Buckling of Composite Sinusoidal Panels under Uniaxial...

Buckling of Composite Sinusoidal Panels under Uniaxial CompressionSACHINTHANI PATHIRANA (Ph.D. STUDENT)

SUPERVISOR: PROFESSOR PIZHONG QIAO

A U G U S T 1 0 , 2 0 1 7

Problem Statement

z

s

H

l

H

y

•Composite Corrugated panels with a sinusoidal configuration are often used as lightweight structures with high out-of-plane stiffness.

•The buckling characteristics of such panels are important in understanding the performance capacity in applications.

•Buckling under UNIAXIAL COMPRESSION is investigated in this study.

•Panel geometry ∶ 𝑧 = 𝐻𝑠𝑖𝑛2𝜋𝑠

𝑙

x direction – perpendicular to screen along axis

y direction – along the circumference

•The radius of curvature is calculated by 𝑅 =1+𝑑𝑧

𝑑𝑠

232

𝑑2𝑧

𝑑𝑠2

•These panels were originally designed to withstand shear forces

•But they have significant uniaxial strength as well

Geometry of panel

Applications

Steel plate shear walls

Webs in I-beams

Sandwich plate cores

Aircraft systems: Wings

FRP (Fiber-Reinforced Polymer) panels : Bridge decks

FRP Bridge decksFRP composites : polymer matrix material with fiber reinforcements

Advantages:

•Rapid installation

•Lightweight

•Corrosion resistance

•Lower life cycle costs

•High strength

•Environmentally friendly with low carbon emissions

BUT high initial cost and complicated design have inhibited the demand for these structures

www.compositeadvantage.com

A Pedestrian Bridge Deck

FRP Bridge Decks

J.F. Davalos, A. Chen, P. Qiao, FRP Deck and Steel Girder Bridge Systems: Analysis and Design, 2013, Taylor and Francis Group, pg. 1-7

Sandwich Construction

• FRP bridge decks can be classified according to type of construction• Sandwich

• Efficient• Ease of manufacturing (more flexibility in

varying geometry)• Adhesively bonded pultruded shapes

• Fixed geometry• Sandwich deck panels with Sinusoidal configuration

were observed to be more beneficial in terms of weight, cost and deflection

• However, the design process is complex• Material – Orthotropic• Shape – Complicated than typically used shapes

• This study hopes to contribute to the design aspect of such composite structuresAdhesively bonded pultruded shapes

Objective•Analyse a single wavelength of the sinusoidal core

•Under uniaxial compression and simply supported (hinges) boundary conditions

•Considering the variability of curvature along surface

•Using Rayleigh-Ritz method

•And Finite Element Analyses (FEA) – ABAQUS 6.13

•Obtain a semi-analytical solution for the Critical Buckling (Local) Load

Honeycomb structure with sinusoidal core geometry

Model: A single wavelength of the sinusoidal panel with uniaxial compression and simple

supports(hinges)

Equivalent Orthotropic Plate Model•The most conservative method of analyzing sinusoidally corrugated panels is modelling it as an equivalent orthotropic flat plate.

•The governing differential equation for flat plates is:

𝐷11𝜕4𝑤

𝜕𝑥4+ 2𝐻

𝜕4𝑤

𝜕𝑥2𝜕𝑦2+ 𝐷22

𝜕4𝑤

𝜕𝑦4+ 𝑁𝑥𝜕2𝑤

𝜕𝑥2+ 𝑁𝑦𝜕2𝑤

𝜕𝑦2+ 2𝑁𝑥𝑦

𝜕2𝑤

𝜕𝑥𝜕𝑦= 0

𝐻 = 2 𝐷12 + 𝐷66

•In uniaxial compression, 𝑁𝑥 = −𝑞, 𝑁𝑦 = 𝑁𝑥𝑦 = 0 and for simply supported boundary conditions, the following assumed buckling displacement function satisfies the requirements:

𝑤 =

𝑚=1

𝑀

𝑛=1

𝑁

𝑊𝑚𝑛𝑠𝑖𝑛𝑚π𝑥

𝑎sin𝑛π𝑦

𝑏

•After substituting the buckling displacement function in the above governing differential equation, the following expression for the buckling load can be obtained:

𝑞 =𝐷11𝑚𝜋𝑎

4+ 2𝐻

𝑚𝜋𝑎

2 𝑛𝜋𝑏

2+ 𝐷22

𝑛𝜋𝑏

2

𝑚𝜋𝑎

2

•The critical buckling load is obtained for the lowest values of m and n satisfying the above equation

Equivalent Orthotropic Plate Model

Bending Rigidities

𝐷11 =𝑙

𝑏

𝐸𝑡3

12 1 − 𝜈2

𝐷12 = 0𝐷22 = 𝐸𝐼

𝐷66 =𝑏

𝑙

𝐸𝑡3

24 1 + 𝜈

𝐷11 =𝑙

𝑏

𝐸𝑡3

12 1 − 𝜈2

𝐷12 = 𝜈𝐷11

𝐷22 =𝐸𝑡𝐻2

2+𝐸𝑡3

12 1 − 𝜈2

𝐷66 =𝐸𝑡3

24 1 + 𝜈

𝐷11 =𝑙

𝑏

𝐸𝑡3

12 1 − 𝜈2

𝐷12 = 𝜈𝐷11

𝐷22 =𝐸𝐼

1 − 𝜈2+𝑙

𝑏

𝐸𝑡3

12 1 − 𝜈2

𝐷66 =𝑏

𝑙

𝐸𝑡3

24 1 + 𝜈

𝐷11 =𝑙

𝑏

𝐸𝑡3

12 1 − 𝜈2

𝐷12 = 𝜈𝐷11

𝐷22 = 𝐸𝐼 +𝑙

𝑏

𝐸𝑡3

12+ 𝜈2𝐷11

𝐷66 =𝑏

𝑙

𝐸𝑡3

24 1 + 𝑣

Seydel (1931) Briassoulis (1986) Xia et al (2012) Ye et al (2014)

• Several studies have been conducted on obtaining the bending rigidities of the sinusoidally corrugated plate with improvements over time as shown below.

• In this study, the expressions proposed by Ye et al (2014), which are the most recent, have been used.

• 𝐼 and 𝑏 are the 2nd moment of area about the x axis and the circumferential (arc) length of a single corrugation.

• Seydel (1931) suggests the formulas 𝐼 =𝐻2𝑡

21 −

0.81

1+0.25𝐻

𝑙

2 and 𝑏 = 𝑙 1 +𝜋2𝐻2

𝑙2,

however these values have been obtained numerically in this study for higher accuracy.

Comparison with FE results• For H=10 mm and l=400 mm,

the maximum difference occurs at the lowest length (14686%)

• The difference decreases as the length increases and is 0.85%at a=1600mm

• For H=40 mm and l=400 mm, the maximum difference occurs at the lowest length and is even higher.

• The minimum difference is 305% at the highest length of panel.

• Therefore, this method is not suitable for short panels with a single corrugation

Energy Method : Buckling of Cylindrical Shells Procedure

•The total potential energy is written in the formΠ = 𝑈𝑚 + 𝑈𝑏 + 𝑈𝑤

•At equilibrium, the total potential energy is minimum:δΠ = 𝛿(𝑈𝑚 + 𝑈𝑏 + 𝑈𝑤) = 0

•The energy integrals are evaluated using linear shell theory – Donnell’s Shell Theory

oThe membrane energy

𝑈𝑚 =𝐸𝑡

2 1 − 𝜈2 0

𝑎

0

𝑏 𝜕𝑢

𝜕𝑥

2

+𝜕𝑣

𝜕𝑦+𝑤

𝑅

2

+ 2𝜈𝜕𝑢

𝜕𝑥

𝜕𝑣

𝜕𝑦+𝑤

𝑅+1 − 𝜈

2

𝜕𝑢

𝜕𝑦+𝜕𝑣

𝜕𝑥

2

𝑑𝑥𝑑𝑦

oThe bending energy

𝑈𝑏 =𝐸𝑡3

24 1 − 𝜈2 0

𝑎

0

𝑏 𝜕2𝑤

𝜕𝑥2

2

+𝜕2𝑤

𝜕𝑦2

2

+ 2𝜈𝜕2𝑤

𝜕𝑥2𝜕2𝑤

𝜕𝑦2+ 2 1 − 𝜈

𝜕2𝑤

𝜕𝑥𝜕𝑦

2

𝑑𝑥𝑑𝑦

oWork done by external loads

𝑈𝑤 = −𝑞𝑡

2 0

𝑎

0

𝑏 𝜕𝑤

𝜕𝑥

2

𝑑𝑥𝑑𝑦

(1)

Buckling Displacement Functions•For Simply Supported Boundary Conditions (SSSS) the following displacement functions were assumed and substituted in eqs. (1)

•The above functions satisfy the following conditions𝑤 = 0 𝑎𝑡 𝑥 = 0, 𝑎 𝑎𝑛𝑑 𝑦 = 0, 𝑏

𝜕𝑤2

𝜕2𝑥=𝜕𝑤2

𝜕2𝑦= 0 𝑎𝑡 𝑥 = 0, 𝑎 𝑎𝑛𝑑 𝑦 = 0, 𝑏

𝑤 =

𝑚=1

𝑀

𝑛=1

𝑁

𝑊𝑚𝑛𝑠𝑖𝑛𝑚π𝑥

𝑎sin𝑛π𝑦

𝑏

𝑢 =

𝑚=1

𝑀

𝑛=1

𝑁

𝑈𝑚𝑛𝑐𝑜𝑠𝑚π𝑥

𝑎𝑠𝑖 𝑛𝑛π𝑦

𝑏

𝑣 =

𝑚=1

𝑀

𝑛=1

𝑁

𝑉𝑚𝑛𝑠𝑖𝑛𝑚π𝑥

𝑎cos𝑛π𝑦

𝑏

Procedure•The buckling displacement functions are substituted in the energy equations

•The derivatives are taken of the total potential energy (Π=U-W) with respect to the coefficients Wmn, Vmn and Umn.

𝑊𝑚𝑛 𝐾1𝑚,𝑛 + 𝑉𝑚𝑛 𝐾2𝑚,𝑛 + 𝑈𝑚𝑛 𝐾3𝑚,𝑛 = 0



•The non-trivial solution is obtained by taking the determinant of the coefficients matrix.

𝑞

=4𝐸𝑡

π2𝑏2(1 − υ2)

𝑚2

𝑎2

𝑚2

𝑎2+𝑛2

𝑏22

[(𝑏

2 𝑏1

𝑏2 𝑠𝑖𝑛2𝑛π𝑦𝑏𝑅2

𝑑𝑦 − 𝑏1

𝑏2 𝑠𝑖𝑛2𝑛π𝑦𝑏𝑅

𝑠𝑖𝑛2𝑛π𝑦𝑏𝑅

𝑑𝑦𝑑𝑦)(2𝑎

𝑚2𝑛

𝑏2 +𝑛

𝑏4𝑎

𝑚4) + (𝑏

2 𝑏1


𝑑𝑦 − υ2 𝑏1



𝑑𝑦𝑑𝑦)

+π2𝐸𝑡3

12 1 − 𝜐2

𝑚2

𝑎2+𝑛2

𝑏22

𝑚2

𝑎2

Comparison with FE results• When compared with

the FE results, for H=10mm the maximum difference is 242%

• When H=40mm the difference is even higher

• This method is not suitable to predict the buckling of sinusoidal panels as it over predicts by a large margin.

• The buckling analysis of cylindrical shells is not suitable as is.

Stability Analysis of Noncircular Cylindrical Shells

•Earliest recorded work of shells with variable curvature was of Marguerre (1951).

•He stated that in such shells, “regions of small curvature caused a weakening effect which exceeded the stiffening of large curvature regions”

•The main concept he used was to approximate the curvature by a Fourier polynomial

1

𝑅=1

𝑅0(1 +

𝑟=1

𝑟∗

2ζ2𝑟 cos2𝑟𝜋𝑠

𝑈)

Where 1

𝑅0is the arithmetic mean of the curvature.

•The linear Donnell equations were used in the analysis

•Conclusion: For elliptic and oval shells, the buckling load depended on 2𝑏

𝑡and 𝑏

𝑎, but not on

the curvature variation in detail

•The solution: 𝑞𝑐𝑟 =Et

3 1−𝜈2

1

rmax

b

a

t

K.Marguerre, “Stability of the Cylindrical Shell of Variable Curvature”, NACA Report 1302, 1951

FE Comparison

95

100

kmin

kmax

• An FE model was constructed in ABAQUS, to observe the comparison with the theoretical prediction of the critical buckling load.

• As predicted, buckling occurs in the region of maximum radius of curvature (minimum curvature) and the difference between the FE and theoretical result is 0.82% which shows good correlation.

E= 69 Gpav= 0.35

𝑞𝑐𝑟 =Et

3 1 − 𝜈2

1

rmax

=𝐸𝑡

3 1 − 𝜈2

95

1002

= 404𝑁

𝑚𝑚

Application to Sinusoidal PanelsThe previous findings can be extended to the study of sinusoidal panels as follows:

1. The radius of curvature may be approximated by a Fourier series in the circumferential variable and adjustments added to simplify the analysis

2. The analysis may be developed based on the assumption that buckling initiates in the region of minimum curvature

Both these options have been explored and are presented in the following slides

1. Sinusoidal Panel: Approximated Curvature

S. Toda, “Buckling of Sinusoidally Corrugated Plates under Axial Compression”, AIAA Technical Notes, Vol. 21, No. 8, 1983

• Toda (1983) studied the buckling of sinusoidal plates whose curvature is approximated by 1

𝑅=1

𝑅𝑚𝑖𝑛cos

2𝜋𝑦

𝑏(Figure 2) where 𝑅𝑚𝑖𝑛 is the radius of curvature at the crests. The term

“quasisinusoidal” was used to refer to these plates.

• Figure 1 shows the curvature calculated by 1R=

𝑑2𝑧

𝑑𝑠2

1+𝑑𝑧

𝑑𝑠

232

and it can be observed that the

approximation is satisfactory.

𝑅𝑚𝑖𝑛 = 58.83

Simplification 𝑚𝑏

𝑎≪ 1 (

1

𝑅=1

𝑅𝑚𝑖𝑛cos

2𝜋𝑠

𝑏)

•Using the energy method described previously with the approximated curvature the following solutions were obtained:

•Without simplification (considering up to n=2 terms)

𝑞 =𝐸𝑡

4𝑅2𝜋2

𝑚𝑎

2

2𝑏

2

+𝑚𝑎

22 +

Et3𝜋2

12 1 − 𝜈2

ma

2+1b

22

ma

2

S. Toda, “Buckling of Sinusoidally Corrugated Plates under Axial Compression”, AIAA Technical Notes, Vol. 21, No. 8, 1983

•This expression significantly over-predicts the buckling load, hence Toda (1983)

suggested a simplification (𝑚𝑏

𝑎) to be made to the expression.

•With simplification

𝑞 =𝐸𝑡

64𝑅2𝜋2𝑏4

𝑍2+𝐸𝑡3𝜋2

12 1 − 𝜈2𝑍2

𝑏4

Where 𝑍 =𝑎

𝑚and with further minimization of q with respect to 𝑍2

𝑞 =𝐸

3 1 − 𝜈2

𝑡2

8𝑅𝑚𝑖𝑛

•This result shows that the critical buckling load is 1/8th of the corresponding critical buckling load of a cylinder with radius Rmin

Simplification 𝑚𝑏

𝑎≪ 1 (

1

𝑅=1

𝑅𝑚𝑖𝑛cos

2𝜋𝑠

𝑏)

Comparison with FE results• In this case, the results are

significantly better than the previous attempts.

• At H=10 mm, the minimum difference is -0.13% at the lowest length and -43% at the highest length.

• However, when H is increased to 40 mm, the results are not satisfactory. (868% difference)

• Therefore, this method is satisfactory as long as the curvature is approximated correctly.

2. Proposed Solution

• By observing the buckled mode shape of the sinusoidal panel in the FEA, it can be seen that buckling initiates in the region of minimum curvature of the sine curve.

• Therefore, it is proposed to identify the range of curvature which affects the buckling load of the panel, by comparing with an extensive collection of FEA results and develop a fitting equation that will predict the range of curvature affecting the buckling load.

Length of panel (a) = 100 mm a = 200 mm a = 400 mm

FE models

Mid-surface height (H) = 10 mm H = 20 mm H = 40 mm

Proposed Solution

• Assumptions• Range of curvature affecting the buckling load - 2𝑙• Since the buckling may be symmetric or antisymmetric, the range of curvature is the same

at all inflection points (points of minimum curvature)

• The gradient at the inflection point is calculated by tan 𝜃 =𝑑𝑧

𝑑𝑠

cos 𝜃 =1

1 + tan2 𝜃

• Then, the arc length along the surface (affecting the buckling load), 𝐿 =𝑙

𝑐𝑜𝑠𝜃

• Inflection points of the sinusoidal curve - 𝑙1and 𝑙2

LL L

L

𝑙 𝑙 𝑙 𝑙

𝜃 𝜃

𝑏1 𝑏2𝑙1 𝑙2

Proposed SolutionDeveloping the fitting equation

◦ In the theoretically obtained equation, the adjustment is made to the limits of the integral within range 𝑏1~𝑏2. The final integral will now be an addition of the integral within the limits 𝑙1 − 𝑙, 𝑙1 + 𝑙 and 𝑙2 − 𝑙, 𝑙2 + 𝑙.

◦ Initially, the range ‘𝑙’ is assumed so that the result from the adjusted expression matches the buckling load from the FEA.

𝑞

=4𝐸𝑡

π2𝑏2(1 − υ2)

𝑚2

𝑎2

𝑚2

𝑎2+𝑛2

𝑏22

[(𝑏

2 𝑏1


𝑑𝑦 − 𝑏1




𝑚2𝑛

𝑏2 +𝑛

𝑏4𝑎

𝑚4) + (𝑏

2 𝑏1





𝑑𝑦𝑑𝑦)

+π2𝐸𝑡3

12 1 − 𝜐2

𝑚2

𝑎2+𝑛2

𝑏22

𝑚2

𝑎2 𝑙1−𝑙𝑙1+𝑙 _ + 𝑙2−𝑙

𝑙2+𝑙 _

Proposed Solution◦ The above step is carried out for panels of wall thicknesses (1, 3, 5, 10 mm)

and wavelengths (100, 200, 400 mm) and lengths of panel (100, 200 400 mm) where the height to the surface alternated between the values 10, 20, 30 and 40 mm.

◦ 𝐿 =𝑙

𝑐𝑜𝑠𝜃is calculated for every instance that the results match

approximately.

◦ Then, the fitting equation is developed using Matlab’s Surface fitting tool with the non-dimensional independent variables taken as (ratio of amplitude to wall-thickness)*(ratio of amplitude to wavelength) (x=(H/t)(H/l) ) and the aspect ratio (y=a/l), and the required output as non-dimensional z= 𝐿/ 𝑎𝐻.

◦ From 𝐿, the range under consideration ‘𝑙’ can be predicted by 𝑙 = 𝐿𝑐𝑜𝑠𝜃.

Proposed Solution-Summary•The fitting equation

𝐿 = 𝑎𝐻 (−64.41 − 9.94𝑒− −21.01𝑦2+ 3.806𝑒− 0.04579𝑥

2+ 0.2734𝑦−0.9925 +

25.54𝑥−0.003862 + 35.18𝑦0.0075 − 0.1041𝑦1.0075 + 0.06167𝑥0.9961 + 0.003634𝑥1.9961 +0.006397 𝑥𝑦 −1)

Adjusted R-squared = 0.9314

𝑞

=4𝐸𝑡

π2𝑏2(1 − υ2)

𝑚2

𝑎2

𝑚2

𝑎2+𝑛2

𝑏22

[(𝑏

2 𝑏1


𝑑𝑦 − 𝑏1




𝑚2𝑛

𝑏2+𝑛

𝑏4𝑎

𝑚4) + (𝑏

2 𝑏1





𝑑𝑦𝑑𝑦)

+π2𝐸𝑡3

12 1 − 𝜐2

𝑚2

𝑎2+𝑛2

𝑏22

𝑚2

𝑎2

𝑙1−𝑙𝑙1+𝑙 __ + 𝑙2−𝑙

𝑙2+𝑙 __

Results• The buckling loads predicted by

incorporating the proposed fitting equation give satisfactory results as can be observed by the graph.

• The aspect ratio (a/l) and the parameter (H/l) is kept constant in this case at 1 and 0.1 respectively.

• The minimum difference is -2.4% and the maximum difference is 14.7%.

Results• Comparing with the previous results

from the approximated curvature, the newly proposed solution gives better results

• The maximum difference between the FE results and the predicted results is reduced to 16% from 26%.

• In this case the parameters (H/t) and (H/l) are kept constant at 4 and 0.2 respectively.

Conclusion• It is concluded that using the conventional method of equivalent orthotropic plate model to predict the buckling of sinusoidal panels is not satisfactory when the number of corrugations is low and the panel length is short.

•While a similar procedure for buckling analysis of cylindrical shells does not give accurate results for the sinusoidal panels (variable curvature), the findings of the buckling analysis of non-cylindrical shells gives satisfactory guidance.

•While the approximated curvature option gave good results at lower curvature, it did not predict the load accurately at higher curvature. This was due to the approximation error of the curvature.

•The final solution obtained in this study with the proposed fitting equation to find the range of curvature affecting the buckling was satisfactory in the ranges of H/t=1-15 , a/l=0.25-4, and H/l=0.025-0.3.

•This result can also be improved further by including more data when developing the fitting equation to further expand the range of application of the solution.

Thank you!Questions?

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