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THERMOMECHANICAL RESPONSE VARIABILITY OF COMPOSITE PANELS WITH CONTINUOUS AND TERMINATED STIFFENERS
Ahmed K. Noor*, James H. Starnes†, and Jeanne M. Peters‡
NASA Langley Research Center Hampton, Virginia 23681 USA
Abstract
A study is made of the variability of the response of stiffened composite panels, associated with variations in the major geometric and material parameters of the structures. The major parameters, which have the most effect on the response quantities of interest, are identified by using a hierarchical sensitivity analysis. The range of variation of the response is determined by using a fuzzy set analysis, with the major parameters treated as fuzzy parameters. Numerical results are presented showing the variability of the response of panels with both continuous and terminated stiffeners associated with variations in the micromechanical, effective layer and geometric parameters. Both flat and curved panels are considered.
Keywords: composite panels, continuous and terminated stiffeners, combined loads, mechanical and thermal loads, nonlinear response, sensitivity analysis, response variability.
_______________
* Eminent Scholar, Professor of Aerospace Engineering, Old Dominion University, Adjunct Professor, University of Florida, and Director, Center for Advanced Engineering Environments, Old Dominion University, Fellow AIAA. † Senior Engineer, Structures and Materials Competency, Fellow AIAA. ‡ IT Specialist, Center for Advanced Engineering Environments, Old Dominion University.
A significant numerical simulation capability now exists for studying the various phenomena associated with the response, failure and performance of multilayered composite panels and shells subjected to combined pressure, mechanical and thermal loads. The phenomena involved cover a wide range of length scales from local to global structural response. The modeling approaches used for multilayered panels include micromechanical models, three-dimensional continuum models, quasi-three-dimensional models, and two-dimensional plate and shell models. Within each category a number of models with several levels of sophistication has evolved. The four categories are described in review papers [1, 2] and monographs [3 – 5]. Despite the extensive literature cited in the aforementioned references, only a few studies have been reported on the effects of stiffness discontinuities, such as those associated with an abrupt stiffener termination or dropped plies, on the response of composite panels (see, for example, [6 – 8]). Stiffener termination is often necessary in composite aerospace structures to satisfy detailed design requirements and, therefore, an understanding and a prediction of its effect on the response and failure of composite panels are desirable. Such a prediction must take into account the fact that current measurement technology does not allow the accurate determination of the material parameters that are used in the analytical models.
The present paper is a step in that direction. The results of a finite element study of the effect of stiffener termination on the response of composite panels subjected to combined pressure, mechanical and thermal loads are presented. The objectives of the study are to: a) develop better understanding of the effects of the stiffness discontinuities and load path eccentricities associated with this structural detail; and b) assess the effects of variability of material and geometric parameters on the response of composite panels with continuous and terminated stiffeners.
The panels considered in the present study have a number of T-shaped continuous or terminated stiffeners
43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Con22-25 April 2002, Denver, Colorado
AIAA 2002-1518
This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.
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(see Figure 1). Both flat panels and panels having cylindrical geometry are analyzed. The panel skin, flange and blade of each stiffener consist of a number of perfectly bonded plies (layers). The individual plies are assumed to be homogenous and anisotropic. The Aboudi cell method is used to evaluate the effective properties of the individual plies.9 A plane of thermoelastic symmetry exists at each point of the skin and the stiffener sections parallel to the reference surface of the section, and the material properties are assumed to be independent of temperature.
Basic Idea of the Approach Used for Assessing the Variability of the Response
The approach used for assessing the variability of the response associated with variations in material and geometric parameters consists of two major phases. In the first phase, hierarchical sensitivity analysis is used to evaluate the sensitivity coefficients with respect to a hierarchy of parameters ranging from micromechanical to component parameters, and to identify the major parameters that have the most effect on the response quantities of interest. In the second phase, the major parameters are taken to be fuzzy parameters, and a fuzzy set analysis is used to determine the range of variation of the response quantities of interest associated with pre-selected variations of the fuzzy parameters. The details of the approach are described in [10].
Numerical Studies
Panels and Loading Conditions Considered
Both flat and cylindrical composite panels with five T-shaped continuous and terminated stiffeners are studied. Each of the panel skin and stiffener blades has eight layers. The number of layers in the stiffener flanges is taken to be four (half that of the blades). The material properties and geometric characteristics for the panels are given in Figures 1 and 2. The material properties, the fiber orientation and the stacking sequence selected for the panels and stiffeners are those typical of panels considered for cryogenic fuel tanks of space transportation vehicles. The loads on each panel consist of a sequence of mechanical and thermal loads: a monotonically increasing uniform pressure, p, up to a maximum principal strain value of 0.008 on the surfaces of the panel, followed by an increasing edge shortening, qe, up to a maximum principal strain of 0.01, and then a uniform temperature change, T, of -412.8ºC. The boundary conditions are shown in Figure
1. The sign convention for the generalized displacements, stress resultants and transverse shear stresses is shown in Figure 3.
Finite Element Models and Computational Strategy
The analytical formulation is based on a first-order shear-deformation. Sanders-Budiansky type shell theory with the effects of large displacements, moderate rotations, average transverse shear deformation through the thickness, and laminated anisotropic material behavior included. A linear Duhamel-Neumann type constitutive model is used and the material properties are assumed to be independent of temperature. The constitutive relations for each of the skin, stiffener blades and flanges are given in Reference 11. A total Lagrangian formulation is used and the panel deformations, at different values of the applied loads, are referred to the original undeformed configuration. Mixed finite element models are used for the discretization of the skin and the blade sections of each stiffener. Each of the stiffener flanges is combined with the adjacent skin (below it) into a single finite element. Biquadratic shape functions are used for approximating each of the generalized displacements, and bilinear shape functions are used for approximating each of the stress resultants. The characteristics of the finite element model are given in Reference 11.
For each load case, global and detailed response quantities are generated. In addition, the hierarchical sensitivity coefficients are evaluated. The hierarchical sensitivity coefficients are derivatives of the different response quantities with respect to subcomponent parameters, laminate stiffnesses, material parameters and fiber angles of individual plies, and the micromechanical parameters (see Figure 4). The hierarchical sensitivity coefficients are used to identify the major parameters, at each hierarchical level, for the response quantities of interest. The major parameters are taken to be fuzzy parameters and a fuzzy set analysis is used to determine the range of variation of the response quantities of interest, associated with preselected variations of the fuzzy parameters. The details of the approach are described in Reference 10.
For the edge shortening case a combination of arc length and displacement incrementation approaches, with small increments, are used to predict the mode changes. To reduce the computational effort, the multiple parameter reduction methods described in References 12 - 14 are used in generating the response and evaluating the sensitivity coefficients. The global response results obtained by an in-house research program are validated by comparing them with those obtained by the STAGS general analysis code15.
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Typical results are presented in Figures 5 - 8 for the response studies, in Figures 9 and 10 for the sensitivity studies, and in Figures 11 and 12 for the variability of the nonlinear responses, and are described subsequently.
Response Studies
Some of the global and detailed response characteristics of the panels considered in the present study are shown in Figures 4 – 7. Plots of the total axial force Nt versus the applied edge shortening qe and the total strain energy U are shown in Figure 5 for both flat and curved panels with continuous and terminated stiffeners. For each of the panels, the variations of the ratios of the strain energy in the skin, stiffener flanges (including adjacent skin) and blades, to the total strain energy of the panel, with load, are shown in Figure 6. Typical contour plots for the transverse displacement w in the skin and stiffener flanges, after each loading, are shown in Figure 7. Through-the-thickness distributions of the transverse shear strain energy density (per unit volume) Ûsh, at the location of the maximum Ūsh (transverse shear strain energy per unit surface area), are shown in Figure 8. An examination of Figures 5 – 8 reveals:
(1) For the given value of the maximum principal surface strain (0.008), the magnitude of the total axial forces generated by application of the pressure load in the panels with continuous stiffeners is higher than the corresponding ones for panels with terminated stiffeners.
(2) The application of the edge shortening qe results in reducing the total axial force for all panels. The reduction is more pronounced in panels with continuous stiffeners.
(3) The values of qe associated with increasing the maximum principal surface strains from 0.008 to 0.01 are considerably smaller for panels with terminated stiffeners than the corresponding values for panels with continuous stiffeners. It is also smaller for the flat panel with continuous stiffeners than for the corresponding curved panel.
(4) For flat panels, the total strain energy associated with the thermal loading condition is considerably higher than that associated with the other two loading conditions. The same is true for curved panels with terminated stiffeners.
(5) For flat panels, the percentage of the strain energy in the stiffener blades is considerably higher than that for the corresponding curved panels.
(6) For flat panels, the percentage of the strain energy in the skin decreases with increasing p, then increases with increasing each of qe and T. By contrast, the percentage of the strain energy carried by the blades increases with increasing p, then decreases with increasing each of qe and T.
(7) For curved panels the percentage of the strain energy in the skin decreases with the increase in each of the loads p, qe, and T. An opposite situation is observed for the percentage of the strain energy in the blades.
(8) For all the panels considered, the maximum transverse shear strain energy per unit volume Ûsh at the location of Ūsh (transverse shear strain energy per unit surface area) occurs after the application of p + qe.
Sensitivity Studies
Sensitivity studies were conducted to identify which of the subcomponent parameters, laminate parameters, effective ply properties, and micromechanical parameters most affect the nonlinear response. Typical results showing the sensitivity of the total strain energy U with respect to the subcomponent parameters for flat panels with continuous and terminated stiffeners are shown in Figure 9. Sensitivity coefficients of U with respect to the fiber angles of the skin, stiffeners flanges, and blades, the effective material properties of the individual plies, and the micromechanical parameters are shown in Figures 9 and 10. The sharp changes appearing in the plots of the sensitivity coefficients versus load (Figure 10) are associated with sharp changes in the deformation patterns at the load values where the sharp changes occur. An examination of Figures 9 and 10 reveals:
(1) For the case of the pressure load p, the total strain energy U in each panel is more sensitive to the variations in the height of the stiffener blades hr than to variations in the other geometric parameters. After application of the edge shortening qe, U in the panel with terminated stiffeners continues to be more sensitive to variations in hr, while the U in the panel with continuous stiffeners becomes more sensitive to variations in the stiffener spacing l.
(2) The total strain energy in each panel is considerably more sensitive to variations in the following parameters than to each of the other parameters in the same category: a) the fiber angles +45º and –45º in the skin; b) the effective elastic modules EL, and for the thermal load, the coefficient of thermal expansion aT and the elastic modules ET; c) the micromechanical parameters vf and E1f, and for the temperature load am, υm, a2f, Em and E2f.
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(3) For panels with terminated stiffeners, the magnitudes of most of the aforementioned sensitivity coefficients, after application of the three loads, are higher than the corresponding ones for panels with continuous stiffeners. This observation is particularly true for the sensitivity coefficients with respect to the fiber angles +45º and –45º in the skin.
(4) The magnitudes of some of the sensitivity coefficients of U increase with the increase in each of the three loads, others decrease with the increase for some of the loads.
Variability of the Response
Studies were conducted to assess the effect of variability of the two major micromechanical parameters, the fiber volume fraction vf and the elastic modulus of the fibers in the longitudinal direction E1f, on the total strain energy U and the transverse shear strain per unit volume Ûsh, at the location of the maximum Ūsh. Each of the two major micromechanical parameters, vf and E1f, was taken as a fuzzy parameter, and their nominal values were changed by 10% and 15%, respectively. The variation of the upper and lower bounds of U with load, due to variation in each of the micromechanical parameters, for flat panels with continuous and terminated stiffeners are shown in Figure 11. The corresponding variations of the through-the-thickness distributions of the upper and lower bounds of Ûsh (at the location of maximum Ūsh), after application of p + qe, are shown in Figure 12.
An examination of Figures 11 and 12 reveals that, after the application of the loads p and qe, the selected variations in vf and E1f result in higher percentage change in Ûsh, and lower percentage changes in Nt and U for panels with continuous stiffeners than for panels with terminated stiffeners.
Concluding Remarks
A study is made of the variability of the response of stiffened composite panels associated with variations in the major geometric and material parameters of the structures. Both flat and cylindrical composite panels with continuous and terminated stiffeners subjected to combined pressure, mechanical and thermal loads are considered. The panel skins and stiffener blades has eight layers. The number of layers in the stiffener flanges is taken to be four. The panel skins, stiffener flanges and stiffener blades are modeled using two-dimensional shear flexible elements. The external loads
applied to each panel consisted of monotically increasing uniform pressure load up to a maximum principal strain value of 0.008 on the surface of the panel, followed by an increasing edge shortening up to a maximum principal strain of 0.01, and then a uniform temperature change of –412.8ºC.
For each panel, both the geometrically nonlinear response, as well as the hierarchical sensitivity coefficients, are generated. The hierarchical sensitivity coefficients measure the sensitivity of the different response quantities to variation in the subcomponent parameters (stiffener dimensions and spacing), as well as to three sets of interrelated parameters; namely, laminate properties, effective ply properties, and micromechanical parameters.
The effect of variations in the major micromechanical parameters on the variability of the total strain energy, and the transverse shear strain energy per unit volume for the panels are described.
Acknowledgements
NASA Cooperative Agreement NCC-1-01-014 and NASA Grant NAG1-01-028 supported the work. The authors gratefully acknowledge the help of Dr. Charles C. Rankin and the staff of the Lockheed Martin Advanced Technology Center in Palo Alto, California, in using the STAGS General Shell Analysis Program.
References
[1] A. K. Noor and W. S. Burton, “Computational Models for High-Temperature Multilayered Composite Plates and Shells,” Applied Mechanics Reviews, ASME, Vol. 45, No. 10, 1992, pp. 419 – 446.
[2] A. K. Noor and J. M. Peters, “Finite Element Buckling and Postbuckling Solutions for Multilayered Composite Panels,” Finite Element in Analysis and Design, Vol. 15, 1994, pp. 343 – 367.
[3] A. K. Noor (ed.), Bucking and Postbuckling of Composite Structures, Proc. Of the Symposium on Buckling and Postbuckling of Composite Structures, ASME Int. Mech. Eng. Congress and Exposition, Chicago, IL, Nov. 6 – 11, 1994, AD Vol. 41/PVP Vol. 293, 1994.
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[4] G. J. Turvey and I. H. Marshall, Buckling and Postbuckling of Composite Plates, Chapman and Hall, London, 1995.
[5] J. N. Reddy, Mechanics of Laminated Composite Plates: Theory and Analysis, CRC Press, Boca Raton, FL, 1997.
[6] B. L. Kemp and E. R. Johnson, “Response and Failure Analysis of a Graphite-Epoxy Laminate Containing Terminating Internal Plies,” AIAA Paper 85-0608, April 1995.
[7] J. M. Curry, E. R. Johnson and J. H. Starnes, Jr., “Effect of Dropped Plies on the Strength of Graphite-Epoxy Laminates,” AIAA Journal, Vol. 30, No. 2, Feb. 1992, pp. 449 – 456.
[8] D. R. Ambur, J. H. Starnes, Jr., C. G. Davila and E. A. Phillips, “Response of Composite Panels with Stiffness Gradients Due to Stiffener Terminations and Cutouts,” AIAA Paper 97-1368, April 1997.
[9] A. K. Noor and C. M. Anderson, “Mixed Models and Reduced/Selection Integration Displacement Models for Nonlinear Shell Analysis,” International Journal for Numerical Methods in Engineering, Vol. 18, 1982, pp. 1429 – 1454.
[10] A. K. Noor, J. H. Starnes, Jr., and J. M. Peters, “Uncertainty Analysis of Composite Structures,” Computer Methods in Applied Mechanics and Engineering, Vol. 185, Nos. 2 – 4, May 2000, pp. 413 – 432.
[11] J. Aboudi, Mechanics of Composite Materials: A Unified Micromechanical Approach, Elsevier, Amsterdam, 1991.
[12] A. K. Noor and J. M. Peters, “Multiple Parameter Reduced Basis Technique for Bifurcation and Postbuckling Analyses of Composite Plates,” International Journal for Numerical Methods in Engineering, Vol. 19, 1983, pp. 1783 – 1803.
[13] A. K. Noor and J. M. Peters, “Recent Advances in Reduction Methods for Instability Analysis of Structures,” Computers and Structures, Vol. 16, Nos. 1 – 4, 1983, pp. 67 – 80.
[14] A. K. Noor and J. M. Peters, “Reduced Basis Technique for Calculating Sensitivity Coefficients of Nonlinear Structural Response,” AIAA Journal, Vol. 30, No. 7, 1992, pp. 1840 – 1847.
[15] F. A. Brogan, C. C. Rankin and H. D. Cabiness, “STAGS Users Manual,” Lockheed Palo Alto Research Laboratory Report LMSC P032594, Palo Alto, CA, 1994.
Figure 1 - Panels and boundary conditions considered in the present study.
Panel DimensionsL1 = 0.508 mL2 = 0.79756 mR = .6096 m (curved panels)h = 0.001176 m
Stiffener dimensions andspacing
b = 0.0381 mhr = 0.03175 mll = 0.159512 m
Boundary ConditionsAt x1 = 0, L1
u1 = ±± qe/2 u2 = w = φφ1 = φφ2 = φφ3 = 0
At x2 = 0, L2u2 = φφ1 = φφ3 = 0
Panel withcontinuous stiffeners
Panel withterminated stiffeners
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3
Generalized displacementsand stress resultants
Transverse shear stresses
Figure 3 - Sign convention for generalized displacements, stress resultants and transverse shear stresses.
Figure 2 - Material properties for the panels used in the present study.
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Figure 4 - Hierarchical sensitivity coefficients for composite structures.
fij
aij
bjk
ShellTheory
LaminationTheory
MicromechanicalModel
StructuralTheories
Fiber, matrix,interface/interphaseproperties
Micromechanical
Component
(e.g.,fuselagebarrelsection)
Frame Spacing,geometriccharacteristicsof section
Subcomponent(e.g., stiffenedpanel)
Stiffenerdimensionsand spacing
Laminate LaminateStiffnesses
Ply (layer)Effectiveplyproperties
eij
sij
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0 0.5 1 1.5-3
-2
-1
0
1
2 x 10
x 10
-5
-3
Nt( E L1
2 )T
qe / L1
0 1 2 3 4-6
-4
-2
0
2 x 10
x 10
-5
-3
qe / L1
0 1 2 3 4 5-6
-4
-2
0
2 x 10
x 10
-5
-7
U / ( E L13 )T
0 0.8 1.6 2.4 3.2-3
-2
-1
0
1
2 x 10
x 10
-5
-7
Nt( E L1
2 )T
U / ( E L13 )T
Curved panelsFlat panels
Figure 5 - Global response characteristics of stiffened panels with continuous and terminated stiffenerssubjected to combined pressure load, edge shortening and temperature change.
Flat panels Curved panels
Panel with DesignationContinuous stiffenersTerminated stiffeners
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Figure 6 - Effect of loading condition on the strain energy ratios in the skin, stiffener flanges and stiffen-er blades. Stiffened panels with continuous and terminated stiffeners subjected to combined pressure
load, edge shortening and temperature change.
0.6 0.7 0.8-6
-4
-2
0
2
Strain energy ratios
x 10-5
0.1 0.2 0.3-6
-4
-2
0
2
Strain energy ratios
x 10-5
0 0.05 0.1 0.15-6
-4
-2
0
2
Strain energy ratios
x 10-5
0.2 0.4 0.6-3-2-1012
Strain energy ratios
NtE L1
2
x 10-5
T
0.1 0.2 0.3-3-2-1012
Strain energy ratios
NtE L1
2
x 10-5
T
0.1 0.2 0.3 0.4 0.5 0.6 0.7-3-2-1012
Strain energy ratios
NtE L1
2
x 10-5
T
c) stiffener blades
a) skin
b) stiffener flanges
Curved panelsFlat panels
Panel with DesignationContinuous stiffenersTerminated stiffeners
Flat panels Curved panels
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Figu
re 7
- N
orm
aliz
ed c
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lots
dep
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oadi
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ng q
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b)curvedpanels
a)flatpanels
panelwith
terminated
stiffeners
panelwith
continuous
stiffeners
panelwith
continuous
stiffeners
panelwith
terminated
stiffeners
1 0
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11A
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Curved panelsFlat panels
a) panels with continuous stiffeners
0 0.6 1.2 1.8 2.4x 10
-6
Ush / E at point 2∧
T
0 0.05 0.1 0.15
1
x3h
TUsh / E at point 1
x 10
∧
-6
0
b) panels with terminated stiffeners
0 3 6 9x 10
-6
Ush / E at point 4∧
T
0 2 4 6
x3h
x 10-6
Ush / E at point 3∧
T
1
0h
x3
Flat panels Curved panels
hx3
pp + qp + q + Te
e
Figure 8 - Effect of loading condition on the through-the-thickness distributions of the transverse shearstrain energy density (per unit volume) Ûsh at the location of the maximum Ush (transverse shear strainenergy per unit surface area). Stiffened panels subjected to combined pressure load, edge shortening and
temperature change.
h
stiffenerblade
x3
h
x3stiffenerblade
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Terminated panelsContinuous panels
-3 -2 -1 0 1 2 3-2.5
-1.25
0
1.25
NtE L1
2T
λ ∂U/∂λ / ( E L13 )
x 10-5
T
x 10-8
-3 -2 -1 0 1 2 3 4-0.8
0
0.8
1.6
λ ∂U/∂λ / ( E L13 )
x 10-8
T
-5x 10
l hrb hr
l bhr b
Panel withContinuous stiffeners
Panel withTerminated stiffeners
Figure 9 - Effect of loading condition on the normalized sensitivity coefficients of the total strain energyU with respect to subcomponent parameters. Flat stiffened panels subjected to combined pressure load,
edge shortening and temperature change.
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a) fiber angles55
-2 -1 0 1 2-3
-2
-1
0
1
λ ∂U/∂λ / ( E L13 )T
θ=-45θ=+45NtE L1
2T
x 10-8
x 10-5
+45 -45
+45-45
-5 -2.5 0 2.5 5-0.8
0
0.8
1.6
λ ∂U/∂λ / ( E L13 )T
θ=+45x 10
-5
x 10-9
θ=-45+45 -45 +45-45
+45 -45
b) effective ply properties
-2 0 2 4 6-0.8
0
0.8
1.6 E TαT
λ ∂U/∂λ / ( E L13 )T
x 10-7
EL
-5x 10
EL
-2 0 2 4 6-3
-2
-1
0
1
E TαTNt
E L12
T
λ ∂U/∂λ / ( E L13 )T
x 10-5
x 10-7
EL
EL
ET
c) micromechanical properties
-1 0 1 2 3 4-0.8
0
0.8
1.6
λ ∂U/∂λ / ( E L13 )T
x 10-7vf
E2f
E1f
-5x 10
E1fν23f Em νmvf
α1f α2f αm
, α1f ,E2f ,Em,α2f ,νm ,αm
-1 0 1 2 3 4-3
-2
-1
0
1 vf
1f
α2fα
mE
Em
ν
1f αmE2f
NtE L1
2T
λ ∂U/∂λ / ( E L13 )T
-5x 10
x 10-7
E1f
ν23f
νm
Emvfα1f
α2fαm
,, ,,
, ,
E2f
Figure 10 - Effect of loading condition on the normalized sensitivity coefficients with respect to fiberangles, effective ply and micromechanical parameters. Flat stiffened panels subjected to combined pres-
sure load, edge shortening and temperature change.
skinflange
skinflangeblade blade
Panel withContinuous stiffeners
Panel withTerminated stiffeners
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Terminated stiffenersContinuous stiffeners
0 0.8 1.6 2.4 3.2-4
-3
-2
-1
0
1
NtE L1
2T
U / ( E L13 )
x 10-5
T
x 10-7
0 1 2 3-2
-1
0
1
2
x 10-7
x 10-5
U / ( E L13 )T
λ λl λr vf .90 1.10 E1f .85 1.15
Figure 11 - Effect of variability in the micromechanical parameters on the total strain energy U. Flatstiffened panels subjected to combined pressure load, edge shortening and temperature change.
Panel withContinuous stiffeners
Panel withTerminated stiffeners
Terminated stiffenersContinuous stiffeners
0 2 4 6 8
T
x 10-6
Ush / E at point 3T
0 0.05 0.1 0.15 0.2
x3h
Ush / E at point 1T
x 10-6
^
1
0
Figure 12 - Effect of variability in the micromechanical parameters on the through-the-thickness distri-bution of transverse shear strain energy density (per unit volume) of Ûsh at the location of the maximum
Ush (transverse shear strain energy per unit surface area). Flat stiffened panels subjected to combinedpressure load and edge shortening.
λ λl λr vf .90 1.10 E1f .85 1.15
hx3
h
stiffenerblade
x3
h
x3stiffenerblade
αα
00
11..00
αα
00
11..00
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