edgewise compressive buckling of flat sandwich panels

UNITED STATES DEPARTMENT OF AGRICULTURE • FOREST SERVICE • FOREST PRODUCTS LABORATORY • MADISON, WIS.

EDGEWISE COMPRESSIVE BUCKLING OF FLAT SANDWICH PANELS: LOADED ENDS SIMPLY SUPPORTED AND SIDES SUPPORTED BY BEAMS

February 1964

FPL-01 9

This Report is One of a Series

Issued in Cooperation with the

CONSTRUCTION FOR FLIGHT VEHICLES

of the Departments of the

AIR FORCE, NAVY, AND COMMERCE

MIL-HDBK -23 WORKING GROUP ON COMPOSITE

The Forest Service, U.S. Department of Agriculture, is dedicated to the principle of multiple use management of the Nation's forest resources for sustained yields of wood, water, forage, wildlife, and recreation. Through for- estry research, cooperation with the States and private forest owners, and management of the National Forests and National Grasslands, it strives--as directed by Congress--to provide increasingly greater service to a growing Nation.

EDGEWISE COMPRESSIVE BUCKLING O F FLAT SANDWICH PANELS:

LOADED ENDS SIMPLY SUPPORTED AND SIDES SUPPORTED BY BEAM&

JOHN J. ZAHN, Engineer and

SHUN CHENG, Engineer

Forest Products Laboratory, Forest Service U.S. Department of Agriculture

2

Abstract

Flat sandwich panels with equal isotropic facings and orthotropic or isotropic cores, under edge compression, are analyzed. The loaded panel ends are assumed to be simply supported and the sides are hinged to simple beams that carry end load. with isotropic core.

Curves of buckling coefficients are presented for sandwich

Introduction

Ordinarily, when actual support conditions do not agree exactly with the math- ematically ideal conditions of perfect fixity or simple support, one assumes for safety that the panel is simply supported, estimate the capacity of the panel to withstand edge compression if the supports are elastic and very soft.

However, this may not under-

This report is an attempt to assess the influence of

1 This research note is another in the series (ANC-23, Item 58-3) prepared and distributed by the Forest Products Laboratory under U.S. Navy, Bureau of Naval Weapons Order No. 19-64-8004 WEPS and U.S. Air Force Contract No. 33(657)63-358. Results here reported are preliminary and may be revised as additional data become available.

2 Maintained at Madison, Wis. , in cooperation with the University of Wisconsin.

FPL-019

By

----

the elasticity of the supports. hinged to beams along the unloaded edges and hinged to rigid supports along the loaded edges.

For simplicity, it is assumed that the panel is

The method of solution follows that presented for a homogeneous plate in Timoshenko's "Theory of Elastic Stability. " 3 Cheng's differential equations

4 are used to describe the bending of sandwich. The approximations of this theory are the same as those of the "tilting" method with equal membrane facings,

5

Notation

a, b -- panel dimensions. "a" is the distance between loaded edges; "b" is the distance between supporting beams.

A -- cross sectional area of beam.

-- displacement coefficients, (9). C 1 C 2

d -- width of edge insert, figure 3.

D -- sandwich stiffness, (3).

E -- Young' s modulus of sandwich facings.

EI -- bending stiffness of beam supports.

f - - thickness of homogeneous plate.

G , G -- shear moduli of core.

h -- centroidal spacing of sandwich faces.

3 Timoshenko, S. , and Cere, J. Theory of Elastic Stability. 2d Ed.

4 Cheng, S. On the Theory of Bending of Sandwich Plates. Proc. Fourth U. S. National Congress of Applied Mechanics, American Society of Mechanical Engineers. 1958.

Under Compressive End Loads. Rpt. No. A. D. 3174, Royal Aircraft Establishment. Yarnborough, England. 194 1

McGraw-Hill, New York. 1961.

5 -Williams, D. , Leggett, D.M.A., and Hopkins, H. G. Flat Sandwich Panels

-2- FPL-019

,

xz yz

I

K

-- moment of inertia of beam.

-- dimensionless critical edge load parameter, (12).

-- (15).

-- (15). M , M

m - - number of half -wavelengths in buckled surface, (9).

M , M , M -- bending and twisting moments in sandwich. x y xy

x N

p

-- critical edge load; force per unit area.

-- lateral load; force per unit area.

Q , Q

t -- face thickness.

-- transverse shear forces in sandwich. x y

c t

w

α

β

e

v

-- core thickness.

-- core shear stiffness parameters, (3).

- - Lagrange multipliers; generalized displacements.

-- displacement in z direction.

-- coordinates, figure 1.

-- relative shape factor. Relates shape of beam cross section to that of sandwich, (25).

-- relative cross-sectional area of beam and. sandwich, (25).

-- (14), (24).

-- define shape of buckle, (9).

-- Poisson' s ratio of faces.

FPL-019 -3-

k 2 I

A =

k , k , k , k 1 2 3 4

-- (15).

L , L 1 2

1 2

V , V x y

u, v

x, y, z

λ , φ

-- aspect ratio of single buckle.

-- Laplacian

Method of Solution

We consider a flat rectangular sandwich panel in the region 0 < x < a,

compressed in the x direction by a load of N

along the edges x = 0, a. b edges and hinged to simply supported beams along the edges y = ± . 2

figure 1. is assumed to equal the compressive stress in the facings. assume

x pound per inch

The panel is simply supported along the loaded

See

The supporting beams are axially compressed by a stress which That is, we

where P is the axial force in one beam and A is its cross-sectional area. The modulus of elasticity of the beam is assumed to be the same as that of the sandwich facing material.

The sandwich construction is assumed to consist of two thin similar isotropic facings of thickness, t, whose centroidal spacing, h, is maintained by an orthotropic antiplane core of thickness, t , figure 2. as membranes.

The facings are treated c

4 According to Cheng, the governing differential equation of the sandwich is,

4 with a slight change of notation from

FPL-019 -4-

ρ = mb

a

b b - < y < 2 2

(1)

(2)

where

3 pounds per inch we take In the case of edge loading of N

x

where N

The boundary conditions are

is considered to be positive for compression. x

FPL-019 -5-

(3)

(4)

(5)

(6)

(7)

where EI is flexural rigidity and A the area of the beams. The last term in (8) arises from the assumption that the facing stress on the loaded edges is continued over the end cross sections of the beam.

Condition (8) is a Kirchhoff boundary condition and uses the notion of edge reaction rather than the boundary value of shear. This permiter us to use biharmonic functions in place of the homogeneous solution of (2). function w, which solves the corresponding problem in homogeneous plate theory, also solves the problem for sandwich except that boundary values are now correct only in the sense of St. Venant. we assume that both beams bend in the same direction and that the buckled surface is symmetric about the x axis and use

Thus the

3 Hence, following Timoshenko-

(9)

This form satisfies (7) and (8) identically. In order that it satisfy (2), we must take φ and λ to be roots of

(10)

-6-

(8)

(11)

(12)

FPL-019

In order that (9) should satisfy (7) and (8), we must have

where

-7- FPL-019

(13)

(14)

(15)

Condition (13) is the stability criterion. presented in figures 4 to 10 for the case V

and V as parameters. (14) and the curves which represent (13). An illustrative example is given under "Curves."

To facilitate its use, curves are = V x y = V showing K vs. ρ with θ

The value of θ must be found by trial from its definition

Reduction to Special Cases

Homogeneous Plate

The solution for sandwich reduces to that for a homogeneous plate if we assume that plane transverse cross sections remain plane. This means there is no shear deformation in the core and hence V Making the inter- pretations

= V = 0. x y

(10) and (11) become

whose solutions are

so that

FPL-019 -8-

(10')

(11')

(16)

(17)

and (13) becomes

which agrees with Timoshenko except for notation:

3 Present Report Timoshenko

Isotropic Core

If V = V = V, some slight simplifications are possible in formulas (15). In x y

particular, k 3 and k 4 may be eliminated by noting that

so that (15) reduce to

FPL-019 -9-

(13')

(18)

(19)

(15')

Beams of Zero Stiffness

(15') cont.

The case of free lateral edges corresponds to θ = 0. The buckled shape in this case is not independent of y, since the free edges curl slightly due to a Poisson' s ratio effect. error and take w to be independent of y. the third of equations (12) we get

However, we can neglect this effect with very little Then λ = θ = 0, and from (11) and

which agrees with we have only used

5 column formula given in Report No. A.D. 3174. So far the fact that λ = 0 to get K.

From this K (20) and (10) we can solve for φ. but this does not correspond to the physically significant branch. The other root is positive and when substituted into (15) and (14) yields θ greater than zero. of a panel which is constrained to buckle cylindrically is not free. there is a panel with very small edge beams whose buckling load is the same as that of a panel without edge beams but which is constrained to buckle cylindric ally.

One root, of course, is φ = 0,

This is to be expected since θ = 0 implies a free edge whereas the edge Hence,

FPL-019 -10-

(20)

2 Consider the case V = 0, for example. substituting this into (10') we get

From equation (20), K = ρ and

2 If we use φ the other root, we can find beams which would provide equivalent edge

constraint. Equation (13') with gives

= 0, we constrain the panel to buckle cylindrically. If we use

from which equivalent beam constraints can be found by using (14). the effect disappears if v = 0.

Note

A panel whose edges are truly free will buckle at a slightly lower load than that given by (20) but the difference is not numerically significant. Hence, (20) was plotted in lieu of θ = 0 and is labeled "Euler" on figures 4 to 10.

Infinitely Rigid Beams

As we approach the case of all edges simply supported. According to equation (13), this corresponds to λ = 1. Then (11) becomes

with A, B, and C given by (12). Solving for K we get

(21)

6 which corresponds with Forest Products Laboratory Report 1583-B.

6 Ericksen, W. S., and March, H. W. Compressive Buckling of Sandwich Panels Having Dissimilar Facings of Unequal Thickness. Forest Prod. Lab. Rpt. 1583-B. Revised, 1958.

FPL-019 -11-

Core Shear Instability

If G is sufficiently low, that is, if V is sufficiently large, local crimping

due to core shear instability yields lower buckling load than that corresponding to general instability, as analyzed above.

xz x

To describe many closely spaced crimps we let m that is, ρ satisfy the differential equation, we must satisfy (10) and (1 1). (10) and (1 1) as follows

To We rewrite

Letting ρ we find

= 0 which φ λ while 0 and 0.

implies

Thus from either (10) or (11) we get ρ ρ

FPL-019 -12-

(10)

(11)

(22)

C 6

ρ

or, using definitions of K (12) and V x (3),

Curves

The use of figures 4 to 10 will be illustrated by several examples. we are given a particular sandwich construction, edge beams, and beam

a spacing, b, and wish to plot K versus aspect ratio .

b among K, ρ, and θ, namely (13) and (14), the former presented graphically in figures 4 to 10. Between these two relations we can eliminate θ and plot

1 1 a K versus where = . From this it is a simple matter to plot K versus ρ ρ mb a

for m = 1, 2, 3, . . . since the curves for m > 2 involve only a change of the b horizontal scale. For any , m must be chosen to minimize K. Unless the

beams are quite stiff, m = 1 will always yield a minimum and the curve of K

Suppose

We have two relations

a b

1 versus is all that is needed. If the beams are stiff, however, m > 2 may

ρ produce the minimum K for certain aspect ratios and the plot of K a -will be cusped. b

A simple graphical method to eliminate θ is as follows: which represent (1 3), plot θ versus K as a family of curves with ρ as a parameter ( V is known). then plot

versus min

From figures 4 to 10

Knowing the sandwich construction, EI, A, and b, one can

on the same sheet as a family of straight lines with a common intercept and negative slopes dependent upon ρ. For each value of ρ there is one straight line (equation (14)) and one curve and their intersection determines K .

Some examples of particular edge supports are presented in figure 11.

FPL-019 -13-

(23)

Here we write

where

where k = I of beam. A

The values for α = 0. 3, 10, 40 and β = 1, 3, 5 are shown in figure 11. A value of α = 0. 3 could represent an edge insert of facing material as the beam support. See figure 3. For such an insert

while β proportional to d. In this case the three curves for β = 1, 3, 5 are all indistinguishable from the Euler curve of the free-edge panel. This shows that the addition of edge inserts to sandwich construction has negligible effect on the buckling stress since it adds little to the stiffness of a construction which is inherently very stiff; the edge inserts do carry end load, however. Although the beams represented by α = 10, 40 are considerably stiffer than edge inserts, we find that m = 1 always yields a minimum and the curves are not cusped. curves for α = 40 exhibit a cusped appearance associated with exceedingly stiff edge beams.

The

FPL-019 -14-

and

2 (26)

APPENDIX

Details of the Mathematics

Method of Solution

We consider a flat rectangular sandwich panel in the region 0 < x < a,

compressed in the x direction by a load of N pound per inch along

the edges x = 0, a. The panel is simply supported along the loaded edges and b hinged to simply supported beams along the edges y = y . 2

supporting beams are axially compressed by a stress which is assumed to equal the compressive stress in the facings.

x

See figure 1. The

That is, we assume

where P is the axial force in one beam and A is its cross-sectional area. The modulus of elasticity of the beam is assumed to be the same as that of the sandwich facing material.

The sandwich construction is assumed to consist of two thin similar isotropic facings of thickness, t, whose centroidal spacing, h, is maintained by an orthotropic antiplane core of thickness, t . treated as membranes.

See figure 2. The facings are c

4 We use Shun Cheng' s differential equation theory. In this theory

-15-

b b 2 2 y < < -

(1)

(A1)

FPL-019

where u and v are generalized displacements. They originate in the theory as Lagrange multipliers. M , M , and M are bending and twisting moments

x y xy and D is the flexural rigidity of the sandwich panel. forces Q and Q

The transverse shear are carried in the core and are given by

x y

Equilibrium

The equilibrium equations of a panel element are

where N is the compressive force per inch in the x direction. x

The above eight equations are reduced to a single governing equation by

in place 3 Timoshenko. The only difference here is that we use

of his distributed loading, ρ. The governing equation is

FPL-019 -16-

(A2)

(A2)

(A3)

Boundary Conditions

x N

2t where EI is the flexural rigidity and A the area of the beams.

the facing stress and it is assumed that this same stress acts over the end cross sections of the beams.

Here is

(A7)

FPL-019 -17-

where

(A4)

(A6)

(A8)

(A9)

We assume that both beams bend in the same direction and that the buckled surface is symmetric about the x axis and use

where

m - number of half-wavelengths in x-direction

f - even function of y.

This identically satisfies boundary condition (A6).

Substituting (A10) into the differential equation (A4), we get

where

Guided by the solution of the corresponding problem in homogeneous plate 3

theory we use

FPL-019 -18-

(A10)

(A12)

(A13)

and (A12) becomes

and cos are linearly (A12) yields two equations because cosh

independent functions.

Noting the form of (A2), we are led to take

where the dimensionless k are so defined. i

4 The following steps are given by Cheng. Solve the first two of equations (A3) for Q and Q , respectively. Substitute for the moments in these expressions

for (A1).

Report No. A. D. 3174, is

x y Then substitute for u and v from equations (A2). The result, from

5

FPL-019 -19-

(A14)

(A15)

(A16)

(A17)

(A18)

(A19)

(A20)

Substitute from the third of equations (A3) into (A19) and get

Substitute from the third of equations (A3) into (A20) and get

Now put (A17) into (A22) and get two equations, one from the coefficients of

sinh sin and one from the coefficients of sin

first equation yields

and the second gives

-20- FPL-019

(A21)

(A22)

sin . The

(A23)

(A24)

Now put (A18) into (A21) and get two equations, one from the coefficients of

and one from the coefficients of cos cos These cosh cos

are then solved for k and k4: 3

Solve (A2) for u and v and substitute into the second of equations (A1) to get

Inserting Q (A17), Q (A18), w from (A10) and (A14), and V and V (A5),

this becomes y x x y

(A26)

y y Note that sin is a factor of M

nations of

boundary condition (A7) is identically satisfied.

Since both M and M are linear combi- x

it follows that sin is also a factor of M . Thus x

FPL-0 19 -21-

(A25)

From the third of equations (A1)

Using (A2), this becomes

Inserting Q (A17), Q (A18), w (A10) (A14), and V x and V y (A5), this becomes x y

We are now ready to apply boundary conditions (A8) and (A9). Using (A26),

(A8) becomes , after canceling

where

FPL-019 -22-

(A27)

(A28)

Using w (A10) (A14), Q y (A17), K (A13), and (A27), (A9) becomes, after

canceling

Define

and rewrite (A30) as

(A32)

FPL-019 -23-

(A30)

(A31)

where

Equations (A28) and (A32) must be satisfied simultaneously. One way to do this is to have C1 = C2 = 0. This is the flat form of equilibrium. A buckled

form of equilibrium can only exist if C and C 1 2 possible only if the determinant of their coefficients is zero:

are nonzero, and this is

Expansion of (A34) yields the condition of instability:

= 0 (A34)

FPL-019 -24- 1. -35

(A33)

(A35)

Figure 1. --Notation for sandwich panel.

Figure 2. --Sandwich construction consisting of two thin similar isotropic facings of thickness, t , whose centroidal spacing, h, is maintained by an orthotropic antiplane core of thickness,

Figure 3. --Cross section of edge support in the form of an insert.

FPL-019 M 125 358

t c

FPL-019

Figure 4. --Edgewise compressive buckling coefficients sandwich panels with loaded ends simply supported an supported by beams.

M 125 359

for flat d sides

Figure 5. --Edgewise compressive buckling coefficients for flat sandwich panels with loaded ends simply supported and sides supported by beams.

M 125 360 FPL-019


M 125 361 FPL-019


M 125 362 FPL-019


M 125 363 FPL-019


M 125 364 FPL-019


M 125 365 FPL-019

M

Fig

ure

ends

125

366

11.

--E

dgew

ise

com

pres

sive

buc

klin

g co

effi

cien

ts

for

flat

san

dwic

h pa

nels

w

ith

load

ed

sim

ply

supp

orte

d an

d si

des

supp

orte

d by

edg

e in

sert

s; V

= 0

.1,

X =

0.

3; β

= 1

, 3,

or

5.

PUBLICATION LISTS ISSUED BY THE

FOREST PRODUCTS LABORATORY

The following lists of publications deal with investigative projects of the Forest Products Laboratory or relate to special interest groups and are available upon request:

Architects, Builders, Engineers, Growth, Structure, and and Retail Lumbermen Identification of Wood

Box, Crate, and Packaging Data Logging, Milling, and Utilization

Chemistry of Wood

Drying of Wood

Fire Protection Laminates, and Wood-Base

Fungus and Insect Defects in

of Timber Products

Mechanical Properties of Timber

Structural Sandwich, Plastic

Components

Forest Products Thermal Properties of Wood

Furniture Manufacturers, Wood Fiber Products Woodworkers, and Teachers of Woodshop Practice Wood Finishing Subjects

Glue and Plywood Wood Preservation

Note: Since Forest Products Laboratory publications are so varied in subject matter, no single catalog of titles is issued. Instead, a listing is made for each area of Laboratory research. Twice a year, December 31 and June 30, a list is compiled showing new reports for the previous 6 months. This is the only item sent regularly to the Laboratory’s mailing roster, and it serves to keep current the various subject matter listings. Names may be added to the mailing roster upon request.

FOREST PRODUCTS LABORATORY

edgewise compressive buckling of flat sandwich panels

Documents