
Substituting this condition in eq 1 gives
= = ∈ ………. (2)
Hence to measure the principle stress it is only necessary to orient a single strain gage
along = tan √ a direction at an angle to the axis and multiply the strain gage reading by
.