The Mathematical model of Shear Beam Load Cell and the best orientation of Strain Gauge

The Mathematical model of Shear Beam Load Cell and the best orientation of Strain Gauge

【Summary】This artice introduces the structure and characteristics of shear beam load cells that are widely used in China, analyzes the mechanical characteristics and boundary influence of the elastic elements of the shear beam, and establishes the elasticity of the rectangular cross-section shear beam with the center of fixed support at both ends under concentrated load. The mathematical model of the element analyzes the stress and strain fields, and obtains the principal strain calculation formula, so as to find the best paste orientation of the resistance strain gauge.

【Key Word】Load cell, Resistance strain gauge, Shear beam load cell, load cell block, Mechanical properties, Strain field

1:Overview

Since the early 1970s, the load cell that uses the tension and compression generated by the shear stress to form dual principal stresses came out. It has completely overcome the inherent shortcomings of the traditional normal stress load cell, and has formed a new development trend. Every year, a variety of structural types of shear load cells are available. Compared with the traditional cylindrical load cell, cylindrical load cell, ring load cell, and curved beam load cell structure of the normal stress load cell, the shear beam load cell has the following characteristics:

(1) The strain zone of the elastic element of the shear beam has no area effect and Poisson effect, the cross-sectional area does not change when loaded, and the inherent linearity is good;

(2) The shear force is constant along the length of the strain beam, so the load cell output is not sensitive to changes in the load point;

(3) The strain beam is an elastic element subjected to bending stress. The stress unit at the neutral axis is in a pure shear state. The absolute values of the principal strains of tension and compression are basically the same, and the non-linear error of the Wheatstone bridge is small. ；

(4) Low profile, good stability, coupled with the symmetrical Wheatstone full bridge circuit composed of resistance strain gauges for measuring positive and negative strains, it has strong anti-eccentricity and lateral load capability;

(5) When the shear strain beam is subjected to both tensile and compressive loads, the sensitivity of the load cell is consistent and can be used as a "tension and compression" load cell;

(6) The shear strain beam is a double-ended fixed-supported beam, and the load introduction point and the two supporting points form a self-balanced force system. There is no need to force the moment balance to be maintained. The boundary support design is greatly simplified;

(7) Between the load introduction points on both sides of the shear strain beam and the two support points, four blind holes are symmetrically processed to form an I-shaped cross-section shear beam, which has more ideal technical and application performance;

(8) The load is introduced by the contact between the steel ball and the bowl, so that the load cell only feels the axial load in the vertical direction but not the transverse load and the transverse torque, which ensures the accuracy of weighing.

The elastic elements of the shear beam load cell are divided into square section, rectangular section, I-shaped section, round I-shaped section, rectangular hollow section and circular hollow section according to the different cross-sectional shapes of the beam strain zone. , And its rated load is mostly 0.5 to 50t.

When selecting the elastic element structure of the load cell, its inherent linearity is very important. The main factors that cause nonlinear errors are elastic components, resistance strain gauges and Wheatstone bridge circuits, and the smaller factor is the manufacturing process. As long as the resistance strain gauge is selected reasonably, its nonlinear error is usually very small; the nonlinear error of the Wheatstone bridge circuit depends on the composition and connection method of the working bridge arm and the strain value of each resistance strain gauge; the nonlinearity of the elastic element The decisive role is the elastic element and the structure of the strain zone. These are issues that the designer must seriously consider. The nonlinear error may come from:

(1) The area effect of the strain zone;

(2) Large deformation and high degree of strain;

(3) Poisson's ratio effect causes the nonlinear relationship between load and stress (P-σ);

(4) Boundary conditions and load splitting of the sealing diaphragm;

(5) The nonlinearity of the material properties of the elastic element, that is, the stress-strain relationship is not completely linear.

These factors should be carefully considered when determining the structure and boundary conditions of the elastic element, selecting metal materials and heat treatment specifications, and determining the structural size and working characteristics of the resistance strain gauge, and the implementation of optimized combination.

2. Mathematical model and stress analysis of elastic element of rectangular section shear beam

The structure of the elastic element of the shear beam load cell is based on a rectangular cross-section beam with fixed ends at both ends and a concentrated load P at the center. The mechanical model is shown in Figure 1, with four-piece and eight-piece resistors. The location of the strain gauge and the bridge circuit are shown in Figure 2 (a) and (b).

Fig. 1 Mechanical model and pasting position of resistance strain gauge

Figure 2 Bridge circuit with four (a) and eight (b) resistance strain gauges pasted

In order to make the shear beam with the center of fixed support at both ends bearing concentrated load only produce shear stress, the following two conditions must be met:

(1) The ratio of the effective distance between the load introduction part of the shear strain beam to the fixed part of the root and the height of the beam section should be as small as possible;

(2) The fixed form of the two ends of the shear strain beam should be a rigid structure. When subjected to an external load, no obvious deformation occurs, that is, the joints at the two ends of the strain beam cannot rotate freely, so the return at A and B The corner is zero, that is

θ_A=θ_B=0

But under the action of shear and bending moment, the strain beam deforms, so its deflection f is not zero.

The double-shear resistance strain gage pasted on the X-Z surface of the elastic element of the rectangular cross-section shear strain beam is in a two-dimensional stress state. The bidirectional stress tensor can be expressed in matrix form:

Where σXZ=σZX is the shear stress on the cross section, σXX is the bending stress, and σZZ is the normal stress.

According to the theoretical calculation assumptions of commonly used beams in engineering, the normal stress σZZ is negligible, and the following equations can prove that σXX and σXZ are related to the bending moment M and the shear force Q:

Where: M-bending moment of shear strain beam

In the formula: J—the moment of inertia of the section

Q—Shear force  

Thus

and

The strain field and the stress field are connected through the elastic modulus E and Poisson's ratio μ of the elastic element material, so the strain field corresponding to the above stress field can be expressed as:

The ideal position of the resistance strain gage in the elastic element should be at the position where the strain amount is the largest and the strain change in the coverage of the strain gage sensitive grid is the smallest, that is the part. Because the strain field in this part is the maximum pure shear stress field, and because M=0 and Z=0, the bending strain is also zero. According to elastic mechanics, in the state of pure shear stress, the direction of the maximum shear stress is consistent with the direction of the neutral axis, and the two principal stress directions and the maximum shear stress direction form an angle of 45⁰ with each other, so the paste resistance strain timer is sensitive The grid center line must be 45⁰ with the neutral axis of the elastic element. The Wheatstone bridge circuit is formed according to the principle that the strain direction of the opposite bridge arm resistance strain gauge is the same, and the strain direction of the adjacent bridge arm resistance strain gauge is opposite.

The principal strain calculation formula is derived as

The sensitivity calculation formula is

From the mathematical model obtained from the above analysis, it is not difficult to conclude that on one or both sides of the rectangular cross-section double-ended fixed-supported beam, paste shear-type resistance strain gauges in a certain orientation to form a Wheatstone full bridge circuit, which can be measured The value is proportional to the output signal of the shear stress (actually the principal stress caused by the shear stress in tension and compression).

However, the resistance strain gauge shown in Figure 1 is attached to one side of the beam, such as the front TA、CA、TB、CB,The bridge circuit is shown in Figure 2(a). The disadvantage of this patch group bridge method is that it is sensitive to lateral load, and additional errors will be introduced when the load introduction point appears eccentrically loaded. The improved method is to use 8 pieces of resistance strain gauges to form a Wheatstone bridge circuit, that is, to paste four pieces of resistance strain gauges on the reverse side of the double-ended fixed beam ,The bridge circuit is shown in Fig. 2 (b). The characteristics of this method are as follows: each arm of the bridge is composed of two pieces of resistance strain gauges in the same strain direction on both sides of the beam in series. When the strain beam is subjected to additional load, the resistance strain gauge can sense the strain of the same size and opposite direction, which has no effect on the output of the bridge, and improves the ability of the load cell to resist eccentric and lateral load.

3:Conclusion

In this paper, the effect of concentrated load P on the center of fixed support at both ends is studied The calculation formula of stress and strain of rectangular cross-section beam (actually a pair of spokes of wheel spoke load cell) can be applied to the design of elastic elements of square section I-beam, circular section I-beam, square hollow section and rectangular hollow section shear beam, so as to calculate the strain zone scale of elastic element It determines the matching mode of each resistance strain gauge in the full bridge circuit. In fact, the scope of mathematical model analysis in the design of load cell is relatively wide. From the simple principle of classical mechanics to the finite element computer model used to determine the optimal size and boundary, it is still the main design and calculation method for the weighing sensor development and production enterprises in China. In recent years, the rapid development of three-dimensional digital design and manufacturing technology will promote the automation and intelligent development and production of weighing sensors, which should also attract extensive attention.