# The Influence of the Layer Arrangement on the Distortional Post-Buckling Behavior of Open Section Beams

^{*}

## Abstract

**:**

## 1. Introduction

^{®}version 18.2 [26]. Additionally, the experimental tests were carried out for a few cases in order to validate the proposed numerical model.

## 2. Experimental Tests

#### 2.1. Laminate Material Properties Determinations

_{1}and in transvers to fiber direction E

_{2}, Poisson ratio ν

_{12}and ultimate tensile stresses in fiber direction X

^{T}and in transvers to fiber direction Y

^{T}have been determined according to ASTM D 3039 standard [28]. Kirchhoff modulus G

_{12}and ultimate shear stress S have been determined in the specimens with layer arrangements ±45° in tensile tests according to ASTM D3518 standard [29]. The ultimate compressive stress in fiber and transvers to fiber direction X

^{C}, Y

^{C}were determined according to ASTM D3410 standard [30]. It should be noted that the most difficult to determine is the ultimate compression stress in fiber direction X

^{C}, due to the possibility of buckling. However, comparing obtained value with cases presented in literature [31,32,33] it was found that X

^{C}/X

^{T}ratio for unidirectional GFRP material varies from 0.5 to 1.

#### 2.2. Bending Tests of Channel Section Beams

_{S}; [45/−45/90/0]

_{S}; [45/−45/45/−45]

_{S}. Nine specimens have been tested—three for each layer layups The cross-section dimensions of all beams have been measured before the test and presented in Table 2 (used notation correspond to those presented in Figure 1).

_{b}(α

_{V}) and M

_{b}(α

_{H}) were determined. Additionally, using a 3D optical system, the deflections of the beam were analyzed.

## 3. Numerical Model

^{®}version 18.2 [26]. The tested beams were subjected to pure bending. The model, i.e., geometry, boundary conditions and the way of load appliance has been assumed as close as possible to those in the experimental test stand (Figure 2). The geometry of the proposed numerical model is very close to those presented in [35].

_{3}= 0, see Figure 3) and lip-channel (b

_{3}≠ 0) section beams, made of laminate with different layer arrangements have been considered.

_{12}= 0.3.

_{1}× b

_{2}and the same thickness t as the composite beam’s specimen. Having in mind that the real grips are very stiff, additionally in cross-section of numerical model where the load was introduced and the structure was supported the diaphragms have been applied (see Figure 4a). It allowed the avoidance of the occurrence of stress concentrations and unexpected deformations. The geometry of the developed model is presented in Figure 4a, where the lines and points taken for the load and boundary condition introduction are depicted. The assumed boundary condition and the method of load introduction are also presented in the discretized model (see Figure 4b). The boundary conditions have been set as: (i) displacement set to zero at all nodes lying on the lower edges of the end cross-section (u

_{y}= 0); (ii) displacement in the horizontal plane in transvers direction to the beam axis at nodes lying in one bottom corner of the end section set to zero (u

_{x}= 0); (iii) displacement towards the beam axis set to zero at node located in the mid-span of the beam lying on the edge of bottom flange and the web (see Figure 4).

_{z}= const.) have been set. The vertical force has been used as a load in case of linear buckling analysis and the vertical displacements loaded the system in case of nonlinear analysis. Both types of the load have been set to one node lying along loading lines (Figure 4).

_{V}—vertical (in plane of load) beam rotation at support, α

_{H}—horizontal (lateral to the load plane) beam rotation.

## 4. Numerical Model Validation by the Results of the Experimental Tests

_{1}= 81 mm, b

_{2}= 40 mm, b

_{3}= 0 and two different thicknesses t = 1.16 mm for beams with layer arrangement [45/−45/90/0]

_{S}and [45/−45/45/−45]

_{S}and t = 1.2 mm for beams with [0/90/0/90]

_{S}lay-ups.

_{S}and [45/−45/45/−45]

_{S}, the buckling modes correspond to the lower buckling load are the same (see Figure 5a)—three half-waves of sine on the upper flange. For the beam with stacking sequence [45/−45/90/0]

_{S}, the first buckling mode has only two half-waves in the longitudinal direction on the upper flange of the beam (cf. Figure 5b). It should be noted that the second buckling mode for this case of layup is characterized by three half-waves, as in the rest of the layer arrangements. Differences between the value of the first and second buckling loads are less than 3% (the second buckling load is 2.8% higher than the first one).

_{S}obtained numerically and experimentally are presented in Figure 6.

_{S}) and 10%, or even 50%, are identical. Considering the beam with layer arrangement [45/−45/90/0]

_{S}, it can be argued that the models of the beams are too stiff if no initial imperfection or too small amplitudes of geometric imperfection are assumed (cf. Figure 7a).

_{V}(Figure 8a,c,e) for all analyzed cases in the pre-buckling state, as well as in close post-buckling range are consistent with the experimental results. This is also true when the course of bending moment vs. angle of rotation α

_{H}is analyzed for the beam with two types of layer arrangement [45/−45/90/0]

_{S}(Figure 8d) and [0/90/0/90]

_{S}(Figure 8f). Only in the case of [45/−45/45/−45]

_{S}the numerically obtained post-buckling stiffness in the lateral direction to the plane of the load (Figure 8b) differ from the results obtained experimentally.

_{S}, the angle of rotation α

_{V}corresponds to the highest bending moment obtained from the numerical calculations and is smaller than that obtained experimentally. The maximal experimental loads and numerical calculations, however, are very close to one another.

_{S}). Nevertheless, the proposed numerical models predict the beam behavior in the full range of the load (except case [45/−45/45/−45]

_{S}layer arrangement—Figure 8a,b), and the obtained results are very similar to those obtained in experimental tests after using the nonlinear numerical model without the progressive damage algorithm, including the worst initial geometric imperfection (NL_N). Additionally, it can be noted that the model NL_N is enough to be employed in the parametric study of layer arrangement influence on buckling load and the post-buckling beam behavior. The above-mentioned model (without the progressive damage algorithm) can exclude the influence of damage in the parametric study. However, It should be noted that in case of [45/−45/45/−45]

_{S}layups in far post-buckling range the increase of deflection (both angles of rotations) with constant load value is observed, what could mean the damages or material nonlinearities typical for such a layer arrangement. For this case the numerical model with the progressive damage algorithm (line PD_N in Figure 8a) was better (closer to experimental results) at describing the real load–deflection relation, but the ultimate load, defined as maximal load obtained during the four-point bending test, can be well estimated even by employing the model without the progressive damage algorithm (cf. NL_N curve in Figure 8a).

## 5. Parametric Study

_{1}= 81 mm, b

_{2}= 40 mm and t = 1.16 mm) and lipped channel section (b

_{1}= 81 mm, b

_{2}= 40 mm, b

_{3}= 6 mm and t = 1.16 mm) with length L = 275 mm. In the case of the lipped channel section, the width of the stiffeners was assumed in such a way that the lowest buckling mode is distortional [38,39]. The angle of the fiber direction at each layer has been considered as a variable parameter.

#### 5.1. Layer Arrangements with Their ABD Laminate Stiffness Matrix

_{x}, N

_{y}, N

_{xy}, M

_{x}, M

_{y}, M

_{xy}—internal forces and moments with indexes correspond to midplane with assumed xy coordinate system; ε

_{x}, ε

_{y}, γ

_{xy}—strains of the plate (beam’s wall) reduces to the mid-surface; κ

_{x}, κ

_{y}, κ

_{xy}—middle-surface curvatures.

_{ij}≠ 0. When A

_{16}, A

_{26}≠ 0, in-plane shear with extension coupling exists, and in the case when D

_{16}, D

_{26}≠ 0 out-of-plane bending and twisting coupling appears. According to the international literature (e.g., [1,2,3,4,40]), the used notations for the ABD laminate stiffness matrix are presented in Table 3. In the parametric study, the layer arrangements were assumed in such a way as to check all possible cases of coupling matrix B presented in Table 3.

- Laminates with symmetrical layer arrangements—the same three cases as those tested experimentally;
- Arbitrary assumed non-symmetric layer arrangements with given angles of fiber inclinations at each layer, denoted as N1 and N2;
- Non-symmetrical layer arrangements denoted as θ (angle of layer orientation with straight fibers in each layer)—the range of θ from −90 to 90 degrees with a 5-degree step was considered.

_{S}B

_{S}D

_{S}, A

_{S}B

_{0}D

_{S}and A

_{S}B

**D**

_{l}_{S}(see Table 4), the change of fiber orientation from positive to negative or vice-versa does not affect laminate behavior. The elements of ABD laminate stiffness matrices with indexes “16” and “26” are antisymmetric with respect to vertical axes (θ = 0) except for D

_{16}and D

_{26}for layup N4 (cf. Figure 12c).

#### 5.2. Layers Arrangement Influence on Load-Deflection Curves

_{V})

_{u}and horizontal (α

_{H})

_{u}are presented in Figure 16, Figure 17, Figure 18, Figure 19, Figure 20 and Figure 21 for channel section beams, and in Figure 22, Figure 23, Figure 24, Figure 25, Figure 26 and Figure 27 for lipped section beams.

- The highest stiffness in the pre-buckling range has been observed for beams with layups denoted as S2, S3, N2 and N2R;
- The highest stiffness in the post-buckling range has been observed for the beam with layup denoted as S2;
- The lowest stiffness in the pre-buckling and post-buckling ranges has been observed for beams with layups denoted as N1 and N1R (the courses of curves are identical);
- The highest ultimate bending moment has been detected for beams with layer arrangement denoted as S2.

- The stiffness of the beams made of laminate with nonsymmetric layups is lower than that of antisymmetric ones;
- All beams with antisymmetric layups have similar pre-buckling and post-buckling stiffness;
- Beams with both cross-section and nonsymmetric layups N5(θ) and N6(θ) have similar stiffness in the pre-buckling and post-buckling range for θ = ±22.5, ±26.5 and ±63.5;
- The lowest stiffness of all compared beams was observed for those made of laminate denoted as N6(±45).

_{H}is increasing significantly when load is around the buckling load, which was also noted in [39].

#### 5.3. Layers Arrangement Influence on Buckling Load and Ultimate Load

_{cr}determined in the linear buckling analysis and ultimate loads M

_{u}from the nonlinear analysis (cf. Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26 and Figure 27) with corresponding angles of rotation in two planes, vertical (α

_{V})

_{u}and horizontal (α

_{H})

_{u}, are presented in Table 5. The extremal values are bolded. It is possible to see that the sign of angle θ does not influence the value of the critical load for both cross-sections. Only in the cases of N4(±22.5) and N5(±22.5) there is a slight difference (cf. Figure 29).

_{V})

_{u}and (α

_{H})

_{u}are the highest, which leads to relatively low ultimate load. The beam with the layer arrangement denoted as N2R generates the lowest critical load with the corresponding lowest (α

_{V})

_{u}. The highest ultimate load is obtained for the N16R(±30) layer arrangement beam, while the lowest one for the N1 and N1R beams. Similar behavior can be noticed in the case of lipped beams, the ultimate load is also the lowest for the N1 and N1R layups. Furthermore, the mentioned beams indicate the lowest critical load. The highest deformations occur in case N3R(±60), which provides the greatest values of (α

_{V})

_{u}and (α

_{H})

_{u}. Both angles are the lowest for beam N16(±30), which has the highest load-carrying capacity. Additionally, it was noted that the angles of rotation in both planes corresponding to load-carrying capacity M

_{u}, in the case of channel section beams for all cases of layer arrangements are in the same relation, i.e., (α

_{H})

_{u}> (α

_{V})

_{u}. For lipped channel section beams, the relation between angles of rotation (α

_{H})

_{u}> (α

_{V})

_{u}depends on layer arrangements and is fulfilled only for cases S1, N4R, N5 and N6.

## 6. Discussion of Obtained Results

_{H})

_{u}/(α

_{V})

_{u}was analyzed. If the relation (α

_{H})

_{u}/(α

_{V})

_{u}is greater than 1, it means that deflection in lateral direction is higher than in the plane of load. It is obvious that the channel section beams are weaker than lipped section beams; thus, it is only in some layer arrangements of laminate used in lipped section beams that the angles of rotation have a relation lower than one. The lowest value of (α

_{H})

_{u}/(α

_{V})

_{u}equal to 0.822 was obtained in the case of layup denoted as N2R, but both angles are higher than 3° (cf. Table 5). This means that not only this relation but also the angle of rotation should be checked. Thus, it can be said that the stiffest beam has a lipped cross-section with the layer arrangement denoted as N16(±30).

_{ij}/(A

_{ij})

_{max}, B

_{ij}/(B

_{ij})

_{max}, D

_{ij}/(D

_{ij})

_{max}. The maximal values presented in Table 6 and Table 7 are bolded and written in red, and the minimal in blue. The analysis of the values of elements of ABD matrices shows that:

- The extremal buckling load depends on the extremal values of the A
_{12}, A_{66}, D_{12}and D_{66}elements. For channel section beams, the highest buckling load was obtained when the mentioned elements had the highest values, and the lowest buckling load when they had the lowest. The opposite was observed for lipped section beams, i.e., the highest buckling load was obtained when the mentioned elements had the lowest values, and the lowest buckling load for the highest value of those elements. - The extremal values of load-carrying capacity for both types of considered cross-sections depend on the value of A
_{11}—if this value is the highest, M_{u}is the highest and if A_{11}is the lowest, M_{u}is the lowest. Additionally, it was found that the maximal ultimate load was obtained for cases when the A_{22}and D_{22}had the lowest value and the extremal A_{16}(maximal negative for channel section and maximal positive for lipped section beams). It has also been found that the minimal ultimate load was obtained in the case when A_{12}, A_{66}, D_{12}and D_{66}had the maximal values. - No influence was found in the case of coupling stiffness matrix elements B
_{ij}, as well as elements D_{16}and D_{26}on the maximal values of buckling and ultimate bending moment. For the minimal ultimate load, it was found that elements B_{16}and B_{26}have extremal negative values, while in the case of minimal buckling load no relations were found.

_{ij}stiffness matrix and the largest percentage of layers with 0 angle orientation across the thickness of the laminate. In the case of channel section beams the presence of layers with 45 degree angle (especially at the outer layers) leads to an increase of the critical load and a decrease of the ultimate load. Such a phenomenon is caused by higher D

_{12}and D

_{66}stiffness matrix elements (S1, N1, N5(±45), N6(±45)).

_{ij}and D

_{ij}are identical, while the elements of matrix B

_{ij}have opposite signs. As such, it is suggested to check how the deflection changes (vertical displacement of node lying in the mid span of the beam on the edge of the upper flange and stiffener) with load increase for different amplitudes of initial geometric imperfection.

**B**laminate stiffness matrix, is “stronger” than deflection connected with the buckling phenomenon. Such a behavior is only observed in this case of layer arrangement, which can be explained by the values of the B

_{12}and B

_{66}elements, which are the highest of all considered layer arrangements—the nondimensional values of laminate stiffness matrix for case N4(90) are as follows:

_{V}for both cases of beam cross-sections (cf. Figure 29) and layups N5(±22.5) need further analysis to explain why they appear. The analysis has been performed considering the lipped section beam, and the results are presented in Figure 31. As can be noted, the course of curves for beams N5(-22.5) and N5(22.5) (cf. Figure 31) depends on beam deflection, which influences its stiffness.

_{11}, B

_{22}, B

_{16}and B

_{26}and the fact that it is only in this case that the extension-bending, extension-twisting and shearing-bending load response coupling exist (

**B**

_{l}

_{t}).

## 7. Conclusions

- The coupling stiffness matrix element B
_{ij}, as well as elements D_{16}and D_{26}, have no influence on the values of maximal buckling load and ultimate load for analyzed beams subjected to pure bending. It was found that only in the case of minimal buckling load and minimal ultimate load elements B_{16}and B_{26}have extremal values (cf. Table 6 and Table 7). - The values of the buckling loads for lipped beams are over three-fold greater than for channel beams. This shows the significant influence of stiffeners on beam behavior. Ultimate load differs around 30%.
- The highest values of load-carrying capacity were obtained for 16-layer beams, which gives more flexibility for tailoring—more flexibility of coupling behavior and HTCS laminate design.
- In the case of the considered type of load, the ABD laminate stiffness matrix is not the only factor to influence the buckling and ultimate loads, but also the considered cross-section.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**(

**a**) Scheme of four-point bending test performed experimentally. (

**b**) Considered cross-section.

**Figure 2.**(

**a**) Experimental test stand. (

**b**) The load and the support in four-point bending test. (

**c**) Aramis software view with depicted points (

**d**) Zoom of points depicted in left grip.

**Figure 3.**(

**a**) The considered laminate beam dimensions with layer arrangement and (

**b**) type of the load.

**Figure 4.**(

**a**) Geometry of numerical model; (

**b**) discrete model with applied boundary condition and load.

**Figure 5.**Buckling modes corresponding to the lowest buckling loads for the beams with the layer arrangement: (

**a**) [0/90/0/90]

_{S}and [45/−45/45/−45]

_{S}; (

**b**) [45/−45/90/0]

_{S}.

**Figure 6.**Comparison of beams deflection for case [45/−45/45/−45]

_{S}obtained (

**a**) numerically with model NL_N; and (

**b**) experimentally; (

**c**) the load-displacements comparison of curves obtained experimentally and numerically.

**Figure 7.**The influence of amplitude of initial geometric imperfection on bending moment vs. angle of rotation α

_{V}, curves for beams with layer arrangement: (

**a**) [45/−45/90/0]

_{S}; (

**b**) [0/90/0/90]

_{S}; (

**c**) [45/−45/45/−45]

_{S}. FE 0—numerical analysis without imperfection, FE −0.01 t—numerical analysis with the imperfection equal to −1% of thickness, FE −0.1 t—numerical analysis with the imperfection equal to −10% of thickness, FE −0.5 t—numerical analysis with the imperfection equal to −50% of thickness, Ex1—experimental results of first specimen, Ex2—experimental results of second specimen.

**Figure 8.**Bending moment vs. angles of rotations M

_{b}(α

_{V}) and M

_{b}(α

_{H}) for all analyzed cases—comparison of numerical results with experimental ones. (

**a**) M

_{b}(α

_{V}) for [45/−45/90/0]

_{S}; (

**b**) M

_{b}(α

_{H}) for [45/−45/90/0]

_{S}; (

**c**) M

_{b}(α

_{V}) for [0/90/0/90]

_{S}; (

**d**) M

_{b}(α

_{H}) for [0/90/0/90]

_{S}; (

**e**) M

_{b}(α

_{V}) for [45/−45/45/−45]

_{S}; (

**f**) M

_{b}(α

_{H}) for [45/−45/45/−45]

_{S}. PD P—progressive damage algorithm analysis with positive imperfection; PD N—progressive damage algorithm analysis with positive imperfection; NL P—nonlinear analysis with positive imperfection; NL N - nonlinear analysis with negative imperfection; Ex1—experimental results of first specimen; Ex2—experimental results of second specimen; Ex3—experimental results of first specimen.

**Figure 15.**Elements (

**a**) [A], (

**b**) [B] and (

**c**) [D] of laminate stiffness matrix value for case N16(θ).

**Figure 16.**Bending moment vs. angles of rotations in (

**a**) vertical and (

**b**) horizontal plane for channel section beams with layups denoted as S1, S2, S3, N1, N1R, N2 and N2R.

**Figure 17.**Bending moment vs. angles of rotation in: (

**a**) vertical; and (

**b**) horizontal planes for channel section beams with layups denoted as N3(θ) and N3R(θ).

**Figure 18.**Bending moment vs. angles of rotation in: (

**a**) vertical; and (

**b**) horizontal planes for channel section beams with layups denoted as N4(θ) and N4R(θ).

**Figure 19.**Bending moment vs. angles of rotations in: (

**a**) vertical; and (

**b**) horizontal planes for channel section beams with layups denoted as A1(θ), A2(θ) and A2R(θ).

**Figure 20.**Bending moment vs. angles of rotations in: (

**a**) vertical; and (

**b**) horizontal planes for channel section beams with layups denoted as N5(θ) and N6(θ).

**Figure 21.**Bending moment vs. angles of rotations in: (

**a**) vertical; and (

**b**) horizontal planes for channel section beams with layups denoted as N16(θ) and N16R(θ).

**Figure 22.**Bending moment vs. angles of rotation in: (

**a**) vertical; and (

**b**) horizontal planes for lipped section beams with layups denoted as S1, S2, S3, N1, N1R, N2 and N2R.

**Figure 23.**Bending moment vs. angles of rotation in: (

**a**) vertical; and (

**b**) horizontal planes for lipped section beams with layups denoted as N3(θ) and N3R(θ).

**Figure 24.**Bending moment vs. angles of rotation in: (

**a**) vertical; and (

**b**) horizontal planes for lipped section beams with layups denoted as N4(θ) and N4R(θ).

**Figure 25.**Bending moment vs. angles of rotation in: (

**a**) vertical; and (

**b**) horizontal planes for lipped section beams with layups denoted as A1(θ), A2(θ) and A2R(θ).

**Figure 26.**Bending moment vs. angles of rotation in: (

**a**) vertical; and (

**b**) horizontal planes for lipped section beams with layups denoted as N5(θ) and N6(θ).

**Figure 27.**Bending moment vs. angles of rotation in: (

**a**) vertical and (

**b**) horizontal planes for lipped section beams with layups denoted as N16(θ) and N16R(θ).

**Figure 28.**Lipped channel section beam deflection for: (

**a**) layups N4(0) and N4(90); (

**b**) layups N4(±22.5); (

**c**) remaining layer arrangements.

**Figure 29.**Differences in the course of curves presenting bending moment vs. angle of rotation α

_{V}for positive and negative angles of fiber inclination for layups denoted as N4(±22.5) and N5(±22.5).

**Figure 31.**(

**a**) Bending moment vs. angle of rotation α

_{V}. (

**b**) Zoom of sub-figure (

**a**). Shape of beam deflection from points depicted in sub-figure (

**b**): (

**c**) from point A1; (

**d**) from point A2; (

**e**) point A3, (

**f**) point A4, (

**g**) point B1, (

**h**) point B2, (

**i**) point B3.

E_{1} | E_{2} | G_{12} | ν_{12} | X^{T} | Y^{T} | X^{C} | Y^{C} | S | |
---|---|---|---|---|---|---|---|---|---|

[GPa] | [GPa] | [GPa] | [-] | [MPa] | [MPa] | [MPa] | [MPa] | [MPa] | |

Data | 39.0 | 9.0 | 2.7 | 0.28 | 1250 | 43 | 620 | 140 | 112 |

SD | 0.4 | 0.7 | 0.1 | 0.003 | 78 | 4 | 62 | 5 | 1 |

Layer Arrangement | [0/90/0/90]_{S} | [45/−45/90/0]_{S} | [45/−45/45/−45]_{S} | ||||||
---|---|---|---|---|---|---|---|---|---|

Specimen No. | B_{1} | B_{2} | T | B_{1} | B_{2} | T | B_{1} | B_{2} | T |

[mm] | [mm] | [mm] | [mm] | [mm] | [mm] | [mm] | [mm] | [mm] | |

1 | 82.1 | 41.0 | 1.18 | 82.3 | 41.1 | 1.15 | 82.4 | 41.2 | 1.15 |

2 | 82.1 | 41.0 | 1.20 | 82.3 | 41.0 | 1.17 | 82.3 | 41.2 | 1.15 |

3 | 82.1 | 41.0 | 1.18 | 82.3 | 41.1 | 1.16 | 82.3 | 41.4 | 1.17 |

average: | 82.1 | 41.00 | 1.19 | 82.3 | 41.1 | 1.16 | 82.3 | 41.3 | 1.16 |

Subscript Notation ESDU (1994) [40] | Description of Load Response Coupling | Stiffness Submatrices |
---|---|---|

A_{S} | simple laminate no in-plane coupling | $\left[\begin{array}{ccc}{A}_{11}& {A}_{12}& 0\\ {A}_{12}& {A}_{22}& 0\\ 0& {A}_{26}& {A}_{66}\end{array}\right]$ |

A_{F} | shear-extension coupling | $\left[\begin{array}{ccc}{A}_{11}& {A}_{12}& {A}_{16}\\ {A}_{12}& {A}_{22}& {A}_{26}\\ {A}_{16}& {A}_{26}& {A}_{66}\end{array}\right]$ |

B_{t} | extension-twisting and shear-bending | $\left[\begin{array}{ccc}0& 0& {B}_{16}\\ 0& 0& {B}_{26}\\ {B}_{16}& {B}_{26}& 0\end{array}\right]$ |

B_{l} | extension-bending | $\left[\begin{array}{ccc}{B}_{11}& 0& 0\\ 0& {B}_{22}& 0\\ 0& 0& 0\end{array}\right]$ |

B_{lt} | extension-bending; extension-twisting; shearing-bending | $\left[\begin{array}{ccc}{B}_{11}& 0& {B}_{16}\\ 0& {B}_{22}& {B}_{26}\\ {B}_{16}& {B}_{26}& 0\end{array}\right]$ |

B_{S} | extension-bending and shear-twisting | $\left[\begin{array}{ccc}{B}_{11}& {B}_{12}& 0\\ {B}_{12}& {B}_{22}& 0\\ 0& 0& {B}_{66}\end{array}\right]$ |

B_{F} | all in-plane with out-of-plane coupling | $\left[\begin{array}{ccc}{B}_{11}& {B}_{12}& {B}_{16}\\ {B}_{12}& {B}_{22}& {B}_{26}\\ {B}_{16}& {B}_{26}& {B}_{66}\end{array}\right]$ |

D_{S} | simple laminate no out-of-plane coupling | $\left[\begin{array}{ccc}{D}_{11}& {D}_{12}& 0\\ {D}_{12}& {D}_{22}& 0\\ 0& 0& {D}_{66}\end{array}\right]$ |

D_{F} | twisting-bending coupling | $\left[\begin{array}{ccc}{D}_{11}& {D}_{12}& {D}_{16}\\ {D}_{12}& {D}_{22}& {D}_{26}\\ {D}_{16}& {D}_{26}& {D}_{66}\end{array}\right]$ |

Case ID | Number of Layers | Layer Arrangement | Laminate Type [40] | Considered θ [deg] |
---|---|---|---|---|

S1 | 8 | 45/−45/45/−45/−45/45/−45/45 | A_{S}B_{0}D_{F} | - |

S2 | 8 | 0/90/0/90/90/0/90/0 | A_{S} B_{0}D_{S} | - |

S3 | 8 | 45/−45/90/0/0/90/−45/45 | A_{S} B_{0}D_{F} | - |

N1 | 8 | 45/45/45/45/−45/−45/−45/−45 | A_{S}B_{t}D_{S} | - |

N2 | 8 | 90/90/90/90/0/0/0/0 | A_{S}B_{l}D_{S} | - |

A1(θ) [41] | 8 | θ/(θ−90)_{2}/θ/−θ/(90−θ)_{2}/−θ | A_{S}B_{t}D_{S} | ±22.5 |

A2(θ) | 8 | 0/90/θ/90−θ/θ−90/−θ/0/90 | A_{S}B_{lt}D_{S} | ±35, ±45, ±55 |

A2R(θ) | 8 | 90/0/−θ/θ−90/90−θ/θ/90/0 | A_{S}B_{lt}D_{S} | −35, −45, −55 |

N3(θ) [1] | 8 | 90/0/θ/−θ/0/90/−θ/θ | A_{S}B_{S}D_{F} | ±30, ±45, ±60 |

N3R(θ) | 8 | θ/−θ/90/0/−θ/θ/0/90 | A_{S}B_{S}D_{F} | −30, −45, −60 |

N4(θ) | 8 | 45/−45/45/−45/−θ/90−θ/θ/θ−90 | A_{S}B_{F}D_{F} | 0, 90, ±22.5 |

N4R(θ) | 8 | θ−90/θ/90−θ/−θ/−45/45/−45/45 | A_{S}B_{F}D_{F} | 90, 22.5 |

N5(θ) | 8 | 45/−45/θ/−θ/θ−90/90−θ/−45/45 | A_{S}B_{lt}D_{F} | ±22.5 |

N6(θ) | 8 | 45/−45/θ−90/θ/−θ/90−θ/−45/45 | A_{S}B_{t}D_{F} | ±26.5, ±45, ±63.5 |

N16(θ) [1] | 16 | −θ/90/θ/0/0/θ/0/0/90/−θ/0/−θ/θ/0/−θ/θ | A_{F}B_{F}D_{F} | ±30, ±45, ±60 |

N16R(θ) | 16 | θ/−θ/0/θ/−θ/0/−θ/90/0/0/θ/0/0/θ/90/−θ | A_{F}B_{F}D_{F} | −30, −45, −60 |

Case ID | θ | C-Section Beam | Lipped Section Beam | ||||||
---|---|---|---|---|---|---|---|---|---|

M_{cr} | M_{u} | (α_{V})_{u} | (α_{H})_{u} | M_{cr} | M_{u} | (α_{V})_{u} | (α_{H})_{u} | ||

[Nm] | [Nm] | [deg] | [deg] | [Nm] | [Nm] | [deg] | [deg] | ||

S1 | – | 78.4 | 288.5 | 2.94 | 3.63 | 190.6 | 352.8 | 4.01 | 4.15 |

S2 | 60.8 | 367.2 | 1.96 | 2.88 | 291.4 | 502.6 | 3.22 | 2.96 | |

S3 | 76.4 | 358.9 | 2.12 | 2.65 | 248.5 | 477.9 | 3.46 | 3.39 | |

A1(θ) | ±22.5 | 60.8 | 335.7 | 2.15 | 3.07 | 223.8 | 449.2 | 3.37 | 3.27 |

A2(θ) | ±35 | 59.7 | 354.6 | 2.16 | 2.89 | 236.3 | 487.7 | 3.94 | 3.6 |

±45 | 59.9 | 350.8 | 2.23 | 2.99 | 233.1 | 475.6 | 3.83 | 3.58 | |

±55 | 59.4 | 354.6 | 2.16 | 2.89 | 238.8 | 485.6 | 3.74 | 3.45 | |

A2R(θ) | ±35 | 59.8 | 354 | 2.16 | 2.96 | 241.2 | 482.7 | 3.55 | 3.34 |

±45 | 59.9 | 350.1 | 2.23 | 3.07 | 238.3 | 475.3 | 3.64 | 3.46 | |

±55 | 59.5 | 354.3 | 2.16 | 2.96 | 243.8 | 485.2 | 3.65 | 3.47 | |

N1 | – | 58.2 | 250.7 | 2.67 | 3.23 | 160.3 | 311.5 | 3.92 | 3.89 |

N1R | 58.3 | 250.8 | 2.67 | 3.19 | 160.3 | 311.6 | 3.92 | 3.86 | |

N2 | 52.4 | 329.3 | 1.87 | 3.09 | 239.4 | 477.9 | 3.15 | 2.96 | |

N2R | 45.8 | 319.2 | 1.81 | 2.56 | 216 | 460.5 | 3.71 | 3.05 | |

N3(θ) | ±30 | 66 | 369.5 | 1.96 | 2.96 | 267.2 | 512.2 | 3.15 | 3.09 |

±45 | 66.2 | 338.8 | 2.12 | 2.92 | 246.8 | 462.7 | 3.46 | 3.43 | |

±60 | 61.6 | 320.5 | 2.26 | 2.96 | 240.4 | 444.1 | 3.98 | 3.79 | |

N3R(θ) | ±30 | 65.7 | 396 | 2.16 | 3.15 | 263.6 | 512.3 | 3.15 | 3.08 |

±45 | 66 | 367 | 2.32 | 3 | 248.6 | 499.4 | 4.1 | 3.89 | |

±60 | 61.5 | 341.4 | 2.42 | 3.11 | 246.9 | 472 | 4.5 | 4.22 | |

N4(θ) | 0 | 58.4 | 357.9 | 2.28 | 3.08 | 234.1 | – | – | – |

90 | 60 | 361.7 | 2.39 | 3.35 | 225.9 | – | – | – | |

22.5 | 71.1 | 340.3 | 2.53 | 3.14 | 220.4 | 436 | 4.01 | 3.96 | |

−22.5 | 69.9 | 336.7 | 2.61 | 3.33 | 219.2 | 434.4 | 4.04 | 3.99 | |

N4R(θ) | 90 | 60.2 | 307.3 | 2.05 | 3.02 | 218.3 | 410 | 3.14 | 3.38 |

22.5 | 70.1 | 308.8 | 2.32 | 3.11 | 221 | 405.3 | 3.46 | 3.65 | |

N5(θ) | ±22.5 | 76.7 | 329.9 | 2.53 | 3.21 | 219.4 | 421.2 | 3.69 | 3.74 |

N6(θ) | ±26.5 | 77.2 | 322.1 | 2.53 | 3.18 | 215.6 | 410.2 | 3.74 | 3.79 |

±45 | 79.1 | 289.6 | 2.94 | 3.6 | 191 | 353.8 | 4.01 | 4.15 | |

±63.5 | 77.6 | 321.1 | 2.53 | 3.15 | 210.6 | 403.7 | 3.74 | 3.82 | |

N16(θ) | ±30 | 71.4 | 385 | 1.82 | 2.76 | 270.2 | 548.6 | 2.87 | 2.76 |

±45 | 73.8 | 370.6 | 1.98 | 2.74 | 262.8 | 529.5 | 3.08 | 2.9 | |

±60 | 68.4 | 361.7 | 2.02 | 2.6 | 260.4 | 518 | 3.15 | 2.95 | |

N16R(θ) | ±30 | 71.2 | 405.5 | 2 | 2.93 | 265.8 | 543.6 | 3.09 | 2.81 |

±45 | 73.7 | 391.4 | 2.16 | 2.92 | 258.9 | 522.2 | 3.46 | 3.24 | |

±60 | 68.3 | 371.4 | 2.03 | 2.42 | 262.8 | 515.9 | 3.74 | 3.52 |

**Table 6.**Nondimensional value of ABD laminate stiffness matrix for laminates used in channel section beam for which the extremal buckling and ultimate load were obtained.

Sub-Matrix | Minimal M_{cr} | Maximal M_{cr} | Minimal M_{u} | Maximal M_{u} |
---|---|---|---|---|

N2R | N6(±45) | N1, N1R | N16R(±30) | |

A | $\left[\begin{array}{ccc}0.84& {0.24}& 0\\ & 0.97& 0\\ & & {0.25}\end{array}\right]$ | $\left[\begin{array}{ccc}{0.56}& {1}& 0\\ & 0.64& 0\\ & & {1}\end{array}\right]$ | $\left[\begin{array}{ccc}{0.56}& {1}& 0\\ & 0.64& 0\\ & & {1}\end{array}\right]$ | $\left[\begin{array}{ccc}{1}& 0.52& {-1}\\ & {0.54}& -0.30\\ & & 0.53\end{array}\right]$ |

B | $\left[\begin{array}{ccc}{-1}& 0& 0\\ & {1}& 0\\ & & 0\end{array}\right]$ | $\left[\begin{array}{ccc}0& 0& 0.12\\ & 0& 0.12\\ & & 0\end{array}\right]$ | $\left[\begin{array}{ccc}0& 0& {-1}\\ & 0& {-1}\\ & & 0\end{array}\right]$ | $\left[\begin{array}{ccc}-0.07& -0.19& 0.06\\ & 0.18& 0.02\\ & & -0.19\end{array}\right]$ |

D | $\left[\begin{array}{ccc}0.81& {0.24}& 0\\ & 0.87& 0\\ & & {0.25}\end{array}\right]$ | $\left[\begin{array}{ccc}0.54& {1}& 0.75\\ & 0.58& 0.62\\ & & {1}\end{array}\right]$ | $\left[\begin{array}{ccc}0.54& {1}& 0\\ & 0.58& 0\\ & & {1}\end{array}\right]$ | $\left[\begin{array}{ccc}0.89& 0.62& -0.05\\ & {0.50}& -0.01\\ & & 0.62\end{array}\right]$ |

**Table 7.**Nondimensional value of ABD laminate stiffness matrix for laminates used in lipped section beam for which the extremal buckling and ultimate load were obtained.

Sub-Matrix | Minimal M_{cr} | Maximal M_{cr} | Minimal M_{u} | Maximal M_{u} |
---|---|---|---|---|

N1, N1R | S2 | N1 | N16(±30) | |

A | $\left[\begin{array}{ccc}{0.56}& {1}& 0\\ & 0.64& 0\\ & & {1}\end{array}\right]$ | $\left[\begin{array}{ccc}0.84& {0.24}& 0\\ & 0.97& 0\\ & & {0.25}\end{array}\right]$ | $\left[\begin{array}{ccc}{0.56}& {1}& 0\\ & 0.64& 0\\ & & {1}\end{array}\right]$ | $\left[\begin{array}{ccc}{1}& 0.52& {1}\\ & {0.54}& 0.30\\ & & 0.53\end{array}\right]$ |

B | $\left[\begin{array}{ccc}0& 0& {-1}\\ & 0& {-1}\\ & & 0\end{array}\right]$ | $\left[\begin{array}{ccc}0& 0& 0\\ & 0& 0\\ & & 0\end{array}\right]$ | $\left[\begin{array}{ccc}0& 0& {-1}\\ & 0& {-1}\\ & & 0\end{array}\right]$ | $\left[\begin{array}{ccc}0.07& 0.19& 0.06\\ & -0.18& 0.02\\ & & 0.19\end{array}\right]$ |

D | $\left[\begin{array}{ccc}0.54& {1}& 0\\ & 0.58& 0\\ & & {1}\end{array}\right]$ | $\left[\begin{array}{ccc}{1}& {0.24}& 0\\ & 0.67& 0\\ & & {0.25}\end{array}\right]$ | $\left[\begin{array}{ccc}0.54& {1}& 0\\ & 0.58& 0\\ & & {1}\end{array}\right]$ | $\left[\begin{array}{ccc}0.89& 0.62& 0.05\\ & {0.50}& 0.01\\ & & 0.62\end{array}\right]$ |

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## Share and Cite

**MDPI and ACS Style**

Kubiak, T.; Urbaniak, M.; Kazmierczyk, F.
The Influence of the Layer Arrangement on the Distortional Post-Buckling Behavior of Open Section Beams. *Materials* **2020**, *13*, 3002.
https://doi.org/10.3390/ma13133002

**AMA Style**

Kubiak T, Urbaniak M, Kazmierczyk F.
The Influence of the Layer Arrangement on the Distortional Post-Buckling Behavior of Open Section Beams. *Materials*. 2020; 13(13):3002.
https://doi.org/10.3390/ma13133002

**Chicago/Turabian Style**

Kubiak, Tomasz, Mariusz Urbaniak, and Filip Kazmierczyk.
2020. "The Influence of the Layer Arrangement on the Distortional Post-Buckling Behavior of Open Section Beams" *Materials* 13, no. 13: 3002.
https://doi.org/10.3390/ma13133002