The Effect of Braid Angle on Hydraulic Hose Geometry

Abstract

Hydraulic hoses are part of most hydraulic systems, from industrial hydraulics with open loop hydraulic systems to mobile hydraulics with closed loop hydraulic systems. The design parameters of hydraulic hoses may influence the duty cycle dynamics of these systems. One of the factors that influence the behavior of a hydraulic hose under pressure loading is the steel braid angle with respect to the hydraulic hose axis. This work aims to determine the effect of the hydraulic hose braid angle on the change in its geometry. The next objective is to determine the forces that occur at the hose ends under pressure loading. The stresses occur when fluid pressure is applied to the inner wall of the hydraulic hose. Consequently, these stresses are transferred to the hose ends through the steel braid or spiral. The phenomenon of the neutral braid angle provides a balance between the stresses generated inside the hydraulic hose. Therefore, hydraulic hose manufacturers try to produce hydraulic hoses with a neutral braid angle, because the lifetime of the hydraulic hose is also related to this. As part of this research work, an experimental device was constructed in order to measuring the properties of hydraulic hoses. When the hose was loaded with fluid pressure, the change in hose geometry was measured and the angle of the hose braid was measured simultaneously. Upon the measurements, the effect of the braid angle on the hose behavior under pressure loading was determined.

1. Introduction

The parts of hydraulic systems are connected to each other by hydraulic lines, which can be formed from hydraulic hoses. There are advantages and disadvantages to using hydraulic hoses as hydraulic lines. The main advantage is the relative movement of the connected parts with respect to each other, Another advantage can be the reduction in pressure peaks during hydraulic shock due to the hydraulic capacity of the hydraulic hoses [1,2]. However, the mentioned influence of the hydraulic capacity of the hose also has the opposite effect; that is, the decrease in the hydraulic system’s stiffness. Wang described the problem of double-acting cylinder position control, where the hydraulic hose influences the system dynamics. Wang compared the PID controller with the ADRC (Active Disturbance Rejection Control), where the ADCR variant of the controller performs better and can compensate for the influence of the hydraulic hoses [3]. Previously, the influence of different parameters on the performance of hoses has been investigated [4]. The hydraulic capacity, bulk modulus [5,6] and viscoelastic properties [7,8] of hydraulic hoses are also related to the design.

A hydraulic hose consists of a rubber inner tube to ensure tightness. The next design component is the braid or spiral, which determines the maximum pressure loading. Hydraulic hoses are available with one or more braids (spirals) depending on the working pressure [9]. With more braids (spirals), the flexibility of the hose is reduced. The working pressure of these hoses is up to 400 bar in common high-pressure hydraulic applications. A hydraulic hose must be able to withstand the maximum working pressure and must not be damaged even by short-term overloading. For this reason, pressure tests of hydraulic hoses are carried out [10]. For high-pressure hydraulic hoses, the braid or spiral is made from steel wire [11,12]. For some applications, the hose braid or spiral can be made from aramid fiber or polyvinyl acetate [13,14]. When a hydraulic hose is loaded with fluid pressure, stresses are generated and transferred through the braid to the hose ends. Tensile forces are applied to the hose fitting if the hose is not installed correctly. This can result in damage to the hose and failure of the entire hydraulic system. Therefore, the importance of the right hose mounting is paramount.

The angle of the hydraulic hose braid has a major influence on the hydraulic hose deformation and the forces acting on the hose ends [15,16,17,18]. When the hose is loaded by fluid pressure, the hose steel braid tends to deform so that the opposing strands of the braid are at a neutral braid angle to each other at which the axial and hoop stress components are in balance. This deformation causes relative friction between the steel braid and the rubber tube or rubber interlayer of the hose. The degree of deformation of a hydraulic hose influences its service life [19]. The solution to this problem is the method of braiding the steel wires at a neutral braid angle φ_N = 54.7356° to the hose axis. In this case, the balance between axial and hoop stresses is ensured. When manufacturing hydraulic hoses, the aim is to maintain the required braid angle. This is ensured by the right combination of the braiding rate and feed rate of the manufacturing machine [20].

The aim of this work is to determine the influence of the braid angle on the deformation of the hydraulic hose and the forces acting on the hose ends. Within the framework of this research, an experimental device was created in order to test hydraulic hoses. The result is the determination of the dependence of the change in hose length and tensile force on the braid angle for hydraulic hoses with different internal diameters and different braid designs. This work serves as a summary of the measured data, which can be used for further research in the field of mathematical 3D modelling, for example in the field of finite element analysis of the deformation stress of hydraulic hoses.

2. Theoretical Background

The definition of the neutral braid angle, at which balance is reached between the stresses generated, is based on the theory of a closed cylinder of radius r and wall thickness s. The working pressure p of the fluid acting on the hose inner wall generates hoop stress σ_O and axial stress σ_A (see Figure 1) in the hose wall [21].

The axial force F_A acting on the closed end of the hose is given below [21]:

$$F_A=\pi \times r^2\times p,$$

(1)

where p is the fluid pressure and r is the inner radius of the hose.

The axial stress σ_A generated in the wall of the hydraulic hose is given by the axial force F_A acting in the cross-sectional area of the hydraulic hose under the condition that r >> s [21]:

$${\sigma }_A=\frac{F_A}{2\times \pi \times r\times s},$$

(2)

where s is the wall thickness of the hose.

We obtain the expression for axial stress σ_A by modifying Equations (1) and (2), as given below [21]:

$${\sigma }_A=\frac{p\times r}{2\times s}.$$

(3)

The hoop force F_O acting on the unit length of the hydraulic hose as given below [21]:

$$F_O=1\times 2\times r\times p,$$

(4)

where 1 is the unit length.

The hoop stress σ_O expressed per the unit length of the hydraulic hose is defined by Equation (5) [21]:

$${\sigma }_O=\frac{F_O}{1\times 2\times s}.$$

(5)

We obtain the expression for hoop stress σ_O by modifying Equations (4) and (5), as given below [21]:

$${\sigma }_O=\frac{p\times r}{s}.$$

(6)

Comparing Equations (3) and (6), it can be seen that the axial stress σ_A is half of the hoop stress σ_O. Figure 2 shows a section of hydraulic hose where the wire tensions acting on the braid wires are indicated. A braid angle φ is given for the braid wire and the longitudinal axis of the hydraulic hose, as shown in Figure 2 on the left and right [21].

When an element of unit length is released, the height of the element is equal to tanφ (see Figure 2). For the balance between the hoop and axial stresses when the hose is subject to internal fluid pressure, Equation (7) for the hoop tension T_O and Equation (8) for the axial tension T_A must satisfy:

$$T_O=T\cdot \mathit{sin}\phi =\frac{p\times r}{s}\times 1\times s,$$

(7)

$$T_A=T\cdot \mathit{cos}\phi =\frac{p\times r}{2\times s}\mathit{tan}\phi \times s,$$

(8)

where 1∙s is the area of the unit length element on which the hoop tension T_O acts and tanφ∙s is the area of the unit length element on which the axial tension T_A acts.

The neutral braid angle φ_N can be defined using the trigonometric [21]:

$$\mathit{tan}{\phi }_N=\frac{T_O}{T_A}.$$

(9)

Substituting Equations (7) and (8) into Equation (9), the expression is as follows [21]:

$$\mathit{tan}{\phi }_N=\frac{2}{\mathit{tan}\phi }.$$

(10)

We obtain the expression for the neutral braid angle by modifying Equation (10) [21]:

$${\phi }_N={\mathit{tan}}^{-1}\sqrt{2}=54.7356°.$$

(11)

Table 1. Geometric changes of the hydraulic hose when the internal fluid pressure increases [22].

Hose Geometry Changes	φ < φN	φ > φN
Length	Decreases	Increases
Diameter	Increases	Decreases
Volume	Increases	Increases

shows the changes in the geometry of the hydraulic hose caused by an internal fluid pressure increase. When the initial braid angle φ becomes greater than the neutral braid angle φ_N, the length of the hydraulic hose increases, and the hose diameter decreases. When the initial braid angle φ is less than the neutral braid angle φ_N, the length of the hydraulic hose decreases, and the hose diameter increases. In both cases, the volume of the hose increases as the internal fluid pressure increases. With a neutral braid angle, there is no change in geometry due to the change in braid angle [22].

The change in the braid angle due to the internal fluid pressure increase in the hose causes the axial force to be applied on the hose ends. In the case of hose shortening, a tensile force F_T should be generated which acts on the hose fitting. In the case of hose extension, on the other hand, a pushing force F_TH should be generated.

Manufacturers of high-pressure hydraulic hoses perform many tests on their products, which may include pressure or temperature tests or chemical resistance tests. One of the tests that are performed on high-pressure hoses is a test where the shortening or elongation of the hose is measured [23]. This test is also part of the SAE J343 standard, which specifies the procedures for performing tests on high-pressure hydraulic hoses. This standard gives detailed instructions for performing the test and specifies allowable values for hose elongation or shortening depending on the hose design and size.

The percentage of hydraulic hose elongation or shortening may vary depending on the standard and the hose. For example, Fitch stated limits that allow a hose to be shortened by 6% of the original length and elongated by 2% of the original length in his publication [23]. The testing of high-pressure hoses for possible elongation or shortening under working pressure is a common practice carried out by manufacturers, but these data are not widely available. The added value in this research will be the simultaneous sensing of the braid angle due to the removal of the rubber cover and the subsequent determination of its effect on hose shortening or elongation.

3. Experiment

For this study, experimental equipment was constructed in order to test hydraulic hoses with different inner diameters and design types (see Figure 3). The left part of the figure shows the equipment design. The middle part of the figure shows the experimental equipment photo, and the right side shows a detail of the hose under test. To measure the braid angle under the fluid pressure, the hydraulic hose cover was removed from the hydraulic hose. The hydraulic hose cover did not affect the pressure capability, but only protected the hose from external influences. In this way, the angle of the hose outer braid could be measured visually. With two and more layers of braids, only the angle of the outer braid could be visually read without damaging the hose. For this reason, only hydraulic hoses with one braid were evaluated. Figure 4 shows on the left side the tested hydraulic hoses. Figure 4 shows on the right side a detail of the hose braid.

Table 2. Technical data of tested hydraulic hoses.

Hydraulic Hose	Inner Diameter din	Outer Diameter dout	Wall Thickness s	Length l	Type of Braid	Maximal Working Pressure pmax
	(mm)	(mm)	(mm)	(mm)	(-)	(bar)
DN12_A	13	19	3	1355	1SC	160
DN12_B	13	19.5	3.25	1345	1SN	160
DN12_C	13	20	3.5	1371	1SN	160
DN16_A	16	22	3	1350	1SC	130
DN16_B	16	23	3.5	1345	1SN	130
DN16_C	16	24	4	1350	1SN	130
DN19_A	19	26	3.5	1345	1SC	105
DN19_B	19	27	4	1345	1SN	105
DN19_C	19	27	4	1345	1SN	105

shows the technical data and geometric dimensions of the tested hydraulic hoses. The table shows two types of braid for the tested hoses, namely SC and SN. Both types of hose braids are suitable for high pressure hydraulics. However, the SC designation defines the possibility of a tighter bend radius for these hoses, which is suitable for installations where space is at a minimum.

Figure 5 shows a simplified scheme of the experimental equipment. The source of the pressurized fluid was a hydraulic power unit which supplies fluid to the channel P. The hydraulic hose H was connected to the pressure line via a ball valve BV. The ball valve was only used to close the pressure line when changing the tested hose H. The top end of the hydraulic hose H was screwed to a top steel plate, which was connected to the aluminum frame. The required pressure value p was set by the pressure proportional relief valve PRV. Working fluid passed through PRV to channel T. The fluid pressure value p was measured by the pressure sensor PS. To measure the tensile force F_T of the hose in the longitudinal axis, the bottom end of the hose was attached to the force sensor FS. This tensile force F_T of the hose increased with increasing the working pressure p. The force sensor FS was attached to a bottom steel plate, which was fixed into the frame structure. To measure the change in length of the hose, the bottom end of the hose was connected to a bracket which was fitted in the linear guides on the sides. This variant allowed one degree of freedom in the longitudinal axis of the hydraulic hose H. As the working pressure p increased, the change in the hydraulic hose length Δl occurred. The length change Δl of the hose was determined by the laser distance sensor LS1. Simultaneously, the diameter d of the hydraulic hose braid was measured by the optical micrometer LS2. A photo of the hose braid with the removed cover was taken with the camera CAM. The photos of the braids were taken in the working pressure range p = (0 ÷ 140) bar. The pressure sensor PS and the force sensor FS were connected to measurement instrument, MS5070, from Hydrotechnik. The working fluid was mineral oil. The used parts are listed in

Table 3. List of used parts.

Symbol	Name	Type (Producer)	Measuring Range	Measuring Accuracy
PRV	proportional relief valve	DBEBE 6X (Rexroth, Hong Kong, China)	-	-
PS	pressure sensor	PR400 (Hydrotechnik, Obergünzburg, Germany)	(0–250) bar	±0.25% of full scale
FS	force sensor	FO 200 (Hydrotechnik)	(0–5) kN	±0.5% of full scale
LS1	laser distance sensor	optoNCDT (Micro epsilon, Hong Kong, China)	(0.5–200) mm	±0.08% of full scale
LS2	optical micrometer	LS-7070 (Keyence, Walnut Creek, CA, USA)	(0.5–65) mm	±3 μm
CAM	camera	FASTCAM MINI UX (Photron, Tokyo, Japan)	-	-

.

4. Results and Discussion

The static properties of nine hydraulic hoses were measured and evaluated. For comparison, hydraulic hoses with one steel braid and different inner diameters (din = 13, 16 and 19 mm) were selected. Two experiments were performed for each hydraulic hose. In the first experiment, the dependence hose length strain ε_l with respect to the working pressure p was determined. In this experiment, the top end of the hose under test was tightly threaded to the frame and the bottom end of the hose was attached to a linear guide that allowed movement in the longitudinal axis of the hose. In this experiment, the working fluid pressure p acting on the inner wall of the hydraulic hose caused the hose length to shorten. In the second experiment, the dependence of the hose tensile force F_T with respect to the working pressure p was determined. The fitting of the bottom end of the hose was performed through the force sensor FS into the bottom fitted steel plate, which was rigidly connected to the frame of the equipment. In this case, there was no shortening of the hydraulic hose as in the first experiment. The working pressure p caused an increase in the tensile force F_T, which was transferred by the braid steel wires to the hydraulic hose ends.

In both experiments, the braid angle of the tested hydraulic hose was simultaneously evaluated for working pressure p_min = 0 bar a p_max = 140 bar. Figure 6 shows the method for the evaluation of the hydraulic hose braid angle. The evaluation of the braid angle was performed using Photron FASTCAM Viewer 4 (PFV4) software. Due to the optical distortion of the angle, the braid wires that crossed relative to each other in the center of the hydraulic hose were evaluated. This was the point where the least optical distortion occurred, which is due to the curvature of the hydraulic hose. Subsequently, the angle α_i was evaluated by using the function “angle 2” with two plotted lines parallel to the braid of the hose. Subsequently, the angle of the braid φ_i with respect to the longitudinal axis of the hydraulic hose was determined using Equation (12):

$${\phi }_i=\frac{180-{\alpha }_i}{2}.$$

(12)

Figure 7 shows the details of the evaluation of the single angles α_i for one measurement. The plotted line follows the selected braid wire along the length at which minimal curvature occurs. To refine the results for this measurement, five angles α₁ to α₅ were evaluated for one image taken. Three measurements of the dependence of length strain ε_l on the working pressure p and three measurements of the dependence of tensile force F_T on the working pressure p were performed for one hydraulic hose. For the measured angle values for a specific working pressure and hose, the arithmetic mean was determined, given by (13):

$$\stackrel{-}{\phi }=\frac{1}{n}\sum _{i=1}^n{\phi }_i,$$

(13)

where n is the number of angle measurements. The measured angle values α_i are included in

Table 4. Measured braid angle values for hydraulic hose DN19_C.

DN19_C	Evaluating the Initial Braid Angle φin for Pressure pmin = 0 bar			Evaluating the End Braid Angle φen for Pressure pmax = 140 bar
	α1 ÷ αn [°]	φ1 ÷ φn [°]	φin [°]	α1 ÷ αn [°]	φ1 ÷ φn [°]	φen [°]
εl = f (p)	75.393	52.30	52.26 ± 0.03	72.7458	53.27	53.44 ± 0.04
Measurement 1	75.4747	52.26		72.7458	53.63
	75.8255	52.09		72.9397	53.53
	75.8728	52.06		73.4766	53.26
	75.3752	52.31		73.0621	53.47
Measurement 2	75.8248	52.09		73.1139	53.44
	75.2628	52.37		72.9507	53.52
	76.0046	52.00		72.7508	53.62
	76.0819	51.96		73.3767	53.31
	75.9478	52.03		72.7527	53.62
Measurement 3	75.6965	52.15		72.5419	53.73
	75.7654	52.12		73.3178	53.34
	74.9611	52.52		73.2721	53.36
	75.5103	52.24		73.5703	53.21
	75.4931	52.25		73.4296	53.29
FT = f (p)	75.7271	52.14		73.704	53.15	53.25 ± 0.07
Measurement 1	75.0143	52.49		72.7037	53.65
	75.7063	52.15		73.7333	53.13
	75.5178	52.24		73.6536	53.17
	75.9032	52.05		73.9443	53.03
Measurement 2	75.1601	52.42		73.3606	53.32
	75.1492	52.43		72.8422	53.58
	75.089	52.46		73.2932	53.35
	75.7168	52.14		73.0096	53.50
	75.3231	52.34		73.6646	53.17
Measurement 3	74.9483	52.53		74.3733	52.81
	75.2163	52.39		73.752	53.12
	75.1989	52.40		72.9056	53.55
	74.9691	52.52		73.3729	53.31
	75.3156	52.34		74.2968	52.85

. The arithmetic mean of the braid angle $\stackrel{-}{\phi }$ is supplemented by the measurement uncertainty type A, which is equal to the sample standard deviation of the arithmetic mean and is given by Equation (14):

$$u_{A\phi }=S_{\stackrel{-}{\phi }}=\sqrt{\frac{\sum _{i=1}^n{\left({\phi }_i-\stackrel{-}{\phi }\right)}^2}{n\left(n-1\right)}}.$$

(14)

The measured values of the angle α_i and the calculated values of the braid angle φ_i for the hydraulic hose DN19_C are included in Table 4. From the calculated values φ₁ to φ_n, the arithmetic mean of the braid angle with type A measurement uncertainty was determined according to Equations (13) and (14). In this way, the initial braid angle φ_in corresponding to the hydraulic hose without working pressure load p was determined. The initial braid angle φ_in was the same for both types of measurements, due to the same initial conditions. The end braid angle φ_en corresponded to the maximum working pressure p_max = 140 bar. The determination of the end braid angle φ_en was performed separately for each type of measurement due to the different conditions (loose/fit hose end).

The braid angles φ_in and φ_en were evaluated in the same way for all hydraulic hoses tested. The change in the braid angle Δφ, given by Equation (15), is the important factor in the hose length change Δl or in the hose tensile force F_T when the fluid pressure p is applied:

$$\Delta \phi ={\phi }_{\mathit{en}}-{\phi }_{\mathit{in}}.$$

(15)

4.1. Evaluation Hose Length Strain with Respect to the Working Pressure

Table 5. Measured values of initial braid angles φ_in, end braid angles φ_en and braid angle changes Δφ of the tested hydraulic hoses for the experiment ε_l = f (p).

Hydraulic Hose	Inner Diameter din	Outer Diameter dout	Wall Thickness s	Initial Braid Angle φin	End Braid Angle εl = f (p) φen	Change in Braid Angle Δφ
	(mm)	(mm)	(mm)	(°)	(°)	(°)
DN12_A	13	19	3	51.01 ± 0.08	52.36 ± 0.06	1.35
DN12_B	13	19.5	3.25	53.05 ± 0.10	53.86 ± 0.10	0.81
DN12_C	13	20	3.5	53.80 ± 0.05	54.12 ± 0.05	0.32
DN16_A	16	22	3	52.51 ± 0.02	53.69 ± 0.06	1.18
DN16_B	16	23	3.5	52.31 ± 0.05	52.85 ± 0.10	0.54
DN16_C	16	24	4	52.77 ± 0.08	53.74 ± 0.02	0.97
DN19_A	19	26	3.5	53.31 ± 0.03	53.96 ± 0.03	0.65
DN19_B	19	27	4	53.03 ± 0.05	53.90 ± 0.07	0.87
DN19_C	19	27	4	52.26 ± 0.03	53.44 ± 0.04	1.18

provides an overview of all initial braid angles φ_in, end braid angles φ_en and braid angle changes Δφ achieved by each hydraulic hose when measuring the dependence of the hydraulic hose length strain ε_l on the working pressure p.

A summary graph of all hydraulic hoses for the experiment ε_l = f (p) can be seen in Figure 8. For evaluating the results, it is important to consider several factors that may affect the individual dependencies. Both the initial braid angle φ_in and the actual change in the braid angle Δφ must be considered. Based on the theory presented in Section 2, the greater the difference between the initial φ_in braid angle and the neutral angle, the greater the potential for geometric change in the hose. While increasing the working pressure p, the braid angle changes from φ_in to φ_en. The greater the change in the braid angle Δφ during the increase in working pressure p, the greater the length strain ε_l will be. The initial braid angle φ_in was measured to be less than the neutral braid angle φ_N for all tested hoses. For all hoses tested, the initial braid angle φ_in was determined to be less than the neutral angle. It can therefore be assumed that all of the hoses tested will experience a length shortening. Other important aspects to be considered in the evaluation are the inner diameter d_in and the wall thickness s of the hose. When the working pressure p is applied to two hydraulic hoses with the same inner diameter d_in but different thickness s, it can be assumed that the hose with the smaller thickness s should achieve a greater length strain ε_l due to the bottom passive resistance due to the deformation of the hose wall. The next aspect that could affect the results is the length of the hose. There is a small difference in the length between the measured hoses, so the hose length change Δl was expressed as a percentage:

$${\epsilon }_l=\frac{l-l_{en}}{l}·100[\%].$$

(16)

where l is the initial hose length and l_en is the end hose length.

The dependencies can also be influenced by the different material properties and manufacturing processes of each hydraulic hose.

Figure 8 shows that the greatest length strain across all hoses is achieved by the hydraulic hose DN16_A, where the length strain was ε_l = 1.57% at the maximum working pressure p_max. The initial braid angle is φ_in = 52.51 ± 0.02° and the change in braid angle is Δφ = 1.18°. A similar length strain was achieved for DN12_A hose where the length strain was ε_l = 1.54% at maximum working pressure p_max and the initial angle φ_in = 51.01 ± 0.08°. This hose achieved the largest change in the braid angle Δφ = 1.35° of all the hydraulic hoses tested. The length strains ε_l versus pressure p curves are similar for these hoses. For comparison, the hydraulic hose DN12_C achieved the smallest length strain ε_l = 0.11% at maximum working pressure p_max. This hose experienced the smallest change in the braid angle Δφ = 0.32° of all the hydraulic hoses tested.

When comparing hydraulic hoses with an inner diameter d_in = 13 mm, it can be seen that the DN12_A hose with the largest deviation of the initial braid angle φ_in = 51.01 ± 0.08° from the neutral braid angle achieves the largest change in braid angle Δφ = 1.35° and the largest length strain ε_l = 1.54%. The hose DN12_B shows a smaller change in braid angle Δφ = 0.81°, which corresponds to a smaller change in length strain ε_l = 0.67%. The smallest change in braid angle Δφ = 0.32° and the smallest change length strain ε_l = 0.11% occurred in hose DN12_C, where the initial braid angle φ_in = 53.80 ± 0.05° was closest to the neutral braid angle. In the evaluation of the dependence ε_l = f (p) for hydraulic hoses DN12_A, DN12_B and DN12_C, all of the hypotheses mentioned above were confirmed.

A similar trend can be observed when comparing hydraulic hoses with an internal diameter d_in = 16 mm. Hydraulic hoses with an inner diameter d_in = 16 mm have similar initial braid angles φ_in, but each achieves a different change in the braid angle Δφ at maximum pressure p_max. The DN16_A hose that achieved the greatest change in braid angle Δφ = 1.18° also achieved the greatest length strain ε_l = 1.57%. It can also be observed that the hose DN16_C has a change in the braid angle Δφ = 0.97° and length strain ε_l = 0.97%. The smallest change in the braid angle Δφ = 0.54° and length strain ε_l = 0.56% was achieved by hose DN16_B. Although DN16_A has a larger initial braid angle φ_in than DN16_B, it achieves a greater change in the braid angle Δφ and a greater length strain ε_l. This could be due to the smaller wall thickness s of DN16_A.

In the comparison of hydraulic hoses with an inner diameter d_in = 19 mm, similar length strains depending on the working pressure can be observed. These hoses have a similar initial braid angle φ_in and show a similar change in the braid angle Δφ. The exception is hydraulic hose DN19_C with a smaller initial braid angle φ_in = 52.26 ± 0.03°, which shows the largest change in braid angle Δφ = 1.18° and the largest length strain ε_l = 0.96%.

From the resulting dependencies of the length strain ε_l with respect to the working pressure p and the measured braid angles, it is evident that the initial braid angle φ_in, the change in the braid angle Δφ and the thickness s of hydraulic hose have significant effects on the length strain ε_l. This can be seen from a comparison of the DN16_A and DN12_A hoses with different internal diameters d_in, where there is a significant length strain ε_l. In addition, the material properties of the steel braid and rubber and the manufacturing technology can also have an effect.

4.2. Evaluation Tensile Force Depending on Working Pressure

The measurement of the dependence of the tensile force F_T on the fluid pressure p was performed. In this case, the bottom end of the hose was connected to the force sensor without the ability to move the hose in the longitudinal axis. The initial braid angle, the end braid angle and the braid angle change were evaluated as in the previous case.

Table 6. Measured values of initial braid angles φ_in, end braid angles φ_en and braid angle changes Δφ of the hydraulic hoses tested for the experiment F_T = f (p).

Hydraulic Hose	Inner Diameter din	Outer Diameter dout	Wall Thickness s	Initial Braid Angle φin	End Braid Angle FT = f (p) φen	Change in Braid Angle Δφ
	(mm)	(mm)	(mm)	(°)	(°)	(°)
DN12_A	13	19	3	51.01 ± 0.08	51.78 ± 0.06	0.77
DN12_B	13	19.5	3.25	53.05 ± 0.10	53.74 ± 0.09	0.69
DN12_C	13	20	3.75	53.80 ± 0.05	53.93 ± 0.02	0.13
DN16_A	16	22	3	52.51 ± 0.02	52.97 ± 0.01	0.46
DN16_B	16	23	3.5	52.31 ± 0.05	52.66 ± 0.07	0.35
DN16_C	16	22.8	3.4	52.77 ± 0.08	53.41 ± 0.02	0.64
DN19_A	19	26	3.5	53.31 ± 0.03	53.62 ± 0.06	0.31
DN19_B	19	27	4	53.03 ± 0.05	53.60 ± 0.06	0.57
DN19_C	19	27	4	52.26 ± 0.03	53.37 ± 0.10	1.11

shows a summary of all the initial φ_in braid angles, the end φ_en braid angles and the Δφ braid angle changes achieved by each hydraulic hose in this measurement. It can be seen in the table that as the working pressure p increased, there was less change in the braid angle Δφ than in the previous experiment (ε_l = f (p)). This is due to the tight fit of both hose ends.

Figure 9 shows the dependencies of the tensile force F_T on the working pressure p for all hydraulic hoses evaluated. The resulting dependencies show significant changes in the tensile force F_T as the working pressure p increases. The initial braid angle φ_in has a major influence. Another factor may be the change in the braid angle Δφ, which is dependent on the increasing working pressure p. The thickness s of the hose itself must also be considered. The last aspect that can have a significant effect on the resulting tensile force F_T is the inner diameter d_in, because as the inner diameter d_in increases, the area on which the working pressure p acts increases. The graph shows that the hydraulic hose DN16_C, which has an initial braid angle of φ_in = 52.77 ± 0.08° and a change in braid angle of Δφ = 0.64°, achieved the highest tensile force F_T = 409 N at the maximum working pressure p_max. The smallest tensile force F_T = 17 N was achieved by hose DN12_C, which had the smallest deviation of the initial braid angle φ_in = 53.80 ± 0.05° from the neutral braid angle φ_N. The change in braid angle Δφ = 0.13° is significantly the smallest of all hoses.

In the comparison of hydraulic hoses with an inner diameter d_in = 13 mm, it can be observed that hose DN12_A with the largest deviation of the initial braid angle φ_in = 51.01 ± 0.08° from the neutral braid angle achieved a change in the braid angle Δφ = 0.77° and a tensile force F_T = 248 N. Hose DN12_B achieved a smaller change in the braid angle Δφ = 0.69°, but the initial braid angle φ_in = 53.05 ± 0.10° was greater. This factor influences the tensile force F_T = 99 N at the maximum working pressure for hose DN12_B. The smallest tensile force F_T = 17 N at maximum working pressure was achieved with hose DN12_C, which has the largest initial braid angle φ_in and the smallest change in braid angle Δφ. In the evaluation of the dependence of F_T = f (p) for hydraulic hoses DN12_A, DN12_B and DN12_C, the above hypotheses were confirmed.

In the comparison of hydraulic hoses with inner diameter d_in = 16 mm, the hose DN16_C achieved the highest tensile force F_T = 409 N at the maximum working pressure p_max. At the same time, this hose experiences a change in the braid angle Δφ = 0.64°, which is the largest in the comparison of hoses with the same inner diameter d_in = 16 mm. Hose DN16_A achieved a tensile force F_T = 344 N at maximum working pressure p_max with the change in the braid angle Δφ = 0.46°. The significantly smaller tensile force F_T = 176 N at maximum working pressure p_max and a change in the braid angle Δφ = 0.35° was achieved with hose DN16_B, and thus the above hypotheses were confirmed.

In the comparison of hydraulic hoses with inner diameter d_in = 19 mm, the hose DN19_B achieved the tensile force F_T = 345 N with a braid angle change Δφ = 0.57° and an initial braid angle φ_in = 53.03 ± 0.05°. Hydraulic hoses DN19_A and DN19_C have a similar trend of dependence of the tensile force F_T on the pressure p. Hose DN19_A achieved a tensile force F_T = 310 N with a change in the braid angle Δφ = 0.31° and an initial braid angle φ_in = 53.62 ± 0.06°. Hose DN19_C achieved a tensile force F_T = 313 N with a change in the braid angle Δφ = 1.11° and an initial braid angle φ_in = 53.62 ± 0.06°. In the evaluation of the dependence F_T = f (p) for hydraulic hoses DN19_A, DN19_B and DN19_C, the above theory was not confirmed. The hose DN19_C did not achieve the highest tensile force F_T due to the largest change in the braid angle Δφ and the smallest initial braid angle φ_in. Furthermore, as already mentioned, the different material properties and the manufacturing process of the hoses may also influence the resulting dependencies.

5. Conclusions

This paper analyzes the effect of the braid angle on the deformation of high-pressure hydraulic hoses under fluid pressure loading. The motivation for this work was previous research in the field of hydraulic hoses, where the properties of hydraulic hoses were investigated using various methods, including their modulus of elasticity and capacity. According to the knowledge obtained from the available sources, it can be said that the braid angle is one of the fundamental factors influencing the behavior of hydraulic hoses under fluid pressure loading. Manufacturers try to maintain a neutral braid angle during production. To verify the effect of the braid angle on the change in the geometry of the hydraulic hose, experimental equipment was constructed. Based on this equipment, the dependence of the change in hose length on the working pressure and the dependence of the hose tensile force on the working pressure was measured. The braid angle of the hydraulic hoses was measured during each measurement. The main objective was to investigate the effect of the braid angle on the behavior of hydraulic hoses under fluid pressure loading. The main contributions of this work include:

(1)

After evaluation of all hydraulic hoses tested, it was found that all tested hoses had a braid angle less than the neutral braid angle. As the working pressure increased, the hydraulic hoses shortened.

(2)

The dependence of the length strain of the tested hydraulic hoses on the working pressure was determined. Due to the partial removal of the rubber cover, the braid angle was detected for each measurement. A significant effect of the initial braid angle on the length strain of the hydraulic hose was confirmed. At the same time, the effect of changing the braid angle on the length strain was determined. Another aspect that influenced the resulting dependence was the wall thickness of the hydraulic hose.

(3)

The dependencies of the tested hydraulic hoses tensile force on the fluid pressure load were determined. When comparing the results, the influence of the initial braid angle on the magnitude of the tensile force was confirmed for most hydraulic hoses. It was found that the wall thickness of the hydraulic hose must also be considered. The manufacturing technology and material properties of the hoses will also affect the length strain and the tensile force.

The equipment assembled in this study can help to evaluate high-pressure hydraulic hoses and provide experimental data that are not available from manufacturers. This work also provided data for the verification of mathematical models of hydraulic hoses.