# Cross-Section Deformation and Bending Moment of a Steel Square Tubular Section

^{*}

## Abstract

**:**

## 1. Introduction

_{zw}: deflection of the inner horizontal wall,

_{zz}: deflection of the outer horizontal wall,

_{yw}: deflection of the inner profile corners,

_{yz}: deflection of the outer profile corners,

_{ym}: maximum lateral displacement of points of the wall beyond the original cross-section.

- FEM numerical modelling during the cold plastic shaping process of a round tube with an external diameter (d = 30 mm) and a wall thickness (g = 2 mm) into a square cross-section pipe.
- FEM numerical modelling of the process of forming a square pipe with a pure bending moment. Model A took into account the technological deformations and residual stresses created during shaping of a square pipe. Model B did not take into account the impact of technological deformations and internal stresses created during shaping of a square pipe.
- Experimental four point bending of a 25 × 25 × 2 mm square pipe was tested to verify the results of numerical calculations.

## 2. Material and Experimental Procedure

_{e}= 388 MPa; ultimate strength R

_{m}= 486 MPa; Young modulus E = 199 GPa; Poisson’s ratio ν = 0.3.

## 3. Numerical Calculations

^{−1}were the bending moment values numerically calculated. Over the wide range of bending curvatures, the bending moment values did not change much, which was partially a result of the pipe material’s slight strain hardening. On the other hand, even a slight strain hardening should increase the value of the bending moment. At the same time, the decrease the cross-section’s moment of inertia value was a result of the walls deflecting, especially the horizontal ones, which completely eliminated the deformation strengthening effect. For bending curvatures κ > 6.0 m

^{−1}, the pipe collapsed locally, which was accompanied by a sharp drop in the bending moment value.

## 4. Conclusions

^{−1}in the initial bending phase, we observed a decrease in the bending moment value. For a curvature of κ > 6.0 m

^{−1}, the pipe collapsed locally. Satisfactory agreement of the experimental results and numerical calculations was obtained by considering the technological deformations and residual stresses created during the cold plastic shaping process of a round tube into a square cross-section pipe.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Characteristic tube dimensions and displacements of the deformed cross section [8].

**Figure 2.**Laboratory four point bending test device [8]: 1—lever, 2—clamping jaws, 3—sample, 4—connector, 5—bending arm, and 6—traverse of the testing machine.

**Figure 3.**Change in the shape of the cross section of the pipe in the subsequent stages of profiling: (

**a**) round pipe, (

**b**–

**d**) intermediate shapes; (

**e**) square pipe.

**Figure 4.**Distribution and values of the total equivalent plastic deformations in the pipe material after cold shaping; value scale: (

**a**) strain range (0–0.55) and (

**b**) strain range (0–0.10).

**Figure 5.**Distribution and values of the Huber–Mises equivalent stresses in the material of the shaped pipe: (

**a**) in the final shaping phase and (

**b**) after the shaping process (internal stress).

**Figure 6.**3D model of the analyzed bending process: 1, bent beam with a fine element (FE) mesh; 2 and 3, surfaces of the clamping jaw; 4 and 5, longitudinal and transverse symmetry planes.

**Figure 7.**The change in the cross-sectional shape of a bent beam (model A) and the distribution of total substitute plastic deformations for: (

**a**) 1/R = 0 m

^{−1}, (

**b**) 1/R = 4 m

^{−1}, (

**c**) 1/R = 6 m

^{−1}, and (

**d**) cross-section templets before (upper) and after bending the local break (lower).

**Figure 8.**Change in cross-sectional shape of the bent beam (model B) and distribution of total substitute plastic deformations for: (

**a**) 1/R = 0 m

^{−1}, (

**b**) 1/R = 4 m

^{−1}, and (

**c**) 1/R = 6 m

^{−1}.

**Figure 9.**The effect of bending curvature when deflecting the horizontal walls of the 25 × 25 × 2 square steel pipe.

**Figure 10.**The effect of bending curvature when deflecting the vertical walls of the 25 × 25 × 2 square steel pipe.

**Figure 11.**The effect of bending curvature on the bending moment of the 25 × 25 × 2 square steel pipe.

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**MDPI and ACS Style**

Kut, S.; Stachowicz, F.
Cross-Section Deformation and Bending Moment of a Steel Square Tubular Section. *Materials* **2020**, *13*, 5170.
https://doi.org/10.3390/ma13225170

**AMA Style**

Kut S, Stachowicz F.
Cross-Section Deformation and Bending Moment of a Steel Square Tubular Section. *Materials*. 2020; 13(22):5170.
https://doi.org/10.3390/ma13225170

**Chicago/Turabian Style**

Kut, Stanisław, and Feliks Stachowicz.
2020. "Cross-Section Deformation and Bending Moment of a Steel Square Tubular Section" *Materials* 13, no. 22: 5170.
https://doi.org/10.3390/ma13225170