The document discusses stress, strain, and strain gages. It defines stress as force per unit area and strain as the ratio of deformation to original length. It describes Hooke's law, which states that stress is proportional to strain for elastic materials. A strain gage works by measuring the increase in electrical resistance of a wire as it is stretched under strain. A Wheatstone bridge circuit is used to precisely measure the small changes in resistance caused by strain.
1. Cylinders are commonly used in engineering to transport or store fluids and are subjected to internal fluid pressures. This induces three stresses on the cylinder wall - circumferential, longitudinal, and radial.
2. For thin cylinders where the wall thickness is less than 1/20 the diameter, the radial stress can be neglected. Equations are derived to calculate the circumferential and longitudinal stresses based on the internal pressure, diameter, and wall thickness.
3. Sample problems are worked out applying the equations to example thin-walled cylinders under internal pressure, finding stresses, strains, and changes in dimensions.
- Shear stress distribution in beams takes a parabolic shape, with the maximum stress at the neutral axis and zero at the ends. In rectangular beams the stress is highest at y=0. In I-beams, most stress is carried by the web in a "top-hat" distribution.
- Circular beams have a shear stress distribution that also follows a parabolic shape, calculated using the area moment of the shaded portion.
- Principal stresses can be determined in beams using the bending and shear stresses. The bending stress is not a principal stress and the principal stresses are found using an equation involving the bending and shear stresses.
This document provides information about various electrical instruments, including multimeters, galvanometers, and ohmmeters. It discusses the d'Arsonval movement used in many of these instruments and how it can be configured to measure different electrical parameters. Specifically, it describes how the basic movement can be converted into a multimeter, multirange milliammeter, multirange voltmeter, and series and shunt type ohmmeters by adding shunt resistors and multipliers to adjust the circuit for different current and voltage ranges. Diagrams are included to illustrate the circuit configurations for these different instrument types.
This document summarizes key concepts in strength of materials including:
- Analysis of pure bending and symmetrical sections bending in a plane of symmetry
- Skew loading and bending about axes other than axes of symmetry
- Eccentric loading introducing both direct stress and bending stress
- The middle third rule and middle quarter rule defining safe load application areas to avoid tension
The document discusses torsion and stresses in circular shafts. It covers topics like net torque due to internal stresses, shear components, shaft deformations, stresses in the elastic range, failure modes, and examples of solving for stress and deformation in statically determinate and indeterminate shafts made of elastic and elastoplastic materials. Sample problems are included to demonstrate calculating stresses, strains, and required diameters for various shaft configurations under applied torques.
This document describes an experiment to determine the stiffness of an open and closed coil spring and the modulus of rigidity of the spring material. A spring testing machine is used to apply loads to the springs and measure the deflection. The stiffness and modulus of rigidity are calculated based on equations that relate the applied load, spring geometry, and measured deflection. Test procedures are outlined for performing compression and tension tests on open and closed coil springs. Measurements of spring dimensions, number of coils, and load-deflection data are collected and used to calculate values for stiffness, shear stress, strain energy, and modulus of rigidity.
This document provides information about the Solid Mechanics course ME 302 taught by Dr. Nirmal Baran Hui at NIT Durgapur in West Bengal, India. It lists four required textbooks for the course and provides a detailed syllabus covering topics like stress, strain, elasticity, bending, deflection, columns, torsion, pressure vessels, combined loadings, springs, and failure theories. The document also includes examples of lecture content on stress analysis, stresses on oblique planes, and material subjected to pure shear.
This document discusses different types of beams and loading conditions used in structural analysis. It defines dead load as the self-weight of building components and live load as external loads on a structure, which can be uniform, varying, or concentrated. Common beam types are described as simply supported, fixed, cantilever, continuous, and overhanging. Load types include concentrated, uniform distributed, uniformly varying, and applied couples. Shear force and bending moment are defined as the algebraic sum of vertical forces and moments acting on a beam cross section. Stress resultants in determinate beams can be calculated from equilibrium equations.
1. Cylinders are commonly used in engineering to transport or store fluids and are subjected to internal fluid pressures. This induces three stresses on the cylinder wall - circumferential, longitudinal, and radial.
2. For thin cylinders where the wall thickness is less than 1/20 the diameter, the radial stress can be neglected. Equations are derived to calculate the circumferential and longitudinal stresses based on the internal pressure, diameter, and wall thickness.
3. Sample problems are worked out applying the equations to example thin-walled cylinders under internal pressure, finding stresses, strains, and changes in dimensions.
- Shear stress distribution in beams takes a parabolic shape, with the maximum stress at the neutral axis and zero at the ends. In rectangular beams the stress is highest at y=0. In I-beams, most stress is carried by the web in a "top-hat" distribution.
- Circular beams have a shear stress distribution that also follows a parabolic shape, calculated using the area moment of the shaded portion.
- Principal stresses can be determined in beams using the bending and shear stresses. The bending stress is not a principal stress and the principal stresses are found using an equation involving the bending and shear stresses.
This document provides information about various electrical instruments, including multimeters, galvanometers, and ohmmeters. It discusses the d'Arsonval movement used in many of these instruments and how it can be configured to measure different electrical parameters. Specifically, it describes how the basic movement can be converted into a multimeter, multirange milliammeter, multirange voltmeter, and series and shunt type ohmmeters by adding shunt resistors and multipliers to adjust the circuit for different current and voltage ranges. Diagrams are included to illustrate the circuit configurations for these different instrument types.
This document summarizes key concepts in strength of materials including:
- Analysis of pure bending and symmetrical sections bending in a plane of symmetry
- Skew loading and bending about axes other than axes of symmetry
- Eccentric loading introducing both direct stress and bending stress
- The middle third rule and middle quarter rule defining safe load application areas to avoid tension
The document discusses torsion and stresses in circular shafts. It covers topics like net torque due to internal stresses, shear components, shaft deformations, stresses in the elastic range, failure modes, and examples of solving for stress and deformation in statically determinate and indeterminate shafts made of elastic and elastoplastic materials. Sample problems are included to demonstrate calculating stresses, strains, and required diameters for various shaft configurations under applied torques.
This document describes an experiment to determine the stiffness of an open and closed coil spring and the modulus of rigidity of the spring material. A spring testing machine is used to apply loads to the springs and measure the deflection. The stiffness and modulus of rigidity are calculated based on equations that relate the applied load, spring geometry, and measured deflection. Test procedures are outlined for performing compression and tension tests on open and closed coil springs. Measurements of spring dimensions, number of coils, and load-deflection data are collected and used to calculate values for stiffness, shear stress, strain energy, and modulus of rigidity.
This document provides information about the Solid Mechanics course ME 302 taught by Dr. Nirmal Baran Hui at NIT Durgapur in West Bengal, India. It lists four required textbooks for the course and provides a detailed syllabus covering topics like stress, strain, elasticity, bending, deflection, columns, torsion, pressure vessels, combined loadings, springs, and failure theories. The document also includes examples of lecture content on stress analysis, stresses on oblique planes, and material subjected to pure shear.
This document discusses different types of beams and loading conditions used in structural analysis. It defines dead load as the self-weight of building components and live load as external loads on a structure, which can be uniform, varying, or concentrated. Common beam types are described as simply supported, fixed, cantilever, continuous, and overhanging. Load types include concentrated, uniform distributed, uniformly varying, and applied couples. Shear force and bending moment are defined as the algebraic sum of vertical forces and moments acting on a beam cross section. Stress resultants in determinate beams can be calculated from equilibrium equations.
This document contains a question bank with multiple choice and numerical problems related to the topic of Strength of Materials for a Mechanical Engineering course. It includes questions related to stress-strain behavior, elastic constants, bending of beams, shear force and bending moment diagrams, torsion, and springs. The questions cover definitions, derivations of equations, and calculations to determine stresses, strains, moduli, loads, dimensions and other mechanical properties. The question bank is divided into three units - Stress-Strain and Deformation of Solids, Beams - Loads and Stresses, and Torsion. It contains both short answer and long numerical type questions for practice and self-assessment of the key concepts in Strength of Materials.
1) Springs are elastic elements that deflect under load and return to their original shape when unloaded. They come in various shapes and are classified by type, with helical springs being most common.
2) Helical springs are used to absorb shocks, store energy, measure forces, and control motion. The main types are compression and extension springs.
3) Springs are designed based on factors like the wire diameter, mean coil diameter, and spring index, which determines stresses and deflection. Proper design ensures springs function reliably under various loads.
1) The document discusses the stability and buckling behavior of columns under axial loading. It introduces Euler's formula for determining the critical buckling load of pin-ended beams and describes how this analysis can be extended to columns.
2) Sample problems demonstrate how to design columns for centric and eccentric axial loads using these analytical methods and by considering stress limits. Design approaches vary based on the column's slenderness ratio.
3) The effects of eccentric loading are evaluated using a secant formula approach, where the eccentric load is modeled as a centric load plus a bending moment. Stress limits and interaction equations are provided.
The document summarizes key concepts related to mechanics of solids, including:
1. Definitions of stress, strain, Hooke's law, shear stress, Poisson's ratio, Young's modulus, and strain energy.
2. Methods for analyzing plane trusses and thin cylindrical shells.
3. Types of beams, loading conditions, shear force and bending moment diagrams.
4. Methods for determining deflection, including double integration, moment area, and Macaulay's method.
The document compares the linearity and sensitivity errors of voltage-fed and current-fed circuits for single strain gages. It finds that voltage-fed bridge circuits provide better linearity by several orders of magnitude and more stable sensitivity than current-fed circuits, especially when initial detuning is present. The relationships between strain, resistance change, and electrical signal are derived, showing that the non-linearity of strain-resistance response is compensated by the opposite non-linearity of voltage-fed circuits, resulting in near-linear output. In contrast, current-fed circuits exhibit much larger linearity and sensitivity errors.
Rankine proposed a hypothesis to overcome limitations in Euler's theory for predicting the buckling load of struts and columns. Rankine's hypothesis assumes the buckling load is equal to the shorter of either the crushing load or Euler's buckling load, plus a factor. This results in an empirical formula known as Rankine's formula that can be used to calculate the buckling load of struts and columns of any slenderness ratio, including short, long, and intermediate lengths. Rankine's formula provides a more practical method of analysis compared to Euler's theory.
1. Alternating current theory describes the sinusoidal waveform used for electric power. The waveform oscillates between positive and negative values with a frequency of 50 or 60 Hz.
2. Sinusoidal waveforms can be represented using phasors, which are vectors that rotate at the frequency of the waveform. Phasor diagrams depict the magnitude and relative phase of sinusoidal waveforms.
3. The phase difference between two waveforms is the angle between their phasors. A waveform leads or lags another if its peak occurs earlier or later relative to the other waveform by the phase difference amount.
This document summarizes an experiment conducted by Vania Lundina to verify how the length of a conductor affects its resistance according to Ohm's Law. The experiment involved measuring the resistance of copper wires of varying lengths (10-35 cm) using a voltmeter, ammeter, and power supply. The results showed that resistance increased with increasing length, supporting the conclusion that resistance is directly proportional to length as predicted by Ohm's Law. Some variability between trials was attributed to inaccuracies in measuring wire length.
Sample lab-report on verfication of ohms lawminteshat
This laboratory report summarizes an experiment to verify Ohm's Law. The experiment used resistors with values of 1.0kΩ and 1.2kΩ in various circuit configurations. Measurements were made of voltage, current, and resistor values using a digital multimeter. The results closely matched the expected theoretical values based on Ohm's Law. This confirmed that Ohm's Law accurately describes the relationship between voltage, current, and resistance in electrical circuits. The experiment also verified the resistor color code system and that total resistance in a series circuit equals the sum of individual resistances.
The document discusses series and parallel circuits. It provides three laws for each:
1) For series circuits, total resistance equals the sum of individual resistances, current is constant, and voltage drops across each resistance.
2) For parallel circuits, total resistance is less than the smallest branch, voltage is the same across each branch, and total current equals the sum of branch currents.
3) Methods for calculating total resistance in parallel circuits include treating each branch separately and using the formula for two resistors in parallel.
This document provides an overview of shear loading and shear flow distribution in beams. It discusses shear in open section beams, closed section beams, and the twist that can occur in closed section beams when loads are not applied through the shear center. Key points covered include:
- Shear flow distribution in open section beams is determined using equations involving shear center location and section properties.
- For closed sections, the section must first be cut to form an open section, then equations are used to determine the unknown shear flow at the cut and resulting basic open section shear flow.
- Twisting can occur in closed sections under shear loads not through the shear center, and an equation is provided to calculate the rate of twist.
-
Sag is the vertical distance between the lowest point of an overhead transmission line conductor and the points of support. There are two types of sag: equal supports and unequal supports. Sag is calculated based on the span length, conductor weight, tension, and differences in support height. Maintaining proper sag is important as it determines safe tension while allowing for factors like wind, ice, and temperature changes. Too little sag causes high tension risks while too much sag compromises safety clearances.
The document discusses Deepak's academic and professional background, including an MBA from IE Business School in Spain and experience founding perfectbazaar.com. It also provides an overview of the topics to be covered in the Strength of Materials course, such as stresses, strains, Hooke's law, and analysis of bars with varying cross-sections. The grading policy and syllabus are outlined which divide the course into 5 units covering various strength of materials concepts.
This document summarizes an experiment on Ohm's Law and finding the resistance of unknown resistors. In part A, the resistance was held constant at 100 ohms while the voltage was varied from 3V to 12V to measure current. Then voltage was held at 10V while resistance was varied to measure current. The measured values aligned well with theoretical calculations using Ohm's Law. In part B, resistors of varying values were used in series and the current and reciprocal of current were measured and graphed. The graph's slope provided the resistance and the y-intercept was zero, confirming Ohm's Law. Minor differences between measured and theoretical values were due to factors like resistor degradation over multiple uses.
The document summarizes an experiment to determine the spring constant of simple extension springs and springs connected in parallel using Hooke's law. Key points:
1) Springs were loaded with weights in increments and the extension was measured to calculate spring constant from the slope of force vs. extension graphs.
2) Theoretical spring constants were also calculated using the springs' material properties and dimensions.
3) For springs in parallel, the total spring constant was calculated as the sum of the individual spring constants, matching the experimental results.
TALAT Lecture 3701: Formability Characteristics of Aluminium SheetCORE-Materials
This lecture describes the fundamental formability characteristics of automotive aluminium sheet metals. It aims at learning about the various methods to characterize the forming behaviour and the forming limits. General background in production engineering and sheet metal forming is assumed.
519 transmission line theory by vishnu (1)udaykumar1106
This document discusses transmission line theory for microwave frequencies. It begins by explaining how power is delivered through electric and magnetic fields along transmission lines at microwave frequencies rather than through wires. It then lists common types of transmission lines and discusses how circuit elements are analyzed as lumped units at microwave frequencies. Key transmission line concepts are also summarized such as characteristic impedance, velocity factor, standing waves, and using transmission lines as filters. The document concludes by discussing the Smith chart and how it can be used to solve problems involving transmission line matching and impedance transformations.
Phys 102 formal simple dc circuits lab reportkgreine
In this lab experiment, the student built both series and parallel circuits containing three resistors each to investigate the relationships between resistance, potential difference, and current. For the series circuit, the student found that the current remains the same throughout while the potential difference varies across each resistor. For the parallel circuit, the current varies across each resistor while the potential difference remains the same. The student's measurements matched well with theoretical calculations, validating the circuit concepts.
This document discusses resistive sensors and their applications. It begins by defining resistive sensors as transducers that convert mechanical changes into electrical signals by changing resistance. Common resistive sensors include potentiometers, strain gauges, thermocouples, photoresistors and thermistors. The document then covers the theory of how resistance changes based on length, area, composition and temperature. It provides examples of specific resistive sensors and their typical applications, such as using light dependent resistors for light switches and strain gauges for sensors in electronic balances. In closing, it discusses how the resistance of sensors varies with changes in factors like temperature, strain or light intensity.
Strain gauges measure strain, which is the deformation of a material under stress. Strain gauges use the piezoresistive property of materials to change electrical resistance proportional to strain. Common types are foil and wire strain gauges, with foil being more prevalent. Choice of strain gauge depends on factors like anticipated strain level, temperature, and specimen material. Strain is calculated from the change in electrical resistance measured by a Wheatstone bridge circuit, using the gauge factor.
This document contains a question bank with multiple choice and numerical problems related to the topic of Strength of Materials for a Mechanical Engineering course. It includes questions related to stress-strain behavior, elastic constants, bending of beams, shear force and bending moment diagrams, torsion, and springs. The questions cover definitions, derivations of equations, and calculations to determine stresses, strains, moduli, loads, dimensions and other mechanical properties. The question bank is divided into three units - Stress-Strain and Deformation of Solids, Beams - Loads and Stresses, and Torsion. It contains both short answer and long numerical type questions for practice and self-assessment of the key concepts in Strength of Materials.
1) Springs are elastic elements that deflect under load and return to their original shape when unloaded. They come in various shapes and are classified by type, with helical springs being most common.
2) Helical springs are used to absorb shocks, store energy, measure forces, and control motion. The main types are compression and extension springs.
3) Springs are designed based on factors like the wire diameter, mean coil diameter, and spring index, which determines stresses and deflection. Proper design ensures springs function reliably under various loads.
1) The document discusses the stability and buckling behavior of columns under axial loading. It introduces Euler's formula for determining the critical buckling load of pin-ended beams and describes how this analysis can be extended to columns.
2) Sample problems demonstrate how to design columns for centric and eccentric axial loads using these analytical methods and by considering stress limits. Design approaches vary based on the column's slenderness ratio.
3) The effects of eccentric loading are evaluated using a secant formula approach, where the eccentric load is modeled as a centric load plus a bending moment. Stress limits and interaction equations are provided.
The document summarizes key concepts related to mechanics of solids, including:
1. Definitions of stress, strain, Hooke's law, shear stress, Poisson's ratio, Young's modulus, and strain energy.
2. Methods for analyzing plane trusses and thin cylindrical shells.
3. Types of beams, loading conditions, shear force and bending moment diagrams.
4. Methods for determining deflection, including double integration, moment area, and Macaulay's method.
The document compares the linearity and sensitivity errors of voltage-fed and current-fed circuits for single strain gages. It finds that voltage-fed bridge circuits provide better linearity by several orders of magnitude and more stable sensitivity than current-fed circuits, especially when initial detuning is present. The relationships between strain, resistance change, and electrical signal are derived, showing that the non-linearity of strain-resistance response is compensated by the opposite non-linearity of voltage-fed circuits, resulting in near-linear output. In contrast, current-fed circuits exhibit much larger linearity and sensitivity errors.
Rankine proposed a hypothesis to overcome limitations in Euler's theory for predicting the buckling load of struts and columns. Rankine's hypothesis assumes the buckling load is equal to the shorter of either the crushing load or Euler's buckling load, plus a factor. This results in an empirical formula known as Rankine's formula that can be used to calculate the buckling load of struts and columns of any slenderness ratio, including short, long, and intermediate lengths. Rankine's formula provides a more practical method of analysis compared to Euler's theory.
1. Alternating current theory describes the sinusoidal waveform used for electric power. The waveform oscillates between positive and negative values with a frequency of 50 or 60 Hz.
2. Sinusoidal waveforms can be represented using phasors, which are vectors that rotate at the frequency of the waveform. Phasor diagrams depict the magnitude and relative phase of sinusoidal waveforms.
3. The phase difference between two waveforms is the angle between their phasors. A waveform leads or lags another if its peak occurs earlier or later relative to the other waveform by the phase difference amount.
This document summarizes an experiment conducted by Vania Lundina to verify how the length of a conductor affects its resistance according to Ohm's Law. The experiment involved measuring the resistance of copper wires of varying lengths (10-35 cm) using a voltmeter, ammeter, and power supply. The results showed that resistance increased with increasing length, supporting the conclusion that resistance is directly proportional to length as predicted by Ohm's Law. Some variability between trials was attributed to inaccuracies in measuring wire length.
Sample lab-report on verfication of ohms lawminteshat
This laboratory report summarizes an experiment to verify Ohm's Law. The experiment used resistors with values of 1.0kΩ and 1.2kΩ in various circuit configurations. Measurements were made of voltage, current, and resistor values using a digital multimeter. The results closely matched the expected theoretical values based on Ohm's Law. This confirmed that Ohm's Law accurately describes the relationship between voltage, current, and resistance in electrical circuits. The experiment also verified the resistor color code system and that total resistance in a series circuit equals the sum of individual resistances.
The document discusses series and parallel circuits. It provides three laws for each:
1) For series circuits, total resistance equals the sum of individual resistances, current is constant, and voltage drops across each resistance.
2) For parallel circuits, total resistance is less than the smallest branch, voltage is the same across each branch, and total current equals the sum of branch currents.
3) Methods for calculating total resistance in parallel circuits include treating each branch separately and using the formula for two resistors in parallel.
This document provides an overview of shear loading and shear flow distribution in beams. It discusses shear in open section beams, closed section beams, and the twist that can occur in closed section beams when loads are not applied through the shear center. Key points covered include:
- Shear flow distribution in open section beams is determined using equations involving shear center location and section properties.
- For closed sections, the section must first be cut to form an open section, then equations are used to determine the unknown shear flow at the cut and resulting basic open section shear flow.
- Twisting can occur in closed sections under shear loads not through the shear center, and an equation is provided to calculate the rate of twist.
-
Sag is the vertical distance between the lowest point of an overhead transmission line conductor and the points of support. There are two types of sag: equal supports and unequal supports. Sag is calculated based on the span length, conductor weight, tension, and differences in support height. Maintaining proper sag is important as it determines safe tension while allowing for factors like wind, ice, and temperature changes. Too little sag causes high tension risks while too much sag compromises safety clearances.
The document discusses Deepak's academic and professional background, including an MBA from IE Business School in Spain and experience founding perfectbazaar.com. It also provides an overview of the topics to be covered in the Strength of Materials course, such as stresses, strains, Hooke's law, and analysis of bars with varying cross-sections. The grading policy and syllabus are outlined which divide the course into 5 units covering various strength of materials concepts.
This document summarizes an experiment on Ohm's Law and finding the resistance of unknown resistors. In part A, the resistance was held constant at 100 ohms while the voltage was varied from 3V to 12V to measure current. Then voltage was held at 10V while resistance was varied to measure current. The measured values aligned well with theoretical calculations using Ohm's Law. In part B, resistors of varying values were used in series and the current and reciprocal of current were measured and graphed. The graph's slope provided the resistance and the y-intercept was zero, confirming Ohm's Law. Minor differences between measured and theoretical values were due to factors like resistor degradation over multiple uses.
The document summarizes an experiment to determine the spring constant of simple extension springs and springs connected in parallel using Hooke's law. Key points:
1) Springs were loaded with weights in increments and the extension was measured to calculate spring constant from the slope of force vs. extension graphs.
2) Theoretical spring constants were also calculated using the springs' material properties and dimensions.
3) For springs in parallel, the total spring constant was calculated as the sum of the individual spring constants, matching the experimental results.
TALAT Lecture 3701: Formability Characteristics of Aluminium SheetCORE-Materials
This lecture describes the fundamental formability characteristics of automotive aluminium sheet metals. It aims at learning about the various methods to characterize the forming behaviour and the forming limits. General background in production engineering and sheet metal forming is assumed.
519 transmission line theory by vishnu (1)udaykumar1106
This document discusses transmission line theory for microwave frequencies. It begins by explaining how power is delivered through electric and magnetic fields along transmission lines at microwave frequencies rather than through wires. It then lists common types of transmission lines and discusses how circuit elements are analyzed as lumped units at microwave frequencies. Key transmission line concepts are also summarized such as characteristic impedance, velocity factor, standing waves, and using transmission lines as filters. The document concludes by discussing the Smith chart and how it can be used to solve problems involving transmission line matching and impedance transformations.
Phys 102 formal simple dc circuits lab reportkgreine
In this lab experiment, the student built both series and parallel circuits containing three resistors each to investigate the relationships between resistance, potential difference, and current. For the series circuit, the student found that the current remains the same throughout while the potential difference varies across each resistor. For the parallel circuit, the current varies across each resistor while the potential difference remains the same. The student's measurements matched well with theoretical calculations, validating the circuit concepts.
This document discusses resistive sensors and their applications. It begins by defining resistive sensors as transducers that convert mechanical changes into electrical signals by changing resistance. Common resistive sensors include potentiometers, strain gauges, thermocouples, photoresistors and thermistors. The document then covers the theory of how resistance changes based on length, area, composition and temperature. It provides examples of specific resistive sensors and their typical applications, such as using light dependent resistors for light switches and strain gauges for sensors in electronic balances. In closing, it discusses how the resistance of sensors varies with changes in factors like temperature, strain or light intensity.
Strain gauges measure strain, which is the deformation of a material under stress. Strain gauges use the piezoresistive property of materials to change electrical resistance proportional to strain. Common types are foil and wire strain gauges, with foil being more prevalent. Choice of strain gauge depends on factors like anticipated strain level, temperature, and specimen material. Strain is calculated from the change in electrical resistance measured by a Wheatstone bridge circuit, using the gauge factor.
The document discusses electricity and magnetism, specifically resistance and heating effects of currents. It explains that resistance depends on the material and structure of a conductor, with tungsten filament lamps having high resistance and copper wires having low resistance. It also covers Ohm's law, defining resistance as the ratio of potential difference to current, and how resistors, circuits, and resistor combinations work based on this relationship. Kirchhoff's laws for analyzing electric circuits are also summarized.
This document provides information about electricity and magnetism, specifically resistance and heating effects of currents. It explains that resistance depends on the material and cross-sectional area of a conductor. It also describes Ohm's law, which states that current is directly proportional to voltage in a conductor. Resistors can be made of nichrome wire or ceramic and carbon. Kirchhoff's laws are introduced to help solve circuit problems using conservation of charge and energy.
This slide introduces the concept of simple strain, a term used in mechanics to describe the deformation of a material under an applied force. The slide includes a diagram illustrating the deformation of a rectangular object under a tensile force, as well as a formula for calculating strain. Simple strain is a fundamental concept in the study of materials and mechanics, and understanding it is essential for many engineering applications
This document discusses strain measurement techniques. It introduces strain gauges as the most common method to measure strain. Strain gauges work by changing electrical resistance proportionally to the amount of strain. To accurately measure small resistance changes from strain, a Wheatstone bridge circuit is used to convert it to a voltage output. The document covers the basic principles of how strain gauges work and are used to measure static, transient, and dynamic strain through small changes in resistance.
Strain gauges are devices used to measure dimensional changes on structural members. They work by converting mechanical strain into electrical resistance changes. There are three main types - mechanical, optical, and electrical. Electrical strain gauges are most common and work by bonding a patterned metal foil to the test surface. Any strain causes a resistance change measured using a Wheatstone bridge circuit. Proper selection, bonding, and calibration allow accurate static and dynamic strain measurement in various applications.
MAHARASHTRA STATE BOARD
CLASS XI AND XII
CHAPTER 9
CURRENT ELECTRICTY
CONTENT
Electric Cell and its Internal resistance
Potential difference and emf of a cell
Combination of cells in series and in parallel
Kirchhoff's laws and their applications
Wheatstone bridge
Metre bridge
Potentiometer – principle and its applications
The document discusses different types of strain gauges used to measure strain. It describes how strain gauges work by changing electrical resistance proportionally to strain. Common types include metallic wire or foil grids bonded to a backing material. Materials like constantan/advance are often used due to properties like self-temperature compensation and linear strain sensitivity. Unbounded wire strain gauges consist of tensioned wires connected to a Wheatstone bridge circuit to measure strain through changes in electrical resistance. Strain gauges are widely applied to experimental stress analysis of structures, machines, vehicles and more.
This document provides details on measuring the resistance of semiconductors using the four probe method and how it varies with temperature. It first explains Ohm's law and the two probe method for measuring resistance. The four probe method is then introduced to overcome issues with contact resistance. The document derives equations to calculate resistivity based on probe spacing and sample thickness/boundaries. Finally, it discusses how the intrinsic conductivity of semiconductors increases with temperature due to more electrons occupying the conduction band, following an exponential relationship, and how carrier mobility decreases with increasing temperature due to more collisions.
1. Electric current is produced when electrons flow through a conducting path from a negatively charged end to a positively charged end.
2. A simple electric circuit consists of a power source, conductor, load, and switch. Current (I) is the rate of flow of electric charge (Q) through a cross-sectional area over time and is measured in amperes.
3. Resistance (R) is a measure of how difficult it is for current to pass through a material and is calculated as the ratio between potential difference (V) and current (I). Resistance depends on the material's length, cross-sectional area, and temperature.
A load cell is an electric transducer that converts force or weight into an electrical signal. It contains a strain gauge, which measures the strain (deformation) on a load cell when a force is applied. The strain gauge uses the piezoresistive effect - where electrical resistance changes with mechanical strain - to convert the strain into a change in electrical resistance. This resistance change is then measured with a Wheatstone bridge circuit and amplified to produce an output voltage proportional to the applied force. Load cells are commonly used to measure weights, forces, pressures and loads in various applications.
This document provides an overview of experimental strain analysis techniques, specifically focusing on strain gages, photoelasticity, and moire methods. It describes how strain gages use changes in electrical resistance to measure strain, and how they are usually connected to a Wheatstone bridge circuit to improve measurement sensitivity. Photoelasticity and moire methods allow full-field displays of strain distributions by exploiting the birefringent properties of certain materials, in which refractive index depends on polarization orientation.
This document summarizes the mechanical properties of materials through stress-strain diagrams. It discusses the differences between stress-strain diagrams for ductile versus brittle materials. For ductile materials, the diagram shows elastic behavior, yielding, strain hardening, necking, and true stress-strain. Brittle materials exhibit no yielding and rupture occurs at a much smaller strain. The document also discusses Hooke's law, Poisson's ratio, axial loading of materials, and provides examples of calculating deformation based on applied loads and material properties.
This document provides information on different electrical concepts including:
- Voltage, current, and resistance definitions.
- Electric power formula using voltage, current, energy, and time.
- Active and passive electronic components and their definitions.
- Ohm's law relating voltage, current, and resistance.
- Current and voltage division rules for circuits with parallel and series resistors.
- Ideal and non-ideal voltage and current sources and their characteristics.
- Examples of calculations using the concepts covered.
This document discusses electrical resistance in conducting wires and Ohm's Law. It explains that resistance depends on the material, length, and cross-sectional area of the conductor. Resistance increases with length or smaller cross-sectional area. Ohm's Law states that voltage is directly proportional to current, and resistance can be calculated as voltage divided by current. The document provides examples and practice problems to illustrate these concepts.
A Strain gauge (sometimes refereed to as a Strain gauge) is a sensor whose resistance varies with applied force; It converts force, pressure, tension, weight, etc., into a change in electrical resistance which can then be measured. When external forces are applied to a stationary object, stress and strain are the result. Learn and Enjoy.
This document provides an introduction to basic electrical concepts including charge, current, voltage, resistors, and capacitors. It defines each concept, provides analogy examples, and explains measurement units. Key points include: charge is carried by electrons and protons; current is the rate of electron flow; voltage is the potential difference required to move charge; resistors limit current and dissipate power as heat; capacitors store electric potential energy and their capacitance depends on plate area, separation, and dielectric material. Diagrams and examples are provided to illustrate circuit connections and component operations.
This document provides an introduction to key concepts related to electricity including charge, current, voltage, circuits, and circuit elements. It defines charge as the fundamental electric quantity carried by electrons and protons. Current is defined as the rate of flow of electrons through a conductor. Voltage is the potential difference required to move charge between two points. Analogies are provided between electric circuits and water flow. Key circuit elements like resistors and capacitors are introduced along with their symbols, units of measurement, and functions. Formulas for resistance, capacitance, and their characteristics are also outlined.
This document discusses different types of temperature sensors including thermocouples, resistance temperature devices (RTDs), and thermistors. It provides details on how thermocouples and RTDs function to convert temperature into an electrical signal. Thermocouples generate small voltages based on the temperature difference between junctions of two dissimilar metals, while RTDs change resistance predictably with temperature. The document compares advantages and disadvantages of different sensor types and constructions, and describes considerations for accurate temperature measurement such as compensating for lead wire resistance effects.
This document discusses traction mechanics and parameters that influence tractor performance. It provides definitions for various traction ratios including travel reduction ratio, net traction ratio, tractive efficiency, gross traction ratio, and motion resistance ratio. These ratios are used to analyze traction data and understand the relationship between pull, weight, speed, and energy loss during traction testing. Regression analysis of traction data and factors that affect traction performance such as soil properties, tire pressure, size, and load are also examined. Equations for predicting soil, tire, and traction properties are provided. The document concludes with a discussion of estimating tractor performance through the use of a tractor performance spreadsheet and how to optimize drawbar performance through tire selection and ballasting.
1) The document discusses strain gages, which are sensing elements that measure surface strain by detecting changes in electrical resistance caused by mechanical elongation or contraction of a metallic resistive foil attached to the object under strain.
2) Strain gages have a grid-shaped metallic foil sensing element laminated between thin plastic and film layers that is bonded to the measurement object. Changes in resistance of the foil correspond to strain in the object.
3) The document provides background on stress, strain, Hooke's law, Young's modulus, and explains that strain is typically on the order of hundreds of micrometers per meter or parts per million and can be tensile or compressive.
This document describes the design, development, and evaluation of a digital tractor dynamometer. A load cell was used as a sensor to measure drawbar pull of tractors. The dynamometer was designed based on a microcontroller, amplifier, analog-to-digital converter, and display unit. Laboratory tests showed high linearity (R^2 = 0.99) between applied force and output voltage. The dynamometer was able to precisely measure drawbar pull of tractors.
Inductive proximity sensors detect metal objects without physical contact through generating a magnetic field. They comprise an oscillator that creates an alternating magnetic field in front of its windings. When a metal object enters this field, it disturbs the oscillation and causes an output signal. Key advantages are contactless detection, high speeds, fast response, and durability in industrial environments due to having no moving parts.
1. The document discusses different types of dynamometers used to measure cutting forces in machining processes like turning, drilling, and milling.
2. It describes the general principle of measurement which involves converting the physical variable like force into a signal using a transducer, conditioning the signal, and reading or recording the signal.
3. Common transducers discussed include strain gauges, which measure elastic deformation from forces, and piezoelectric transducers, which generate voltage proportional to applied pressure.
This document provides an overview of key soil properties that soil scientists evaluate when characterizing soils, including: color, texture, structure, consistence, shrink-swell potential, bulk density, porosity, permeability, infiltration, drainage, available water holding capacity, reaction, cation exchange capacity, and landscape position. It discusses how each property influences other physical and chemical characteristics of the soil and describes methods used to assess properties in the field.
This document discusses modeling abrasive flow machining (AFM) to determine stress levels, depth of indentation, and material removal rate. AFM uses an abrasive particle-filled viscoelastic polymer that is forced through a workpiece to improve its surface finish. The summary is as follows:
(1) Computational fluid dynamics (CFD) analysis using ANSYS software was used to model AFM of mild steel with a convergent-divergent nozzle.
(2) The CFD simulation results provided values for axial stress, radial stress, normal stress, depth of indentation, and material removal rate.
(3) Modeling equations were presented for calculating the normal force on abrasive particles
This document provides formulas and information for calculating properties of helical gears, including beam strength, torque, horsepower capacity, tooth dimensions, and factors. It includes the Lewis formula to calculate beam strength based on tooth load, material stress, face width, tooth form factor, and diametral pitch. It also provides formulas for relating transverse and normal diametral pitch, circular pitch, lead, and conversions between transverse and normal dimensions. Tables list safe static stress values for various materials and tooth form factors for 45-degree helix angle gears.
Optimizing Gradle Builds - Gradle DPE Tour Berlin 2024Sinan KOZAK
Sinan from the Delivery Hero mobile infrastructure engineering team shares a deep dive into performance acceleration with Gradle build cache optimizations. Sinan shares their journey into solving complex build-cache problems that affect Gradle builds. By understanding the challenges and solutions found in our journey, we aim to demonstrate the possibilities for faster builds. The case study reveals how overlapping outputs and cache misconfigurations led to significant increases in build times, especially as the project scaled up with numerous modules using Paparazzi tests. The journey from diagnosing to defeating cache issues offers invaluable lessons on maintaining cache integrity without sacrificing functionality.
Redefining brain tumor segmentation: a cutting-edge convolutional neural netw...IJECEIAES
Medical image analysis has witnessed significant advancements with deep learning techniques. In the domain of brain tumor segmentation, the ability to
precisely delineate tumor boundaries from magnetic resonance imaging (MRI)
scans holds profound implications for diagnosis. This study presents an ensemble convolutional neural network (CNN) with transfer learning, integrating
the state-of-the-art Deeplabv3+ architecture with the ResNet18 backbone. The
model is rigorously trained and evaluated, exhibiting remarkable performance
metrics, including an impressive global accuracy of 99.286%, a high-class accuracy of 82.191%, a mean intersection over union (IoU) of 79.900%, a weighted
IoU of 98.620%, and a Boundary F1 (BF) score of 83.303%. Notably, a detailed comparative analysis with existing methods showcases the superiority of
our proposed model. These findings underscore the model’s competence in precise brain tumor localization, underscoring its potential to revolutionize medical
image analysis and enhance healthcare outcomes. This research paves the way
for future exploration and optimization of advanced CNN models in medical
imaging, emphasizing addressing false positives and resource efficiency.
Embedded machine learning-based road conditions and driving behavior monitoringIJECEIAES
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
1. Stress, Strain, and Strain Gages, Page 1
Stress, Strain, and Strain Gages
Author: John M. Cimbala, Penn State University
Latest revision: 24 October 2013
Introduction
Stress and strain are important aspects of Mechanical Engineering, especially in structural design.
In this learning module, we discuss stress and strain and their relationship, and how to measure them.
Definitions
Stress
o When a material is loaded with a force, the stress at some location in the material is
defined as the applied force per unit of cross-sectional area.
o For example, consider a wire or cylinder, anchored at the top, and hanging down. Some
force F (for example, from a hanging weight) pulls at the bottom, as sketched, where A is
the original cross-sectional area of the wire, and L is the original wire length.
o In this situation, the material experiences a stress, called an axial stress, denoted by the
subscript a, and defined as a
F
A
.
o Notice that the dimensions of stress are the same as those of pressure – force per
unit area.
Strain
o In the above simple example, the wire stretches vertically as a result of the force.
Strain is defined as the ratio of increase in length to original length.
o Specifically, when force is applied to the wire, its length L increases by a small
increment L, while its cross-sectional area A decreases, as sketched.
o In the axial direction (the direction of the applied force), axial strain a is defined
as a
L
L
.
o The dimensions of strain are unity – strain is a nondimensional quantity.
Hooke’s law
o It turns out that for elastic materials, stress is linearly proportional to strain.
o Mathematically, this is expressed by Hooke’s law, which states a aE , where E = Young’s modulus,
also called the modulus of elasticity.
o Young’s modulus is assumed to be constant for a given material.
o Hooke’s law breaks down when the strain gets too high. On a typical
stress-strain diagram, Hooke’s law applies only in the elastic stress
region, in which the loading is reversible. Beyond the elastic limit (or
proportional limit), the material starts to behave irreversibly in the
plastic deformation region, in which the stress vs. strain curve deviates
from linear, and Hooke’s law no longer holds, as sketched.
o In this learning module, only the elastic stress region is considered.
Wire resistance
The electrical resistance R of a wire of length L and cross-sectional area A is given by
L
R
A
, where is
the resistivity of the wire material. (Do not confuse with density, for which the same symbol is used.)
The electrical resistance of the wire changes with strain:
o As strain increases, the wire length L increases, which increases R.
o As strain increases, the wire cross-sectional area A decreases, which increases R.
o For most materials, as strain increases, the wire resistivity also increases, which further increases R.
The bottom line is that wire resistance increases with strain.
In fact, it turns out that at constant temperature, wire resistance increases linearly with strain.
Mathematically, a
R
S
R
, where S is the strain gage factor, defined as
/
a
R R
S
.
S is typically around 2.0 for commercially available strain gages. S is dimensionless.
F
A L
F
L + L
L
L
a
a
Elastic limit
Elastic stress
region
Yield
stress
2. Stress, Strain, and Strain Gages, Page 2
Strain gage
The principle discussed above, namely that a wire’s resistance increases
with strain, is key to understanding how a strain gage works.
The strain gage was invented by Ed Simmons at Caltech in 1936.
A strain gage consists of a small diameter wire (actually an etched metal
foil) that is attached to a backing material (usually made of plastic) as
sketched. The wire is looped back and forth several times to create an
effectively longer wire. The longer the wire, the larger the resistance, and
the larger the change in resistance with strain.
Here, four loops of metal foil are shown, providing an effective total foil
length L that is eight times greater than if a single wire, rather than a
looping pattern, were used. Commercially available strain gages have
even more loops than this. The ones used in our lab have six loops.
The direction of the applied strain is indicated on the sketch. The
connecting wires or leads go to an electronic circuit (discussed below)
that measures the change in resistance.
Consider a beam undergoing axial strain; the strain is to be measured.
A strain gage is glued to the surface of the beam, with the long sections of the etched metal foil aligned with
the applied axial strain as sketched below left (the strain gage is mounted on the front face of the beam).
As the surface stretches (strains), the strain gage stretches along with it. The resistance of the strain gage
therefore increases with applied strain. Assuming the change in resistance can be measured, the strain gage
provides a method for measuring strain.
Other practical applications are shown below – a strain gage glued (rather sloppily) onto a cylindrical rod,
and a strain gage mounted on a re-bar, which is then encased in concrete, used to measure shrinkage and to
monitor the strain on structural components in bridges, buildings, etc.
F
Strain
gage
Beam
Typical strain gage values
Here are some typical values for resistance, strain gage factor, and strain, along with the predicted values of
change in resistance:
o The electrical resistance R of a commercial strain gage (with no applied strain) is typically either 120
or 350 .
o The most widely used commercially available strain gages have R = 120 .
o The strain gage factor S of the metal foil used in strain gages is typically around 2.0.
o In typical engineering applications with metal beams, the range of axial strain is 10-6
< a < 10-3
.
Using these limits and the above equation for change in resistance as a function of strain and strain gage
factor, aR RS , and the typical range of R is 6 3
120 2.0 10 120 2.0 10R
, or
0.00024 < R < 0.24 .
Notice how small R is!
For a typical 120 strain gage, the range of fractional change in resistance is 2 10-6
< R/R < 2 10-3
.
This is the main problem when working with strain gages: We cannot use a simple ohm meter to measure the
change in resistance, because R/R is so small. Most ohm meters do not have sufficient resolution to measure
changes in resistance that are 3 to 6 orders of magnitude smaller than the resistance itself.
Etched
metal foil
Backing
material
Connecting wires
(leads)
Direction
of strain
Solder
terminal
3. Stress, Strain, and Strain Gages, Page 3
Strain gage electronics
Since R/R is very small and difficult to measure directly, electronic circuits must be designed to measure the change
in resistance rather than the resistance itself. Fortunately, there are circuits available to do just that.
The Wheatstone bridge
A clever circuit to measure very small changes in resistance is called a Wheatstone bridge.
A schematic diagram of a simple Wheatstone bridge circuit is shown to
the right.
As seen in the sketch, a DC supply voltage is supplied (top to bottom)
across the bridge, which contains four resistors (two parallel legs of two
resistors each in series).
The output voltage is measured across the legs in the middle of the
bridge.
In the analysis here, it is assumed that the measuring device (voltmeter,
oscilloscope, computerized digital data acquisition system, etc.) used to measure output voltage Vo has an
infinite input impedance, and therefore has no effect on the circuit.
Output voltage Vo = Vo
+
– Vo
–
is calculated by analyzing the circuit. Namely,
3 1 4 2
o
2 3 1 4
s
R R R R
V V
R R R R
.
[This equation is “exact” – no approximations of small change in resistance were made in its derivation.]
How does the Wheatstone bridge work? Well, if all four resistors are identical (R1 = R2 = R3 = R4), the bridge
is balanced since the same current flows through the left leg and the right leg of the bridge. For a balanced
bridge, Vo = 0.
More generally (as can be seen from the above equation), a Wheatstone bridge can be balanced even if the
resistors do not all have the same value, so long as the numerator in the above equation is zero, i.e., if
3 1 4 2R R R R . Or, expressed as ratios, the bridge is balanced if 1 4
2 3
R R
R R
.
In practice, the bridge will not be balanced automatically, since “identical” resistors are not actually
identical, with resistance varying by up to several percent. Thus, a potentiometer (variable resistor) is
sometimes applied in place of one of the resistors in the bridge so that
minor adjustments can be made in order to balance the bridge.
o An arrow through the resistor indicates that its resistance can vary,
as sketched to the right.
o In this circuit, resistor R2 was arbitrarily chosen to be replaced by a
potentiometer, but any of the four resistors could have been used
instead.
Quarter bridge circuit
To measure strain, one of the resistors, in this case R3, is replaced by the
strain gage, as sketched to the right. (Note that one of the other resistors
may still be a potentiometer rather than a fixed resistor, but that will not
be indicated on the circuit diagrams to follow.)
Again, an arrow through the resistor indicates that its resistance can
vary this time because R3 is an active strain gage, not a potentiometer.
With only one out of the four available resistors substituted by a strain
gage, as in the above schematic, the circuit is called a quarter bridge
circuit.
The output voltage Vo is calculated from Ohm’s law, as previously,
3 1 4 2
o
2 3 1 4
s
R R R R
V V
R R R R
.
Let R1 = R2 = R4 = 120 , and let the initial resistance of the strain gage (with no load) be R3,initial = 120 .
The bridge is therefore initially balanced when R3 = R3,initial, since R3,initialR1 – R4R2 = 0, and Vo is thus zero.
Vs = supply
voltage
R1 R2
R4 R3
Vo
+
+
Pot
Vs = supply
voltage
R1 R2
R4 R3
Vo
+
+
R3 = strain gage
Vs = supply
voltage
R1 R2
R4 R3
Vo
+
+
Vo
–
Vo
+
4. Stress, Strain, and Strain Gages, Page 4
Unbalanced quarter bridge circuit - to measure strain
In normal operation, the Wheatstone bridge is initially balanced as above. Now suppose strain is applied to
the strain gage, such that its resistance changes by some small amount R3. In other words, R3 changes from
R3,initial to R3,initial + R3.
Under these conditions the bridge is unbalanced, and the resulting output voltage Vo is not zero, but can be
calculated as
3,initial 3 1 4 2
o
2 3,initial 3 1 4
s
R R R R R
V V
R R R R R
.
We simplify the numerator by applying the initial balance equation, R3,initialR1 – R4R2 = 0, yielding
3 1
o
2 3,initial 3 1 4
s
R R
V V
R R R R R
. [This equation is exact only if the bridge is initially balanced.]
We simplify the denominator by recognizing, as pointed out previously, that the change in resistance of a
strain gage is very small; in other words, R3/R3,initial << 1. This yields
3 1
o
2 3,initial 1 4
s
R R
V V
R R R R
.
We apply the relationship derived earlier for change in resistance of a strain gage as a function of axial strain,
resistance, and strain gage factor, namely, 3 3,initial aR R S . After some algebra,
2
2 3,initialo
2 3,initial
1
a
s
R RV
V S R R
.
Furthermore, since R2 = R3,initial (e.g., both are 120 ), this reduces to o 1
4a
s
V
V S
or o
4
a
s
S
V V
.
The significance of this result is this:
For constant supply voltage Vs and constant strain gage factor S, axial strain at the location of the strain gage
is a linear function of the output voltage from the Wheatstone bridge circuit.
Even more significantly:
For known values of S and Vs, the actual value of the strain can be calculated from the above equation after
measurement of output voltage Vo.
Example:
Given: A standard strain gage is used in a quarter bridge circuit to measure the strain of a beam in tension.
The strain gage factor is S = 2.0, and the supply voltage to the Wheatstone bridge is Vs = 5.00 V. The
bridge is balanced when no load is applied. Assume all resistors are equal when the strain gage circuit is
initially balanced with no load. For a certain non-zero load, the measured output voltage is Vo = 1.13 mV.
To do: Calculate the axial strain on the beam.
Solution:
o We apply the above equation for axial strain for a quarter bridge circuit, yielding
o 1 1.13 mV 1 1 V
4 4
5.00 V 2.0 1000 mV
a
s
V
V S
= 0.000452.
o Since strain is such a small number, it is common to report strain in units of microstrain (strain),
defined as the strain times 106
. Note that strain is dimensionless, so microstrain is a dimensionless unit.
o The unit conversion between strain and microstrain, expressed as a
dimensionless ratio, is (106
microstrain/strain). Thus,
6
10 strain
0.000452
strain
a
= 452 strain.
o Finally, keeping to two significant digits (since S is given to only two digits),
450 straina .
Discussion: It is also correct to give the final answer as 0.00045a .
Half bridge circuit
Suppose we mount two active strain gages on the beam, one at the front and one
at the back as sketched to the right.
Also suppose that both strain gages are put into the Wheatstone bridge circuit, as
shown in the circuit diagram below, noting that resistors R1 and R3 have been
Front
strain
gage
Beam
Rear
strain
gage
F
5. Stress, Strain, and Strain Gages, Page 5
replaced by the two strain gages.
Since half of the four available resistors in the bridge are strain gages, this is called a half bridge circuit.
After some algebra, assuming that both strain gage resistances change identically as the strain is applied, it
can be shown that
2
2 3,initialo
2 3,initial
1
2
a
s
R RV
V S R R
.
Furthermore, since R2 = R3,initial = 120 , the above equation reduces to
o 1
2a
s
V
V S
or o
2
a
s
S
V V
.
Compared to the quarter bridge circuit, the half bridge circuit yields
twice the output voltage for a given strain. We say that the sensitivity of
the circuit has improved by a factor of two.
You might ask why R1 (rather than R2 or R4) was chosen as the resistor
to replace with the second strain gage. It turns out that R1 is used for the
second strain gage if its strain is of the same sign as that of R3.
To prove the above statement, suppose all four resistors are strain gages
with initial values R1,initial, R2,initial, etc. The corresponding changes in resistance due to applied strain are R1,
R2, etc. It can be shown (via application of Ohm’s law, and neglecting higher-order terms as previously) that
the output voltage varies as
2,initial 3,initialo 31 2 4
2
1,initial 2,initial 3,initial 4,initial2,initial 3,initials
R RV RR R R
V R R R RR R
. [This equation is
approximate – assumes initially balanced bridge and small changes in resistance.]
As can be seen, the terms with R1 and R3 are of positive sign, and therefore contribute to a positive output
voltage when the applied strain is positive (strain gage in tension).
However, the terms with R2 and R4 are of negative sign, and therefore contribute to a negative output
voltage when the applied strain is positive (strain gage in tension).
In the above beam example, in which both strain gages measure the same strain, it is appropriate to choose R1
for the second strain gage. If R2 or R4 had been chosen instead, the output voltage would not change at all as
strain is increased, because of the signs in the above equation. (The change in resistance of the two strain
gages would cancel each other out!)
Example a cantilever beam experiment
As an example, consider a simple
lab experiment. A cantilevered
beam is clamped to the lab bench,
and a weight is applied at the end
of the beam as sketched to the
right. A strain gage is attached on
the top surface of the beam, and
another is attached at the bottom surface, as shown.
As the beam is strained due to the applied force, the top strain gage is stretched (positive axial strain), but
the bottom strain gage is compressed (negative axial strain).
If the beam is symmetric in cross section, and if the two strain gages are identical, the two strain gages have
approximately the same magnitude of change in resistance, but opposite signs, i.e., Rbottom = Rtop.
In this case, if R1 and R3 were chosen for the two strain gages in the bridge circuit, the Wheatstone bridge
would remain balanced for any applied load, since R1 and R3
would cancel each other out.
In this example, the half bridge circuit should be constructed with
pairs of resistors that have opposite signs in the above general
equation – the choices are R1 and R2, R1 and R4, R2 and R3, or R3
and R4 as the two resistors to be substituted by the strain gages.
An example circuit for this simple experiment uses R3 for the top
strain gage and R4 for the bottom strain gage, with the Wheatstone
bridge circuit wired as sketched to the right.
Bench
Clamp
Strain gages (on top and bottom)
F
Cantilevered beam
Vs = supply
voltage
R1 R2
Vo
+
+
R3 = top
strain gage
R4 = bottom
strain gage
Vs = supply
voltage
R1 R2
R4 R3
Vo
+
+
R1 = strain gage
R3 = strain gage
6. Stress, Strain, and Strain Gages, Page 6
Circuit analysis for this case yields o 1
2a
s
V
V S
or o
2
a
s
S
V V
.
Compared to the quarter bridge circuit, the voltage output of this half bridge circuit (with two active strain
gages) is twice that of the quarter bridge circuit (with only one active strain gage), all else being equal.
For any system, sensitivity is defined as (change in output) / (change in input). In this case, the output is the
voltage Vo, and the input is the axial strain being measured.
Thus, we conclude: The sensitivity of a half bridge Wheatstone bridge circuit is twice that of a quarter bridge
Wheatstone bridge circuit.
Full bridge circuit
If we substitute strain gages for all four resistors in a Wheatstone
bridge, the result is called a full bridge circuit, as sketched to the right.
Warning: You need to be very careful with the signs when wiring a full
bridge circuit!
If the wiring is done properly (e.g., R1 and R3 have positive strain, while
R2 and R4 have negative strain), the sensitivity of the full bridge circuit
is four times that of a quarter bridge circuit, o 1
a
s
V
V S
or o a sV SV .
In general, we define n as the number of active gages in the Wheatstone bridge:
o n = 1 for a quarter bridge
o n = 2 for a half bridge
o n = 4 for a full bridge
Then the strain can be generalized to o4 1
a
s
V
n V S
or o
4
a s
n
V SV .
One cautionary note: In derivation of the above equation, it is assumed that positive strain gages (R1 and R3)
are chosen for positive strain (tension), and negative strain gages (R2 and R4) are chosen for negative strain
(compression). If instead we were to wire the circuit such that the positive gages are in compression and the
negative gages are in tension, a negative sign would appear in the above equation.
On a final note, it is not always necessary to initially balance the bridge. In other words, suppose there is
some initial non-zero value of bridge output voltage, namely Vo,reference 0. This voltage represents the
reference output voltage at some initial conditions of the experiment, which may not necessarily even be
zero strain.
We can still calculate the strain by using the output voltage difference rather than the output voltage itself,
o o,reference4 1
a
s
V V
n V S
or o o,reference
4
a s
n
V V SV .
In the lab, we use voltmeters with a “REL” button, which stands for “relative” voltage. [On some voltmeters,
the REL button is indicated by triangular symbol “” instead.]
Under conditions of zero strain with a slightly unbalanced bridge, the reference output voltage is not zero
(Vo,reference 0). However, pushing the REL button causes the voltmeter to read all subsequent voltages
relative to Vo,reference. In other words, the voltmeter reads Vo – Vo,reference instead of Vo itself. We effectively
“trick” the voltmeter into showing zero voltage for zero strain.
Warning: The REL button can be dangerous if you forget to turn it off, because all subsequent voltages are
displayed relative to the voltage input at the time the REL button was pushed.
Vs = supply
voltage
R1 R2
R4 R3
Vo
+
+