1. Introduction
When establishing power transmission and communication towers on soft soil near rivers and lakes, a flexible foundation slab is a common foundation form [
1,
2]. The form of foundation reaction force distribution greatly influences the internal force response of the foundation base plate. In engineering, a concentrated load is a common form of load used between the foundation and building, and the contact stress distribution between the thin slab and substrate with different width-to-height ratios is usually assumed to be linear [
3,
4,
5]. However, actual contact stresses are generally nonlinearly distributed and can be saddle-shaped, parabolic, or anti-parabolic, which is mainly determined by the substrate bed factor or soil type and the material properties of the sheet [
6]. The three most common nonlinear contact stress distributions described in the generalized Winkel foundation model are saddle-shaped, parabolic, and anti-parabolic [
7,
8]. Scholars have conducted studies on the contact stresses of foundation footings under concentrated loads, and Wang et al. [
9] found that the measured field test values of foundation contact stress distribution present a convex parabolic form with small, middle, and large edges and the distribution of reaction forces does not vary much for different soils. Similarly, a large number of field measurements of foundation contact stress distribution in engineering practice show a similar parabolic variation law [
10,
11,
12]. To account for this nonlinear variation law, Wang et al. [
13] used a parabolic surface function to describe the foundation contact stress distribution and used K to control the adjustment function, where K > 1 for sandy foundations, K < 1 for clay foundations, and K = 1 for uniform distribution of reaction forces. Let the expression for the contact stress at a specific location in the foundation be represented by P
x=x0 = K · P
0, where K is the control adjustment coefficient function. The relationship of K can be determined through experimental simulations. However, this parabolic function does not accurately reflect the nonlinear law of the stresses in the thin plate and the substrate, and a more accurate functional model is needed to characterize it.
The Technical Provisions for the Design of Power Transmission Line Foundations (DL/T5219-2005) [
14] and the Code for the Design of Building Foundations (GB50007-2002) [
15] assume linear contact stress distribution when the aspect ratio of foundation steps is less than or equal to 2.5 to simplify calculations. However, when the step width-to-height ratio exceeds 2.5, no calculation method for the internal force of the thin plate is provided in the specification. Determining the distribution of contact stresses between thin slabs and footings for calculating internal forces in foundation footings when the aspect ratio is greater than 2.5 is crucial for optimizing thin slab designs and reducing engineering costs [
16]. The problem of contact stress distribution between the thin plate and the substrate has been studied by many scholars, and various methods have been developed to solve this problem. Solving the contact stress distribution of a four-sided free plate on an elastic substrate is important for calculating the analytical solution of displacement and internal forces in the plate [
17]. Numerous scholars at home and abroad have studied the methods for solving the bending problem of a rectangular thin plate with four free sides on an elastic foundation, including the numerical method, analytical method, and semi-analytical method with semi-numerical values [
18,
19,
20]. Each solution method must satisfy not only the fourth-order differential bending equation for the plate [
21] but also the geometric and internal force boundary conditions on the four free edges [
22]. However, the geometric and internal force boundary conditions related to the form of distribution of contact stresses [
23,
24] are essential for the bending equations [
25]. Therefore, accurately describing the distribution form of contact stresses through model studies is necessary. In engineering and scientific fields, measuring contact stress is vital for evaluating material performance, designing new products, and optimizing production processes.
Many mechanical systems’ durability and performance are significantly impacted by contact stress, a crucial factor. The deformation, wear, and failure of these parts can all be significantly influenced by the distribution of stresses that take place at the interface between two contacting bodies. Therefore, for designing and optimizing mechanical systems, it is crucial to comprehend how contact stress arises and how it can be measured and controlled. Guan investigated the creation of tri-axial stress measuring and sensing technology for tire-pavement contact surfaces [
26] and described a cutting-edge method for determining the tri-axial stress distribution at the tire–pavement contact surface using a sensor array [
27].
First of all, it has been hard to quantify contact stress. Secondly, the distribution of contact stress for thin plates is primarily linear and parabolic, which is relatively conservative for the input conditions of internal force calculation. In this paper, we introduce an innovative type of nonlinear distribution of contact stress called the exponential function model. In order to determine contact stresses in the substrate of thin plates with different aspect ratios, this paper performs film pressure measurement tests. This method takes into account the problems posed by the aspect ratio of the base plate as well as the phenomenon of nonlinear stress distribution in thin plates and substrates. We investigate how aspect ratio affects contact stress distribution and propose a nonlinear distribution model of contact stresses in the substrate of thin plates under concentrated loading. The results of our study have important theoretical and engineering value by providing more precise external boundary conditions for designing and calculating the strength and stiffness of thin plates with different aspect ratios.
2. Experimental Programs
This section first introduces the component composition and working principle of the thin film pressure distribution test system, as well as the data acquisition method and process. Then, an experimental plan is developed to obtain the real contact stress distribution of the thin plate substrate under graded loading by changing the aspect ratio of the thin plate used in the test and based on the test of the thin film pressure distribution measurement system.
2.1. Introduction to Thin Film Pressure Distribution Measurement System
The film pressure distribution measurement system is divided into a hardware part and a software part [
28]. The hardware part consists of a polyester film sensor, an Analog to Digital Converter (A/D) conversion circuit for the sensor (handle), and software based on a PC using C language to complete an intelligent display system on a microcontroller.
In the test device,
Figure 1a shows a 5076-P1-35611DT1-50 type thin film pressure sensor. The area of the thin film pressure sensor is 83.8 mm wide × 83.8 mm high. The pressure sensor consists of two very thin polyester films, as shown in
Figure 1b, where the inner surface of one film is laid with a number of band conductors arranged in rows, and the inner surface of the other film is laid with a number of band conductors arranged in columns conductors. The intersection of the rows and columns is the pressure sensing unit, which has a total of 44 rows and 44 columns, forming a uniform distribution of 1936 measurement points. The conductors are made of conductive material with a certain width, and the distance between rows can be changed according to requirements. Therefore, polyester film sensors are available in a variety of sizes and shapes and can be used in different contact conditions. The outer surface of the conductor in the sensing area is coated with a special pressure-sensitive semiconductor material. When two films are combined into one, the intersection of a large number of transverse and longitudinal conductors forms an array of stress-sensing points. When an external force is applied to the sensing unit, the resistance value of the conductor changes linearly with the change in the external force, thus reflecting the stress value at the sensing point. The resistance value is maximum when the pressure is zero and decreases as the pressure increases. This linear change in voltage can reflect the magnitude and distribution of the pressure between the two contact surfaces. The small thickness and high flexibility of the thin-film pressure sensor have no influence on the contact environment, so it can directly measure the real contact stress distribution under different conditions and has the characteristics of high measurement accuracy.
The connection diagram between the conductive handle and the sensor is shown in
Figure 1a. The connection part in the handle has a conductive interface that can transmit the electrical signal from the sensor film to the computer.
The handle is the connection device between the computer software and the sensor and is also the A/D converter. When an external force is applied to the sensing point, the resistance value of the semiconductor changes proportionally to the change in the external force. This change in electrical signal is transmitted to the control circuit on the handle, which is then input to the software and displayed on the computer screen to reflect the stress value and its distribution at the sensing point.
The width and spacing of the conductors within the sensor determine the number of sensing points per unit area and the spatial resolution, which can be determined as needed. The value of the spatial resolution on the sensor area can meet a variety of measurement requirements. The sensors can be fabricated in various dimensions and configurations, exhibiting a stress measurement capacity ranging from 0.1 to 175 MPa. The unique feature of this grid sensor is that the sensing area is completely insulated from the non-sensing area. Knowing the spatial size distribution of the sensing points, the force applied to the sensor on a certain area can be intelligently digitized and displayed with Light Emitting Diode (LED) or Liquid Crystal Display (LCD) using the software.
The software part of the thin film pressure distribution measurement system is used to process the measured 2D matrix voltage data and convert it into 2D graphics or 3D graphics display. When a calibration file is not added, the initial values are displayed without units. When the calibration file is added, the initial values are converted into force values and divided into 17 color levels to display the measured stress distribution and the magnitude of the combined force values, indicated from small to large, in a blue-to-pink gradient 2D display graphic. Each pixel point corresponds to the intersection of the band conductors in the film, where the color reflects the measured data at the level set by the corresponding calibration file. In addition, the calibration file is different, and the same force value will be displayed in different colors.
2.2. Experimental Program
Square glass sheets of the same width and different thicknesses are used as test specimens, and the widths of the specimens are all 75 mm, and the thicknesses are 4 mm, 5 mm, 8 mm, 10 mm, and 12 mm, respectively. Finally, the test molds with aspect ratios of 18.75, 15, 9.375, 7.5, and 6.25 are obtained, as shown in
Figure 2a,b.
The actual stress distribution of the square plate when subjected to the concentrated load is observed by applying a graded concentrated load on the center of the square glass plate and placing a thin-film pressure transducer at the bottom. The width of the glass plate is kept constant in the test, and the thickness of the glass is varied to achieve a change in the aspect ratio. A schematic diagram of the test assembly is shown in
Figure 3 below. The overall picture of the experimental assembly can be visualized according to
Figure 3.
The loading device for the test is a microcomputer-controlled electronic universal testing machine, as shown in
Figure 4. The loading speed should not be too fast to avoid irregular discrete damage by too rapid force, and 10 N/s is used. Glass plates with different aspect ratios are placed on the upper part of the pressure film and loaded by 250 N, 500 N, 750 N, 1000 N, and 1250 N in a graded manner. The film pressure distribution system is used to measure the real contact stress distribution of the thin plate substrate, and finally, the system-recorded data is displayed.
4. Modeling of Exponential Function Distribution of Contact Stress under Concentrated Load
An analytical model of the contact stress in the base of the thin plate is established, as shown in
Figure 7, where a is the side length of the square thin plate,
is the thickness of the square thin plate, and the aspect ratio is defined as
. A concentrated load
F is applied at point
O (
a/2,
a/2) in the center of the thin plate. The x-axis and y-axis represent the plane where the test plate is located, and the z-axis indicates the direction of the applied load and the direction of the contact stress.
In the introduction, Wang [
13] mentioned that parabolic functions can be used to reflect the distribution form of contact stresses in the base slab, and the contact stress distribution function at the bottom of the slab is assumed to be
The five parameters
A1,
A2,
A3,
A4, and
A5 in Equation (1) are coefficients to be determined, which can be determined by a stress balance equation and a continuous condition of the reaction force distribution at four corner points. However, the process of solving is complicated. In addition, in the actual situation, the distribution form of contact stress is not one kind of paraboloid, and there are other common nonlinear distribution cases, such as saddle plane and bell shape. In order to better describe the nonlinear distribution of contact stresses in the base of a square thin plate with four free sides under concentrated loading, an exponential function contact stress distribution model is proposed in this paper [
31].
Equation (2) can simultaneously describe a variety of nonlinear contact stress distribution forms, including parabolic, saddle surface, and bell-shaped, among other nonlinear shapes, where A, B, and C are coefficients to be determined.
The magnitude of the contact stress in the center of the thin plate is defined as the average force under the plate multiplied by the corresponding aspect ratio coefficient p, where p = kp0, p0 = F/a2. p and p0 are the actual central contact stress of the thin plate and the uniform force of the thin plate, respectively. “k = k(λ)” is the aspect ratio coefficient of the thin plate. According to the equilibrium equation, the symmetry of the square thin plate structure and the central contact stress can determine the coefficients A, B, and C of Equation (2).
According to the previous definition, the contact stress at the center of the thin plate is the mean contact stress multiplied by the aspect ratio factor:
Integration along the region leads to the equilibrium equation:
For a square thin plate, the concentrated load acts at the center of the plate, and the square thin plate structure are symmetrical. Therefore, the contact stress distribution model of the substrate is also symmetrical.
The corresponding coefficients to be determined can be obtained by solving the system of equations from (3) to (5).
Different external conditions may have an effect on the initial contact stress value of the contact stress at the base of the thin plate, but under the test conditions, the environment is stable, the 4 sides are free, and the boundary contact stress value is 0. In order to make the model meet the external conditions when building the model, the value of
A is, therefore, taken as 0.
Substituting the center coordinates (
a/2,
a/2) in Equation (2), Equation (7) is obtained.
The values of
A and
B that have been determined are substituted into the contact stress distribution model (2) and integrated along the region to obtain Equation (8).
F is calculated as follows:
Using the polar coordinate transformation, let
, where the integration regions of
,
coincide exactly with the original integration region (the integration region is shown in
Figure 8). Equation (9) is obtained. The transformation of the integration region can be better understood in
Figure 8.
The Procedure for calculating Equation (9) is as follows:
The simplification yields Equation (11).
Since the integration region and the integration function are the same as for
satisfies
), Equation (13) is obtained.
Equation (13) is approximately equal to Equation (14) and finds the solution of
C.
Finally, the contact stress distribution model, including the aspect ratio factor for a square thin plate under a central concentrated load, is derived as Equation (15).
6. Conclusions
Multiple measurement tests based on experimental measurements of the film pressure distribution measurement system are used to confirm the repeatability of the system. For measuring the base cohesion values and contact stress distribution forms of thin plates with various aspect ratios, the multiple tests also provide precise experimental support. The contact stress distribution can be studied more thoroughly, and the impact of aspect ratio on it can be confirmed by controlling the base force value within a tolerable error range. A graded loading test scheme is created for various load forms based on the thin film pressure distribution measurement system. The thin film pressure transducer directly measures the real contact stresses in the substrate of the sheet with various aspect ratios and material properties to produce the nonlinear situation of the real contact stress distribution.
The form of the contact stress distribution at the bottom of thin plates with various aspect ratios is examined in this paper, and a model for the contact stress distribution with an exponential function and aspect ratio coefficients is suggested. The model, which can more accurately describe the nonlinear situation of the actual distribution of contact stresses at the bottom of the substrate and is supported by the experimental data, has a straightforward form and only requires the determination of an aspect ratio coefficient. The theoretical value of the exponential function model, the experimentally measured value of the real contact stress, and the nonlinear model of the current contact stress distribution are compared. Additionally, the corresponding error analysis is derived to confirm the accuracy of the model in this paper. The exponential function distribution model of contact stress, which is more accurate and consistent with the actual nonlinear distribution, is proposed in accordance with the characteristics of the real test values of the contact stress distribution and model distribution. The exponential function contact stress distribution model can provide nonlinear load input and more precise external force boundary conditions for the internal force calculation of thin slabs with aspect ratios of 6~8 or more, improving the strength and stiffness calculation of foundation base slabs. This model is more accurate than the original linear model of contact stress distribution and the parabolic model.
Only the contact stress distribution in the base of thin plates with different aspect ratios under concentrated loading is taken into account by the exponential function contact stress distribution model proposed in this paper. However, the subsequent work will look into the thin plate’s material characteristics as well as additional influencing factors like the type of loading action.