# EXPERIMENTAL STRESS ANALYSIS PDF

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Experimental Stress Analysis has been traditionally applied—through a direct or forward approach—for solving structural mechanical problems. A large number of problems where experimental stress analysis techniques have The brittle-lacquer technique of experimental stress analysis relies on the. Experimental Stress Analysis - James W Dally_William F Riley 3Ed - Ebook download as PDF File .pdf), Text File .txt) or view presentation slides online.

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CHAPTER 1 STRESS. Introduction. 3. Definitions. 3. Stress at a Point. 5. Stress Equations of Equilibrium. 7. Laws of Stress Transformation. 8. Experimental Stress Analysis by: James W. Dally, William F. Riley Stress & Experimental Analysis of Simple and Advanced Pelton Wheel. Read more. MODERN EXPERIMENTAL STRESS ANALYSIS completing the solution of partially speciﬁed problemsJames F. Doyle Purdue Unive.

When looking at convergence, it is necessary to change the mesh in a systematic way. The other meshes are variations of this mesh, obtained by uniformly changing the number of modules. The results are shown in Figure 1. It is pleasing to see that it gives good results even for relatively few modules. The performance of the DKT element for the moments is also very good. These reported moments are nodal averages. Finite Element Methods 0. Convergence study for displacements and moments.

The notations for the shell are identical to those of the space frame. The assemblage procedure is the same as for the frame. Applied Body and Traction Loads The applied loads fall into two categories. The others arise from distributed loads in the form of body loads such as weight and applied tractions such as pressures; these are the loads of interest here.

We want to develop a scheme that converts the distributed loads into equivalent concentrated forces and moments. In this process, this adds another level of approximation to the FEM solution; but keep in mind that any desired degree of accuracy can always be achieved by increasing the number of nodes.

This is at the expense of computing time, but has the advantage of being simple. The question now is as follows: We will investigate two approximate ways of doing this.

To get the equivalent nodal loads, we will equate the virtual work of the nodal loads to that of the distributed load. The virtual work is the load times the virtual displacement, 1. Consider the beam shown in Figure 1. Replacement of an arbitrary load distribution by equivalent nodal values. Finite Element Methods It is interesting to note the presence of the moments.

The next section, when dealing with the mass matrix, shows examples of using linear interpolations on the body forces.

Traction distributions are treated similar to the body forces except that the equivalent loads act only on the loaded section of the element.

Consider the triangle shown in Figure 1. Let the tractions at the end points be tx1 , ty1 and so on, and let them be linearly interpolated with the same interpolation functions Py 2 Px2 2 2 qi x 1 1 Py 1 Px1 Figure 1. Replacement of an arbitrary traction load by equivalent nodal values. Because the weighting functions are the shape functions for the element displacements this representation is called the consistent load representation. Notice that the consistent loads in the case of the beam have moments even though the applied distributed load does not.

The consistent loads are a statically equivalent load system. This can be shown in general, but we demonstrate it with the following special cases.

As shown in the example of Figure 1. There is also another reason for preferring the lumped approach: The categorizing of problems is generally associated with the type of applied loading. Note that the integrations are over the original geometry. When the shape functions [ N ] are the same as used in the stiffness formulation, the mass and damping matrices are called consistent.

The assemblage process for the mass and damping matrices is done in exactly the same manner as for the linear elastic stiffness. As a result, the mass and damping matrices will exhibit all the symmetry and bandedness properties of the stiffness matrix. This is a diagonal matrix. Note that this relation is not likely to hold for structures composed of different materials. However, for lightly damped structures it can be a useful approximation.

We will establish the mass matrix for the beam. For the membrane behavior, we use the shape functions associated with the constant strain triangle.

It is useful to realize that because the mass matrix does not involve derivatives of the shape function, we can be more lax about the choice of shape function than for the stiffness matrix. We show some examples here. The simplest mass model is to consider only the translational inertias, which are obtained simply by dividing the total mass by the number of nodes and placing this value of mass at the node. Finite Element Methods respectively. Generally, these contributions are negligible and the above are quite accurate especially when the elements are small.

There is, however, a very important circumstance when a zero diagonal mass is unacceptable and reasonable nonzero values are needed. First consider the frame. We will use the diagonal terms of the consistent matrix to form an estimate of the diagonal matrix. This scheme has the merit of correctly giving the translational inertias. Types of Linear Dynamic Problems For transient dynamic problems, the applied load P t are general functions of time.

Typically, the equations require some numerical scheme for integration over time and become computationally intensive in two respects. First, a substantial increase in the number of elements must be used in order to model the mass and stiffness distributions accurately.

The other is that the complete system of equations must be solved at each time increment. When the applied load is short lived, the resulting problem gives rise to free wave propagation. This is the discrete approximation of the dynamic structural stiffness. It is therefore frequency dependent as well as complex.

This system of equations is now recognized as the spectral form of the equations of motion of the structure. One approach, then, to transient problems is to evaluate the above at each frequency and use the FFT [31] for time domain reconstructions. A case of very special interest is that of free vibrations. When the damping is zero this case gives the mode shapes that are a very important aspect of modal analysis [76, 82].

Note that the larger the number of elements for a given structure , the larger the system of equations; consequently, the more eigenvalues we can obtain.

In the remainder of this section, and in some of the forthcoming chapters, we will be primarily concerned with the transient dynamic problem. The algorithm for the step-by-step solution operates as follows: The algorithm operates very similar to the explicit scheme: These are used to obtain current values of the right-hand side.

Finite Element Methods Numerical integration schemes are susceptible to numerical instabilities, a symptom of which is that the solution diverges at each time step. For the central difference algorithm, we require that [71] T 2 1. Overlap of effectiveness of implicit and explicit integration methods. Incremental Solution Scheme Consider the loading equation 1. Simple pinned truss with a grounded spring.

## Manual on Experimental Stress Analysis Fifth

To make the ideas explicit, consider the simple truss whose geometry is shown in Figure 1. The members are of original length Lo and the unloaded condition has the apex at a height of h. The two ends are on pinned rollers. When h is zero, this problem gives rise to a static instability buckling ; nonzero h acts as a geometric imperfection.

This problem is considered in greater depth in Reference [71]. These results, for different values of h, are shown in Figure 1. The effect of a decreasing h is to cause the transition to be more abrupt. The second matrix is called the geometric stiffness matrix because it arises due to the rotation of the member; note that it depends on the axial load F o. Table 1. In order for this simple scheme to give reasonable results, it is necessary that the increments be small.

This can be computationally prohibitive for large systems because at each step, the tangent stiffness must be formed and decomposed. Incremental results using simple stepping. What we can do, however, is repeat the above process at the same applied load level until we get convergence; that is, with i as the iteration counter, we repeat solve: Combined incremental and iterative results are also given in Figure 1. We see that it gives the exact solution. Iteration results for a single load step equal to 0.

We see that convergence is quite rapid, and the out-of-balance forces go to zero. It is worth pointing out the converged value above Pcr in Figure 1. Such a situation would not occur physically, but does occur here due to a combination of linearizing the problem i. Newton—Raphson iterations for a load step 0. The cost of this can be quite prohibitive for large systems. Thus, effectively, the computational cost is like that of the incremental solution with many steps.

It must be realized, however, that because of the quadratic convergence, six Newton—Raphson iterations, say, is more effective than six load increments. This generally requires more iterations and sometimes is less stable, but it is less computationally costly. Both schemes are illustrated in Figure 1. Clearly, however, a good quality tangent stiffness will give more rapid convergence as well as increase the radius of convergence.

In this, a local coordinate system moves with each element, and constitutive relations and the like are written with respect to this coordinate system. Consequently, for linear materials, all of the nonlinearities of the problem are shifted into the description of the moving coordinates.

We will review just the essentials of the scheme here. The main objectives are establishing the relation between the local variables and the global variables, from which we can establish the stiffness relations in global variables. To help in the generalization, assume that each element has N nodes with three components of force at each node.

There are 2N position variables we are interested in: The corotational concept. However, the spin is not independent of the displacements because we require that the local spin be zero since it is rotating with the element. Discussions of the projector matrix can be found in References [, , ]. We will keep that in mind when we develop the stiffness relations. The latter contribution which is relatively complex is the geometric stiffness matrix.

We therefore recognize the global stiffness matrix as the components of the local stiffness matrices transformed to the current orientation of the element.

However, it is not just the local stiffness itself but the local stiffness times the projector matrix. The remaining set of terms gives the geometric contribution to the stiffness matrix.

An important point learned from the earlier simple results is that the more accurate the tangent stiffness, the better the convergence rate, but a consequence of using Newton—Raphson equilibrium iterations is that it is not essential that the actual exact tangent stiffness be used.

Consequently, if it is convenient to approximate the tangent stiffness, then the basic nonlinear formulation is not affected, only the convergence rate and radius of convergence of the algorithm is affected. We will see that the explicit scheme is simpler to implement primarily because it does not utilize the tangent stiffness matrix and therefore is generally the more preferable of the two.

The applied loads in this case include the inertia and damping forces. Furthermore, there are no equilibrium iteration loops as will be needed in the implicit schemes. A very important feature of this scheme is that the structural stiffness matrix need not be formed [22]; that is, we can assemble the element nodal forces directly for each element.

For example, Reference [] uses a cohesive stress between element sides in order to model the dynamic fracture of brittle materials. In this context, we can also model deformation dependent loading effects such as follower forces or sliding friction without changing the basic algorithm. As indicated in Figure 1. Since this encompasses nearly static problems, we see that we must use some sort of implicit integration scheme that utilizes the tangent stiffness [71]. The implicit integration scheme, when applied to nonlinear problems, has a step size restriction much more restrictive than for the corresponding linear problem [71].

The best that can be done at present is to always test convergence with respect to step size and tolerance.

These results have an important implication for our choice of time integration scheme. If we choose an explicit integration scheme, then the stiffness matrix is not readily available and we must use a mass proportional damping. Consequently, the high-frequency components will not be attenuated. Finite Element Methods 1. We elaborate on both of these in this section. The key to the FEM formulation for nonlinear materials is to write the element stiffness relations for the increment of strain rather than the total strain.

In this sense, it has a good deal in common with the incremental formulation of the geometrically nonlinear problem. These are obtained as part of the updating after the new displacements are computed.

The constitutive description is simplest in the corotational context because the rotations are automatically accounted for and that is the one used here. We consider both nonlinear elastic and elastic-plastic materials and the task is to establish [ k ].

Hyperelastic Materials Hyperelastic materials are elastic materials for which the current stresses are some nonlinear function of the current total strains; a common example is the rubber materials.

For these materials, the stresses are derivable from an elastic potential, which can be written from Equation 1. Examples of nonlinear material behavior. Even this simple example shows a characteristic of nonlinear materials and nonlinear problems in general , namely, the possibility of static instabilities. We need it to form the tangent stiffness matrix for an element, but the stresses when needed will be computed from the total strain relationship. Plasticity is a phenomenon that occurs beyond a certain stress level called the yield stress.

We would like to have a similar effective measure of the plastic strain. The basic assumptions of the Prandtl—Reuss theory of plasticity [99, ] is that plasticity is a distortion no volume change dominated phenomenon and that the plastic strain increments are proportional to the current stress deviation. At each stage of the loading, the increment of stress is related to the increment of strain.

To establish this incremental stiffness relation, we need to replace the plastic strain increments with total strains. From a programming point of view, it is necessary to remember the state of stress and elastic strain from the previous step. It is not intended to be a primer on how to program these codes—this is best left to specialist books.

Indeed, as we will see in the ensuing chapters, we will develop inverse methods whereby we use FEM programs as stand-alone external processes such that FEM code is not part of the inverse methods. In this way, essentially, if an FEM program can handle a certain nonlinear material behavior, then the inverse method also can handle that material behavior and the particular implementation details are not especially relevant.

This also applies to special material implementations such as those for composite materials.

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These are of immense importance in measurement because, generally, it happens that the original quantity strain, say to be measured is too awkward or too small to handle in its original form and instead an intermediate quantity is used. In modern measurements, the most common intermediate quantity is the electric voltage. While there are a great many sensors available to the experimental engineer, only two sensors are focused on in this chapter: In contrast to point transducers, the optical methods give information that are distributed continuously over an area; Figure 2.

References [43, 48, ] give an in-depth treatment of the experimental methods considered here, and also cover a host of other methods. Chip Tool Motion Specimen Figure 2. Photoelastic fringes formed during the machining of a polycarbonate specimen. Experimental Methods Electrical Filter Circuits All electrical measurement systems are inherently dynamic.

This is due primarily to the feedback loops necessary to control the unwanted electrical noise. As will be seen, these are characterized by the increasing order of differential relations. All electrical components have these properties, so it is of interest to see how they affect dynamic circuits. The RC differentiator circuit is shown in Figure 2.

Figure 2. It is apparent that as RC is decreased, there can be a good deal of distortion. This is characteristic of dynamic circuits and as to understanding them is essential for accurate measurements. The RC integrator circuit is shown in Figure 2. This circuit can also cause a distortion in the output voltage, but the distortion is different than that for the differentiator circuit.

Two RC circuits. Response of RC circuits to a rectangular input voltage history.

Forced Frequency Response To get further insight into the functioning of the circuits, we will now consider response to sinusoidal inputs. The output signal is distorted in two ways: This means that at a given frequency, the relation between the driving signal and output signal is as shown in Figure 2.

It is evident from Figure 2. Frequency response of RC circuits. Experimental Methods As seen from Figure 2. Generally, it is easier to characterize the performance of circuits and hence devices in the frequency domain.

This, however, is a more direct route. One such simple circuit is discussed here. The circuit parameters and equation variables are related by the following expressions: Typically, the damping range is between 0. In this range, magnitude errors are small, and phase angle shift is nearly linear with respect to frequency.

This latter property is needed so that complex signals maintain their wave shape. Once digitized, the signal can be stored in memory, transmitted at high speed over a data bus, processed in a computer, or displayed graphically on a monitor.

These two parameters, in turn, depend on the conversion technique; we therefore begin by considering some conversion techniques. Binary Code Representation and Resolution Each bit of a digital word can have two states: Consider a digital word consisting of a 3-bit array; these can be arranged to represent a total of eight numbers as follows: When all bits are 0, [], the equivalent count is 0. The word length implies the number of parts that the voltage range of a conversion can be divided into.

For a converter that produces a bit word, there are parts into which the input voltage can be divided. The uncertainty illustrated in Figure 2.

This equation shows that analogto-digital conversion results in a maximum uncertainty that is equivalent to 12 LSB. Our interest is in analog-to-digital conversion, but we start with the simpler digital-toanalog conversion to illustrate the design of the circuits.

A reference voltage, ER , is connected to a set of precision resistors 10, 20, 40, 80; i. The switches are operated by digital logic, with 0 representing open and 1 representing closed. Discretization of an analog signal showing the uncertainty of a measured signal. Consider, as an example, the digital code [] or 5 with the circuit shown in Figure 2.

Conversion can be made at 10 to MHz, which gives sampling times of to 10 ns. Each comparator is connected to the unknown voltage Eu. The reference voltage is applied to a resistance ladder so that when it is applied to a given comparator it is 1 LSB higher than the reference voltage applied to the lower adjacent comparator. As these are latching-type comparators, they hold high or low until they are downloaded to a system of digital logic gates that converts them into a binary coded word.

This is why it is the most expensive method. The method of successive approximation is shown schematically in Figure 2. The bias voltage, Eb , is compared to a sequence of precise voltages generated by an 2. We now describe the conversion process in more detail. Since this overestimates Eb , the MSB is locked at 0.

The voltage comparison shows Eo Eb , bit 3 is also locked on.

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The speed of this conversion process depends primarily on the number of bits. Analog-to-digital conversion by integration is the slowest of the three methods because it is based on counting clock pulses. Experimental Methods The comparator monitors the output voltage Ei when it goes to zero. If a counter is initiated by the switch on the integrator, then the counter will give a binary number representing the unknown voltage Eu.

The integration method has several advantages. First, it is very accurate, as its output is independent of R, C, and clock frequency because these quantities affect the up and down ramps in the same way.

This is too slow for generalpurpose data-acquisition systems, and certainly for any dynamic recordings. However, it satisfactory for digital voltmeters and smaller data-logging systems. Some allow analog-to-digital conversion, digital-to-analog conversion, digital input, digital output, external and internal triggering, direct-memory access and internal and external clocking of the processes. The input end consists of several sections as shown in Figure 2.

Spectral Analysis of Signals As discussed earlier in this chapter, an arbitrary time signal can be thought of as the superposition of many sinusoidal components, that is, it has a distribution or spectrum of components. Working in terms of the spectrum is called spectral analysis or sometimes frequency domain analysis. A schematic of a typical data-acquisition board. However, the numerical manipulation of signals requires discretizing and truncating the distribution in some manner—in recent years, the discrete Fourier transform DFT and it numerical implementation in the form of the fast Fourier transform FFT has become the pervasive method of analyzing signals.

We now give an introduction to some of their characteristics, especially as to how it applies to signal processing. Fourier Transforms The continuous transform is a convenient starting point for discussing spectral analysis because of its exact representation of functions.

We now look at discretized forms. The Fourier series gives the exact values of the continuous transform, but it does so only at discrete frequencies. This, 96 Chapter 2. Experimental Methods of course, is the continuous transform limit. These integrations are replaced by summations as a further step in the numerical implementation of the continuous transform.

It is interesting to note that the exponentials do not contain dimensional quantities; only the integers n, m, N appear. In this transform, both the time and frequency domains are discretized, and as a consequence, the transform behaves periodically in both domains. There are other possibilities for these scales found in the literature.

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To put this point into perspective, consider the following: Thus it is the continuous Fourier transform of a periodic signal. The lesson is that by choosing a large signal sample length, the effect due to the periodicity assumption can be minimized and the discrete Fourier transform approaches the continuous Fourier transform. This is not a different transform — the numbers obtained from the FFT are exactly the same in every respect as those obtained from the DFT. The following examples serve to show the basic procedures used in applying the FFT to transient signals.

The FFT is a transform for which no information is gained or lost relative to the original signal. However, the information is presented in a different format that often enriches the understanding of the information. In the examples to follow, we will try to present that duality of the given information. Transforms of Time-Limited Functions The complementarity of information between the time and frequency domains is illustrated in Figure 2.

The top three plots are for triangles. The third trace shows a smoothed version of the second triangle; in the frequency domain, it is almost identical to the second, the only difference being the reduction in the high-frequency side lobes. The remaining two plots are for modulated signals — in this case, a sinusoid modulated multiplied by a triangle. The sinusoid function on its own would give a spike at 15 kHz, whereas the triangular modulations on their own are as shown for the top traces.

The product of the two in the time domain is a convolution in the frequency domain. The convolution tends to distribute the effect of the pulse. A point of interest here is that if a very narrow-banded signal in the frequency domain is desired, then it is necessary to extend the time domain signal as shown in the bottom example.

The modulated waves are called wave packets or wave groups because they contain a narrow collection of frequency components. Digital Processing of Experimental Signals The signals we collect will be the functions of time. Even nominally static problems will have time-varying signals due, at least, to the noise.

Spectral analysis can give insight into how all are treated. Comparison of some pulse-type signals in both the time and frequency domains.

Experimental Methods I: Consider an event with a characteristic frequency of Hz and the need to monitor eight channels of analog data. Assume that the converter can handle eight channels of input and requires a clock pulse for each conversion.

Further, assume that we want to have 20 samples during each period of the incoming signal. Measurement Techniques for Transients Testing associated with transients arising from impact and stress wave propagation is different from that of vibration testing.

Some of the characteristics are: The number of data recording channels are usually limited 2—8. There is quite a range of measurement techniques available for studying wave propagation and high-frequency structural dynamics problems, but we restrict ourselves to strain gages and accelerometers, both of which are discussed in the next sections. Historically, oscilloscopes were the primary means of recording and displaying the waveform but nowadays digital recorders are used. For high-speed recording, we use the Norland []; this has four channels, each storing 4 K data points, with up to ns per sample.

The DASH [] is an example of the newer breed of recorders; it is like a digital strip chart recorder and notebook computer rolled into one.

The data is continuously recorded and stored to the computer hard drive. The DASH can record up to 18 channels and has 2. There are also relatively inexpensive recorders available that can be installed in desktop computers.

The Omega Instruments DAS data-acquisition card is an example; this is a bit card capable of 1-MHz sampling rate and storing up to 1 M data points in the on-board memory.

Smoothing The measured signal can contain unwanted contributions from at least two sources. This is especially prevalent for dispersive signals as the recording is usually extended to capture as much as possible of the initial passage of the wave.

There are many ways of smoothing digital data but the simplest is to use moving averages of various amounts and points of application. This section reviews the major effects that should be considered. Effect of moving-average smoothing. Experimental Methods through the window. Examples of time windows commonly used are the rectangle and the Hanning window which is like a rectangle with smoothed edges ; see Figure 2.

None of the windows usually used in vibration studies seem particularly appropriate for transient signals of the type obtained from propagating waves. The estimated spectrum is given only at discrete frequencies and this can cause a problem known as leakage. For example, suppose there is a spectral peak at 17 kHz and the sampling is such that the spectrum values are only at every 2 kHz. In the analysis of the impact of structures, the signals generated generally do not exhibit very sharp spectral peaks.

The simplest means of doing this is by padding. There are various ways of padding the signal and the one most commonly used is that of simply adding zeros. Parenthetically, it does not matter if the zeros are added before the recorded signal or after. Rectangle Exponential Gaussian Figure 2.

## Experimental Stress Analysis - James W Dally_William F Riley 3Ed

Some common windows. Aliasing caused by coarse sampling; the sampled high-frequency signal appears as a lower frequency signal.

This phenomenon of aliasing can be avoided if the sampling rate is high enough. The principle of its action is that the electrical resistance of a conductor changes proportionally to any strain applied to it. Thus, if a short length of wire were bonded to the structure in such a way that it experiences the same deformation as the structure, then by measuring the change in resistance, the strain can be obtained. While this principle was known by Lord Kelvin back in , it was not until the late s that Simmons and Ruge made practical use of it; in spite of its simple concept, there are many features of its behavior that must be understood in order to obtain satisfactory results.

Concept of the electrical resistance strain gage. Thus, the unit change in resistance can be related to the imposed strain on the conductor. Correspondingly, two types of gages are on the market, differentiated by the contribution that is emphasized. An actual gage, however, is not a straight length of wire but is in the form of a grid as shown in Figure 2. The original wire gages were wound back and forth over a paper former so as to maintain a large resistance but small size.

The actual gage sensitivity may not, therefore, be the same as the material sensitivity. This is known as the transverse sensitivity error. It does not measure shear strain directly—this must be obtained by deduction from a number of extension measurements.

This is considered later when we discuss strain gage rosettes. Temperature Effects The discussion of the last section made no reference to the temperature effects. This topic must be treated separately because it is potentially the source of most serious errors in the measurement of static strains. When the thermal environment changes, both the gage and specimen are affected and this must be taken into account in deriving the expression for the change of resistance because these changes could be misinterpreted as strain.

Specimen elongates causing extension of gage: Gage grid elongates: Resistivity of gage material changes: Thus, the apparent strain depends not only on the nature of the gage but also on the material to which it is bonded. However, these gages usually have a very narrow range of application with regard to both temperature and specimen material. Temperature compensation can also be effected by special circuitry design, and this will be discussed later.

Table 2. Properties of some gage materials. Experimental Methods Gage Construction and Mounting In practice, the active sensing element of the gage is bonded to the specimen so as to form an integral part of it. In this way, the deformation in the specimen is experienced exactly by the gage. The core of the gage is the sensing element, which may be of three basic types. These gages have been superseded by foil gages that are formed by photoetching thin sheets usually less than 0.

They also can be obtained in sizes ranging from 0. The newer form of gage is of the piezoresistive type, which uses a doped crystal element. They exhibit large resistance changes, but are very temperature sensitive and cannot be self-compensated.

The backing or carrier material is primarily a handling aid; it allows proper alignment and supports the lead wires. It also serves the function of insulating the gage from the specimen. Adhesives capable of maintaining a shear strength of about 10 MPa psi are generally acceptable for strain gage work.

The major adhesives are acrylic or epoxy based. The junction between the lead-in wire and the gage is particularly vulnerable and for this reason a step-down approach is always used. Protective coatings are often used to prevent mechanical or chemical damage to the gage installation. Strain gage construction and mounting. Leads 2. The manufacturer generally lists them with designations such as the following from Micro-Measurements: EABG where the above letters and numbers indicate, respectively: There are also many special purpose gages that can be welded, or have high elongation or can be embedded in concrete.

When ordering gages, the individual manufacturers, handbook should be consulted. Depending on the type and amount of impurity diffused into the pure silicon, the sensitivity can be either positive or negative—Boron is used in producing P-type positive while Arsenic is used to produce N-type negative gages.

They also have problems when mounted on curved surfaces. Since one strain gage cannot measure shear directly, three gages must be arranged to give three independent extension measurements and the transformation equations used to determine the shear component.

This combination of three sometimes four gages is called a strain gage rosette. In practice, rosettes are usually mounted with respect to a reference axis not necessarily the principal strain axes. It is possible to mount three separate gages on the specimen but this can be awkward if the surface area is small. Having the three separate gages oriented correctly on the same backing allows all three gages to be easily mounted simultaneously.

Note, however, that each gage must have its own circuit and be wired separately. To see how the data reduction is achieved, consider the three gages shown in Figure 2. Since there are three equations and three unknowns, these equations can be solved, but rather than doing this in general, consider the following special cases. Strain gage orientations for a general rosette.

The problem is that any ohmmeter would utilize all its accuracy on the What is needed is to measure the difference directly. The main circuits for accomplishing this will be reviewed here. Referring to Figure 2. The Wheatstone bridge. This is a highly nonlinear relation and it requires computing differences. It is seen from Equation 2. It is also noted that some of the resistance changes are positive while others are negative. It is this occurrence that will be used to a great advantage as another means of temperature compensation.

Circuit Parameters and Temperature Compensation There are many possibilities in utilizing the Wheatstone bridge, so it is of advantage now to establish a number of parameters that will aid in the evaluation of the different circuits.

In practice, however, there is an upper limit to r since the voltage required to drive the circuit becomes excessive. The ability of the gage to carry a high current is a function of its ability to dissipate heat, which depends not only on the gage itself but also on the specimen structure.

Currents may reasonably be restricted to 10 milliamps mA for long-term work on metallic specimens, but up to 50 mA are used for dynamic transient work.

On poor heat sinks such as plastics, the currents are correspondingly decreased. Gage resistances are grouped around values such as 60, , , , and ohms. The higher resistances are not necessarily preferable to the lower resistances unless they can also carry currents such that IR is high. Often, several low-resistance gages on a specimen can be placed in series and thus dissipate larger amounts of heat.

The above parameters can be used to evaluate the circuits shown in Figure 2. The nominal is when all resisters have the same values; we will use this as a reference. Case b has an active gage in position 1 and dummy gage in position 4. Sample circuits. Experimental Methods The superscripts S and T refer to strain and temperature effects, respectively. Thus, the temperature effects of the two gages cancel each other leaving a voltage change that is due only to the strain.

This is an effective way of providing temperature compensation. Consider a similar case with an active gage in position 1 and dummy gage in position 2. While this circuit is also capable of temperature compensation, its maximum sensitivity is only half that of case b and therefore we could conclude that b is the better circuit design.

Case c has two active and two dummy gages. Note that for some arrangements, the resistance changes cancel. This can be utilized in two ways as shown in Figure 2. Suppose a specimen is in simple bending with the top in tension and bottom in compression, then placing the gages at the top and bottom will double the output. Furthermore, if this is put in a transducer, then any unwanted axial components of strain would be eliminated. Note that making R3 instead of R2 active allows axial strains to be measured while canceling any unwanted bending strains.

Potentiometer and Constant-Current Circuits The Wheatstone bridge circuit is adept at measuring small resistance changes; and further, it can easily be adapted for temperature compensation without adversely affecting circuit performance. In some cases, however, even simpler circuits can be used. This is especially true of dynamic problems and two useful circuits will now be reviewed. Some circuit arrangements. Potentiometer circuit. However, if the event is dynamic and the output connected to the ac option on an oscilloscope or an RC differentiator circuit, the constant dc component, E, can be blocked.

In this way, only the variable part which is due to the change in resistance will be recorded. So far, the power sources in the circuits have been constant voltage sources. However, from the analysis of the circuits it seems that the current and not the voltage is the relevant quantity, and so a constant current source seems more appropriate.

To see this, reconsider the potentiometer circuit. With the advent of technology that Chapter 2. The function of the op-amp is to essentially maintain a zero potential across ab and thus allow all the current to go through Rg which will then be the same as in R2. This circuit, however, cannot be used for temperature compensation but it was concluded that the potentiometer circuit was poor in this respect anyway. Also, temperature compensation is generally not as important in dynamic testing especially if selfcompensated gages can be used.

It is also possible to use a constant-current source in a regular Wheatstone bridge for static testing, but not much except a slight decrease in circuit nonlinearity is achieved. Commercial Strain Indicators Perhaps the most commonly used bridge arrangement for static strain measurements is that of the reference bridge. In this arrangement, the bridge on one side contains the strain gage or gages, and the bridge on the other side contains variable resistors.

When gages are attached, the variable resistance is adjusted to obtain initial balance between the two bridges. Strains producing resistance changes cause an unbalance between the two bridges, which can then be read by a meter.

In some cases, this meter reading is obtained by further nulling the adjustments of the variable resistors. The advantage of the reference-bridge arrangement is that the gage bridge is left free for the strain gages and all adjustments for the initial and null balance are performed on the other bridge.

Recent advances in electronics have lead to the development of direct-reading Wheatstone bridge arrangements. Often, it is required that many strain gages be installed and monitored several times during testing. In these circumstances, it would be very costly to use a separate recording instrument for each gage.

An effective economic solution is to use a single recording instrument, and let the gages be switched in and out of this instrument. Two different methods of switching are commonly used in multiple-gage installations. One method involves switching each active gage, in turn, into arm R1 of the bridge as shown in Figure 2. Since the switch is located within arm R1 of the bridge, a high-quality switch must be used.

The advantage of this arrangement is that it is simple and inexpensive, the disadvantages are that the gages must be nominally the same, only a single compensating gage can be used, and the gage outputs cannot be individually nulled.

The second method switches the complete bridge. The advantage is that the switch is not located in the arms of the bridge, and hence switching resistance is not so important.

Other advantages are that each gage can be different and can be individually nulled. However, the arrangement is clearly more expensive to implement. Experiments using Strain Gages Most types of electrical resistance strain gages are usually adequate for dynamic work and since the events are of short duration, temperature compensation is usually not a serious issue.

Schematic of the switching unit. A typical gage of length 3 mm 0. Other aspects of the use of strain gages in dynamic situations are covered in References [27, ]. Either constant-current, potentiometer or Wheatstone bridge gage circuits can be used.

Experiment I: Nonlinearity of a Wheatstone Bridge The Wheatstone bridge is an inherently nonlinear device and this must be taken into account when large resistance changes as with semiconductor gages are expected. Hence, semiconductor gages are usually restricted to small-strain situations. The set up is essentially that of References [65, ]. As the wave propagates, the pulse alternates in sign with a gradually decreasing amplitude due to various dissipation mechanisms.

Clearly, there is a consistent difference between the compressive and tensile responses with the former overestimating and the latter underestimating the average voltage. Bridge nonlinearity observed in a wave propagation problem. Experiment II: Force Transducer for Static Measurements Figure 2. The use of a ring design converts the axial load into a bending load and therefore increases the strain. Additionally, since there are multiple tension and compression points on the inside of the ring, a complete four-arm bridge arrangement can be used.

This further increases the sensitivity of the load cell, provides temperature compensation, and avoids the need for completion resistors. The C-shaped design is a little simpler. It also induces both tension and compression strains due to bending and therefore has attributes similar to the ring. Note that if a commercial strain indicator is used, then completion resistors are supplied automatically.

For this simple geometry it is possible to calculate the transducer sensitivity, but nonetheless, a calibration is usually in order. Typically, a commercial strain indicator is used; therefore a strain gage factor must be set.

The gage factor need not be the actual one, however, each time the transducer is connected up, the same one must be used. Two force transducers and their gage layouts.

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Experimental Methods 2. Pressure and force transducers have the same mechanical characteristics as seismic transducers and hence we will treat them together here.

Additional details on the material covered here can be found in References [52, ]. Basic Theory The mechanical behavior of force, pressure, and motion-measuring instruments can be modeled using a single degree of freedom mechanical model with viscous damping as shown in Figure 2. The sensing element measures the relative motion between the base and the seismic mass. To this end, consider the free vibration of the system with viscous damping and look for particular solutions of M Housing um K C Quartz crystals Seismic mass u t Conductor a P t Base b Figure 2.

Schematic of a seismic transducers. This is the situation of most interest to us in structural analysis, however, in instrument design, we set it much higher. Consider the motion of the mass after it is displaced from its initial position and released. Hence, for small amounts of damping this is essentially the undamped natural frequency.

Motion and Force Transducers Seismic displacement transducers measure the displacement of the base relative to the seismic mass. This type of transducer behavior can be illustrated by assuming that the excitation force P t is zero. Experimental Methods u t. Damped response due to initial displacement. The magnitude ratio is seen to approach a value of unity regardless of the amount of damping. These results show that the natural frequency should be as low as possible so that the operating frequency range is as large as possible under most measurement situations.

For high frequencies, Equation 2. Thus, a large rattle space with soft springs is required to accommodate the large relative motion that takes place within the instrument. Seismic velocity transducers are obtained by employing velocity-dependent magnetic sensing elements in displacement transducers to measure the relative velocity between the base and the seismic mass.

Differentiation of Equation 2. Frequency response function. The corresponding FRF is identical to that shown for the displacement transducer in Figure 2. Seismic accelerometers are constructed by using displacement sensors with a stiff spring to give a high natural frequency. For frequencies far below the natural frequency of the transducer, Equation 2.

There are no magnitude peaks for damping ratios greater than 0. This also shows that the high natural-frequency requirement is detrimental to obtaining large sensitivities since the relative motion is inversely proportional to the square of the natural frequency. Fortunately, an accelerometer with a piezoelectric sensing element can simultaneously provide the required stiffness and sensitivity. Force and pressure transducers require stiff structures and high natural frequencies.

Consequently, they are seismic instruments that respond to base acceleration as well as force or pressure. The FRF for these transducers is given by Equation 2. Thus, force and pressure transducers have characteristics that are very similar to accelerometers. Construction of Accelerometers Figure 2.

Piezoelectric accelerometers function to transfer shock and vibratory motion into Chapter 2. The built-in electronics operate over a coaxial or two- conductor cable; one lead conducts both signal and power. The signal is separated from the bias voltage in the power units.

Good high-frequency transmissibility into the base of the accelerometer is dependent upon intimate contact between the base and the mounting surface. For more permanent installations, a thin layer of epoxy between mating surfaces will ensure good high-frequency performance without fear of the accelerometer becoming loose during the use.

Petro wax is a convenient temporary mounting method for small accelerometers used in place of studs and permanent adhesives. Test results show that frequency responses comparable to stud mounting is achievable with a thin coating.

Accelerometers for Structural Testing A wide variety of small accelerometers are available. There is a problem with accelerometers, however, in that if the frequency is high enough, ringing will be observed in the signal. This can often occur for frequencies as low as 20 kHz. Generally speaking, accelerometers do not have the necessary frequency response of strain gages, but they are movable, thus making them more suitable for larger, complex structures.

The accelerometers used are PCB A and []. These are light weight 2 and 4 g, respectively and have good sensitivity 1. Force Transducer for Impact Measurements Figure 2. The back mass is necessary in order to register the force.

The front mass was machined with a near-spherical cap, so that the impacts would be point impacts. The transducer was made part of a light weight but stiff pendulum system of about mm 20 in. Because of the attached masses, it is necessary to recalibrate the transducer. To this end, a mass with an attached accelerometer was freely suspended by strings and used as the impact target. The accelerometer Kistler, Piezotron model has a sensitivity of 9.

The weight of the combined mass and accelerometer was 0. Metal-on-metal contact can generate forces with a high-frequency content; however, both the force transducer and accelerometer have an upper resonance limit that prevents them from accurately registering these signals.

It was therefore necessary to place a masking tape on the mass so as to soften the impact. The recorded accelerometer and force voltage output are shown in Figure 2. PCB A05 Sensitivity: Schematic of the construction of a force transducer for impact studies. Voltage Force: Calibration experiment for force transducer. The accelerometer output was converted to force as above and used for the horizontal axis of Figure 2.

This compares to the manufacturer-reported value of 1. The difference is that the quasistatic calibration involved forces at both ends of the transducer, whereas the dynamic calibration involves a force only at the front end.

The location of the active part of the transducer near the mid-length is such that it experiences only about half of the load applied at the tip. This has had a profound effect on the optical methods of stress analysis, and therefore we begin with a discussion of the digital analysis of images.

Concept of Light and Color As shown in Figure 2. It is this periodic variation that allows light to also be considered as a wave phenomenon. Reference [90] is a very readable introduction to all aspects of optics. Dual aspect of light. Some light sources and the wavelength of light produced.

The occurrence of such a wide range can be explained by the need to carry different amounts of energy. An idea of the range can be seen in the following table: The wavelengths of some light sources is given in Table 2. Any color can be matched by mixing suitable proportions of any three independent colors.

List the various limitations of a mechanical strain gauge. Write the advantages of Foil type strain gauge over Wire type strain gauge? What is a Strain rosette? What are the advantages of an acoustical gauge? What are the limitations of an optical gauge? Write a short account of the various types of strain gauges. Give their special advantages and limitations. Which factors should be considered 3. What are the various types of Mechanical strain gauges? Explain Huggenberger tensometer in detail.

What are the various types of optical strain gauges? Explain the Tuckerman gauge in detail. Explain the construction and working of Acoustical strain gauge.

What are the different types of electrical strain gauges? Describe a capacitance strain gauge and give its uses and limitations. Name the different types of electrical strain gauges. Name the different types of electrical resistance strain gauges. Differentiate between bonded and unbounded gauges. What is a weldable gauge? What are its advantages?

Name the materials used for material gauges. What is meant by Temperature compensation? Why Wheatstone bridge circuits are preferred over potentiometer circuits in static strain measurements? What are the requirements of strain gauge materials? What are the limitations of potentiometer circuit? Define a strain rosette. Write short notes on the following : i Electromagnetic strain gauge 6 ii Weldable strain gauge 6 2. What is the necessity of temperature compensation?

How this can be achieved? What do you understand by a strain rosette? What are the different types of strain rosette configurations currently in use? Discuss their uses and limitations. Discuss the various methods of calibrating a strain gauge. What are the essential requirements of a balancing technique? Discuss the different ways in which you can balance a bridge. What is the photoelastic effect?

Differentiate between ordinary and monochromatic light. Define Stress optic law. Define isoclinics and isochromatics. What do you mean by compensation? What is a stress trajectory? Define sensitivity index and figure of merit of a photoelastic material. What is the necessity of calibration of a photoelastic material? Why Tardy method of compensation is preferred over all other methods?

Who discovered the photoelastic effect and when?Forced Frequency Response To get further insight into the functioning of the circuits, we will now consider response to sinusoidal inputs. First, a substantial increase in the number of elements must be used in order to model the mass and stiffness distributions accurately.

One of the most important characteristics of an element is its convergence properties, that is, it should converge to the exact result in the limit of small element size. It is apparent that as RC is decreased, there can be a good deal of distortion. The advantage of this arrangement is that it is simple and inexpensive, the disadvantages are that the gages must be nominally the same, only a single compensating gage can be used, and the gage outputs cannot be individually nulled.

Once digitized, the signal can be stored in memory, transmitted at high speed over a data bus, processed in a computer, or displayed graphically on a monitor. Combined incremental and iterative results are also given in Figure 1.

Experimental stress analysts also have a vital role to play in this. Experimental Methods Blue M: It is an advantage if the area size can be selected because processing the delimited area is faster than processing the whole area.