*2.3. Experimental Study*

The developed test stand enabled obtaining reproducible measurement results at a short duration of a single set's test. It consists of five essential assemblies, shown in Figure 5a.

**Figure 5.** (**a**) Test stand. (**b**) Test set 6 × 60.

A computer set (1) enables data saving and processing. The twisting of the test sets starts at the drive assembly (2), where it is forced with a mechanical lever and a screw-nut transmission powered by an electric motor. It ensures continuous moment and torsion measurement. The rotational motion is transferred by the assemblies of the torque meter (3) and a double-joint shaft (4) to the test set (5), where the tested profile packages are installed; the torque meter and double-joint shaft assemblies are connected in series.

The test sets are mounted on supports. One end of each test set is permanently connected, while the other (connected with the double-joint shaft) enables rotation along the profile's lengthwise axis and compensation of its length change resulting from expansion and twisting. A tilt sensor is also installed there. The sets are expanded with spacer wedges, from 5 mm to 30 mm thick, with a 5 mm stroke. Pressure plates press them from the outside.

The sensors used during the tests included:


Additional calibration of torque meters was conducted by applying the particular static force to an arm of known length.

The test sets consist of two flat bars of the same length. In the central part of each flat bar, there are two folds of the inner edges that enable their use in subsequent tests. The profiles were made of 50 HF spring steel plate and connected at the ends by welding. Then they were quenched to ca. 46–48 HRC and tempered. The profiles are shown in Figure 5b.

The system operation was checked after installing the test packages on the test stand. Test twisting was performed, and the operation of the limit switches and tension of fixing bolts were checked. The repeatability of the test was verified by preliminary torque comparison for minimum and maximum profiles twist. The preliminary twisting was repeated three times for each analyzed profile set to eliminate the clearance. Subsequently the consistency was verified by torque curves comparison. After the preliminary inspection, the actual measurement was carried out. The profiles were twisted from 0◦ to 22◦, while recording the torque and twist angle. After reaching the desired twist angle, the system switched off automatically and saved the data. The unloading measurement was performed in the same way. A reduction in the maximum recorded torsion was observed as the stiffness of the test sets increased. The changing stiffness forced the use of three torque meter types, adapting the measurement range in this way.

The test results were processed. Noise and interferences were removed, and the measurement range adapted (from 0◦ to 20◦). Diagrams of the torque change depending on the torsion were developed for the data prepared this way. Trend lines were plotted for the obtained wavelengths. The best representation was obtained for the fourth-order polynomial. Based on these, the torsional moment was calculated and its results were used for determining the K stiffness increase index.

#### **3. Results**

#### *3.1. Numerical Tests*

A change in the moment depending on the test package twisting, and expansion was analyzed during the numerical tests for two groups—3 mm and 6 mm thick. The plotted diagrams were used for subsequent analyses. Sample diagrams (plotted for the cross-section extreme dimensions: H × B—3 × 60 and 6 × 30) are shown in Figure 6. The color intensity of each curve corresponds to successive expansions levels—from 0 mm (the lightest) to 30 mm (the darkest).

It was observed that a non-linear waveform characterized the packages made of thin and broad profiles (3 × 60). The more evident, the more expanded the test sets were. The diagrams (Figure 6a) reveal a decreasing moment gradient between successive torsion levels. The described effect confirms the growing share of the flexural moment Mt,w in the

entire profile torsional moment Ms. The observed bending of the curves in the plot reveals the non-linear dependence of Mt,w on the torsion.

**Figure 6.** Change in the torsional moment of the 3 × 60 and 6 × 30 test sets during the numerical twisting test. The following expansions are marked with colors from the lightest to the darkest: 0, 5, 10, 15, 20, 25, and 30 mm. (**a**) 3 × 60 package. (**b**) 6 × 30 package.

The results of numerical tests are summarized in Table 1. The first column on the left includes the dimensions of a single profile cross-section dimensions (H × B; height (thickness) × breadth). The top row shows the successive expansion values from 0 (no expansion) to 30 mm. In order to facilitate understanding, each level is preceded by the R index. The entered values represent the torsional moment for a 20◦ twist.

In both analyzed thickness groups (3 mm and 6 mm), reproducible regularities can be observed. The torsional moment increases depending on the expansion. The lowest values occur for unexpanded test profiles, while the highest ones are achieved for the maximum expansion amounting to 30 mm.

The 3 mm thick sets are characterized by a measurement range from 30.4 Nm to 188.6 Nm. For the 6 mm thick packages, the values range from 214.4 Nm to 919.8 Nm.

The results of the numerical experiment helped to determine the profiles' stiffening under bending. A per cent K stiffness increase index was developed according to formula (2) as a quotient of torsional moment read for expanded profiles Ms,R to torsional moment read for not expanded profiles Ms,0:

$$\mathbf{K} = \left[ (\mathbf{M}\_{\rm s,R} / \mathbf{M}\_{\rm s,0}) \times 100 \right] - 100 \left[ \% \right] \tag{2}$$

The index shows the value by which the torsional stiffness of the bar's working part changed. For instance, a 10% increase means that the test set generates a 10% higher response moment than before the expansion. An increase by 100% signifies a two-fold increase in the torsional stiffness. The calculated values of the K index are summarised in Table 2, which shall be interpreted strictly in the same manner as Table 1.


**Table 1.** Results of numerical simulations. Torsional moment [Nm] for 20◦ twist, depending on the package's cross-section dimensions (H × B) and the expansion R [mm].

**Table 2.** Numerical simulation. Summary of the percentage value of the K stiffness increase index, depending on the test package (H × B) and expansion R [mm].


In both analysed thickness groups (3 mm and 6 mm), repetitive regularities can be observed. The index value increases from 0% (for unexpanded profiles) to the maximum value (achieved for 30 mm expansion), and the growing breadth of the test packages contributed to higher stiffness gains.

Thin sets (3 mm) responded to expansion most sensitively. The maximum value of 171.3% was obtained for the 3 × 60 package, while the minimum of 77.0% was obtained for the 3 × 30 package. Thick sets (6 mm) were less susceptible to expansion. The maximum value amounting to 82.6% was obtained for the 6 × 60 package, while the minimum was 27.5% (for the 6 × 30 package).

Experiments confirmed the results of numerical simulations.

#### *3.2. Experiments*

The experiments covered the same test sets as the ones used in numerical simulation. The packages consisted of two profiles with a single flat bar thickness of 3 and 6 mm. The expansion was performed within a 0 mm to 30 mm range, with a 5 mm stroke.

Hysteresis between the profiles' loading and unloading was observed during the tests. The hysteresis resulted from inner friction in the test package material [40] and flexibility of the test stand elements.

The results were processed to remove the noise and interferences. A diagram of the torque change during twisting was prepared for each test set. Only the curves obtained while twisting the profiles were used for their analysis. The comparison between the particular curves obtained for the same sample sets revealed that the coincidence was satisfactory (less than 0.5% for different twist values) and did not require more precise data processing. Sample waveforms for 3 × 60 and 6 × 30 sets are shown in Figure 7. The color intensity of each curve corresponds to successive expansion levels, from 0 mm (the lightest) to 30 mm (the darkest).

**Figure 7.** Experiment. Torsional moment (torque) change depending on the test set torsion. The following expansions are marked with colors, from the lightest to the darkest: 0, 5, 10, 15, 20, 25 and 30 mm. (**a**) 3 × 60 package (**b**) 6 × 30 package.

The waveforms of the unexpanded profiles were nearly linear. The expanded sets, similar to the results of the numerical experiment, were characterised by non-linearity. It was most evident for thin (3 mm), broad (mostly at 60 mm) and expanded (mostly at 30 mm) profiles. In each of these situations, the moment's gradient between subsequent torsion levels revealed decreasing values. Trend lines were plotted for the obtained wavelengths. The best representation was obtained for the fourth-order polynomial. The functions enabled the moment calculation at a 20◦ twist of the test sets. The results are summarised in Table 3.

**Table 3.** Results of the bench experiment. Torsional moment Ms [Nm] for 20◦ twist, depending on the package's cross-section dimensions (H × B) and expansion R [mm].


Similar to the numerical experiment, the torsional moment values were observed to increase with the test set's thickness, breadth, and expansion. The lowest values in each group were read for narrow, unexpanded profiles. Analogically, the highest ones applied to broad profiles at maximum expansion.

The 3 mm thick sets are characterised by the measured moment range from 30.5 Nm to 168.0 Nm. For 6 mm test packages, it was from 215.8 Nm to 846.8 Nm.

The results of the experiment carried out in the test stand enabled the determination of the profile's stiffening under bending. The per cent K stiffness increase index was calculated according to formula (1). The index values are summarised in Table 4.


**Table 4.** Results of the bench experiment. Summary of the percentage value of the K stiffness increase index, depending on the test package (H × B) and expansion R [mm].

Repetitive regularities can be observed in both analysed thickness groups (3 mm and 6 mm). The index value rises from 0% (for unexpanded profiles) to the maximum value (achieved for 30 mm expansions). The increasing breadth of the test packages causes higher stiffness gains.

The profiles responded most intensively to expansion in thin sets (3 mm). A maximum of 133.3% was achieved for the 3 × 60 mm package, while a minimum of 79.8% was achieved for the 3 × 30 package. Thick sets (6 mm) demonstrated lower susceptibility to expansion. A maximum of 69.3% was achieved for the 6 × 60 package, while a minimum of 29.7% was achieved for the 6 × 30 package.

#### **4. Discussion**

#### *4.1. Accurancy Analysis*

To provide good quality results discussion, it is helpful to know the measurement error. Simulation and experimental uncertainty were estimated.

#### 4.1.1. Experiment Test

The errors in experiments include:

1. A systematic error (δ<sup>i</sup> of the inclinometer, δ<sup>m</sup> of the torque meter and δad of the measurement results proximation). It amounts to:

$$
\delta\_\mathrm{P} = \pm \sqrt{\left(\delta\_\mathrm{i}\right)^2 + \left(\delta\_\mathrm{m}\right)^2 + \left(\delta\_\mathrm{ad}\right)^2} = \pm 2.63\,\%\,\mathrm{s}
$$


$$\delta\_{\rm g} = \pm \sqrt{\left(\delta\_{\rm cd}\right)^2 + \delta\_{\rm sp}}^2 + \delta\_{\rm mp}\,^2 + \delta\_{\rm ks}\,^2 + \delta\_{\rm nm}\,^2\mathrm{s} = \pm 4.77\% \,\mathrm{s}$$

3. The δbp random error was assumed as:

$$
\delta\_{\mathsf{b}\mathsf{p}} = \pm \mathsf{S}^{\alpha} \lhd\_{\mathsf{b}}.
$$

#### 4.1.2. Numerical Simulations

It is hard to estimate a numerical simulation error. It is affected by divergences between the experiment and simulations and FEM errors. Based on experience and professional literature [41], the following accuracy of the computational model was assumed:

$$
\delta\_s = \pm \mathbb{S}^o\_{\mathbb{C}}
$$

#### 4.1.3. Accuracy of Measurements

The δ<sup>e</sup> experimental method's accuracy was determined as:

$$
\delta\_{\rm e} = \pm (\delta\_{\rm P} + \delta\_{\rm g} + \delta\_{\rm bP}) = \pm 12.40\%
$$

Therefore, the maximum difference between the results obtained on the test stand and the results of numerical simulations is:

$$\delta\_{\rm c} = \pm (\delta\_{\rm c} + \delta\_{\rm s}) = \pm 17.40\%$$

### *4.2. Analysis of the Moment Twisting the Unexpanded Test Sets*

Unexpanded test sets in each measurement series generated the lowest twisting moment. It represents the reference value necessary to determine the per cent K stiffening increase index for flat bars. Table 5 summarises the results of three test methods obtained for the profiles twisted by 20◦. The values were similar. The knowledge of the differences will help determine the accuracy of the tests.

**Table 5.** Summary of the results of calculating the torsional moment [Nm] of unexpanded test sets using analytical, numerical and experimental methods.


Based on Table 5, the numerical and experimental method error versus the analytical method were determined. The obtained values were summarised as a diagram in Figure 8. The values lower than one (ordinate axis) meant higher torque values than those suggested by analytical calculations and vice versa for the lower ones.

**Figure 8.** Per cent error of the numerical and experimental methods versus the analytical method.

An evident declining trend characterised all waveforms. Its occurrence can be explained by material loss in the central section of the flat bars, occurring when the edges are milled to mount the expanding mechanism. Residual stress in the material (occurring during heat treatment) can also contribute to the observed curves' waveform; they are larger for broader profiles.

Similar error values characterised the 6 mm thick sets in each test. They ranged from +0.7% to +5.5%. The differences between the experimental and numerical methods revealed negligibly low differences (max. 1.6% for the 6 × 55 package).

Divergences can be observed for the 3 mm thick packages. The highest error was observed during the experiment (3 × 55 package) and amounted to 8.3%. The comparison of the results obtained in the simulation and the experiment reveals significant differences. A variable error ranging from 6.5% to 6.9% was observed between the 3 × 45, 3 × 50 and

3 × 60 packages. It can be concluded that test sets with low stiffness (3 mm) are more susceptible to the impact of the test stand execution inaccuracies and mounting errors. This was observed in the three presented tests.

The influence of limited deplanation on the entire torsional moment Ms was considered in the calculations. It's percent share is marked in orange in Figure 9 (the right ordinate axis). Grey corresponds to the Ms torsional moment (the left ordinate axis), while blue corresponds to the restrained warping moment (the left ordinate axis).

**Figure 9.** Limited deplanation share in the Ms total torsional moment affecting the unexpanded test sets.

The diagram analysis reveals that the limited deplanation moment increases with the test sets' breadth. Its per cent shares take similar values, regardless of the thickness of the flat bars in the set. They range from 5.3% to 10.1%. The per cent differences between the corresponding values (e.g., 5.3% corresponds to 5.5% for the 3 × 60 and 6 × 30 sets) do not exceed 3.5%. This is helpful information for a designer who constructs a solution based on a similar principle. Instead of time-consuming calculations of limited deplanation, the result obtained for pure torsion can be increased by λ deplanation dimensionless value. The formula (3) can be used for this purpose:

$$\mathbf{M}\_{\rm s} = \mathbf{M}\_{\rm t,s} \ast \left[ (100/(100-\lambda)) \right] \quad \text{[Nm]} \tag{3}$$

*4.3. Comparison of the Simulation and Experimental Results*

4.3.1. Torsional Moment Research

Non-Linearity of the Torsional Moment's Wavelength

The tests were carried out to determine the stiffening after twisting the set by 20◦. The influence of the non-linear waveform of the moment when a non-expanded set is twisted must not be neglected in order to understand the phenomenon comprehensively. For a nearly linear waveform (for the unexpanded package), it causes the variability of the K stiffness increase index.

An additional analysis of the K stiffness increase index change was performed for the 3 × 60 set (based on the experiment results). The scope of the study was limited to the torsion range from 3◦ to 20◦. Interferences disturbing the observations occurred below the minimum value. The maximum strengthening for the most significant expansion (R = 30 mm) amounts to 320.3% (at twisting by ϕ = 3◦), whereas for twisting by ϕ = 20◦, it was only 133.3%. Similarly, though not as high, differences can be read for other expansions. The results are summarised in the diagram in Figure 10.

**Figure 10.** Variable value of the K stiffness increase index depending on the test set twisting.

The analysis above suggests the possibility of changing the K stiffness increase index value by changing the twisting range. For the 3 × 60 package, the K index increased by 2.4 times.

#### Results of Torsional Moment Tests

The results obtained with the numerical and empirical methods were compared. An error versus the tests performed in the test stand was calculated and is summarised in Table 6. A three-colour scale was used for better understanding. The stronger colour indicates a higher per cent difference between the compared moments. Red (positive values) indicates a moment lower than the one obtained in the experiment, while blue indicates the opposite. White stands for the same value in both measurement methods.

**Table 6.** Summary of the results of calculating the torsional moment [Nm] of unexpanded test sets using analytical, numerical and experimental methods.


An analysis of the error distribution helps us to notice that the numerical model stiffening trend dominates. As expansion increases, red fades and turns into blue. This means that the simulation model becomes stiffer earlier than the experimental one.

The differences exceeding 10% were obtained for the broadest set (3 × 60) at two measurement points—at 25 mm (11.4%) and 30 mm (12.3%) expansion. In the group of packages made of 6 mm thick steel plate, the most significant deviation (for the 6 × 60 test set) amounted to 8.6%. The quoted values do not exceed the calculated measurement difference (±17.4%).

The differences between the experiment and simulation results are caused by inaccurate test stand execution, stiffness, the technology of making the test sets, non-homogenous material structure, and divergent numerical simulations versus real conditions. High deviation values observed for the broadest packages may result from a simplified method of applying the nodes in the fixing areas (FEM simulation). The area of the mounting elements in the test stand was wavy, whereas in the numerical method, it was a plane. This could contribute to the occurrence of micro-motions or local material upsetting during the test, which reduced the value of the read torsional moment in real conditions.

The mean relative error amounted to:


The mean result for the relative error lower than 5% confirms that the FEM model was correctly prepared. None of the calculated deviations exceed the determined maximum measurement difference of 17.4%.

#### 4.3.2. K Stiffness Increase Index

The K stiffness increase index informs about the quantitative increase in the torsional moment affecting the test set under expansion. The torque measured for non-expanded profiles is the reference value, hence in this case, the coefficient value is always 0%. It was calculated for the measurements made with the numerical and experimental method.

Two factors were hampering the analysis of the K stiffness increase index. Firstly, the moments read for each measurement method had different values. It resulted from measurement differences (mentioned in Section 4.1). Secondly, the comparison of values suffering from deviations independent of one another causes a risk of their overlapping and consequently strengthening or weakening.

Due to the above-mentioned factors, it was decided to not compare numerical and experimental K index results. Such results would not be a reliable source of knowledge about achievable strengthening. Still, they can be used to estimate the strengthening achievable for the particular test set.

#### **5. Conclusions**

This paper is devoted to the study of change in the torsional stiffness of expanded rectangular profiles connected permanently on both ends. Two bending stress states were analysed: first, twisted flat bars, and second, expanded and twisted flat bars. The first state was tested with analytical, numerical, and experimental methods; the other was based on FEM simulation and doing an experiment at the test stand. Moreover, a measurement error was analysed, the share of limited deplanation in the total torsional moment investigated, and the stiffness increase index change during twisting determined.

The analysis of the study works leads to the following conclusions:


Positive results obtained for test sets made of spring steel suggest that the use of modern composite materials would allow us to obtain a higher stiffness increase index.

According to the authors, the study results can be used for designing a car stabiliser that actively changes its stiffness. The decreasing torque gradient during twisting of the expanded working part can positively influence travelling comfort and safety, fulfilling the function of informing the driver about approaching the steerability limit.

### **6. Patents**

Dmitrova Z, Kaszuba S, Macikowski K. Systeme Anti-Devers A Raideur Variable Comportant Des Series De Barres Qui S'ecartent Entre Elles. FR3 057813, 2018.

**Author Contributions:** Conceptualization, K.M.; methodology, K.M.; software, K.M.; validation, K.M., B.W., G.M., Z.D. and D.B.; formal analysis, K.M.; investigation, K.M.; resources, K.M. and G.M.; data curation, K.M.; writing—original draft preparation, K.M.; writing—review and editing, K.M., B.W., G.M., Z.D. and D.B.; visualization, K.M. and G.M.; supervision, B.W. and G.M.; project administration, D.B.; funding acquisition, K.M. and G.M. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data is contained within the article.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

