Efficient Component-Wise Finite Elements for the Dynamic Response Analysis of Metallic and Composite Structures Marco Petrolo, Erasmo Carrera, Ibrahim Kaleel Department of Mechanical and Aerospace Engineering, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
[email protected]
Abstract This paper presents a novel component-wise (CW) approach for the finite element, dynamic response analysis of composite structures. In a CW model, each component of a complex structure can be modelled through a refined 1D model based on the Carrera Unified Formulation (CUF). The CUF allows the use of any order 1D structural models in a unified manner. Finite element matrices are obtained using a set of a few fundamental nuclei that are independent of the structural model order. The adoption of only 1D finite elements to model complex structures improves the multi-dimension coupling capabilities and reduces the computational costs to a great extent. The CW leads to finite element models in which fibres, matrix, and plies can be modelled through the 1D CUF models. Furthermore, the CW can be seen as a physics-based approach. In fact, a detailed physical description of a real structure can be obtained placing the problem unknown variables on the physical surfaces of the real 3D model. No artificial surfaces or lines are needed, and the CAD-FEM coupling is facilitated. Each component can be modelled with its material characteristics; that is, no homogenization techniques are required. In this paper, the CW is exploited for the dynamic response analysis of structures is presented. The CW capabilities are compared to those of classical techniques and commercial codes to prove the CW high accuracy and low computational cost. Keywords: FEM, Beam, Composites, CUF, Component-Wise
Introduction The dynamic analysis of structures is of primary importance in many applications, such as impact problems, damage detection, and health monitoring. Such analyzes can be challenging tasks. Computational models have to detect very accurate displacement, strain and stress fields. Moreover, local effects, multi-scale and multi-dimension structural components (e.g. layers and fibers, panels and stringers), and anisotropy have to be considered to obtain reliable results. Currently, most of the techniques that have been developed for these tasks are based on very cumbersome numerical models, such as the 3D solid finite elements. The accurate structural analysis of complex structures is almost impossible due to the enormous number of degrees of freedom that is required. This paper presents the dynamic analysis of metallic and composite structures via refined structural models and the finite element modeling. In particular, 1D advanced structural models were used. Classical 1D models, or beams, were provided by Euler-Bernoulli [1, 2], and Timoshenko [3], hereafter referred to EBBT and TBT, respectively. These models are computationally cheap and, to some extent, reliable for many structural mechanics problems. However, EBBT and TBT cannot detect many mechanical behaviors; such as out-of-plane warping, in-plane distortions, torsion, coupling effects, or local effects. These effects are usually due to small slenderness ratios, thin walls, geometrical and mechanical asymmetries, and the anisotropy of the material [4]. Due to their computational efficiency, 1D models with advanced capabilities have been developed over the last decades. Some of the most relevant are based on the use of higher-order displacement fields [5, 6], the Variational Asymptotic
Method [7-9], the Generalized Beam Theory [10], and the Carrera Unified Formulation (CUF) [11]. Some works focused on the development of advanced 1D models for the dynamic response analysis [12-15]. This paper makes use of the advanced 1D models based on the CUF. The CUF [16, 17] provides refined 1D and 2D structural theories that are extremely accurate and computationally cheap. Recently, the 1D CUF has led to the development of the component-wise approach (CW) [18]. Figure 1 shows an example of CW modeling for a layered composite plate. The CW can model the macro (layers) and microscale components (fibers and matrix) using 1D models only. All these components can be coupled straightforwardly by imposing the displacement continuity at the interfaces. A detailed, physical description of composites can be obtained since the problem unknowns can be placed on the physical surfaces of the real 3D model. Moreover, each component is modeled using its material characteristics, that is, no homogenization techniques are required.
Figure 1: CW modeling of composites
The CW can be exploited to model other types of complex structures, such as aircraft wings [19, 20]. Recently the authors have extended the CW to the analysis of damaged isotropic, thin-walled structures [21]. This paper evaluates the enhanced capabilities of the CW for the dynamic response analysis of metallic and composite structures. Free vibration and dynamic response analyses are carried out. The results are compared with those of 3D solid finite elements.
1. Carrera Unified Formulation The cross-section of the beam lies on the xz-plane, and it is denoted by Ω, whereas the boundaries over y are 0≤y≤L. Within the framework of the CUF, the 3D displacement field is expressed as
𝒖(𝑥, 𝑦, 𝑧; 𝑡) = 𝐹𝜏 (𝑥, 𝑧)𝒖𝜏 (𝑦; 𝑡), 𝜏 = 1,2, … , 𝑀
(1)
Where 𝑭𝝉 are the functions of the coordinates 𝑥 and 𝑧 on the cross-section. 𝒖𝜏 is the vector of the generalized displacements. 𝑴 stands for the number of the terms used in the expansion, and the repeated subscript, 𝜏 indicates summation. LE (Lagrange Expansion) 1D CUF models exploit 2D Lagrange polynomials to model the displacement field of the structure above the cross-section. For instance, the displacement field of an L9 LE model can be expressed as
𝑢𝑥 = 𝐿1 𝑢𝑥1 + 𝐿2 𝑢𝑥2 + ⋯ + 𝐿9 𝑢𝑥9
(2)
For the sake of brevity, only the x-component of the displacement field is reported. The L9 model has 27 displacement variables that coincide with the three displacement components of the 9 Lagrange nodes. Two or more Lagrange elements can be conveniently assembled to discretized cross-sections, and improve the accuracy of the model. Figure 2 shows a typical cross-section modelling in which a finer modelling is used in the proximity of the applied load.
Figure 2: Multiple L9 discretization
A compact form of the virtual variation of the strain energy can be obtained as shown in [17],
𝛿𝐿𝑖𝑛𝑡 = 𝛿𝒒𝑇𝑠𝑗 𝑲𝑖𝑗𝜏𝑠 𝒒𝜏𝑖
(3)
Where 𝑲𝑖𝑗𝜏𝑠 is the stiffness matrix written in the form of the fundamental nuclei whose components can be found in [17]. 𝛿indicates the virtual variation. 𝒒𝜏𝑖 is the nodal vector. Superscripts indicate the four indexes exploited to assemble the matrix: 𝑖 and 𝑗 are related to the shape functions, 𝜏 and 𝑠 are related to the expansion functions. The fundamental nucleus is a 3 × 3 array that is formally independent of the order of the beam model. It should be underlined that the formal expression of 𝑲𝑖𝑗𝜏𝑠 does not depend on the expansion order and on the choice of the 𝐹𝜏 expansion polynomials. All the other FEM matrices can be obtained in a similar manner. The advanced capabilities of CUF 1D models can be particularly convenient in the case of multicomponent structures (MCS). The Component-Wise approach exploits LE 1D elements to model each component of a structure separately and independently of their geometrical and material characteristics. In other words, each 1D, 2D, 3D or micro and macro component can be modeled via LE 1D models with no need for ad hoc coupling and interface techniques. Figure 1 shows a typical CW strategy for a composite plate; 1D LE models can be simultaneously adopted to model layers (macroscale), matrix and fibers (microscale). This methodology can be very powerful when, for instance, detailed stress fields are required in a specific portion of the structures. Similar strategies can be used for aircraft structures as in [20].
2. Numerical Example A clamped-clamped, thin-walled, isotropic cylinder was considered to highlight the enhanced capabilities of the present formulation. The outer diameter d is equal to 0.1 (m), the thickness is equal to 0.001 (m), and the span-to-diameter ratio (L=d) is equal to 10. The material is aluminum (E = 69 GPa, ν = 0:33, ρ = 2700 kg/m3). Four points were considered over the mid-span cross-section as shown in Fig. 3. Four concentrated forces were applied as time-dependent sinusoids with amplitude Pz0 = 10000 (N) and a phase shift,
𝑃𝑧𝐴 (𝑡) = 𝑃𝑧0 sin(𝜔𝑡 + 𝜑𝐴 ), 𝑃𝑧𝐵 (𝑡) = 𝑃𝑧0 sin(𝜔𝑡 + 𝜑𝐵 ), 𝑃𝑧𝐶 (𝑡) = 𝑃𝑧0 sin(𝜔𝑡 + 𝜑𝐶 ) , 𝑃𝑧𝐷 (𝑡) = 𝑃𝑧0 sin(𝜔𝑡 + 𝜑𝐷 ), 𝜑𝐴 = 0, 𝜑𝐵 = 30, 𝜑𝐶 = 60, 𝜑𝐷 = 90
Figure 3: Thin walled cross-section
Where the angular frequency is 𝜔 = 100 rad/s. The dynamic response of the structure was evaluated over the time interval [0; 0.025] s. The Newmark integration scheme was exploited. Table 1 shows the transverse displacements of point A at t = 0 s. A 44 L9 model was used in this paper. The configuration at the final time instant t = 0.025 s is shown in Fig. 4 (mid-span cross-section). The results show how the present 1D formulation can detect severe cross-section deformations and match 3D solid results with much lower computational costs.
Table 1: Transverse displacement of point A
CW (44 L9) Solid DOFs
24552
268440
uzA
-9.5388
-9.8840
Figure 4: Deformation of the mid-span circular cross-section, t = 0:025s, 44 L9
3. Conclusions This paper has presented a brief overview of the Component-Wise approach (CW) for the high-fidelity dynamic analysis of metallic and composite structures. The CW is based on the 1D CUF models. Such
structural models are computationally efficient and accurate. 1D CUF, in fact, can provide 3D-like accuracy with 10-100 times fewer degrees of freedom. A numerical example has been carried out on a thin-walled structure undergoing a dynamic load. The present approach could be useful for impact problems in which high accuracy is needed, and a low computational cost is desirable.
Acknowledgements This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 642121.
References [1] D. Bernoulli, Commentarii Academiae Scientiarum Imperialis Petropolitanae, chap. De vibrationibus et sono laminarum elasticarum, Petropoli, 1751. [2] L. Euler, De curvis elasticis, chap. Methodus Inveniendi Lineas Curvas Maximi Minimive Proprietate Gaudentes, Sive Solutio Problematis Isoperimetrici Lattissimo Sensu Accept, Bousquet, 1744. [3] S.P. Timoshenko, On the corrections for shear of the differential equation for transverse vibrations of prismatic bars, Philosophical Magazine, Vol. 41, pp. 744-746, 1922. [4] E. Carrera, A. Pagani, M. Petrolo, and E. Zappino, Recent developments on refined theories for beams with applications, Mechanical Engineering Reviews, Vol. 2, No. 2, 2015. [5] K. Kapania and S. Raciti, Recent Advances in Analysis of Laminated Beams and Plates, Part I: Shear Effects and Buckling, AIAA Journal, Vol. 27, No. 7, pp. 923-935, 1989. [6] K. Kapania and S. Raciti, Recent Advances in Analysis of Laminated Beams and Plates, Part II: Vibrations and Wave propagation, AIAA Journal, Vol. 27, No. 7, pp. 935-946, 1989. [7] V. Berdichevsky, E. Armanios, and A. Badir, Theory of anisotropic thin-walled closed-cross-section beams, Composite Engineering, Vol. 2, No. 5-7, pp. 411-432, 1992. [8] W. Yu and D. and Hodges, Generalized Timoshenko Theory of the Variational Asymptotic Beam Sectional Analysis, Journal of the American Helicopter Society, Vol. 50, No. 1, pp. 46-55, 2005. [9] Q. Wang and W. Yu, A Variational Asymptotic Approach for Thermoelastic Analysis of Composite Beams, Advances in Aircraft and Spacecraft Sciences, Vol. 1, No. 1, pp. 93-123, 2014. [10] N. Silvestre and D. Camotim, Shear Deformable Generalized Beam Theory for the Analysis of Thin-Walled Composite Members, Journal of Engineering Mechanics, Vol. 139, pp. 1010-1024, 2013. [11] E. Carrera and M. Petrolo, Refined One-Dimensional Formulations for Laminated Structure Analysis, AIAA Journal, Vol. 50, No. 1, pp. 176-189, 2012. [12] S. Ramalingerswara Rao and N. Ganesan, Dynamic response of tapered composite beams using higher order shear deformation theory, Journal of Sound and Vibration, Vol. 187, pp. 737-756, 1995. [13] S. Marur and T. Kant, On the performance of higher order theories for transient dynamic analysis of sandwich and composite beams, Computers and Structures, Vol. 65, pp. 741-759, 1997. [14] S. Na and L. Librescu, Dynamic response of elastically tailored adaptive cantilevers of non-uniform cross section exposed to blast pressure pulses, International Journal of Impact Engineering, Vol. 25, pp. 847-867, 2001. [15] E. Carrera and A. Varello, Dynamic response of thin-walled structures by variable kinematic one-dimensional models, Journal of Sound and Vibration, Vol. 331, No. 24, pp. 5268-5282, 2012. [16] E. Carrera, G. Giunta, and M. Petrolo, Beam Structures: Classical and Advanced Theories, John Wiley & Sons, 2011. [17] E. Carrera, M. Cinefra, M. Petrolo, and E. Zappino, Finite Element Analysis of Structures through Unified Formulation, John Wiley & Sons, 2014. [18] E. Carrera, M. Maiarù, and M. Petrolo, Component-Wise Analysis of Laminated Anisotropic Composites, International Journal of Solids and Structures, Vol. 49, pp. 1839-1851, 2012. [19] E. Carrera, A. Pagani, and M. Petrolo, Classical, Refined and Component-wise Theories for Static Analysis of Reinforced-Shell Wing Structures, AIAA Journal, Vol. 51, No. 5, pp. 1255-1268, 2013. [20] E. Carrera, A. Pagani, and M. Petrolo, Component-wise Method Applied to Vibration of Wing Structures, Journal of Applied Mechanics, Vol. 80, No. 4, 2013. [21] M. Petrolo, E. Carrera, and A.S.A.S. Alawami, Free vibration analysis of damaged beams via refined models, Advances in Aircraft and Spacecraft Sciences, Vol. 3, No. 1, pp. 95-112, 2016.
Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
EFFICIENT COMPONENT-WISE FINITE ELEMENTS FOR THE DYNAMIC RESPONSE ANALYSIS OF METALLIC AND COMPOSITE STRUCTURES Marco Petrolo*, Erasmo Carrera* and Ibrahim Kaleel* *Department of Mechanical and Aerospace Engineering, Politecnico di Torino
Full 1st International Conference on Impact Loading of Structures and Materials, ICILSM2016 19 Nov 2015, Torino (Italy)
Marco Petrolo, Erasmo Carrera and Ibrahim Kaleel CARRERA UNIFIED FORMULATION - COMPONENT-WISE 1D MODELS - POLITECNICO DI TORINO (ITALY) - WWW.MUL2.COM
Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
MUL2 - Our Research Group
Marie Curie Project on Composites
Full
www.fullcomp.net
Marco Petrolo, Erasmo Carrera and Ibrahim Kaleel CARRERA UNIFIED FORMULATION - COMPONENT-WISE 1D MODELS - POLITECNICO DI TORINO (ITALY) - WWW.MUL2.COM
Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
The FULLCOMP project FULLy integrated analysis, design, manufacturing and health-monitoring of COMPosite structures The FULLCOMP project is funded by the European Commission under a Marie Sklodowska-Curie Innovative Training Networks grant for European Training Networks (ETN). The FULLCOMP partners are: 1 2 3 4 5 6 7 8 9
Politecnico di Torino (Italy) - Coordinator University of Bristol (UK) Ecole Nationale Superieure d’arts et Metiers (Bordeaux, France) Leibniz Universitaet Hannover (Germany) University of Porto (Portugal) University of Washington (USA) RMIT (Australia) Luxembourg Institute of Technology Elan-Ausy, Hamburg, (Germany)
FULLCOMP has recruited 12 PhD students who will work in an international framework to develop integrated analysis tools to improve the design of composite structures. The full spectrum of the design of composite structures will be dealt with, such as manufacturing, health-monitoring, failure, modeling, multiscale approaches, testing, prognosis, and prognostic. The FULLCOMP research activity is aimed at many engineering fields, e.g. aeronautics, automotive, mechanical, wind energy, and space. www.fullcomp.net Marco Petrolo, Erasmo Carrera and Ibrahim Kaleel CARRERA UNIFIED FORMULATION - COMPONENT-WISE 1D MODELS - POLITECNICO DI TORINO (ITALY) - WWW.MUL2.COM
Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
Overview 1
Description of the Carrera Unified Formulation for refined models (CUF).
2
Main 1D CUF capabilities overview (1D Taylor- and Lagrange-based models).
3
Numerical examples dealing with different applications (aerospace and civil structures, composites, Bio Structures).
4
Introduction to the Component-Wise approach (CW).
5
Dynamic response analysis : Sandwich beam structure
Marco Petrolo, Erasmo Carrera and Ibrahim Kaleel CARRERA UNIFIED FORMULATION - COMPONENT-WISE 1D MODELS - POLITECNICO DI TORINO (ITALY) - WWW.MUL2.COM
Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
Brief Overview of Beam Refinement Methods and Contributors 1
Shear correction factors (Timoshenko, Sokolnikoff, Cowper, Gruttmann, etc.).
2
Warping functions and Saint-Venant solutions (El Fatmi, Ladéveze, etc.).
3
Variational asymptotic method (Berdichevsky, Hodges, Yu, etc.).
4
Generalized beam theory (Schardt, Camotim, Silvestre, etc.).
5
Higher-order models (Washizu, Reddy, Kapania, Carrera, etc.).
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Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
1D Advanced Structural Models Actual Wing Our Model
1D Carrera Unified Formulation, CUF - FEM Version
Ni(y) Classical 1D FE
+
Fτ(x,z) Cross-Section Functions
Marco Petrolo, Erasmo Carrera and Ibrahim Kaleel CARRERA UNIFIED FORMULATION - COMPONENT-WISE 1D MODELS - POLITECNICO DI TORINO (ITALY) - WWW.MUL2.COM
Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
Fundamental Nucleus Equations by CUF Finite Element Formulation (FEM) i , j : Shape function indexes u = Fτ (x , z )Ni (y ) uτi (1D ) (depend on the FE u = Fτ (z )Ni (x , y ) uτi (2D ) discretization). ij τs δLint = δqTτi K qsj τ, s : Expansion function ij τs ¨ sj δLine = δqTτi M q indexes (depend on the model T order). δLext = Pδu Assembly Technique 3 x M x M τs-Block
{ 3 x 3 Nucleus
{ Kxx Kxy Kxz Kyx Kyy Kyz Kzx Kzy Kzz
}
}
}
s=1
s = 2, M-1
s=M
}
τ=1
}
τ = 2, M-1
}
τ=M
Fundamental Nucleus
ij τs
Kxx =
˜ 22 C
Z
Z Ω
Fτ,x Fs ,x d Ω
Ni Nj dy + ...
l
Marco Petrolo, Erasmo Carrera and Ibrahim Kaleel CARRERA UNIFIED FORMULATION - COMPONENT-WISE 1D MODELS - POLITECNICO DI TORINO (ITALY) - WWW.MUL2.COM
Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
The CUF as a Second Generation of Theory of Structures Reduction of 3D problems to 2D and 1D (not only structural problems). All the governing equations can be written via only two formal statements. Complete 3D stress fields via 1D and 2D models. Some 10-100 times less DOFs than 3D models. Any structural theory can be implemented as a particular case via a hierarchical formulation and the free choice of the expansion order and type. An arbitrary rich displacement field could lead to more and more accurate results independently of the problem characteristics (Washizu).
Marco Petrolo, Erasmo Carrera and Ibrahim Kaleel CARRERA UNIFIED FORMULATION - COMPONENT-WISE 1D MODELS - POLITECNICO DI TORINO (ITALY) - WWW.MUL2.COM
Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
1D CUF Applications Overview - Books Thin-Walled and Reinforced Structures
Buckling, Free Vibration and Dynamic Response Analysis
Composite Structures
FGM Structures
Variable Kinematics Models
Axiomatic/Asymptotic Analyses and Best Theory Diagrams
Aeroelasticity
Load Factors and Non-Structural Masses
Rotors and Rotating Blades
Biomechanics
Multifield Analysis
Nanostructures
Analysis of Aerospace Structures via the Component-Wise Approach
Analysis of Civil Structures via the Component-Wise Approach
Component-Wise Approach for the Multiscale Analyses of Composites
FEM
2D and Smart Structures
1D
Marco Petrolo, Erasmo Carrera and Ibrahim Kaleel CARRERA UNIFIED FORMULATION - COMPONENT-WISE 1D MODELS - POLITECNICO DI TORINO (ITALY) - WWW.MUL2.COM
Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
The Taylor CUF 1D Models, TE ux = uy = uz =
ux1 uy1 uz1
|{z} N=0 τ=1 3 DOFs
+x ux2 + z ux3 +x uy2 + z uy3 + x uz2 + z uz3 | {z } N=1 τ = 2, τ = 3
+x 2 ux4 + xz ux5 + z 2 ux6 + ... +x 2 uy4 + xz uy5 + z 2 uy6 + ... + x 2 uz4 + xz uz5 + z 2 uz6 +... | {z } N=2 τ = 4, τ = 5, τ = 6
9 DOFS
18 DOFs
Classical models, such as Timoshenko, can be obtained as particular cases of the linear models. Assembly Technique 3 x M x M τs-Block
{ 3 x 3 Nucleus
{ Kxx Kxy Kxz Kyx Kyy Kyz Kzx Kzy Kzz
}
}
}
s=1
s = 2, M-1
s=M
}
τ=1
}
τ = 2, M-1
}
τ=M
Fundamental Nucleus ij τs
Kxx =
˜ 22 C
Z
Z Ω
Fτ,x Fs ,x d Ω
Ni Nj dy + ...
l
Marco Petrolo, Erasmo Carrera and Ibrahim Kaleel CARRERA UNIFIED FORMULATION - COMPONENT-WISE 1D MODELS - POLITECNICO DI TORINO (ITALY) - WWW.MUL2.COM
Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
Wave propagation in a Thin-Walled Cylinder
E. Carrera, A. Varello, Dynamic Response of Thin-Walled Structures by Variable Kinematic One-Dimensional Models Journal of Sound and Vibration, 331(24), pp. 5268-5282, 2012.
N = 8 beam model, 6000 DOFs
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Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
Arterial cross-section in a human external iliac artery
E. Carrera, A. Varello, Nonhomogeneous atherosclerotic plaque analysis via enhanced 1D structural models Smart Structures and Systems, In Press.
N = 20 beam model, 22000 DOFs SOLID, 760000 DOFs Marco Petrolo, Erasmo Carrera and Ibrahim Kaleel CARRERA UNIFIED FORMULATION - COMPONENT-WISE 1D MODELS - POLITECNICO DI TORINO (ITALY) - WWW.MUL2.COM
Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
Lagrange-Based 1D Models, LE ux = Lτ uxτ uy = Lτ uyτ uz = Lτ uzτ
3D Geometry from CAD
LE Modeling
L9 polynomials - Isoparametric Lτ =
1 2 (r + r rτ )(s 2 + ... 4
Beam element Beam node Lagrange node above the first beam node cross-section Lagrange node above the second beam node cross-section DOFs: pure displacements of each Lagrange node (3 DOFs per Lagrange node)
Lagrange nodes can be placed above the physical surface of the structure
Cross-Section Elements Computational Model
Cross-Section nodes Disconnected nodes
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Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
The Component-Wise Approach
Only displacements as unknowns. Each component of the structure is modeled via beams only. No need of reference surfaces. This might be useful in a CAD-FEM interface scenario. No need of homogenization techniques. Marco Petrolo, Erasmo Carrera and Ibrahim Kaleel CARRERA UNIFIED FORMULATION - COMPONENT-WISE 1D MODELS - POLITECNICO DI TORINO (ITALY) - WWW.MUL2.COM
Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
Aircraft Wing Shell-like modal shapes by means of a BEAM MODEL
Z X
c
E. Carrera, A. Pagani and M. Petrolo, Component-wise Method Applied to Vibration of Wing Structures, Journal of Applied Mechanics, 80(4), 2013. doi:10.1115/1.4007849.
Natural frequencies [Hz] Mode 1 2 3 4 5 6 7
1D-LE 4.23 21.76 25.15 31.14 59.26 66.65 74.23
SOLID 4.22 21.69 24.78 29.18 56.12 62.41 68.77
Computational costs CUF LE ≈ 20000 DOFs; SOLID ≈ 190000 DOFs
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Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
CW for Multiscale Analyses of Composite Structures
E. Carrera, M. Maiaru, M. Petrolo, Component-Wise Analysis of Laminated Anisotropic Composites International Journal of Solids and Structures, 49, 1839-1851, 2012.
Marco Petrolo, Erasmo Carrera and Ibrahim Kaleel CARRERA UNIFIED FORMULATION - COMPONENT-WISE 1D MODELS - POLITECNICO DI TORINO (ITALY) - WWW.MUL2.COM
Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
Evaluation of Failure Parameters Failure Indexes LE - 7533 DOFs
5
1.6 FI Max Stress - LE
FI Max Strain - LE
Integral Quantities
1.4 1.2 1
4
3
0.8 0.6 0.4
2
1
0.2 0
0
SOLID - 268215 DOFs
Marco Petrolo, Erasmo Carrera and Ibrahim Kaleel CARRERA UNIFIED FORMULATION - COMPONENT-WISE 1D MODELS - POLITECNICO DI TORINO (ITALY) - WWW.MUL2.COM
Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
Damage Analysis via CUF Motivations and Ideas 1D CUF can deal with multi-component structures by means of low computational costs. We can easily obtain very accurate stress fields (to predict failure) or introduce damages (to simulate the mechanical response of a damaged structure). We can evaluate the effect of damage in terms of modal shapes and static and dynamic responses.
Damage Modeling Damage was introduced in a portion of the structure. In the damaged zone we have Ed ∗ = (1 − d ) × E with 0 < d < 1 (e.g. E0.9 = 0.9 × E).
Damaged
1D CUF:L-elements discretizing the cross-sections of each component
Mid-span assembled cross-section Reinforced-shell structure
DAMAGE
Component-wise approach
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Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
Fiber/Matrix Cells- 1 Fiber Damaged
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Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
Fiber/Matrix Cells - Frequencies - Solid vs CW
CW ≈ 10000 DOFs. SOLID ≈ 125000 DOFs.
Marco Petrolo, Erasmo Carrera and Ibrahim Kaleel CARRERA UNIFIED FORMULATION - COMPONENT-WISE 1D MODELS - POLITECNICO DI TORINO (ITALY) - WWW.MUL2.COM
Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
Composite Sandwich Structure Composite sheet 1.9 34.8
1 -D CUF Lagrange elements for the cross-section
1.9 260
Foam
59.9
(all dimensions in mm)
Mechanical properties of Composite sheet EL [GPa]
ET = EZ [GPa]
GLT = GLZ [GPa]
GTZ [GPa]
νLT =νLZ
νTZ
ρ [Kg /m3 ]
159.380
14.311
3.711
5.209
0.2433
0.2886
1300
Mechanical properties of Foam Core E [GPa]
G [GPa]
ν
ρ [Kg /m3 ]
0.056
0.022
0.27
60
CUF ≈ 34,000 DOFs
A sinusoidal line load of 105 N /m with a frequency of 25.0 rad/s is applied at mid span (bottom and top)
SOLID ≈ 260,000 DOFs
Dynamic response over the time period [0,1.0] s
Marco Petrolo, Erasmo Carrera and Ibrahim Kaleel CARRERA UNIFIED FORMULATION - COMPONENT-WISE 1D MODELS - POLITECNICO DI TORINO (ITALY) - WWW.MUL2.COM
Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
Composite Sandwich Structure - Displacement field CUF-CW +2.01e−04 +1.51e−04 +1.00e−04 +5.01e−05 −2.02e−07 −5.05e−05 −1.01e−04 −1.51e−04 −2.01e−04
ABAQUS +2.07e−04 +1.55e−04 +1.04e−04 +5.18e−05 +7.28e−12 −5.18e−05 −1.04e−04 −1.55e−04 −2.07e−04
×10-4
CUF - CW ABAQUS
Ux [m]
2 0 -2 0×10-4
0.1
0.2
0.3
0.4
Ux [m] Uy [m]
-2 0
×10-4
-4 0×10 2
0.8
0.9
0.2
0.3
0.4
0.1
0.2
0.3
0.4
0.5
Time [s] 0.5
1
CUF - CW ABAQUS
0.6
0.7
0.8
0.9
0.6
0.7
0.8
0.9
Time [s]
CUF - CW ABAQUS1 1 CUF - CW ABAQUS
0 -2 0 ×10-4
0×10 2
Uy [m]
1
Uz [m]
0.1
2 ×10-3 1 0
0 -2
-1 +6.61e−04 +4.96e−04 +3.31e−04 +1.65e−04 −4.07e−10 −1.65e−04 −3.31e−04 −4.96e−04 −6.61e−04
0.7
0 -2
0
+6.49e−04 +4.87e−04 +3.25e−04 +1.62e−04 +2.91e−11 −1.62e−04 −3.25e−04 −4.87e−04 −6.49e−04
0.6
2 0
Ux [m]
+1.73e−04 +1.29e−04 +8.63e−05 +4.32e−05 +7.28e−12 −4.32e−05 −8.63e−05 −1.29e−04 −1.73e−04
Uy [m] Uz [m]
+1.68e−04 +1.26e−04 +8.41e−05 +4.20e−05 +7.28e−12 −4.20e−05 −8.41e−05 −1.26e−04 −1.68e−04
0.5
Time [s]
2 ×10-4
0
-3
0.1
0.1
0.2
0.2
0.1
0.3
0.3
0.2
0.3
0.4
0.4 0.4
0.5
Time 0.5 [s]
Time [s] 0.5
0.6
0.7
0.8
0.7
0.8
0.9
0.6
0.7
0.8
0.9
0
0.1
0.2
0.3
0.4
-1
Uz [m]
1
Time [s] 0 -2
0
CUF - CW CUF - CW1 ABAQUS ABAQUS1
0.9
0.6
1
×10-3
0.1
0.2
0.3
0.4
0.5
Time [s] 0.5
0.6
0.6
0.7
0.7
0.8
0.8
CUF - CW CUF - CW ABAQUS 0.9 ABAQUS 1 0.9
1
Time [s]
0
CUF - CW ABAQUS
-1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Time [s]
Displacement field at time 0.36s Marco Petrolo, Erasmo Carrera and Ibrahim Kaleel CARRERA UNIFIED FORMULATION - COMPONENT-WISE 1D MODELS - POLITECNICO DI TORINO (ITALY) - WWW.MUL2.COM
1
Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
Composite Sandwich Structure - 3D Stress fields +3.007e+07 +2.449e+07 +1.891e+07 +1.333e+07 +7.750e+06 +2.171e+06 −3.407e+06 −8.986e+06 −1.457e+07 −2.014e+07
+6.208e+08 +4.843e+08 +3.477e+08 +2.112e+08 +7.466e+07 −6.187e+07 −1.984e+08 −3.349e+08 −4.715e+08 −6.080e+08
σxx in midspan of beam at time 0.36s +2.659e+07 +2.068e+07 +1.477e+07 +8.863e+06 +2.954e+06 −2.954e+06 −8.863e+06 −1.477e+07 −2.068e+07 −2.659e+07
σyy at time 0.36s
σyz in midspan of beam at time 0.36s Marco Petrolo, Erasmo Carrera and Ibrahim Kaleel CARRERA UNIFIED FORMULATION - COMPONENT-WISE 1D MODELS - POLITECNICO DI TORINO (ITALY) - WWW.MUL2.COM
Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
Main Conclusions and Perspectives
1
1D CUF structural models are powerful tools to analyze structures for different applications, including aerospace and civil structures and composites.
2
The Component-Wise approach is able to analyze complex structures by means of a unique structural formulation.
3
CW 1D Models require at least 10 times less DOFs than Solid models.
Future extensions CUF models can provide 3D strain and stress fields with low computational cost and 3D-like accuracy. Extension of CUF models for non-linear problems, where accurate stress fields are necessary. For instance : Progressive damage model, plasticity etc.
Marco Petrolo, Erasmo Carrera and Ibrahim Kaleel CARRERA UNIFIED FORMULATION - COMPONENT-WISE 1D MODELS - POLITECNICO DI TORINO (ITALY) - WWW.MUL2.COM
Introduction
CUF
Taylor Models
Component-Wise Approach
Dynamic Analysis
Conclusion
EFFICIENT COMPONENT-WISE FINITE ELEMENTS FOR THE DYNAMIC RESPONSE ANALYSIS OF METALLIC AND COMPOSITE STRUCTURES Marco Petrolo*, Erasmo Carrera* and Ibrahim Kaleel* *Department of Mechanical and Aerospace Engineering, Politecnico di Torino
Full 1st International Conference on Impact Loading of Structures and Materials, ICILSM2016 19 Nov 2015, Torino (Italy)
Marco Petrolo, Erasmo Carrera and Ibrahim Kaleel CARRERA UNIFIED FORMULATION - COMPONENT-WISE 1D MODELS - POLITECNICO DI TORINO (ITALY) - WWW.MUL2.COM
