HyperSizer Analysis - Local Postbuckling - Compression.hme

METHODS & EQUATIONS

LOCAL POSTBUCKLING, COMPRESSION

Figure 1, Representative load-strain curve of compression panel allowed to postbuckle. Skin local buckling occurs before the collapse load causing a non-linear reduction of overall panel stiffness.

October 7, 2010

2010 Collier Research Corp. Methods And Equations Compression Postbuckling Date: 2010-10-07 2

Table of Contents

1 SUMMARY ..........................................................................................................................................................3

2 SYMBOLS ............................................................................................................................................................4

3 INTRODUCTION ..................................................................................................................................................5

4 TRADITIONAL METHODS ..................................................................................................................................7

4.1 ANALYTICAL POSTBUCKLING EXAMPLE ....................................................................................................7

4.2 VON KRMN EFFECTIVE WIDTH FORMULAS ...........................................................................................9

4.3 EXAMPLES ................................................................................................................................................12

5 HYPERSIZER METHOD ...................................................................................................................................15

5.1 EFFECTIVE WIDTH ....................................................................................................................................15

5.2 LOAD-DEPENDENT ITERATION .................................................................................................................15

6 USAGE ..............................................................................................................................................................17

6.1 EFFECTIVE WIDTH OUTPUT ......................................................................................................................17

6.2 REQUIRED MARGINS.................................................................................................................................17

6.3 BACKDOOR OPTIONS ................................................................................................................................18

6.4 DIAGNOSTICS ............................................................................................................................................19

6.5 SUMMARY .................................................................................................................................................20

7 REFERENCES ....................................................................................................................................................22


1 Summary Aerospace metallic and composite laminate stringer-stiffened panels are designed to support load beyond local buckling of the skin (Figure 2). This design approach requires a method to quantify the postbuckled collapse strength of the panel. This document presents an accurate and computationally efficient analysis method intended for broad industry usage based on a rapid, iterative convergence of the effective width of the unbuckled skin and then convergence of the updated overall panel stiffness and resulting internal load redistribution.

Traditional effective width methods employed in industry are formulated in terms of stringer stress. These formulas implicitly assume that the panel is metallic and is undergoing a uniform end shortening load. The classic formulas are therefore not directly applicable in a general-purpose analysis code. In HyperSizer, the traditional effective width concept has been extended to handle any general loading and material properties. The HyperSizer method degenerates into the classic methods for metallic panels under pure compression.

This document describes the compression postbuckling methods only. The shear postbuckling methods are located in a separate HME document. Verifications to published solutions and validations to test data for both shear and compression postbuckling are included in a single HVV document.

Figure 2, Z stiffened panel with a depicted local buckling mode shape.


2 Symbols a plate length

b plate width

eb effective width

xS stiffener spacing

t thickness of plate

E modulus of elasticity

,c ck K buckling coefficient

cr critical stress of a plate

eff stress in effective portion of plate

st stress in stringer

cr critical strain of a plate

MS margin of safety


3 Introduction Stiffened panels have high buckling stability and are efficient at supporting compression loading. The lowest failure mode of a stiffened panel is usually initial buckling of the skin between stiffeners. Local buckling of the skin at operating loads occurs due to a relatively wide span of the skin (high b/t ratio). Local postbuckling of the skin is permissible if the panel can be shown to support additional load beyond the first occurrence of buckling (bifurcation point). In aerospace designs, skins are normally allowed to local buckle before reaching limit load usually around 50% limit load.

Illustrated in Figure 3 is the sequence of local buckling. First, an initial bifurcation local buckling mode occurs - green line. Second, additional load causes the mode shape amplitude to be greater, blue line. Finally, as the full local postbuckling strength is realized, the mode shape becomes its largest, red line. We see that as the load is increased, the buckling mode shape becomes larger and larger.

The stress analysis problem in the local postbuckling range is highly nonlinear. In traditional aerospace design the concept of the effective width be is used to approximate postbuckling behavior. It is based on the physical observation that the buckled portion of the skin cannot transmit any additional load because it is now in bending. Any additional load is assumed to redistribute towards the remaining unbuckled (effective) portions of the skin and the fully effective stringer.

Figure 3, Different local panel mode amplifications due to progressive compression loading and the resulting remaining stable effective width be.

As another example, consider a metallic Zee stiffened panel that is loaded in uniform compression (Figure 4). In a typical postbuckling analysis, uniform compression is more accurately described as an uniform end shortening since the loaded panel edges are usually required to remain straight as in a test fixture. At the onset of skin local buckling the analysis is linear elastic, and all of the panel objects such as the skin, stiffener web, and stiffener flange are all at the same stress level. This level of stress is depicted as the horizontal dashed line in Figure 4c.

As additional load is applied to the panel, the buckled skin between stiffeners remains at the same stress (constant bifurcation load) and the additional load is picked up by the stiffener and the remaining effective width be of the skin. As more load is applied, the effective width becomes more narrow and the remaining stable cross section of skin and stringer carries a higher stress until either the material reaches compressive yield or the stiffener cripples. For effective width methods, the actual continuous state of stress distribution of Figure 4d is modeled with a rectangular step function, as shown in Figure 4e.


Figure 4, The effective width of the skin is included with the stiffener in the calculation of the remaining panel stable cross section. As load increases the remaining effective width becomes narrower.


4 Traditional Methods 4.1 Analytical Postbuckling Example

Effective widths can be derived from analytical postbuckling solutions for single plates. This section will discuss a solution for a very simple case in order to illustrate the physical significance of an effective width.

The analytical solution for the case of a thin, square, isotropic plate under uniaxial loads can be found in Singer (1998). The plate is simply supported on all edges and no imperfections are specified. All edges of the plate are required to remain straight. This last restriction that the plate edges must remain straight dictates a uniform end-shortening type load. The Nx load distribution across the width of the plate for this simple case is described by the following equation and is plotted in Figure 5.

, ,12 1 1 cos

2 2x app x app

x crcr cr

Et yNb

= + 2 +

(4.1)

Figure 5, Load distribution across the width of a square postbuckled plate as a function of overall applied strain x.

Several data series are plotted in Figure 5 corresponding to different load levels. In this case, load level is described by the overall strain in the x-direction. The same plot could also be constructed using net load, P. At strains equal to or below the buckling strain x,app cr, the load distribution is constant. As the overall strain is increased above the critical strain, the Nx load redistributes across the width of the plate. The center portion of the plate remains at a constant load and additional load is absorbed at the edges of the plate.


To find the net load P at a given load level, the Nx line load is integrated across the face of the plate,

( )

2

2

,2

b

xb

x app cr

P N dy

Etb

=

= +

(4.2)

The effective width is defined as the width of plate required to transmit the entire net load P assuming the effective width has a uniform stress level equal to the edge stress of the actual postbuckled distribution.

In terms of equations, the definition of the effective width is,

, ,orx edge e x edge eP N b P b t= = (4.3)

Graphically, the effective width concept can be illustrated as shown in Figure 6. The piece-wise effective width distribution is bounded by the dashed lines, and the area under the effective distribution is shaded. By definition the area under the effective distribution will be equivalent to the area under the actual postbuckled load distribution. Of course, both of these areas will be equal to the net load P.

Figure 6, Effective width for load level of 10 times the critical buckling load. The area under the effective width distribution is shaded. Both the area under the effective distribution and the actual distribution will be equal to the net applied load.

To solve for the effective width, the edge stress or line load is needed. The edge line load Nx occurs at y / b = -0.5. Solving for Nx at the edge


, ,

,

,

2 1 02

x app x appx edge cr

cr cr

x app

EtN

Et

= +

=

(4.4)

Substitute Equations (4.4) and (4.2) into (4.3) to solve for the effective width be

( ) ( )

,

, ,2

x edge e

x app cr x app e

P N bEtb Et b

=

+ =

,

12

cre

x app

bb

= +

(4.5)

Equation (4.5) represents the effective width for this case of a square isotropic plate. The effective width will vary for plates under different loadings, boundary conditions, aspect ratios, and degrees of material anisotropy. Exact effective width solutions are more difficult solve for more general conditions.

4.2 von Krmn Effective Width Formulas

As discussed in the previous section, the effective width can be derived for certain cases using analytical solutions. However, closed-form analytical solutions are rare and numerical solutions are difficult to solve. Also, in some cases it is easier and more accurate to derive the effective width relationship experimentally (by measuring the stress distribution directly). It is for these reasons that complex solutions are typically not used in design.

Instead, simplified effective width expressions are used. The common effective width equation used in industry is sometimes attributed to von Krmn (1932). The expression is derived for a simply supported, isotropic plate. The main assumption that von Krmn makes is that the effective width is at a stress level equal the critical buckling load of a simply supported plate of width be. In other words, the remaining strip of effective width must be at the point of buckling stability assuming simple supported conditions

. In terms of the buckling equation,

( )22

212 1c

effe

k E tb

=

(4.6)

Solving Equation (4.6) in terms of effective plate width,

( )

2

2

112 1

ce

eff

k Eb t

=

(4.7)

This expression has a lot of history in the aerospace industry and can easily be made to correlate with test data. Note that kc is function of the number of buckling halfwaves and is equal to 4 for long, simply-supported plates.

At first glance, the von Krmn assumption of the simply-supported effective strip seems somewhat arbitrary, but it is logical. Consider Figure 7 below. Though the figure shows a stiffened panel, it behaves as a single plate if simple-supports are assumed at the stringer. As the load is increased to some intermediate load step beyond initial buckling, the effective width becomes more narrow and denoted as be(i), red color.


It is logical to assume that the remaining effective width be(i) must be stable given some assumed boundary conditions. By inspection, the boundary condition of the effective width span width is difficult to comprehend. It appears as if it were some complicated combination or rotational and translational restraint. However, an appropriate boundary condition would be to perform local buckling on the half-span of be(i) that is be(i)/2. The outer edge (away from web) would be constrained against rotation (as a slider), and the central edge would be simply supported.

Figure 7, The effective width of a panel stiffener can be analyzed using simple boundary conditions regardless of its relative width to the unbuckled span.

The top of Figure 8 shows the assumed boundary conditions of the effective width. The sliders act as symmetry boundary conditions for the half-span. This is equivalent to modeling the full span with simple supports as shown in the bottom of Figure 8. Therefore, simple-simple boundary conditions on the full effective width can be used to determine the stability of the effective width.

Figure 8, (Top) Symmetry boundary conditions on the edges of the effective width half-span. (Bottom) Equivalent boundary conditions of simple supports on the full

4.2.1 von Krmn Effective Width Forms

effective width span. The latter boundary conditions are used to check stability of the effective strip.

The effective width Equation (4.7) is reported in many different forms. This section will derive the different versions found in commonly used design references.

Nius Stress Analysis and Sizing (1997) defines buckling coefficients as,


( )

2

212 1c

ckK

=

(4.8)

Equation (4.7) becomes the form applicable to metallic materials found as Equation 14.2.1 in Niu.

ceeff

K Eb t

= (4.9)

The stress term in Equation (4.9) requires some explanation. In some references, the stress term is denoted as st for the stringer. As we saw in Section 4.1, the stress in the effective portion is assumed to be the edge stress of the actual distribution. When discussing metallic panels under uniform end-shortening, the edge stress in the skin is equal to the stress in the stringer (which is uniform). This leads to the common form,

cest

K Eb t

= (4.10)

If we set kc = 4 and = 0.3 Equation (4.10) can be reduced to the form found in Bruhn (Eq. C7.15) and Singer (Eq. 8.3).

( )3.62

est

Eb t

= (4.11)

1.9est

Eb t

= (4.12)

Some authors recommend knocking down the constant of 1.9 to 1.7 in order to match test data for panels with light stringers. This is the form found in Flabel (1997) as Equation 6-6.

1.7est

Eb t

= (4.13)

A final form of the von Krmn effective width equation will be derived. First, define the initial critical buckling load of the plate as,

( )

( )

22

2

22

2

12 1

12 1

ccr

ccr

k E tb

k E bt

=

=

(4.14)

Assuming that the buckling coefficient kc of the original plate is equivalent to that of the effective plate, substitute the left-hand side of Equation (4.14) into (4.7) to get the expression found in Singer (Eq. 3.110) and Brush (Eq. 3.86).


2 1

e creff

bb tt

=

(4.15)

creeff

b b

= (4.16)

Again, eff is the stress level in the effective portion of the plate. This form is convenient to use can be used as a rough extension to composite materials and biaxial loads. Equation (4.16) is plotted non-dimensionally in Figure 9.

Figure 9, The von Krmn effective width expression, Equation (4.16), as a function of applied stress.

4.3 Examples

This section computes the effective width under three different applied loadings: applied stress, applied strain, and applied load. The objective is to clarify how the notion of an effective width relates to the applied load. For this example, consider a long metallic plate with the dimensions shown in Table 1. The critical buckling stress is computed below.

Table 1, Plate properties of example problem. See calculation of critical stress below.

E t a b cr 10 Msi 0.3 0.1 40 10 3,615 psi


( )( )( )

( )

22

2

2 6 2

2

12 1

4 10 10 0.11012 1 0.3

3,615 psi

ccr

k E tb

=

=

=

4.3.1 Applied Stress

Find the effective width at a stress level of 10 ksi.

Using Equation (4.16),

361510 6.01"10,000

cre

eff

b b

= = =

Applying stress to a postbuckled structure is not physically realistic because the net load will vary with the changing effective area. For example the net load under 10 ksi stress for a fully effective plate is,

( )( )10,000 10 .1 10,000 lbfeb b

P = = =

and the net load using the effective width is,

( )6.01 10,000 6.01 .1 6,010 lbfebP = = =

Though it is physically unrealistic, the applied stress way of thinking is convenient if the crippling stress of the stringer is known and the skin effective width is needed to in order to determine the total load that the combined stringer+skin section can support. Typically, textbooks phrase postbuckling problems in terms of stress because crippling margins or Johnson-Euler column margins of safety are being solved.

4.3.2 Applied Strain

Find the effective width at an applied strain of 0.001 in/in.

First find the stress is the effective portion,

( )610 10 0.001 10,000 psieff E = = =

The effective width is the same as the first example,

361510 6.01"10,000

cre

eff

b b

= = =

4.3.3 Applied Load

Find the effective width and effective stress at an applied load of 6,010 lbf.


The effective portion of the plate must support 6,010 lbf. Rearrange Equation (4.16) in terms of load,

( ), /

cre

eff

x cr

e

b b

N tb

P b t

=

=

,2 x creN

b bP

= (4.17)

Now we have an expression for effective width in terms of the applied load P. Nx,cr is the critical line load of the fully effective plate.

, 361.5 lbf/inx cr crN t= =

Before we solve for effective width, we must verify that the plate will buckle under the applied load.

,6,010 601 lbf/in

10x appN = =

The applied line load (assuming fully effective plate) is greater than the critical load therefore we must use Equation (4.17) to find the effective width.

,2 2 361.510 6.01"6,010

x cre

Nb b

P= = =


5 HyperSizer Method 5.1 Effective Width

The traditional equations are used whenever applicable. The traditional equations do not lend themselves directly to a general-purpose analysis code. Firstly, an extension to composite materials is not obvious considering that the elastic modulus of a laminate is no longer sufficient to described buckling resistance (D terms are required). Also, biaxial loadings must be accounted for.

5.1.1 Metallic

For metallics Equation (4.16) is used in terms of Nx.

,

,

x cre

x eff

Nb b

N= (5.1)

Plate properties

Initial local buckling Nx,cr is calculated using the following assumptions:

o Fastened skin only o Bonded skin+bonded combo smeared stiffness

Plate width b o spacing span Sx

Boundary conditions o simply-supported

Loading o Ny load is considered if applicable

The HyperSizer effective width solution will match the classic methods if the panel is fastened, metallic, under uniaxial loads, and the skin has a Poissons ratio of 0.3.

5.1.2 Composite

For composites, Equation (5.1) is no longer applicable. Instead, an effective width is iterated using the von Krmn assumption of a buckling-critical simply-supported effective strip. The logic of the iteration is described in the next section.

5.2 Load-Dependent Iteration

The HyperSizer effective width solution is a function of the applied loads and thus needs to be iterated to equilibrium. This load-dependent iteration is performed at two levels: (1) panel stiffness and (2) effective width stability. Panel stiffness iteration models how the local change in stiffness of the skin affects the global stiffness of the entire panel. Effective width stability enforces the von Krmn assumption that the remaining strip of effective width must be at a load level equal to the buckling load of the strip.

5.2.1 Panel Stiffness

The reduction in stiffness of the skin will reduce the global stiffness of the panel. In HyperSizer, the global response of each panel is described with [A], [B], and [D] matrices that quantify the membrane, membrane-bending, and bending stiffness. A panel with a postbuckled skin will have reduced A11 stiffness and will therefore


have more panel strain than a fully effective panel. This in turn redistributes the loads in each object. This load path redistribution is equivalent to running the HyperFEA process in order to converge the load paths in the global loads FEM.

When the effective width of an object is determined, the remaining stable cross section is used to determine updated panel stiffness matrices [A], [B], and [D], including corresponding thermal coefficients for membrane, bending, and membrane-bending coupling. These updated stiffnesses and thermal coefficients will cause an update in forces of all objects, which in turn will change the margins of safety.

5.2.2 Effective Width Stability

The second iteration loop determines whether or not the remaining effective width is stable under the applied load (von Krmn assumption). Simply-supported boundary conditions are used to check the buckling load of the effective width. This iteration is necessary only for composites because a closed-form solution is not used (as opposed to metallics).

Figure 10, Schematic of load-dependent iteration for postbuckling analysis. The loop in the blue box is the logic necessary to iterate the effective width such that the load in the effective width is at or near the buckling load of the effective width (von Krmn assumption).

Skin object local buckled

?

done

Is effective width stable

?

update effective width

yes

no

no

yes

computed object loads

update panel stiffness & object loads


6 Usage This section will describe how to execute a postbuckling analysis in HyperSizer. Understanding the available options and output is critical to running and interpreting postbuckling analyses. To enable postbuckling analyses, toggle the local postbuckling checkbox in the Buckling Tab as shown in Figure 11.

6.1 Effective Width Output

The Failure Tab will display LPB ON if postbuckling was executed. Also, an effective width will appear in the Sizing Form.

Consider the load-strain curve in Figure 1. The stiffness is linear up until local buckling. After local buckling, the panel softens as additional load is applied. Eventually the panel fails at some mode crippling, material strength etc.

Because the effective width depends on the load level, HyperSizer computes the effective widths at four load levels:

Because the postbuckling response is non-linear, the stiffness (effective width) is dependent on load level.

Applied limit load Applied ultimate load Failure load at lowest limit margin of safety Failure load at lowest ultimate margin of safety

The effective width reported in the Buckling Tab of the Sizing Form is the effective width at the controlling applied load and controlling failure load

. For example, if ultimate loads are controlling, the be shown in the buckling tab is the effective width at the applied ultimate load. This is consistent with how the object loads are reported.

Figure 11, Advanced window of the Buckling Tab. The checkbox enables local postbuckling. The effective width reported corresponds to the effective width at the either the limit or ultimate load depending on which is controlling.

In order to save computation time, the effective width is not iterated to each individual failure mode. Instead, the effective width is iterated to the lowest limit and the lowest ultimate failure mode

6.2 Required Margins

. Each subsequent limit and ultimate margin is computed using the effective width from the most critical mode.

In HyperSizer, a component is said to have failed if the margin of safety is less than the required margin (default zero). Required margins can be set in the Failure Form by right-clicking the cell of a particular failure mode.

If then failurei requiredMS MS<

As discussed previously, the skin of compression panels are typically allowed to local buckle at some fraction of the design limit load. For purposes of discussion say the requirement is 0.5 times the limit load. Therefore, the required margin of safety for skin local buckling is:


, 0.5 1 0.5req local buckMS = =

In order to design with postbuckling requirements, required margins of safety must be applied. If the local buckling margins are set to zero, any negative margin will be interpreted as failure

6.3 Backdoor Options

.

Backdoor data options are found by right-clicking the name of a project or workspace in the database explorer and selecting Backdoor Options. Two options are available for compression postbuckling.

6.3.1 Empirical Correction Factor

This option applies a constant factor to the effective width solution. This can be used to correlate the effective width to test data.

ee b e

b k b = (6.1)

6.3.2 Include Load From Non-Effective Skin

The uneffective portion of the skin is traditionally defined as having zero load as in the left side of Figure 12. This is the default behavior (False). By activating this flag (True), the uneffective portion of the skin will have a load level equal to the initial buckling load (right side of Figure 12)


Figure 12, (Left) Include Load in Non-Effective Skin =FALSE. This is the default behavior. (Right) Include Load in Non-Effective Skin = TRUE.

6.4 Diagnostics

Diagnostic data can be found in two temporary output files. Due to the large amounts of data, these files are generated only when a single component is analyzed. These output values will eventually be formally included in the HyperSizer Stress Reports.

6.4.1 Material and Analysis Detail *.MTL

This file can be accessed directly from the Sizing form, Options Material and Analysis Detail. This file contains the effective width information only at the lowest limit and ultimate failure load. Here you can see how the overall panel membrane stiffness is reduced (A11) during the iteration process.

6.4.2 Postbuckling Detail *.PCD

This file can only be accessed in the Output directory of the Temp folder.

\HyperSizer Data\Projects\Temp\\Output\.PCD

This file contains the effective widths at all four load levels.


6.5 Summary

o Example: Skin buckling is acceptable at C times the limit load. Set the required limit margins for

skin local buckling to C 1.0.

Set skin local buckling margins to less than 1.0.

Toggle the Local Post Buckling checkbox in the Buckling Tab.


The effective width in the sizing form corresponds to the controlling applied load (limit or ultimate) and to the controlling failure load.

o Material and Analysis Detail file

Diagnostics

Sizing Form Options Material and Analysis Detail o *.PCD temp file

\HyperSizer Data\Projects\Temp\\Output\.PCD


7 References Bruhn, E. (1973). Analysis and Design of Flight Vehicle Structures. Jacobs Publishing, Inc.

Brush, D., & Almroth, B. (1975). Bucklng of Bars, Plates, and Shells. New York: McGraw-Hill Book Company.

Collier, C., Yarrington, P., & Gustafson, P. B. (2009). Local Post Buckling: An Efficient Analysis Approach for Industry Use. AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. Palm Springs, CA: AIAA.

Flabel, J. (1997). Practical Stress Analysis for Design Engineers. Hayden Lake: Lake City Publishing Company.

Niu, M. C. (1997). Airframe Stress Analysis and Sizing. Conmilit Press Ltd.

Niu, M. C. (1988). Airframe Structural Design. Conmilit Press Ltd.

Singer, J., Arbocz, J., & Weller, T. (1998). Buckling Experiments: Experimental Methods in Buckling of Thin-Walled Structures. New York: John Wiley & Sons, Inc.

von Krmn, T., Sechler, E., & Donnell, L. (1932). The Strength of Thin Plates in Compression. ASME Applied Mechanics Transactions , 54, 53-57.

1 Summary2 Symbols3 Introduction4 Traditional Methods4.1 Analytical Postbuckling Example4.2 von Krmn Effective Width Formulas4.2.1 von Krmn Effective Width Forms

4.3 Examples4.3.1 Applied Stress4.3.2 Applied Strain4.3.3 Applied Load

5 HyperSizer Method5.1 Effective Width5.1.1 Metallic5.1.2 Composite

5.2 Load-Dependent Iteration5.2.1 Panel Stiffness5.2.2 Effective Width Stability

6 Usage6.1 Effective Width Output6.2 Required Margins6.3 Backdoor Options6.3.1 Empirical Correction Factor6.3.2 Include Load From Non-Effective Skin

6.4 Diagnostics6.4.1 Material and Analysis Detail *.MTL6.4.2 Postbuckling Detail *.PCD

6.5 Summary

7 References

HyperSizer Analysis - Local Postbuckling - Compression.hme

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