*Article* **Change in the Torsional Stiffness of Rectangular Profiles under Bending Stress**

**Krzysztof Macikowski 1,\*, Bogdan Warda 1, Grzegorz Mitukiewicz 1, Zlatina Dimitrova <sup>2</sup> and Damian Batory <sup>1</sup>**


**\*** Correspondence: kmacikowski@gmail.com; Tel.: +48-728528189

**Abstract:** This article presents the results of research on the change in torsional stiffness of two rectangular profiles, arranged one on top of the other, which were permanently connected at their ends. The flat bars were expanded in the middle of their active length. The test involved determining the increase in the stiffness of a twisted test set before and after expanding. The authors present an analysis of the structure load and compare the results of tests carried out using analytical (for selected cases), numerical and experimental methods, obtaining satisfactory compliance. The analytical calculations included the influence of limited deplanation in the areas of the profile's restraint. The ANSYS package software was used for calculations with the Finite Element Method. A change in the stiffness increase index at torsion was determined. The obtained results showed that expanding the test sets in their middle causes an increase in torsional stiffness, which is strongly dependent on the design parameters such as bending deflection, torsion angle and dimensions of the cross-section of the flat bar in the package.

**Keywords:** rectangular profile; torsional stiffness; stiffness increase; research; finite element method

**Citation:** Macikowski, K.; Warda, B.; Mitukiewicz, G.; Dimitrova, Z.; Batory, D. Change in the Torsional Stiffness of Rectangular Profiles under Bending Stress. *Materials* **2022**, *15*, 2567. https://doi.org/10.3390/ ma15072567

Academic Editor: Krzysztof Schabowicz

Received: 25 February 2022 Accepted: 25 March 2022 Published: 31 March 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

#### **1. Introduction**

The torsion of rectangular profiles is a load state in which the profiles are affected by the torsional moment in their cross-section plane. If the bars are restrained, their condition is described as restrained torsion. Torsional stiffness is the measure of the profiles' resistance to torsion and it depends on the material's mechanical characteristics and geometrical dimensions of the twisted profile [1].

The results of torsion tests are applied in many areas. In the automotive sector, they help design, e.g., torsion bars of variable stiffness. In civil engineering, they are used in the load-bearing systems of buildings [2–8] and other structures [9].

Ribeiro and Silveira [10] investigated the influence of changes in torsional stiffness by changing the distance between a car stabiliser's sleeves on the car body tilt while driving along an arch. The researchers determined that by increasing the test span from 100 mm to 300 mm, the tilt can be reduced from 2.76◦ to 2.65◦. Without modifying the other structural parameters of the bar, they changed its stiffness from 7.87 kN/m to 9.35 kN/m.

Owing to the use of a hydraulic BMW Dynamic Drive hydraulic mechanism [11,12] which changes the torsional moment load of the middle part of the cross stabiliser (by counter-moment generation), the body tilt can be controlled depending on the vehicle motion conditions. For a vehicle that moves along a 40 m radius arch, it was reduced from ca. 3◦ to ca. 0.5◦ (at the lateral acceleration of 5 m/s2).

Contrary to the BMW hydraulic system, Buma [13] used an electrical mechanism in the central part of the bar, connected with a toothed gear that enables reaching high torque values. The torque rotates the torsional bar in the opposite direction; the bar consists of two coupled parts. The test car body's tilt was reduced from ca. 2.1◦ to 1◦ at the lateral acceleration of 5 m/s2.

According to the results obtained by Doody [14] and Husen and Naniwadekar [15,16], the use of a metal sleeve (instead of a rubber one) for mounting the stabiliser increases its stiffness (including torsional stiffness of the central part) by 35% and 37%, respectively.

Shokry [17] investigated the strength of a rectangular beam made from polyester composite reinforced with glass fibre. The addition of nanoparticles was aimed at increasing the sample's bending strength. Other researchers also focused on the issue of bent profiles used for car leaf springs [18,19].

The results of experimental tests on reinforcing narrow beams with carbon fiber-reinforced laminates were reported by Bakalarz et al. [20]. The authors obtained 11% and 7% increase in the global modulus of elasticity in the bending and stiffness coefficients, respectively.

Wan and Jung [2,21,22] examined an LSB (LiteSteel beam) with a C-shaped crosssection, used in floor systems as ground beams or supporting beams, whose centre of gravity does not overlap with the shear centre. The results revealed that the torsional moment rises as the shear centre eccentricity increases, which leads to a significant decrease in the beams' resistance to the bending moment. Other researchers who carried out tests on beams with similar shapes obtained parallel results [23–25].

Structures used in everyday life are often exposed to complex states of stress. Simultaneous twisting and bending represent one of the most challenging cases. The literature review did not reveal any examples of two flat bars simultaneously subjected to such loads. In the presented paper, the numerical models of torsional stiffness of rectangular profiles under bending stress were determined. Additionally, the laboratory torsional tests of various sets of profiles with different cross-section shapes were conducted for the validation of the developed numerical models. The comparison of the results revealed good agreement between the registered numerical and experimental data, giving an opportunity of optimization of the profile shape and stiffening method of the analyzed sets in terms of achieving large changes in torsional stiffness with slight bending of the profiles.

#### *1.1. Analysis of a Unexpanded Test Package*

The presence of the cross-section limited deplanation results in auxiliary normal (tensile) stress occurring in addition to static (torsional) [26–30] stress in the profiles, contrary to pure torsion. This kind of torsion is called flexural torsion, and the total moment Ms necessary to twist the test set is a sum of two components: Mt,s and Mt,w. The first corresponds to pure torsion (Saint Venant), while the other corresponds to flexural torsion moment.

$$\mathbf{M}\_{\mathfrak{s}} = \mathbf{M}\_{\mathfrak{t},\mathfrak{s}} + \mathbf{M}\_{\mathfrak{t},\mathbf{w}} \tag{1}$$

A literature analysis [31–35] reveals that the Mt,s and Mt,w shares change in a nonlinear way along the calculated element. The first ones have a constant value, while the latter ones decay rapidly as the distance to the fixing points increases. The flexural torsion moment Mt,w depends on several factors:


The outcomes of previous research [26,31] indicate that the torsional strength of opensection thin-wall profiles increases when torsional moments are taken over not only by unrestrained torsion but also by the forces warping the flanges in different directions, i.e., by flexural torsion stress. In other words, an increase in the Mt,w share in the total Ms

makes a structure stiffer. In order for the strengthening effect to occur, the following two prerequisites have to be fulfilled:


#### *1.2. Analysis of an Expanded Test Package*

As a result of the profiles' expanding, additional bending stress occurs in the middle of the active length. The stress distribution changes, and so do the shares of Mt,s and Mt,w in the Ms transferred torsional moment. It can be evidence of the change in the torsional stiffness under expansion (bending stress).

The available literature misses examples of calculations for expanded flat bars. An analogy can be found in construction, where calculations for a twisted I-section present a similar case. If the analogy is used, the web thickness shall be assumed as zero. Unfortunately, the analogy can be applied only to flat bars whose shape did not change under expansion, i.e., distant from one another (e.g., for unexpanded profiles' calculations). A sample calculation for I-sections is presented in EUROCODE 3 [34]. The method consists in the determination of a substitute warping moment called bimoment. To that end, the bending center shall be identified, and warping inertia determined.

The flexural torsion phenomenon and its influence on the torsional resistance of I-sections was explicitly presented in the SCI study [32], where the torsional stiffness coefficient was determined, along with its change depending on the beam length. Computational procedures were also presented for load cases common in engineering.

The literature hardly provides other examples to gain knowledge for a more detailed analysis of profiles under bending stress. There is no information about expanding arched beams. Moreover, there is a lack of information on bending or twisting elements with initial stress (analogy to stress that occurs when the profiles are expanded).

To summarize, it shall be concluded that it is hard to twist expanded bars. An approximate solution can be obtained in the best-case scenario, requiring long-lasting and labour-consuming analytical efforts.

This study includes experiments and simulation tests on two flat bars arranged one onto another, having a fixed length and variable cross-section dimensions. The flat bars were durably connected at the ends and expanded in the middle of their active length, ranging from 0 to 30 mm, and then twisted. The results of the tests can be applied in a design of a vehicle stabiliser of variable stiffness [36].

#### **2. Materials and Methods**

The tests were carried out on a package composed of two rectangular profiles connected permanently at the ends. The profiles were expanded in the middle of their active length and then twisted. The influence of bending on the change in test package torsional stiffness was analyzed.

The test set (Figure 1), with the twisted active length L amounting to 528 mm, consisted of one pair (two pieces) of flat bars, with the possible extension of the package with successive pairs. The tests were planned for 51CrV4 spring steel plate packages, 3 mm and 6 mm thick (H) and 30 mm to 60 mm broad (B). With regard to the mechanical resistance, including but not limited to thicker (higher) test sets, the maximum expansion (R) was limited to 30 mm. The maximum torsion value (ϕ) was assumed to amount to 20◦. The assumed values result from an analysis of stabilizer bars used in typical passenger vehicles. An occurring space limitation and required resistance for twist (a torque needed to twist) in a middle part of the bar cause natural constraint for using shapes not exceeding the given values.

**Figure 1.** Schematic diagram of a test set (unexpanded on the top and expanded on the bottom).

A load analysis enables the determination of two characteristic states: torsion of unexpanded profiles and torsion of expanded profiles. It was determined that the test package was exposed to the restrained torsion phenomenon and that the deplanation capacity of the flat bars in the restraint areas was limited. Moreover, the profiles were identified to be subject to a complex stress state that depends on the analysed form of the test set (unexpanded and twisted or expanded and twisted). The following types of stress were distinguished:


#### *2.1. Analytical Study*

Analytical calculations of an unexpanded test package were carried out based on the material strength knowledge, including but not limited to thin-walled bars. The course of the calculations is not presented in this paper. Further sections include the obtained results that are compared with the results of numerical simulations and experiments.

#### *2.2. Numerical Study*

A simulation using the Finite Element Method was performed in the ANSYS environment [37,38]. The computational model's structure reflected the real one in the restraining and loading method. Solid elements were used. This way, the edge folds used for mounting the bending mechanism were taken into consideration. Every profile was divided into five sections (Figure 2). The outer sections (1) and (5) corresponded to durable restraint. The middle section (3) was expanded. The other two sections, (2) and (4), were free, and their strain during simulation resulted from the restraining method and working conditions of the rectangular bars in the test stand.

A permanent bond was used at the contact planes of the flat bars' outer edges. It corresponded to the processing capabilities of the test sets' preparation (i.e., welding of the six available edges—see Figure 1). All degrees of freedom were taken on the outer planes of section (5). Friction between the rectangular bars (on Sections 2–4) was taken into consideration. For the unexpanded sets, the coefficient of friction amounting to μ = 0.2 was applied. For the expanded sets, it was μ = 0. This way, the mutual penetration of the profiles during twisting was avoided.

**Figure 2.** Numerical model. Successive sections are marked with digits from (1) to (5).

The model's discretization was carried out in the Ansys Meshing tool, using non-linear mechanics settings. The Multizone algorithm was applied. HEXA20 and WED15 type cubic elements were generated. The first one was a dominant type. A single profile's thickness in the test set was adopted as the mesh size. The number of the nodes changed depending on the model size and ranged from ca. 20,000 to ca. 80,000. The number of elements ranged from ca. 3500 to ca. 15,000. The mesh quality was checked with Orthogonal Quality indices (ca. 0.7–0.9), Skewness (ca. 0.2–0.5) and Aspect Ratio (ca. 1.5–5). The obtained values, depending on the test model's dimensions, were sufficient for the simulation. No inflation or local concentration of the mesh was used. Quadratic element order function was applied just as it was done by Jafari [39].

The analysis included a non-linear material model (Figure 3). Its parameters were obtained owing to a static tensile test. It was carried out for all tested flat bars' thicknesses. The results were converted into real stress. Material stiffening under significant strain was also taken into account. The force, torque, displacement and rotation margins were assumed (from 0.05% to 0.1%). It enabled obtaining reproducible results with low sensitivity to the mesh size changes.

**Figure 3.** Non-linear model of a 6 mm thick material. The course of actual stress depends on the material deformation. The conventional yield point amounts to Re = 1417 MPa.

Bending and twisting were performed using the remote displacement function. It enabled to define necessary degrees of freedom, leaving the other ones non-defined. The displacement along axis *Y* (from 0 to ±15 mm) was applied to the inner planes of section (3). Twisting (by 20◦) around axis *Z* and displacement equal 0 mm along axis *X* and *Y* was applied to the outer planes of section (1)—Figure 2.

The simulation was performed in two stages using the remote displacement function (Figure 4). In the first stage, the profiles were expanded, while in the second one they were twisted. The reactive moment was measured on the outer planes of section (1). In a purpose of strength control the reduced stress was checked.

**Figure 4.** (**a**): test set expanded by 30 mm. (**b**): test set expanded by 30 mm and twisted by 20◦.

The prepared model helped to perform numerical tests in the whole planned range. The tested profiles were 3 mm and 6 mm thick and 30 mm to 60 mm broad. They were expanded within the 0–30 mm range with a 5 mm stroke. The maximum twist amounted to 20◦. The results obtained for unexpanded sets were compared with the results of analytical calculations. They were characterised by high convergence, ranging from −3% to +5.5%. It was then assumed that the simulation had been correctly prepared, and the results obtained for the expanded test packages were correct and confirmed by the experiment results.
