MAE 244Electrical Resistance Strain GagesLab-2 a.1
Electrical Resistance Strain Gages - Applications to Beam Bending
Theoretical Background
Definition of Normal (Extensional) Strain
When an elastic body is subjected to external loading, adjacent particles in the body may undergo different displacements, so that a relative motion occurs between them. This relative motion is called elastic deformation, and it is measured in terms of STRAIN, which is a non-dimensional parameter defined as follows:
Figure 1.1 Uniaxial Loading Creates a State of Normal Strain.
Measurement of Strain STRAIN GAGES
Strain is never measured directly, but it is derived from direct measurements of other parameters, that may rely on one of the following principles:
(1)Mechanical extensometers can measure directly the change in the gage length of the test specimen. The corresponding strain is then obtained from the above formula, by using the initial size of the gage length.
(2) Optical Some optical phenomena (e.g. birefringence) are related to strain changes caused by loading in certain optical patterns provide an indirect measurement of the corresponding strains.
(2)Electrical: The electrical properties of some materials change as a result of elastic deformations this effect provides another approach to measuring strain.
Basic Principles of Electrical Resistance Strain Gages
The electrical resistance of a conducting wire changes when the length of the conductor changes through elastic deformation (stretching). If such a wire is bonded to the surface of an elastic body that undergoes a deformation, it becomes a means of measuring the extent of the corresponding strain, as an electrical analog of the associated change in resistance. This is the basis of the electrical resistance strain gage, which is probably the most pervasive device for measuring strain. The actual strain gage, however, is not constructed as a straight segment of conducting wire, but is wound back and forth many times (very much like a tight serpentine), so as to reduce its overall size while maintaining the capability to generate a large change in electrical resistance for a relatively small change in the total length of the wire.
Most modern gages are of the foil type, where the conducting wire is printed to the desired pattern on an expendable foil, with an acid resistant ink. When dipped in acid the foil is dissolved away, and the conductor remains attached to thin, flexible, supporting backing. Samples from a batch of new strain gages are then calibrated in a known strain field by noting the corresponding changes in resistance. The Gage Factor, or the Sensitivity Factor of the strain gage is, subsequently, calculated from the measured pair of changes in electrical resistance and length:
(1)
Since the SENSITIVITY Factor of every particular gage is known in advance when the gage is attached to a structure subjected to elastic deformation, the strain induced in the structure is measured by the gage indirectly, according with the following relationship:
(2)
NOTE: An one-element strain gage measures only the NORMAL (Extensional) strain component in one, chosen direction. The measurement of shear strain components by using strain gages requires multiple measurements of normal strain components in different directions, at the location where the shear strain is sought.
Strain Gage Circuits
FUNCTION – closed electric circuits of resistors and DC power supply, whose function is to convey a small change in electrical resistance into a large change in voltage that can be readily measured, displayed, and recorded.
Consider a foil gage mounted on a bar of steel (Young's Modulus, E=30x106 psi), experiencing a stress of 10,000 psi at the location of the gage. The corresponding change in electrical resistance is very small, as indicated below for a strain gage with a gage factor, Sg=2.
This change in resistance is a very small quantity, but fortunately simple circuitry can be constructed that measures it accurately. The main types of such circuits are reviewed here.
Wheatstone Bridge
The Wheatstone bridge was the first circuit designed to measure R and is very accurate because it is based on a null principle. Its configuration is sketched below, where Ohm's Law, relating voltage to current and to resistance, V = I*R, can be applied to express the currents in the various arms of the circuit as follows:
(4)
The corresponding voltage drops across the various arms are:
(5)
The difference in electric potential between connections (nodes) B and D is therefore
(6)
If the four resistances in the circuit change by various levels, R1, R2, R3, R4, respectively, the voltage drop between the nodes B and D will also change, by an amount equal to E:
(7)
In a balanced bridge, E=0, so that Eq.(6) yields the following relationships between the resistances in the circuit:
(8)
When the bridge is balanced,, r = 1, and Eq. (7) yields:
(9)
The Wheatstone bridge is an inherently nonlinear device, since the relationship between the changes in resistances, Ri , and the change in voltage, E, is not linear. When large resistance changes are expected, such as in the case of "Semiconductor Gages", the full, non-linear relationship must be considered. However, the relationship between the resistance and the voltage changes can be linearized in the particular case of small resistance changes.
If the following condition is met:
then Eq.(9) yields:
(10)
This is the fundamental equation for analyzing the Wheatstone bridge. It is important to note that the measured voltage is, in general, E+E, but E = 0 when the bridge is initially balanced, and the voltage change E caused by the changes in resistances, is measured directly. It is also noted that some resistance changes in the expression of E (Eq.(10)) are positive, while others are negative. This alternation of signs is used to great advantage as a possible approach to temperature compensation of strain measurements.
Single Active Gage: If only one of the FOUR arms of a Wheatstone bridge contains an active resistor, suppose that R1 is the only active gage, then R2 = R3 = R4 = 0, and the Wheatstone bridge equation, Eq. (10), reduces to
(11)
where the resistance change R1 is related to the length change of that resistor through the Gage Factor of the corresponding strain gage:
(12)
where V is the excitation voltage and E is the measured output voltage., i.e the CHANGE in the voltage drop across the bridge, E, associated with a change R1 in the ACTIVE gage.
Strain Indicators
A convenient tool for determining strain is the strain indicator. The basic function of a strain indicator is to determine the strain directly, without hand calculations, by using the Wheatstone bridge equation. A full, half or quarter bridge arrangement may be used with a strain indicator. The gage factor must be set on the indicator and the bridge must be balanced. Note the procedures for connection for each bridge arrangement before you start your experimental setup.
Temperature Compensation and Special Circuits
An examination of Equation (10), reveals that the output from the Wheatstone bridge is dependent on the number of active gages, as well as on their placement in the circuit. For example, if two identical gages register the same R in the arms "1 and 2" of the bridge, there will be no voltage change, E = 0, whereas if the same identical gages are placed in positions "1 and 3" of the bridge, then the output will double, i.e. the voltage change E will be doubled. This principle can be used to our advantage. For example, we can add an active gage to position 1 and a dummy gage to position 2 of the bridge, in order to compensate for temperature. A dummy gage is a strain gage mounted on the same material as the active gage but it is not loaded. Thus, any strain component associated with expansion or contraction due to a shift in temperature would cause identical resistance changes in the active and the dummy gages, which will cancel each other in accordance with the bridge equation, Eq.(10). The value of E measured through such a bridge configuration would, therefore, represent only the effect of loading on the active gage, R1 without any contamination from changes in the ambient temperature.
One may notice, indeed, that R2 =0, since the dummy gage is placed on a separate specimen made of the same material, but not loaded, while R1 = R2 in a balanced bridge, and R3 and R4 are equal to zero, since the corresponding resistors are not strained (inactive). The bridge equation (10) reduces again to Eq. (11), as in the case of a single active gage, with no temperature effects (used in a setting where the environment is controlled to maintain constant temperature).
Special Configurations for Multi-Axial States of Stress
Similarly to the temperature compensation approach, certain stress components can be removed from the strain measurement through appropriate arrangements of the Wheatstone bridge circuit. Consider the following options of placing strain gages in the various four arms of Wheatstone bridge circuits: Circuit (a) - will result in a doubling of the output; Circuit (b) - output will include axial and transverse strains added and doubled; Circuit (c) has two active gages and will eliminate axial but double output due to bending; Circuit (d) has two active gages that will eliminate bending but double output due to axial strain.
Fig. 1.2 – Special Arrangements of Wheatstone Bridge Circuits
Beam Bending Theory for Linear, Isotropic Materials
The elementary beam theory for symmetric bending of linear, isotropic materials leads to the simple "flexure formula" for the maximum axial (normal) stresses at the top and the bottom boundaries of the beam, which in the case of a cantilever beam may be expressed as follows:
(13)
The symbol "P" in the above equation denotes the magnitude of a concentrated force applied at the free end of the beam, where the distance "x" from the clamped end is equal to the length of the beam, x=L (see Fig. 1.3). The variable "x" defines the cross-section of the beam where the Bending Moment, "M", and the maximum stresses, , are to be determined. The distance from the applied load to the cross-section where the bending moment is calculated is denoted by “d” in Fig.1.3, "c" is the distance from the neutral axis to the top (or bottom) of the beam (normally defined as half the height "t" of the beam, c= t/2), and "I" is the cross sectional moment of inertia, (I = (1/12) b t3), where "b" is the width of the beam).
NOTE: The expression of the maximum normal stress, , in Eq.(13) above is valid for any loading scenario, since the single loading parameter that it involves is the bending moment, "M", in the cross-section of interest. The formula shown in Eq.(13) for the bending moment "M"(x) depends on the specific loading scenario, boundary conditions, and geometry of the beam; the form shown in the above equation is valid only for the case of a point-load, "P", applied at the free end of a cantilever beam.
The UNIAXIAL Hooke's law for linear, isotropic materials yields:
(14)
where E is the Young's modulus (stiffness constant) of the material. Solving for the strain, in Equation (14) and substituting into Equation (13), the following equation is obtained:
(15)
The tip deflection, , of a cantilever beam is, on the other hand, given by the following formula:
(16)
where "L" is the overall length of the beam.
Solving Equation (16) for the load "P", and substituting the result into Equation (13), the bending stress, "", can be expressed in terms of "" at any position along the beam as defined by the distance parameter "d" (see Fig.1.3), and Equation (14) can be solved for strain and rewritten in terms of the tip deflection as follows:
(17)
Equation (17) can be used now to calculate a "theoretical" value of strain at the any distance, "d" from the tip, for a measured tip deflection, and compare it to the strain measured by a strain gage at that location along the beam.
Figure 1.3 – Parameters of Cantilever Beam Instrumented with Axial Strain Gage.
Approximate Analysis of a Composite Beam
Beam bending experiments performed on homogeneous, isotropic beams are common, and they conform, usually, with prescribed, standard guidelines and procedures, as those specified in the standard D-790 of ASTM (the American Society for Testing and Materials). When the tested beams are made of composite materials, the above formulas must be modified to account for the non-homogeneous and directional dependent properties of such materials. The internal structure of composite beams consists, usually, of separate layers (plies) reinforced in different directions by continuous fibers, and stacked on top of each other in a certain pattern (lay-up configuration) to form a laminated material. The theory discussed above for homogeneous, isotropic beams will be used as an approximation to the bending beam theory of composite structures.
Error Analysis of Multimeter Measurements of Strain
Consider the fundamental equation of a single-element strain gage, Eq. (2), that allows the calculation of strain from the change in resistance of an electrical resistance strain gage, i.e.
Let the gage factor, Sg, and the relative change in electrical resistance, R/R, be allowed to vary within preset limits, to account for possible errors in the experimental set-up:
Sg = 2.0 ± .01 R/R = 520 ± 10
The expected variation in the sensitivity of the strain gage is specified by the manufacturer, whereas the variation in R/R is related, primarily, with the last digit readings of the multimeter. For example, if the multimeter has five digits (e.g. 120.53 ohm), then the fifth digit is estimated and the error in resistance measurement could be as high as ±0.01 ohm. Combining the possible variations in "Sg and R/R", the maximum and minimum boundaries on the associated strain measurements can be estimated as follows:
or, we can represent the strain data as:
= 260 ± 6.33
Material Information – Equivalent Engineering Constants:
Composite Beam:AS4/3501-6 Graphite Epoxy
Lay-up geometry:[0/90/0/0/0/0/90/0/0/0/0]s
Apparent Elastic Modulus:Ex = 16.71 Msi
Poisson's Ratio:xy = 0.0925
Aluminum Beam:7075-T6
Elastic Modulus:Ex = 10.4 Msi
Poisson's Ratio:xy = 0.3
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