Abstract:The cantilever shear beam type load cell has become an important basic component of electronic scales due to its simple and compact structure, small height, low center of gravity, no end effect, strong lateral load resistance, and convenient installation and debugging. This paper introduces the mechanical characteristics of square section and circular section cantilever shear I-beam type load cells, and the calculation of shear stress, strain, principal stress and strain in the strain zone; Evaluation of lateral load resistance, calculation and analysis of lateral load resistance. And a brief introduction was given to the many factors that generate lateral load effects, in order to help applications further understand the metrological performance and application characteristics of such load cells.

** 【Keywords】:**load cell; Shear beam; Lateral load; Resistance to lateral loads; Lateral load bearing capacity

**1:Mechanical characteristics of cantilever shear beam type ****load cell****s**

The cantilever shear beam type load cell can be divided into two structural types according to the cross section shape of the shear beam: square section I-beam and circular section I-beam. The structural diagram and mechanical model of its elastic elements are shown in Figure 1 and Figure 2.

Figure1: Square I-beam type elastic element and mechanical model

Fig.2:Circular I-beam type elastic element and mechanical model

The rated range of the cantilever shear beam type load cell is usually 100kg~30t. Round section I-beam structure, mostly used for small range load cells, adopts blind hole sealing and bellows welding for double sealing; The square section I-beam structure is used for large scale load cells, and the blind hole filling and circular diaphragm welding are used for double sealing. Both structures are on the center line of the I-beam web in the blind hole, and two pieces of double shear resistance strain gauges are symmetrically pasted. It is known from Linear elasticity that the resistance strain gauge pasted on the web is in the state of two-dimensional stress, and its stress tensor can be expressed in matrix form:

In the equation: σ Normal stress in the x-direction;

σ Y-normal stress in the y-direction;

τ Xy – Shear stress on the cross section, x represents τ The action plane of xy is perpendicular to the x-axis, where y represents τ Xy along the y-axis direction;

τ Yx – Shear stress on the cross section, y represents τ The action plane of yx is perpendicular to the y-axis, where x represents τ Yx

Along the x-axis direction.

The I-shaped section of an elastic strain beam can be regarded as composed of several rectangles, using the Juravsky formula,have

*Q**xy*__ __*S**z*__ __

*τ** xy* =

*J*

*z*

*b*When y=0, Shear stress τ If xy is the largest, then

*τ** xy* (

*) =*

*wax*

*J*

*z**(3)*

*b*In the formula: Qxy – shear force on any section perpendicular to the y-axis;

Sz – the static moment of the part on the cross-section outside the horizontal line y from the neutral axis z to the neutral axis z; Jx – moment of inertia of the transverse plane towards the neutral axis;

B – width of I-beam web.

The strain field corresponding to the above stress field is:

In the equation: ε X ε Y – normal strain;

γ Xy – shear strain;

E – elastic modulus of I-beam metal materials;

μ— Poisson’s ratio of I-beam metal materials.

After analyzing the stress state of I-beam web under pure shear under Plane stress state, it can be concluded that the length change along the neutral axis of the beam in the direction of 45 ° is the main force under pure shear stress state

In the stress direction, the principal stress is equal to the maximum Shear stress; The principal strain is equal to half of the maximum shear strain, i.e

According to the principle that the strain direction of the relative bridge arm is the same and the strain direction of the adjacent bridge arm is opposite, the four piece resistance strain gauges form the Wheatstone Bridge circuit circuit, and carry out accurate circuit compensation and adjustment to complete the weighing and metering task.

**2:Calculation of stress and strain under forward load**

2.1: Square cut I-shaped cross-section elastic element

See Figure 1 for the I-shaped section and size parameters of the elastic element. It is composed of three rectangles. In order to obtain the Shear stress and shear strain, first calculate the static moment and inertia moment of the section.

The shear force Q is a constant along the length direction of the beam, i.e. Q=Py.

The static moment of the part outside the horizontal line y from the neutral axis z on the cross-section towards the neutral axis z is:

* S* =

*B*

*H*

__−__

*(*

*h*

*+*

*h*

*H*

__−__

*) +*

*h*

*b*

*h*

*=*

*h*

*B*__(__

*H*

__2__

__−__

*h*__2__

__)__

__+__

*bh*__2__

The moment of inertia of the transverse plane towards the neutral axis is:

Substitute the shear force, static moment and inertia moment into formula (2) to obtain the Shear stress and shear strain on the I-beam web:

The principal strain in the direction of 45 ° to the neutral axis is:

The circular cross-section and size parameters of the elastic element are shown in Figure 2. It consists of two arches and a rectangle in the middle, and its static moment and inertia moment are the sum of the static moments and inertia moments of the three sections of the two shapes. A circular arched section formed by two straight lines in the 45 ° direction between the center of the circle and the z-axis can be used to calculate the area and centroid of the arch height, chord length, and arc length to obtain the moment of rest and moment of inertia. After calculation, it can be concluded that the static moment of the circular I-shaped section is:

Circular I-shaped section static moment:

Moment of inertia of a circular I-section:

The shear force Q is a constant along the length direction of the beam, i.e. Q=Py.

Substitute the shear force, static moment and moment of inertia into formula (2) to get the Shear stress on the web of the round section I-beam and the shear strain:

The principal strain in the direction of 45 ° to the neutral axis is:

3:Evaluation of lateral load resistance of cantilever shear beam type load cells

As an important basic component of electronic scales, strain type weighing load cells are subjected to isotropic loads that can be summarized as six force and torque components, namely, the forces in three directions are Px, Py, Pz, and the torque in three directions

For Mx, My, and Mz, the force distribution of their elastic components is shown in Figure 3.

Figure 3 Schematic diagram of the force acting on the elastic element of a cantilever shear beam

For weighing load cells, except for the forward load that needs to be measured, all other components can be considered as lateral loads. The forward load of the cantilever shear beam type weighing load cell is Py, while the remaining components Px, Pz, Mx, My, Mz are all lateral loads. In terms of lateral loads, discussing the effects of Pz and Px is of the most practical significance, as z and x lateral loads are the main mechanical interference quantities of cantilever shear beam type weighing load cells. In the 1980s, the former load cell Research Laboratory of the Chinese Academy of Metrology proposed through experimental research to evaluate the lateral load capacity of cantilever shear beam type weighing load cells using lateral load bearing capacity. The so-called lateral load carrying capacity refers to the lateral load carrying capacity of the load cell when the maximum stress value on a specific section of the elastic element under a certain value of lateral load is equal to the corresponding maximum stress values under the forward rated load. The ratio of the lateral load value to the forward rated load value is called the lateral load carrying capacity of the load cell. It reflects the difference between the stress-strain value of the weighing load cell under lateral load and the stress-strain value under forward load. It describes the capacity of the load cell to withstand lateral loads under normal (non overload) and safe (non destructive) conditions.

4:Analysis of the lateral load resistance ability of cantilever shear beam type weighing load cells

4.1. Square I-beam type load cell

(1) Z-direction load Pz bearing capacity

The cantilever shear beam type weighing load cell produces lateral bending of the elastic strain beam under the action of the z-direction load Pz. To ensure accurate evaluation of lateral load resistance, three sections C, B, and A in the elastic strain beam are taken for calculation. The specific cross-sectional positions of the elastic elements are shown in Figure 4, and the mechanical model under the z-direction load Pz is shown in Figure 5.

Figure 4 Specific section on elastic element of square section I-beam

Fig. 5 Mechanical Model of I-beam under z-direction Load cells

① Fixed end C-C section of cantilever shear beam

The maximum bending stress on section C-C under the action of Pz is:

In the formula: Wcy – section C – bending section modulus of section C for the y-axis;

OC – The distance between section O-O and section C-C.

Similarly, under the action of load Py, the maximum bending stress on section C-C is:

order σ Cz max= σ Cy max can be used to obtain the z-direction load Pz bearing capacity calculated according to the specific cross-section C-C:

② Elastic strain beam and transition section B-B near fixed end

By using the same method as the C-C section, it can be obtained that

③ Elastic strain beam strain zone A-A section

First, calculate the maximum Shear stress with equation (2)

According to the mechanical model of the square section I-beam elastic element under Pz load in Figure 5, the static moment and inertia moment of the section can be obtained

Substitute the static moment and moment of inertia of the section into equation (16) to obtain

Simultaneous equations (6), (17), and τ Xzmax= τ Xymax

The maximum principal stress in the strain zone

There are also

(2) X direction load Px bearing capacity

The mechanical model of the strain shear beam under the x-direction load Px is shown in Figure 6.

Fig. 6 Mechanical Model of I-beam under X-direction Load

Under the action of Pz, the strain beam undergoes axial tensile or compressive deformation, and only the tensile and compressive normal stresses need to be calculated. The normal stress on the cross-section of the strain zone is:

The load Px bearing rate in the x direction can be obtained by comparing the estimated value of the average Shear stress on the web of the I-beam under the positive load Py with equation (22). Since the shear force is a constant along the length of the strain beam, the Shear stress is basically evenly distributed, and more than 95% of the Shear stress on the section is borne by the web, so it is easy to get the estimated value of the average Shear stress on the web, that is

Because of the tensile allowable stress of the elastic component material[ σ] And shear allowable stress[ τ] There is an approximate relationship as follows:

In the formula, f ≈ 0.6~0.8.

Simultaneous equations (22), (23), and (24), with f σ Ax= τ Xy, get it

Formula (25) can be used to calculate the lateral load bearing rate of the square section I-beam load cell.

**Round I-beam type load cell**

(1) Z-direction load Pz bearing capacity

Since the section of the fixed end of the round section I-beam load cell and the section of the strain zone are the same circular elasticity Element, therefore, under the action of loads Px and Py, the stress state on the fixed end section is equivalent, so there is no need to Three specific sections of the above square section I-beam are selected for calculation. Obviously, its z-direction load bearing rate is 100%. For the cross section at the center of the strain zone, since the bending moment is zero, it is not necessary to calculate the normal stress on the section, only the maximum Shear stress can be calculated. Mechanics of elastic elements of circular section I-beam under z-direction load Pz

The model is shown in Figure 7.

Fig. 7 Mechanical Model of Circular I-beam under z-direction Load

Similarly, using equation (2), there are

The static moment of the cross-section on the y-axis is:

(2) X direction load Px bearing capacity

Under the action of load Px, the circular I-beam is subject to axial tension and compression deformation, so only the normal stress needs to be calculated

The mechanical model of force is shown in Figure 8.

Fig. 8 Mechanical Model of Circular I-beam under X-direction Load

**Calculation of load bearing capacity of Pz and Px in z-direction**

Taking the square section I-beam with a range of 5000kg and 10000kg and the round section I-beam with a range of 2000kg as examples, the load bearing rate of Pz in z direction and Px in x direction is calculated.

(1) 5000kg square section I-beam load cell

The elastic element of the square section cantilever shear beam with a rated range of 5000kg, and the sectional dimension parameters of the I-beam in the strain zone are:

Py=5000kg H=40mm h=30mm B=48mm b=9mm

Substitute the above I-shaped section size parameters into equations (6), (7), and (8) to obtain

τ Xymax=17.211kg/mm2

γ Xymax=2131 × 10-6

ε 45 ° max=± 1065 × 10-6

Pz load bearing capacity in z-direction:

Substitute the size parameters of the I-shaped section into equation (21) to obtain

Px load bearing capacity in x direction:

Substitute the size parameters of the I-shaped section into equation (25) and take f=0.7 to obtain

(2) 10000kg square section I-beam load cell

The elastic element of the square section cantilever shear beam with a rated range of 10000kg, and the sectional dimension parameters of the I-beam in the strain zone are:

Py=10000kg H=60mm h=44mm B=70mm b=12mm

Substitute the above I-shaped section size parameters into equations (6), (7), and (8) to obtain

τ Xymax=17.156kg/mm2 γ Xymax=2124 × 10-6 ε 45 ° max=± 1062 × 10-6

Pz load bearing capacity in z-direction:

Substitute the size parameters of the I-shaped section into equation (21) to obtain

Px load bearing capacity in x direction:

Substitute the size parameters of the I-shaped section into equation (25) and take f=0.7 to obtain

(2):10000kg square section I-beam weighing load cell

The elastic element of the square section cantilever shear beam with a rated range of 10000kg, and the sectional dimension parameters of the I-beam in the strain zone are:

Py=10000kg H=60mm h=44mm B=70mm b=12mm

Substitute the above I-shaped section size parameters into equations (6), (7), and (8) to obtain

τ Xymax=17.156kg/mm2 γ Xymax=2124 × 10-6 ε 45 ° max=± 1062 × 10-6

Pz load bearing capacity in z-direction:

Substitute the size parameters of the I-shaped section into equation (21) to obtain

Through theoretical calculation and experimental measurement, it is proved that the lateral load bearing rate of the square section cantilever shear I-beam type load cell can reach as high as 50%~100% in z-direction; 400%~600% in x-direction; The lateral load bearing rate of the circular cross-section cantilever shear I-beam type load cell is lower: 40%~60% in z-direction; 300%~500% in x direction. The ability to resist lateral loads is significantly superior to other structural load cells, without the need for any lateral load suppression measures. It is worth noting that the installation stiffness of the root of the cantilever shear beam type weighing load cell must be strictly ensured. The specific requirements are: the fastening screws should have sufficient strength and vertically press the cantilever shear beam weighing load cell to form a rigid fixation as much as possible; When using multiple screws for fastening, it is necessary to apply a pressure equalizing pad to eliminate local stress effects, or adjust the pre tightening force of each screw one by one to make it as consistent as possible; The root fixing device should have the function of preventing lateral displacement or rotation of the cantilever shear beam; The installed cantilever shear beam weighing load cell must ensure that its forward load coincides with the loading axis.

**5:Analysis of Factors Influencing Lateral Load of Cantilever Shear Beam Type Weighing load cell**

Since the double shear resistance strain gauge is symmetrically pasted along the neutral axis of the I-beam web, and the sensitive grid is 45 ° to the neutral axis, the resistance strain gauges sensing positive and negative strains on both sides of the I-beam web are the opposite arms of the Wheatstone bridge respectively. Under ideal conditions, the load cell is used for lateral load, and the bridge output is zero.

Under the z-direction load Pz, the resistance strain gauge sensing positive and negative strain values on both sides of the I-beam web is subject to tensile stress on one side and compressive stress on the other. Due to the same amplitude of tensile and compressive stress, the output of the bridge remains unchanged.

Under the x-direction load Px, the resistance strain gauges sensing positive and negative strain values on both sides of the I-beam web are subject to tensile or compressive stress at the same time. Each resistance strain gauge is subject to the same positive or negative strain, and the output of the bridge is unchanged.

In fact, there is a certain degree of deviation and various factors affecting the manufacturing and application of cantilever shear beam type weighing load cells. When the cantilever shear beam type weighing load cell is subjected to lateral load, there will still be a small output signal, and its value can be obtained through calibration. These deviations and influencing factors can be summarized as follows: dimensions and geometric tolerances of elastic components; Inhomogeneity of elastic component materials; Heterogeneity of heat treatment process; The connection, frictional contact, fastening or any non integral state influence of elastic components and their accessories; The deviation between the actual neutral plane and the ideal neutral plane of the elastic element when subjected to lateral bending; The dispersion and orientation deviation of the resistance values of each resistance strain gauge; Compensate and adjust the temperature gradient effect on the resistor area using a resistance strain gauge or various circuits; The inconsistency between the curing and post curing states of strain adhesive.

**6:Conclusion**

From the above calculation and analysis, it is not difficult to find that the I-shaped section web in the strain zone of the cantilever shear beam type load cell is in the state of Plane stress, and the principal stress plane with zero Shear stress and the maximum Shear stress plane with zero principal stress multiply each other by an angle of 45 °. To improve the ability of cantilever shear beam type weighing load cells to resist lateral loads

To verify the accuracy of resistance strain gauge positioning, double shear resistance strain gauges should be used as much as possible to ensure that the sensitive gate is at a strict 45 ° angle with the neutral axis of the elastic element. The calculation formulas of Shear stress, principal stress and principal strain of I-section shear beams derived from pure bending theory are basically consistent with the exact solution of Linear elasticity, which can be used in practical design. The former load cell Research Office of the Chinese Academy of Metrology proposed that it is feasible to evaluate the lateral load capacity of a cantilever shear beam type load cell using lateral load bearing capacity, which has reference value for the design and calculation of load cells and the evaluation of lateral load capacity.