Influence of stiffeners on the buckling behaviour of composite panels subjected to non-uniform edge loads

Azimi, Sasha; Rajanna, T; Asha, K; Ravindra, P.K; Kalgutkar, Akshay Prakash

doi:10.1590/1679-78258093

Sasha Azimi

methodology

investigation

validation

writing original draft

Department of Civil Engineering, BMS College of Engineering, Bengaluru 560 019, India. Email: sashaasimi@bmsce.ac.in, t.rajanna@gmail.com, asha.civ@bmsce.ac.in

https://orcid.org/0000-0001-9933-450X

T Rajanna ^**Corresponding author

conceptualization

writing reviewing and editing

supervision

Department of Civil Engineering, BMS College of Engineering, Bengaluru 560 019, India. Email: sashaasimi@bmsce.ac.in, t.rajanna@gmail.com, asha.civ@bmsce.ac.in

https://orcid.org/0000-0001-6056-3946

K Asha

methodology

investigation

validation

writing original draft

Department of Civil Engineering, BMS College of Engineering, Bengaluru 560 019, India. Email: sashaasimi@bmsce.ac.in, t.rajanna@gmail.com, asha.civ@bmsce.ac.in

https://orcid.org/0000-0002-8792-7698

P.K Ravindra

conceptualization

writing reviewing and editing

supervision

Department of Civil Engineering, Government Engineering College, Ramanagar-562159, India. Email: ravindrapkravindra@ymail.com

https://orcid.org/0009-0009-3162-7651

Akshay Prakash Kalgutkar

conceptualization

writing reviewing and editing

supervision

Department of Civil Engineering, Indian Institute of Technology, Mumbai 400 076, India. Email: akshay_kalgutkar@iitb.ac.in

https://orcid.org/0000-0002-9626-5147

*Corresponding author

Editor: Marco L. Bittencourt

No.	Parameter	Stiffened/unstiffened panels
No.	Parameter	Isotropic	Composite
1	Non-dimensional frequencies ( $\bar{ω}$ )	$ω b^{2} \sqrt{ρ h / D}$	$ω b^{2} \sqrt{ρ / E_{22} h^{2}}$
2	Non-dimensional buckling load (γ_cr)	$P_{c r} b^{2} / D$	$P_{c r} b^{2} / E_{22} h^{3}$
where $D = E h^{3} / 12 (1 - ν^{2})$ , ω and P_cr are the absolute frequencies and absolute critical loads, respectively.

b/h	2 layers (45/-45)				8 layers (45/-45/45…)
	Present results			(Reddy and Phan 1985Reddy, J.N., Phan, N.D., (1985). Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear deformation theory. Journal of Sound and Vibration, 98(2):157-170. https://doi.org/10.1016/0022-460X(85)90383-9. https://doi.org/10.1016/0022-460X(85)903... )	Present results			(Reddy and Phan 1985Reddy, J.N., Phan, N.D., (1985). Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear deformation theory. Journal of Sound and Vibration, 98(2):157-170. https://doi.org/10.1016/0022-460X(85)90383-9. https://doi.org/10.1016/0022-460X(85)903... )
	9-NHE	9-NLE	8-NSE		9-NHE	9-NLE	8-NSE
5	10.335	10.244	10.243	10.335	12.892	12.863	12.862	12.892
10	13.044	12.975	12.975	13.044	19.289	19.235	19.235	19.289
20	14.179	14.154	14.153	14.179	23.259	23.225	23.225	23.259
25	14.338	14.322	14.321	14.338	23.924	23.899	23.899	23.924
50	14.561	14.557	14.556	14.561	24.909	24.902	24.901	24.909
100	14.618	14.617	14.617	14.618	25.176	25.174	25.174	25.176

Load pattern (α)	Source	h/b = 0.01	h/b = 0.05	h/b = 0.1
0.5	Zhong and Gu (2007)Zhong, H., Gu, C., (2007). Buckling of symmetrical cross-ply composite rectangular plates under a linearly varying in-plane load. Composite Structures, 80(1):42-48. https://doi.org/10.1016/j.compstruct.2006.02.030. https://doi.org/10.1016/j.compstruct.200...	47.267	41.075	29.432
	9–Heterosis	47.261	40.939	29.228
	9–Lagrangian	47.259	40.930	29.225
	8–Serendipity	47.256	40.921	29.120

1.0	Zhong and Gu (2007)Zhong, H., Gu, C., (2007). Buckling of symmetrical cross-ply composite rectangular plates under a linearly varying in-plane load. Composite Structures, 80(1):42-48. https://doi.org/10.1016/j.compstruct.2006.02.030. https://doi.org/10.1016/j.compstruct.200...	64.982	56.705	40.999
	9–Heterosis	64.975	56.486	40.525
	9–Lagrangian	64.962	56.478	40.514
	8–Serendipity	64.960	56.457	40.500

1.5	Zhong and Gu (2007)Zhong, H., Gu, C., (2007). Buckling of symmetrical cross-ply composite rectangular plates under a linearly varying in-plane load. Composite Structures, 80(1):42-48. https://doi.org/10.1016/j.compstruct.2006.02.030. https://doi.org/10.1016/j.compstruct.200...	91.374	80.336	47.708
	9–Heterosis	91.355	79.989	48.300
	9–Lagrangian	91.338	79.957	48.291
	8–Serendipity	91.331	79.930	48.285
2.0	Zhong and Gu (2007)Zhong, H., Gu, C., (2007). Buckling of symmetrical cross-ply composite rectangular plates under a linearly varying in-plane load. Composite Structures, 80(1):42-48. https://doi.org/10.1016/j.compstruct.2006.02.030. https://doi.org/10.1016/j.compstruct.200...	129.785	114.837	47.872
	9–Heterosis	129.758	114.167	48.720
	9–Lagrangian	129.730	114.152	48.660
	8–Serendipity	129.725	113.980	48.514

φ/b	40-layers (45/-45/0/90)_5S, b/h = 75		8-layers (0/90)_2S, b/h = 100
φ/b	Present	Jain and Kumar (2004)Jain, P., Kumar, A., (2004). Postbuckling response of square laminates with a central circular/elliptical cutout. Composite Structures, 65(2):179-185. https://doi.org/10.1016/j.compstruct.2003.10.014. https://doi.org/10.1016/j.compstruct.200...	Present	Ghannadpour et al., (2006)Ghannadpour, S.A.M., Najafi, A., Mohammadi, B., (2006). On the buckling behavior of cross-ply laminated composite plates due to circular/elliptical cutouts. Composite Structures, 75(1):3-6. https://doi.org/10.1016/j.compstruct.2006.04.071. https://doi.org/10.1016/j.compstruct.200...
0.0	19.16 (19.16)	19.23	13.82 (13.83)	13.79
0.1	18.41 (18.43)	18.56	12.83 (12.84)	12.80
0.2	16.94 (16.96)	17.10	10.84 (10.85)	10.82
0.3	15.40 (15.41)	15.30	8.98 (08.99)	8.97
0.4	14.63 (14.63)	14.63	7.52 (07.52)	7.51
0.5	13.96 (13.97)	13.84	6.40 (06.40)	6.39

$\bar{β}$	η	Non-dimensional buckling load
		Neglecting T and e				Considering T and e
		Timoshenko and Gere (1961)Timoshenko, S.P., Gere, J.M., (1961). Theory of elastic stability, McGraw-Hill, New York.	Mukhopadhyay and Mukherjee (1990)Mukhopadhyay, M., Mukherjee, A., (1990). Finite element buckling analysis of stiffened plates.Computers and Structures, 34(6): 795-803. https://doi.org/10.1016/0045-7949(90)90350-B. https://doi.org/10.1016/0045-7949(90)903...	Hamedani and Ranji (2013)Hamedani, S.J., Ranji, A.R., (2013). Buckling analysis of stiffened plates subjected to non-uniform biaxial compressive loads using conventional and super finite elements. Thin-Walled Structures, 64:41-49. https://doi.org/10.1016/j.tws.2012.12.004. https://doi.org/10.1016/j.tws.2012.12.00...	Present (heterosis)	Hamedani and Ranji (2013)Hamedani, S.J., Ranji, A.R., (2013). Buckling analysis of stiffened plates subjected to non-uniform biaxial compressive loads using conventional and super finite elements. Thin-Walled Structures, 64:41-49. https://doi.org/10.1016/j.tws.2012.12.004. https://doi.org/10.1016/j.tws.2012.12.00...	Present (heterosis)
5	0.05	12.00	11.72	11.80	11.80	11.91	11.86
	0.10	11.10	10.93	10.98	10.98	11.09	11.27
	0.20	9.72	9.70	9.69	9.69	9.86	10.01

10	0.05	16.00	16.00	15.97	15.97	18.16	17.91
	0.10	16.00	16.00	15.97	15.97	16.97	17.20
	0.20	15.80	15.44	15.67	15.67	15.04	15.77

15	0.05	16.00	16.00	15.97	15.97	20.41	20.30
	0.10	16.00	16.00	15.97	15.97	20.41	20.33
	0.20	16.00	16.00	15.97	15.97	19.36	20.33

\begin{array}{l} ε_{x} = {(\frac{\partial {\bar{u}}^{}}{\partial x})}^{8} + \frac{1}{2} {(\frac{\partial u^{}}{\partial x})}^{2} + \frac{1}{2} {(\frac{\partial v^{}}{\partial x})}^{2} + \frac{1}{2} {(\frac{\partial w^{}}{\partial x})}^{2} + \frac{1}{2} z^{2} {[(\frac{\partial θ_{x}^{}}{\partial x})}^{2} + {(\frac{\partial θ_{y}^{}}{\partial x})}^{2}] \\ ε_{y} = {(\frac{\partial {\bar{v}}^{}}{\partial y})}^{8} + \frac{1}{2} {(\frac{\partial u^{}}{\partial y})}^{2} + \frac{1}{2} {(\frac{\partial v^{}}{\partial y})}^{2} + \frac{1}{2} {(\frac{\partial w^{}}{\partial y})}^{2} + \frac{1}{2} z^{2} {[(\frac{\partial θ_{x}^{}}{\partial y})}^{2} + {(\frac{\partial θ_{y}^{}}{\partial y l})}^{2}] \\ γ_{x y} = (\frac{\partial {\bar{u}}^{}}{\partial y} + \frac{\partial {\bar{v}}^{}}{\partial x}) + \frac{\partial u^{}}{\partial x} \frac{\partial u^{}}{\partial y} + \frac{\partial v^{}}{\partial x} \frac{\partial v^{}}{\partial y} + (\frac{\partial w^{}}{\partial x}) (\frac{\partial w^{}}{\partial y}) + z^{2} [\frac{\partial θ_{x}^{}}{\partial x} \frac{\partial θ_{x}^{}}{\partial y} + \frac{\partial θ_{y}^{}}{\partial x} \frac{\partial θ_{y}^{}}{\partial y}] \end{array}\}

\{\begin{matrix} N_{x}^{} \\ N_{y}^{} \\ N_{x y}^{} \\ M_{x}^{} \\ M_{y u}^{} \\ M_{x y}^{} \\ Q_{x z}^{} \\ Q_{y z}^{} \end{matrix}\} = [\begin{matrix} A_{11}^{} & A_{12}^{} & A_{16}^{} & B_{11}^{} & B_{12}^{} & B_{16}^{} & 0 & 0 \\ A_{12}^{} & A_{22}^{} & A_{26}^{} & B_{12}^{} & B_{22}^{} & B_{26}^{} & 0 & 0 \\ A_{16}^{} & A_{26}^{} & A_{66}^{} & B_{16}^{} & B_{26}^{} & B_{66}^{} & 0 & 0 \\ B_{11}^{} & B_{12}^{} & B_{16}^{} & D_{11}^{} & D_{12}^{} & D_{16}^{} & 0 & 0 \\ B_{12}^{} & B_{22}^{} & B_{26}^{} & D_{12}^{} & D_{22}^{} & D_{26}^{} & 0 & 0 \\ B_{16}^{} & B_{26}^{} & B_{66}^{} & D_{16}^{} & D_{26}^{} & D_{66}^{} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & S_{44}^{} & S_{45}^{} \\ 0 & 0 & 0 & 0 & 0 & 0 & S_{45}^{} & S_{55}^{} \end{matrix}] \{\begin{matrix} u_{, x}^{u} \\ v_{, y}^{u} \\ u_{, y}^{} + v_{, x}^{u} \\ θ_{x, x}^{u} \\ θ_{y, y}^{u} \\ θ_{x, y}^{u} + θ_{y, x}^{u} \\ w_{, x}^{u} + θ_{xp} \\ w_{, y}^{u} + θ_{yp} \end{matrix}\}

\{\begin{matrix} N_{r H}^{} \\ M_{r H}^{} \\ M_{r t}^{} \\ Q_{r z}^{H} \end{matrix}\} = [\begin{matrix} {\bar{A}}_{11}^{} & {\bar{B}}_{11}^{} & {\bar{B}}_{16}^{} & 0 \\ {\bar{B}}_{11}^{} & {\bar{D}}_{11}^{} & {\bar{D}}_{16}^{} & 0 \\ {\bar{B}}_{16}^{} & {\bar{D}}_{16}^{} & {\bar{D}}_{66}^{} & 0 \\ 0 & 0 & 0 & {\bar{A}}_{44}^{} \end{matrix}] \{\begin{matrix} u_{, r H}^{} \\ θ_{r, r}^{} \\ θ_{t, r}^{} \\ w_{, r}^{} + H θ_{r}^{} \end{matrix}\}

\{\begin{matrix} \begin{matrix} \{N\} \\ \{Q\} \end{matrix} \end{matrix}\} = [\begin{matrix} [D_{b}] & [0] \\ [0] & [D_{s}] \end{matrix}] \{\begin{matrix} \{ε\} \\ \{γ\} \end{matrix}\}

[k_{e s}^{}] = b_{s} \int_{- 1}^{+ 1} {[T_{e}^{}]}_{}^{T k} {[T_{o}^{}]}_{}^{T k} {[B_{s}]}_{}^{T k} [D_{s}] [B_{s}] [T_{o}^{}] [T_{e}^{}] |J_{s}| d ξ

[k_{G s}^{k}] = b_{s} \int_{- 1}^{+ 1} {[T_{e}^{}]}_{}^{T k} {[T_{o}^{}]}_{}^{T k} {[B_{G s}^{}]}_{}^{T k} {[S_{s}]}_{_{}} {[B_{G s}^{}]}_{} {[T_{o}^{}]}_{} {[T_{e}^{}]}_{} |J_{s}| d ξ

[u m_{s}] = b_{s} \int_{- 1}^{+ 1} {[T_{e}^{}]}_{}^{T} {[T_{o}^{}]}_{H}^{T k} {[s {\bar{N}}_{s}]}_{H}^{T k} {[{\bar{I}}_{s}]}_{H} {[s {\bar{N}}_{s}]}_{H} {[s T_{o}^{}]}_{H} {[s T_{e}^{}]}_{H} |J^{s}| d ξ H s

[1] *Corresponding author

Brasil

Brasil

Influence of stiffeners on the buckling behaviour of composite panels subjected to non-uniform edge loads

Abstract

Graphical Abstract: