Research on the Characteristics of Jacks Used to Rectify Tilted Buildings

Abstract

One method of rectifying tilted buildings is by lifting them unevenly using hydraulic jacks. These jacks are loaded both monotonically and cyclically during the rectification process. It has been shown that the change in jack length is the sum of the change in the piston slide out and the change in the jack’s cylinder length, which is supported by a parallelepiped element. Laboratory tests were conducted to investigate the piston slide out and the change in the jack’s cylinder length under both monotonically and cyclically loaded conditions. The results indicated that the piston slide out forms a hysteresis loop. In contrast, the change in the jack’s cylinder length does not exhibit a hysteresis loop and is a non-linear function of the load. A structural model of the jack was proposed, consisting of three components: a linearly elastic component connected in parallel to the component where the frictional force occurs, and a component with non-linear elastic characteristics connected in series with them. Displacements of the linear elastic component, characterized by a constant stiffness, occur as long as the external load exceeds the internal frictional force. The value of the frictional force in this model increases with the load. The stiffness of the non-linear elastic component increases proportionally to the load.

1. Introduction

The tilt of structures from the vertical is a common phenomenon observed worldwide [1,2,3,4]. It can affect various types of objects, including residential buildings [5,6], industrial structures [7], historic towers [8,9,10], chimneys [11], churches [12], tanks [13], transmission towers [14], grain elevators [15], tall buildings [16], bridge supports [17], piles [18], high-voltage poles [19], and oil platforms [20] under construction. The most common causes of building deflection are uneven settlement of the deformable subsoil [21,22] and the uneven lowering of the ground surface due to underground mining [23,24]. Tilt can lead to damage to buildings [25] and even construction disasters [26].

A significant number of tilted buildings are stabilized by reinforcing the foundations with various types of piles, such as pressed-in micro piles [27], steel piles [28], and reinforced concrete piles [29], or through soil reinforcement [8]. If individual structural elements become deflected, they are then stabilized in a new position [30]. In some cases, the decision is made to demolish the structure. In special cases, tilting objects can be rectified by returning them to a vertical position. Occasionally, it may be necessary to move the object before rectifying it [31].

Rectification through uneven lifting is achieved using hydraulic jacks that are built into the building structure (Figure 1a). Once all the jacks are installed, a force denoted as Q_mean is stabilized in each jack. During the rectification process, each jack can operate as either an active or passive jack. In an active jack, the piston extension (u_ext) is forced, resulting in displacement of the object and a change in the force values in the jack (Figure 1c). In a passive jack, as a result of this forcing there is a change in the force value by ΔQ, relative to the Q_mean value. Additionally, there is a piston slide out (u_pist) relative to the initial position and a change in the length of the jack body (Δl_cyl). It is often necessary to expand the working range of the jacks during rectification [32], which is achieved by placing stacks of parallelepiped elements beneath the jacks (Figure 1b). To embed these stacks, the jacks are unloaded in a monotonically decreasing manner until they reach zero force. Subsequently, after embedding the stacks, they are loaded in a monotonically increasing manner up to Q_max. Therefore, the jacks are subjected to both monotonic and cyclic loading during the rectification process.

The study of jacks as supports for rectifying buildings has not been extensively researched to date. A preliminary, ad hoc study was carried out on-site during the experimental rectification of a five-story residential building [33]. However, there have been studies conducted on stacks of parallelepiped steel elements with imperfections [34]. Understanding the characteristics of jacks, when supported by parallelepiped steel elements, is crucial for the design and implementation of building rectification. The parallelepiped elements used in the rectification process have imperfections that impact the rigidity of the jacks’ support and the change in their length during rectification. Therefore, laboratory tests were conducted to simulate the actual behavior of the jack. The tests were designed for both monotonic loading and cyclical loading. Monotonic loading involves gradually increasing the load value (Q) from zero to Q_max and then decreasing it back to zero. Cyclical loading, on the other hand, involves alternating between decreasing and increasing the force in the support by a value of ΔQ, with Q_mean being the average force.

During the tests, a jack was used that had the capability to mechanically block the movement of the piston. This feature is crucial for ensuring the safety of rectified structures in the event of an unexpected and unmeasured drop in cylinder oil pressure.

The main objectives of this research were to propose a mathematical model for the jack under cyclic loading and to define an algorithm for calibrating the parameters of the model. The model presented in this paper has been specifically developed for the purpose of building rectification design, which is a pioneering approach within the field of civil engineering. One of the major strengths of this model lies in its universal applicability, as it can be easily adapted for use with other types of jacks.

2. Materials and Methods

The hydraulic jack (Figure 2a) is supplied with oil from an external pump. It consists of a body (1) that rests on a parallelepiped element (2), a piston (3), and a valve system. The jack is supplied with oil, under p_oil pressure from the external pump. A pressure sensor is located in the control box, between the pump and the jack valves, to measure the oil pressure (p_oil) during testing (Figure 2b). The jack body has an outer diameter of 209 mm and a height of 440 mm. The piston, with a diameter of 169 mm, has a nominal extension of 200 mm. During testing, the piston was extended to a length of 100 mm. The arrangement shown in Figure 2b replicates a real-world scenario where the jack is placed on a stack of parallelepiped elements, and the parallelepiped element is characterized by geometric imperfection.

Under the influence of an external load change (Q), the jack piston slides out relative to its initial position, causing a change in the upstroke of the piston (u_pist). Additionally, the length of the jack body that rests on the parallelepiped element (Δl_cyl) also changes. This change refers to the distance between the upper edge of the jack body and the surface of the parallelepiped element. It takes into account any deformations that occur at the interface between the jack body and the parallelepiped element. These deformations primarily occur because of imperfections within the parallelepiped element. The change in the length of the jack (Δl_jack) is the sum of the change in piston slide out (u_pist) and the change in body length (Δl_cyl).

The force transmitted by the press to the jack (Q) was measured using an electro-resistance dynamometer positioned between the press and the jack piston (Figure 2a). The change in piston slide out (u_pist) was determined by averaging the measurements from two Peltron LVDT displacement transducers (u_pist,r and u_pist,l) positioned on opposite sides of the piston. Similarly, the change in Δl_cyl was determined by averaging the measurements from two linear LVDT displacement transducers (Δl_cyl,r and Δl_cyl,l) mounted on opposite sides of the jack’s body. The LVDT sensors had a measurement range of 10 mm and an error of less than 0.5%. The oil pressure in the jack (p_oil) was measured using a WIKA A-10 electronic pressure sensor with a measurement range of 1000 bar and an error of less than 0.5%. The sensor was mounted in the control box at the exit of the oil supply hose to the jack. The sensor readings were recorded at a frequency of 2 Hz using an HBM QuantumX MX840A measurement amplifier, which included 24-bit analog-to-digital cards.

Tests were conducted on the jack, involving both monotonic and cyclic press loading. Monotonic loading consisted of gradually increasing the load from Q = 0 to Q = Q_max at a speed of 10 kN/s, and then relieving the load back to Q = 0 at the same speed. The tests were carried out for Q_max values of 50 kN, 100 kN, 200 kN, 300 kN, 400 kN, 500 kN, 600 kN, 700 kN, 800 kN, and 900 kN. Each 0-Q_max-0 loading cycle was repeated three times.

The cyclic tests involved increasing and decreasing the load by ΔQ relative to the Q_mean value at a rate of 10 kN/s. The Q_mean values were 100 kN, 200 kN, …, 800 kN, and the ΔQ values were 5 kN, 10 kN, 20 kN, 50 kN, 100 kN, 200 kN, and 300 kN. Similar to the monotonic load, each cyclic load on the jack was repeated three times.

3. Results

The research yielded extensive results, and some of them are presented below. Figure 3a shows the Q–p_oil relationships obtained by monotonically loading the jack from zero to Q_max, and then monotonically unloading it back to zero. From the analysis of the figure, it is evident that each monotonical loading process, in (Q, p_oil) coordinates, is represented by a line that originates at the coordinate system’s origin. However, each monotonical unloading process is represented by two segments in this figure: the first one is vertical and parallel to the Q axis, while the second one points towards point (0, 0). The sections representing the second phase of unloading also form a straight line, but with a smaller inclination angle compared to the line representing loading. Consequently, the Q–p_oil relationships form hysteresis loops that share a common apex at point (0, 0).

Figure 3b illustrates the Q–p_oil relationships obtained during cyclic loading of the jack. The load is varied between (Q_mean − ΔQ) and (Q_mean + ΔQ) for Q_mean equal to 400 kN and different values of ΔQ equal to 5 kN, 10 kN, 20 kN, 50 kN, 100 kN, 200 kN and 300 kN. For small ΔQ values (5 kN and 10 kN), the load change cycle is represented by a vertical straight line. At other ΔQ values, hysteresis loops are formed in the (p_oil, Q) coordinates. Both the loading and unloading processes are divided into two sections. In the first loading phase, the Q–p_oil line is vertically oriented and equidistant from the Q axis. Further loading is represented by an inclined line (second phase of loading). In the first phase of unloading, the Q–p_oil line is vertical, and in the second phase, it is inclined. The sections of the hysteresis loop that represent the second phase of loading, with different values of ΔQ, are all found on a single straight line. Similarly, the sections of the hysteresis loop that represent the second phase of unloading, corresponding to different values of ΔQ, are also located on a single straight line.

Figure 3c displays the p_oil-u_pist relationships obtained by loading the jack monotonically from zero to Qmax, and then unloading it monotonically back to zero. Similar to the previous figure (Figure 3a), each monotonic loading process, in the coordinate system (p_oil, u_pist), is represented by a line that originates from the coordinate system’s origin. The monotonic unloading process can be divided into two parts: the first segment is vertical and parallel to the p_oil axis, while the second segment reaches the point (0, 0). On the other hand, the p_oil-u_pist relationship shown in Figure 3d was obtained through cyclic loading of the jack. In this case, the relationship, just like in Figure 3b, is represented by four segments: two under increasing load and two during load reduction.

Figure 4a,b show the relationships between Q and u_pist for the same load changes as illustrated in Figure 3a,b. The hysteresis loops in Figure 4a,b have the same shapes as the corresponding loops in Figure 3a,b. Specifically, when the load increases monotonically from zero to a value of Q_max and then decreases back to zero, the hysteresis loop is characterized by the fact that the u_pist value is also zero when Q is zero. In contrast, for the cyclic loading ranging from (Q_mean − ΔQ) to (Q_mean + ΔQ), closed hysteresis loops were observed. These loops are characterized by two sections representing the loading and unloading cycles: vertical in the first loading and unloading phase and inclined to the axis in the second phase. The similarities between the Q–u_pist and Q–p_oil diagrams described here indicate that the oil pressure, p_oil, is linearly correlated with the change in u_pist of the piston slide out.

In contrast, the Q–Δl_cyl lines in Figure 5, which represent the change in Δl_cyl length of the jack body supported by a parallelepiped, have a different shape. When a monotonic load varies from zero to Q_max and then decreases to zero, the lines representing load and relief do not perfectly coincide. As a result, the hysteresis loop formed has a much smaller area compared to the loops in Figure 3a and Figure 4a. Unlike all the lines in Figure 3 and Figure 4, the lines in Figure 5a do not start as straight lines. Instead, they have a shape similar to that of an exponential function graph. In the case of cyclic loading varying from (Q_mean − ΔQ) to (Q_mean + ΔQ), the line representing loading (in approximation) coincides with the line representing unloading (Figure 5b).

4. Discussion

To analyze the obtained results, the jack model was first defined. Then, the characteristics of the model were adopted, and subsequently, the values of these characteristics were determined.

4.1. Model of the Jack

When explaining the deformation of the jack caused by the load Q, we assumed that the change in its length (Δl_jack) is a result of the extension of the piston (u_pist) and the changes in the length of the jack body supported by parallelepiped steel elements (Δl_cyl)—as shown in Figure 6a. Therefore, the jack model consists of elements connected in series, representing the piston and the body supported by a parallelepiped element. The piston is represented by two elements connected in parallel: a linear-elastic element and an element with friction. On the other hand, the body supported by the parallelepiped is represented by a non-linear elastic element (Figure 6b).

We considered the mechanism for the piston sliding out (u_pist) and the changes in the length of the jack body supported by the parallelepiped (Δl_cyl) separately. We took this approach to ensure clarity and flow in the analysis.

We assumed that the quasi-static external load Q is balanced by two internal forces acting on the piston in the jack: N_pist-el resulting from the oil pressure (p_oil) and the internal friction force N_pist-fr (Figure 6c). The value of the force N_pist-el is calculated as the product of the oil pressure (p_oil) and the piston area (k_oil)

$$N_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}-\mathrm{e}\mathrm{l}}=p_{\mathrm{o}\mathrm{i}\mathrm{l}}{k}_{\mathrm{o}\mathrm{i}\mathrm{l}}.$$

(1)

The sense of the internal friction force is opposite to the sense of the change in piston slide (Figure 6c), indicating that it has the opposite sign as the function sign (u_pist).

$$N_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}-\mathrm{f}\mathrm{r}}=-\mathrm{s}\mathrm{g}\mathrm{n}\left(u_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}}\right)·\left|N_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}-\mathrm{f}\mathrm{r}}\right|.$$

(2)

Tests conducted on a jack that is loaded monotonically show (see Figure 3a and Figure 4a) that the piston returns to its original position after being unloaded. This means that when N_pist-el is zero, the internal friction force (N_pist-fr) is also zero.

As the load Q increases, the oil pressure in the piston also increases, causing the internal forces N_pist-el and N_pist-fr to act in an upward direction (see Figure 6c). This phase of loading is referred to as Q_load (see Figure 7a). The value of Q during this phase is

$$Q_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}}=N_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}-\mathrm{e}\mathrm{l}}+N_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}-\mathrm{f}\mathrm{r}}.$$

(3)

When the external load reaches its maximum value of Q_max, there is a maximum oil pressure, p_oil-max, in the jack resulting in a force of N_pist-el-max, along with an internal friction of N_pist-fr-max. At the beginning of the load-reducing process, the piston remains stationary and there is still a p_oil-max oil pressure present. This phase, represented by the line Q_unl-I (Figure 7a), is characterized by a vertical direction of the Q–p_oil line. A change in the piston slide can only occur after the static friction force has been overcome and its maximum value has been surpassed. Further reduction in load is accompanied by an upward movement of the piston and a downward movement of the internal friction force. In Figure 7a, this phase of the cycle is shown by the line Q_unl-II, while the load value is

$$Q_{\mathrm{u}\mathrm{n}\mathrm{l}-\mathrm{I}\mathrm{I}}=N_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}-\mathrm{e}\mathrm{l}}-N_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}-\mathrm{f}\mathrm{r}}.$$

(4)

The Q–p_oil diagram forms a hysteresis loop with an area called ψ_oil. Within this area, the internal frictional force N_pist-fr performs work on the sections Q_load and Q_unl-II. The maximum value of this force, corresponding to p_oil-max, was determined from the relationship defining the area ψ_oil

$$N_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}-\mathrm{f}\mathrm{r}-\mathrm{m}\mathrm{a}\mathrm{x}}=\frac{{\psi }_{\mathrm{o}\mathrm{i}\mathrm{l}}}{p_{\mathrm{o}\mathrm{i}\mathrm{l}-\mathrm{m}\mathrm{a}\mathrm{x}}}.$$

(5)

The k_oil constant, which is equal to the surface area of the piston, has been identified

$$k_{\mathrm{o}\mathrm{i}\mathrm{l}}=\frac{Q_{\mathrm{m}\mathrm{a}\mathrm{x}}-N_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}-\mathrm{f}\mathrm{r}-\mathrm{m}\mathrm{a}\mathrm{x}}}{p_{\mathrm{o}\mathrm{i}\mathrm{l}-\mathrm{m}\mathrm{a}\mathrm{x}}}.$$

(6)

In the adopted model, the Q–u_pist relationship is represented by three lines on the graph (Figure 7b): Q_load for the load path, while Q_unl-I and Q_unl-II for the unload path. These lines form a hysteresis loop with an area of ψ_pist. The loop is formed due to the work done by the internal frictional force N_pist-fr, which changes from N_pist-fr-max to its opposite value on the vertical section Q_unl-I.

The internal elastic force N_pist-el is calculated by multiplying the piston’s stiffness k_pist with the change in its extension u_pist, as shown in equation

$$k_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}}=\frac{N_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}-\mathrm{e}\mathrm{l}}}{u_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}}}.$$

(7)

The maximum value of the internal elastic force, N_pist-el-max, is related to the maximum internal friction force, N_pist-fr-max, and the maximum load, Q_max, according to the following:

$$Q_{\mathrm{m}\mathrm{a}\mathrm{x}}=N_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}-\mathrm{e}\mathrm{l}-\mathrm{m}\mathrm{a}\mathrm{x}}+N_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}-\mathrm{f}\mathrm{r}-\mathrm{m}\mathrm{a}\mathrm{x}}.$$

(8)

By substituting the maximum values of N_pist-el and u_pist into Equation (7), we can determine the stiffness of the piston, k_pist. The energy absorption coefficient, χ_pist, is defined as the ratio of the energy ψ_pist absorbed during one cycle (represented by the horizontally hatched area in Figure 7b) to the change in potential energy ΔE_p-pist (represented by the vertically hatched area) as shown in equation

$${\chi }_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}}=\frac{{\psi }_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}}}{\Delta E_{\mathrm{p}-\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}}},$$

(9)

where

$$\Delta E_{\mathrm{p}-\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}}=\frac{k_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}}{\left(u_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}}\right)}^2}{2}.$$

(10)

In addition, the secant stiffness of the piston, ${\overline{k}}_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}}$, was defined. This stiffness is calculated by dividing the change in load force by the change in extension. The value of this stiffness can be determined using relation (11), which is derived from the designations in Figure 8

$${\overline{k}}_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}}=k_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}}\frac{\Delta Q}{\Delta Q-N_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}-\mathrm{f}\mathrm{r}}}, \mathrm{g}\mathrm{d}\mathrm{y} \Delta Q>N_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}-\mathrm{f}\mathrm{r}}.$$

(11)

Relation (11) is applicable when the change in load surpasses the friction force.

In order to describe the second component of the change in the jack length, which is the change in the body length based on a parallelepiped element (Δl_cyl), it is observed that the relationship Q–Δl_cyl is not linear (Figure 5a,b). However, the lines representing loading and unloading almost coincide, allowing us to disregard inelastic forces. As a result, the load and internal elastic force are considered equal (N_cyl-el = Q). However, the internal elastic force N_cyl-el does not vary linearly with Δl_cyl. The stiffness of the jack body, based on a parallelepiped element, increases. This increase is mainly due to an increase in the contact area between the jack base and the element on which the jack rests. The increase in contact area with increasing load is a result of imperfections in the parallelepiped element. Based on these assumptions, it is presumed that the stiffness k_cyl increases linearly with the value of the internal elastic force N_cyl-el

$$k_{\mathrm{c}\mathrm{y}\mathrm{l}}={\beta }_{\mathrm{c}\mathrm{y}\mathrm{l}}{N}_{\mathrm{c}\mathrm{y}\mathrm{l}-\mathrm{e}\mathrm{l}},$$

(12)

where β_cyl is the proportionality factor. Therefore, the change in length Δl_cyl,max, caused by the load varying from zero to Q_max, is given by the integral.

$${\Delta l}_{\mathrm{c}\mathrm{y}\mathrm{l}-\mathrm{m}\mathrm{a}\mathrm{x}}=\underset{0}{\overset{N_{\mathrm{c}\mathrm{y}\mathrm{l}-\mathrm{e}\mathrm{l}-\mathrm{m}\mathrm{a}\mathrm{x}}}{\int }}\frac{\mathrm{d}{N}_{\mathrm{c}\mathrm{y}\mathrm{l}-\mathrm{e}\mathrm{l}}}{{\beta }_{\mathrm{c}\mathrm{y}\mathrm{l}}{N}_{\mathrm{c}\mathrm{y}\mathrm{l}-\mathrm{e}\mathrm{l}}}.$$

(13)

Using Equation (13), the value of β_cyl is determined as

$${\beta }_{\mathrm{c}\mathrm{y}\mathrm{l}}=\frac{\mathrm{l}\mathrm{n}\left(N_{\mathrm{c}\mathrm{y}\mathrm{l}-\mathrm{e}\mathrm{l}-\mathrm{m}\mathrm{a}\mathrm{x}}\right)}{{\Delta l}_{\mathrm{c}\mathrm{y}\mathrm{l}-\mathrm{m}\mathrm{a}\mathrm{x}}}.$$

(14)

The secant stiffness of the jack, ${\overline{k}}_{\mathrm{j}\mathrm{a}\mathrm{c}\mathrm{k}}$, which is the combination of the body stiffness connected in series with the jack stiffness, is calculated as

$${\overline{k}}_{\mathrm{j}\mathrm{a}\mathrm{c}\mathrm{k}}=\frac{{\overline{k}}_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}}{k}_{\mathrm{c}\mathrm{y}\mathrm{l}}}{{\overline{k}}_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}}+k_{\mathrm{c}\mathrm{y}\mathrm{l}}}.$$

(15)

4.2. Estimating the Characteristic Values of the Jack Model

In the upcoming sections, we will determine the values of the jack characteristics that were defined in Section 4.1.

4.2.1. Stiffness k_pist and Energy Dissipation Caused by Piston Slide Out

Based on tests of jacks loaded with a monotonically increasing force Q that varies from zero to Q_max and then monotonically decreases to zero, the following values were recorded for each loading process: Q_max, p_oil-max, u_pist-max, and Δl_cyl-max. These values are summarized in

Table 1. The characteristic values of the jack model.

Hysteresis Loop	Qmax, kN	Δlpist-max, mm	ψoil, MPa·mm	poil-max, MPa	Npist-fr-max, kN	koil 10−3·m2	Npist-el-max, kN	kpist, MN/m	ψpist, kN·mm	χpist, -
50 kN	48.43	0.336	16.85	1.79	9.78	22.43	38.65	115	2.932	0.376
100 kN	87.35	0.728	15.86	3.70	3.75	19.78	83.60	115	3.506	0.095
200 kN	194.67	1.430	67.61	8.13	7.81	21.58	186.86	131	12.615	0.089
300 kN	288.04	2.093	163.61	11.71	13.34	22.39	274.71	131	30.562	0.101
400 kN	389.77	2.699	270.84	15.88	16.47	22.70	373.30	138	47.560	0.094
500 kN	484.47	3.257	416.51	19.77	20.46	22.79	464.01	142	70.510	0.096
600 kN	587.73	3.857	551.73	24.01	22.41	22.96	565.31	147	90.600	0.088
700 kN	683.32	4.386	668.15	28.01	23.37	23.09	659.95	150	106.995	0.080
800 kN	788.81	5.002	780.57	32.36	23.69	23.22	765.12	153	122.680	0.071
900 kN	896.75	5.935	801.35	37.07	21.31	23.28	875.44	148	130.160	0.053

. The hysteresis loop area (ψ_oil) was subsequently determined based on the recorded Q–p_oil relations using the following relationship

$${\psi }_{\mathrm{o}\mathrm{i}\mathrm{l}}=\frac{1}{2}\sum _i{Q}_i\left(p_{\mathrm{o}\mathrm{i}\mathrm{l},\mathrm{i}+1}-p_{\mathrm{o}\mathrm{i}\mathrm{l},\mathrm{i}-1}\right).$$

(16)

Then, the following values were determined: N_pist-fr-max from Equation (5), k_oil from Equation (6), N_pist-el-max from Equation (8), and k_pist from Equation (7). These values are shown in Table 1.

Figure 9a illustrates the maximum values of the internal friction force acting on the piston (N_pist-fr-max) as a function of the maximum load value (Q_max). The value of N_pist-fr-max increases as Q_max increases. For Q_max equal to 100 kN, the value of N_pist-fr-max is 3.75 kN, and for Q_max = 800 kN, it is 23.69 kN.

The internal frictional force, N_pist-fr, represents approximately 3.0% to 6.4% of the external load (Q). The stiffness, k_pist, remains relatively constant regardless of changes in Q, ranging from 115 MN/m for Q_max = 100 kN to 153 MN/m for Q_max = 800 kN (Figure 9b). The average stiffness of k_pist, for a load Q between 100 kN and 800 kN, is 138 MN/m.

Based on the determined parameter values, graphs can be drawn to illustrate the changes in the piston slide out. Figure 10a shows the hysteresis loop of the model, depicting the change in u_pist under a monotonically varying load from zero to Q_max. In the (Q, u_pist) coordinate system, the relationships corresponding to different values of Q_max are plotted with solid lines. When the load increases monotonically from zero to a value of Q_max, the Q–u_pist relationship is described by Equation (3). When the load decreases from Q_max to a value of Q_max − 2⋅N_pist-fr-max, the Q–u_pist relationship is represented by straight lines parallel to the Q axis. For load values less than Q_max − 2⋅N_pist-fr-max, Equation (4) applies. In Figure 10a, the dotted lines show the corresponding relationships obtained from the tests and derived from Figure 4a. This demonstrates the correspondence between the developed model and the test results.

In contrast, Figure 10b shows the hysteresis loop of the model, depicting the change in u_pist in the case of the cyclic load for Q_mean = 400 kN and different values of ΔQ. It is important to note that the piston slide out does not occur if the change in ΔQ is small. This is because there are internal frictional forces that need to be overcome for the piston to slide out. The magnitude of these forces increases as the value of Q_mean increases. In Figure 10b, the dotted line shows the corresponding relationships obtained from the tests and derived from Figure 4b. Again, the correspondence between the developed model and the test results can be observed.

4.2.2. Stiffness k_cyl

The stiffness of the jack body resting on a parallelepiped element, k_cyl, is determined by multiplying β_cyl by the internal elastic force, N_cyl-el (12). In our model, we assume that the value of N_cyl-el is equal to the load, Q, as there are no frictional forces acting in the direction of the load where the jack is supported. The values of β_cyl and k_cyl, calculated using Equations (12) and (14), are summarized in

Table 2. Values for the parameter β_cyl and the stiffness k_cyl.

Load Range	Ncyl-el,max, kN	Δlcyl,max, mm	βcyl, 1/mm	kcyl, MN/m
0–50–0	48.43	0.553	3.05	152
0–100–0	87.35	0.356	5.46	546
0–200–0	194.67	0.557	4.11	822
0–300–0	288.04	0.616	4.00	1199
0–400–0	389.77	0.684	3.79	1516
0–500–0	484.47	0.725	3.71	1853
0–600–0	587.73	0.758	3.65	2192
0–700–0	683.32	0.777	3.65	2553
0–800–0	788.81	0.791	3.66	2930
0–900–0	896.75	0.867	3.41	3065

. The value of β_cyl remains relatively constant regardless of the load, with an average value of 3.85. On the other hand, the stiffness of k_cyl depends on the load. For a load of Q = 100 kN, the value of k_cyl is 546 MN/m, while for a load of Q = 800 kN, the value of k_cyl is 2930 MN/m. This shows that the stiffness of k_cyl increases as the force loading the jack increases (Figure 11a). Since β_cyl remains approximately constant, the increase in k_cyl stiffness is a linear function of the load, Q. The change in length, Δl_cyl, as a function of Q and using the determined characteristics, was calculated according to Equation (13) by assuming N_cyl-el-max = Q_max and medium value β_cyl equal to 3.85. The result is graphically presented in Figure 11b. In the same figure, the results of the Q–Δl_cyl relationship obtained from tests and derived from Figure 5a are plotted. We can observe slight differences between the results of the model calculations and the test results in this figure. These differences can be attributed to the use of the mean value as β_cyl.

In Figure 12, the change in Δl_cyl for Q_mean values of 50, 100, 200, …, 900 kN and various load changes, ΔQ, is illustrated. It is clear that as the Q_mean value increases, the corresponding change in length, Δl_cyl, decreases for the same value of ΔQ.

4.2.3. Jack Stiffnesses k_jack

Above, in Section 4.1, we defined the structural model of the jack as a combination of piston and body models based on a parallelepiped element (Figure 6b). Therefore, the change in jack length (Δl_jack) is the sum of the change in piston slide out (u_pist) and the change in body length (Δl_cyl). Figure 13a illustrates the Δl_jack changes induced by different values of ΔQ for various Q_mean values. In each of the plotted relations in Figure 13a, two phases can be observed. In the first phase, when the Δl_jack length changes are up to approximately 0.01 mm–0.02 mm, only the jack body resting on the parallel-walled element is deformed. Piston extension does not occur due to the internal friction forces not being overcome. Hence, the lines Δl_jack–ΔQ have a steep slope, with a greater slope corresponding to larger Q_mean values. In the second phase, when the piston stroke has already occurred, the slope of the lines in Figure 13a becomes smaller and almost identical for all Q_mean values. The secant stiffnesses ${\overline{k}}_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}}$, determined according to (11), are included in Figure 13b.

The graph illustrates that when there is a slight change in the force value in the jack (ΔQ ≤ N_pist-fr), the secant stiffness ${\overline{k}}_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}}$ becomes extremely high. However, as the force value increases by about 60 kN, the ${\overline{k}}_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}}$ values approach the stiffness values of k_pist. The secant stiffnesses of the jacks (${\overline{k}}_{\mathrm{j}\mathrm{a}\mathrm{c}\mathrm{k}}$), determined according to Equation (14), are shown in Figure 14. The change in these stiffnesses follows a similar pattern as the change in the secant stiffness ${\overline{k}}_{\mathrm{p}\mathrm{i}\mathrm{s}\mathrm{t}}$, except that for ΔQ ≤ N_pist-fr, the ${\overline{k}}_{\mathrm{j}\mathrm{a}\mathrm{c}\mathrm{k}}$ value is not unlimited and takes on the value of k_cyl. Based on this information, we can conclude that the stiffness of the jacks used for rectification is primarily influenced by the stiffness of the piston, which extends due to the external load Q. The piston extension is only possible if the change in the jack load exceeds the value of the internal friction force.

When defining the model of the jack subjected to cyclic loading, we considered its actual structure. This allowed for a physical interpretation of the elastic and inelastic internal forces. However, in some cases, it is not possible to take this approach. In those situations, rheological material models [35,36,37] and numerical models [38,39] are used to describe the internal forces.

5. Conclusions

The change in length of the hydraulic jack used to rectify buildings tilted from vertical is determined by two factors: the slide out of the piston and the change in length of the jack’s body resting on parallelepiped steel elements. During the rectification process, this jack is subjected to both monotonic and cyclic loading. Monotonic loading refers to the gradual increase in the load from zero to its maximum value, and then a subsequent decrease back to zero. On the other hand, cyclic loading involves the fluctuation of the load relative to a constant load value. The constant load value results from the weight of the building, while the change in load is caused by the impact of neighboring jacks during the rectification process.

Based on laboratory tests, it was determined that under monotonic loading, the relationship between load and piston slide can be divided into three segments. The load phase, represented in the load force—piston slide out coordinates—begins at the origin of the coordinate system. On the other hand, the unloading process consists of two segments: the first segment is vertical and parallel to the vertical load axis, while the second segment tends toward the origin of the coordinate system. The shape of the hysteresis loop remains the same, regardless of the maximum load value.

During cyclic loading, which occurs in a constant force environment, hysteresis loops are also formed in the load–unload relationships. The loading and unloading processes are represented by two sections. The first loading phase and the first unloading phase are represented by vertical sections. Subsequent loading and unloading are represented by segments inclined with respect to the axis of the coordinate system. The segments of the hysteresis loop that represent the second loading phase and the second unloading phase are parallel to each other.

In contrast, the lines depicting load–jack body length changes have a different shape. Regardless of the type of load, these lines resemble the graph of an exponential function. With monotonic loading, these curves originate from the origin of the coordinate system.

A structural model of the jack, composed of three elements, has been proposed. It consists of a linearly elastic element connected in parallel with a frictional force element, as well as a nonlinear elastic element connected in series with them. Displacements of the linear elastic element occur when the external load surpasses the frictional force, while displacements of the nonlinear elastic element occur regardless of the external load value.

The stiffness of the linear elastic element, which changes in length to represent the piston’s slide out, remains approximately constant with an average of 137 MN/m. The value of the friction force varies depending on the load, ranging from 9.78 kN at a load of 50 kN to 21.31 kN at a load of 900 kN. The stiffness of the non-linear elastic element also varies with the load, ranging from 152 MN/m at a load of 100 kN to 3065 MN/m at a load of 900 kN.

This defined structural model of the jack serves as the basis for further research and the development of a method to rectify vertically tilted buildings by unevenly lifting them with jacks. One of the major strengths of this model lies in its universal applicability, as it can be easily adapted for use with other types of jacks.