Phil. U., M Eng Dep., Measurements, Chap#12 This chapter considers force and torque measuring methods and relates it to basic strain measurement. Force is represented mathematically as a vector with a point of application. According to Newton’s first law: F = ma Torque (T) is represented as a moment vector by the cross product of a force and a radius (r) vector. Thus: T = r  F = r F sin  Force, Torque, and Strain Measurements

Phil. U., M Eng Dep., Measurements, Chap#12 Elastic elements are frequently used to furnish an indication of the magnitude of an applied force through a displacement measurement. The spring is an example of this type of force-displacement transducer. In this case: F = ky where k is the spring constant and y is the displacement from the equilibrium position. Elastic Elements for Force Measurements

Phil. U., M Eng Dep., Measurements, Chap#12 Another example is the simple bar shown in figure 10.3 in which: F = (AE/L) y where E is the modulus of elasticity for the bar material Moreover, the cantilever beam shown in figure 10.4 can be used to measure the force as: F = (3EI/L )y where I (= bh /12) is the moment of inertia of the beam about the centroidal axis in the direction of deflection. Elastic Elements for Force Measurements 3 3

Phil. U., M Eng Dep., Measurements, Chap#12 Typically, the above three example are used coupled with displacement measuring device such as: the differential transformer, the capacitance transducer, etc. Also, the surface strain, of the elastic elements in the above three examples, may be taken as an indication of the impressed force. Elastic Elements for Force Measurements

Phil. U., M Eng Dep., Measurements, Chap#12 Torque, or moment, may be measured by observing the angular deformation of a bar or hollow cylinder, as shown in Figure the moment is given as: M =  G [(r o - r i )/2L]  where  is the angular defecktion in radian, and G is the shear modulus of elasticity as: G = E/2(1+  )  is the Poisson’s ratio. Torque Measurements 44

Phil. U., M Eng Dep., Measurements, Chap#12 Also, the induced strain, taken from the stain gages attached at 45 , can be used to measure the moment as:  45 =  Mr o /  G(r o - r i ) Prony brake is one of the oldest devices for torque and power measurement (figure 10.8). The torque (T) needed to be measured is balanced by a force (F) that has an arm L as: T = FL Also, the power dissipated in the brake is given by: P = 2  TN/33,000 hp Torque Measurements 44

Phil. U., M Eng Dep., Measurements, Chap#12 Consider figure If the load T is applied to the bar such that the stress does not exceed the elastic limit of the material, the axial strain is given by:  = (T/A)/E =  a /E and the unit axial strain (axial deformation per unit area) is defined as:  a = dL/L The ratio of the unit strain in the transverse direction to the unit strain in the axial direction is defined as Poisson’s ratio as:  = -  t /  a = - (dD/D)/(dL/L) Strain measurement

Phil. U., M Eng Dep., Measurements, Chap#12 A simple method of strain measurement is to place some type of grid marking on the surface of the workpiece under zero-load condition and then to measure the deformation of this grid when the specimen is subjected to a load. Strain measurement

Phil. U., M Eng Dep., Measurements, Chap#12 On the other hand, the electrical-resistance strain gage is the most widely used device for strain measurement. Its operation is based on the principle that the electrical resistance of a conductor changes when it is subjected to mechanical deformation. Typically, an electrical conductor is bonded to the specimen under no-load conditions. A load is then applied, which produces a deformation in the specimen and the resistance element. This deformation is an indicated through a measurement of the change in resistance of the element. The calculation procedure described below. Strain measurement

Phil. U., M Eng Dep., Measurements, Chap#12 The resistance of the conductor is: R =  (L/A) where: L: length A: cross-sectional area  : resistivity of the material By differentiating the above equation, we get: dR/R = d  /  + dL/L – dA/A Also: dA/A = 2 dD/D Strain measurement

Phil. U., M Eng Dep., Measurements, Chap#12 By using the definition of  a and , we have: dR/R =  a (1 + 2  ) + d  /  The gage factor, F, is defined as: F = (dR/R)/  a therefore: F =  + (1/  a )(d  /  ) Thus, we may express the local strain in terms of the F, R, and  R, as:  = (1/F)(  R/R) Strain measurement

Phil. U., M Eng Dep., Measurements, Chap#12 The value of F and R are usually specified by the manufacturer so that the user only needs to measure  R in order to determine the local strain (  ). For most gages, F is constant over a rather wide range of strains. Also, it is the same for both compressive and tensile strains. Three common types of resistance strain gages are shown in figure Strain measurement