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The Info List - Sandwich Theory


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Sandwich theory[1][2] describes the behaviour of a beam, plate, or shell which consists of three layers—two facesheets and one core. The most commonly used sandwich theory is linear and is an extension of first order beam theory. Linear
Linear
sandwich theory is of importance for the design and analysis of sandwich panels, which are of use in building construction, vehicle construction, airplane construction and refrigeration engineering. Some advantages of sandwich construction are:

Sandwich cross sections are composite. They usually consist of a low to moderate stiffness core which is connected with two stiff exterior face-sheets. The composite has a considerably higher shear stiffness to weight ratio than an equivalent beam made of only the core material or the face-sheet material. The composite also has a high tensile strength to weight ratio. The high stiffness of the face-sheet leads to a high bending stiffness to weight ratio for the composite.

The behavior of a beam with sandwich cross-section under a load differs from a beam with a constant elastic cross section. If the radius of curvature during bending is large compared to the thickness of the sandwich beam and the strains in the component materials are small, the deformation of a sandwich composite beam can be separated into two parts

deformations due to bending moments or bending deformation, and deformations due to transverse forces, also called shear deformation.

Sandwich beam, plate, and shell theories usually assume that the reference stress state is one of zero stress. However, during curing, differences of temperature between the face-sheets persist because of the thermal separation by the core material. These temperature differences, coupled with different linear expansions of the face-sheets, can lead to a bending of the sandwich beam in the direction of the warmer face-sheet. If the bending is constrained during the manufacturing process, residual stresses can develop in the components of a sandwich composite. The superposition of a reference stress state on the solutions provided by sandwich theory is possible when the problem is linear. However, when large elastic deformations and rotations are expected, the initial stress state has to be incorporated directly into the sandwich theory.

Contents

1 Engineering sandwich beam theory 2 Linear
Linear
sandwich theory

2.1 Bending
Bending
of a sandwich beam with thin facesheets

2.1.1 Constitutive assumptions 2.1.2 Kinematics 2.1.3 Stress-displacement relations 2.1.4 Resultant forces and moments 2.1.5 Bending
Bending
and shear stiffness 2.1.6 Relation between bending and shear deflections 2.1.7 Governing equations 2.1.8 Temperature dependent alternative form of governing equations

3 Solution approaches

3.1 Analytical approach 3.2 Numerical approach

4 Practical Importance 5 See also 6 References 7 Bibliography 8 External links

Engineering sandwich beam theory[edit]

Bending
Bending
of a sandwich beam without extra deformation due to core shear.

In the engineering theory of sandwich beams,[2] the axial strain is assumed to vary linearly over the cross-section of the beam as in Euler-Bernoulli theory, i.e.,

ε

x x

( x , z ) = − z  

d

2

w

d

x

2

displaystyle varepsilon _ xx (x,z)=-z~ cfrac mathrm d ^ 2 w mathrm d x^ 2

Therefore, the axial stress in the sandwich beam is given by

σ

x x

( x , z ) = − z   E ( z )  

d

2

w

d

x

2

displaystyle sigma _ xx (x,z)=-z~E(z)~ cfrac mathrm d ^ 2 w mathrm d x^ 2

where

E ( z )

displaystyle E(z)

is the Young's modulus
Young's modulus
which is a function of the location along the thickness of the beam. The bending moment in the beam is then given by

M

x

( x ) = ∫ ∫ z  

σ

x x

 

d

z

d

y = −

(

∫ ∫

z

2

E ( z )  

d

z

d

y

)

 

d

2

w

d

x

2

=: − D  

d

2

w

d

x

2

displaystyle M_ x (x)=int int z~sigma _ xx ~mathrm d z,mathrm d y=-left(int int z^ 2 E(z)~mathrm d z,mathrm d yright)~ cfrac mathrm d ^ 2 w mathrm d x^ 2 =:-D~ cfrac mathrm d ^ 2 w mathrm d x^ 2

The quantity

D

displaystyle D

is called the flexural stiffness of the sandwich beam. The shear force

Q

x

displaystyle Q_ x

is defined as

Q

x

=

d

M

x

d

x

  .

displaystyle Q_ x = frac mathrm d M_ x mathrm d x ~.

Using these relations, we can show that the stresses in a sandwich beam with a core of thickness

2 h

displaystyle 2h

and modulus

E

c

displaystyle E^ c

and two facesheets each of thickness

f

displaystyle f

and modulus

E

f

displaystyle E^ f

, are given by

σ

x x

f

=

z

E

f

M

x

D

  ;    

σ

x x

c

=

z

E

c

M

x

D

τ

x z

f

=

Q

x

E

f

2 D

[

( h + f

)

2

z

2

]

  ;    

τ

x z

c

=

Q

x

2 D

[

E

c

(

h

2

z

2

)

+

E

f

f ( f + 2 h )

]

displaystyle begin aligned sigma _ xx ^ mathrm f &= cfrac zE^ mathrm f M_ x D ~;~~&sigma _ xx ^ mathrm c &= cfrac zE^ mathrm c M_ x D \tau _ xz ^ mathrm f &= cfrac Q_ x E^ mathrm f 2D left[(h+f)^ 2 -z^ 2 right]~;~~&tau _ xz ^ mathrm c &= cfrac Q_ x 2D left[E^ mathrm c left(h^ 2 -z^ 2 right)+E^ mathrm f f(f+2h)right]end aligned

Derivation of engineering sandwich beam stresses

Since

d

2

w

d

x

2

= −

M

x

( x )

D

displaystyle cfrac d^ 2 w dx^ 2 =- cfrac M_ x (x) D

we can write the axial stress as

σ

x x

( x , z ) =

z   E ( z )  

M

x

( x )

D

displaystyle sigma _ xx (x,z)= cfrac z~E(z)~M_ x (x) D

The equation of equilibrium for a two-dimensional solid is given by

σ

x x

∂ x

+

τ

x z

∂ z

= 0

displaystyle frac partial sigma _ xx partial x + frac partial tau _ xz partial z =0

where

τ

x z

displaystyle tau _ xz

is the shear stress. Therefore,

τ

x z

( x , z ) = ∫

σ

x x

∂ x

 

d

z + C ( x ) = ∫

z   E ( z )

D

 

d

M

x

d

x

 

d

z + C ( x )

displaystyle tau _ xz (x,z)=int frac partial sigma _ xx partial x ~mathrm d z+C(x)=int cfrac z~E(z) D ~ frac mathrm d M_ x mathrm d x ~mathrm d z+C(x)

where

C ( x )

displaystyle C(x)

is a constant of integration. Therefore,

τ

x z

( x , z ) =

Q

x

D

∫ z   E ( z )  

d

z + C ( x )

displaystyle tau _ xz (x,z)= cfrac Q_ x D int z~E(z)~mathrm d z+C(x)

Let us assume that there are no shear tractions applied to the top face of the sandwich beam. The shear stress in the top facesheet is given by

τ

x z

f a c e

( x , z ) =

Q

x

E

f

D

z

h + f

z  

d

z + C ( x ) =

Q

x

E

f

2 D

[

( h + f

)

2

z

2

]

+ C ( x )

displaystyle tau _ xz ^ mathrm face (x,z)= cfrac Q_ x E^ f D int _ z ^ h+f z~mathrm d z+C(x)= cfrac Q_ x E^ f 2D left[(h+f)^ 2 -z^ 2 right]+C(x)

At

z = h + f

displaystyle z=h+f

,

τ

x z

( x , h + f ) = 0

displaystyle tau _ xz (x,h+f)=0

implies that

C ( x ) = 0

displaystyle C(x)=0

. Then the shear stress at the top of the core,

z = h

displaystyle z=h

, is given by

τ

x z

( x , h ) =

Q

x

E

f

f ( f + 2 h )

2 D

displaystyle tau _ xz (x,h)= cfrac Q_ x E^ f f(f+2h) 2D

Similarly, the shear stress in the core can be calculated as

τ

x z

c o r e

( x , z ) =

Q

x

E

c

D

z

h

z  

d

z + C ( x ) =

Q

x

E

c

2 D

(

h

2

z

2

)

+ C ( x )

displaystyle tau _ xz ^ mathrm core (x,z)= cfrac Q_ x E^ c D int _ z ^ h z~mathrm d z+C(x)= cfrac Q_ x E^ c 2D left(h^ 2 -z^ 2 right)+C(x)

The integration constant

C ( x )

displaystyle C(x)

is determined from the continuity of shear stress at the interface of the core and the facesheet. Therefore,

C ( x ) =

Q

x

E

f

f ( f + 2 h )

2 D

displaystyle C(x)= cfrac Q_ x E^ f f(f+2h) 2D

and

τ

x z

c o r e

( x , z ) =

Q

x

2 D

[

E

c

(

h

2

z

2

)

+

E

f

f ( f + 2 h )

]

displaystyle tau _ xz ^ mathrm core (x,z)= cfrac Q_ x 2D left[E^ c left(h^ 2 -z^ 2 right)+E^ f f(f+2h)right]

For a sandwich beam with identical facesheets and unit width, the value of

D

displaystyle D

is

D

=

E

f

w

− h − f

− h

z

2

 

d

z

d

y +

E

c

w

− h

h

z

2

 

d

z

d

y +

E

f

w

h

h + f

z

2

 

d

z

d

y

=

2 3

E

f

f

3

+

2 3

E

c

h

3

+ 2

E

f

f h ( f + h )   .

displaystyle begin aligned D&=E^ f int _ w int _ -h-f ^ -h z^ 2 ~mathrm d z,mathrm d y+E^ c int _ w int _ -h ^ h z^ 2 ~mathrm d z,mathrm d y+E^ f int _ w int _ h ^ h+f z^ 2 ~mathrm d z,mathrm d y\&= frac 2 3 E^ f f^ 3 + frac 2 3 E^ c h^ 3 +2E^ f fh(f+h)~.end aligned

If

E

f

E

c

displaystyle E^ f gg E^ c

, then

D

displaystyle D

can be approximated as

D ≈

2 3

E

f

f

3

+ 2

E

f

f h ( f + h ) = 2 f

E

f

(

1 3

f

2

+ h ( f + h )

)

displaystyle Dapprox frac 2 3 E^ f f^ 3 +2E^ f fh(f+h)=2fE^ f left( frac 1 3 f^ 2 +h(f+h)right)

and the stresses in the sandwich beam can be approximated as

σ

x x

f

z

M

x

2 3

f

3

+ 2 f h ( f + h )

  ;    

σ

x x

c

≈ 0

τ

x z

f

Q

x

4 3

f

3

+ 4 f h ( f + h )

[

( h + f

)

2

z

2

]

  ;    

τ

x z

c

Q

x

( f + 2 h )

2 3

f

2

+ h ( f + h )

displaystyle begin aligned sigma _ xx ^ mathrm f &approx cfrac zM_ x frac 2 3 f^ 3 +2fh(f+h) ~;~~&sigma _ xx ^ mathrm c &approx 0\tau _ xz ^ mathrm f &approx cfrac Q_ x frac 4 3 f^ 3 +4fh(f+h) left[(h+f)^ 2 -z^ 2 right]~;~~&tau _ xz ^ mathrm c &approx cfrac Q_ x (f+2h) frac 2 3 f^ 2 +h(f+h) end aligned

If, in addition,

f ≪ 2 h

displaystyle fll 2h

, then

D ≈ 2

E

f

f h ( f + h )

displaystyle Dapprox 2E^ f fh(f+h)

and the approximate stresses in the beam are

σ

x x

f

z

M

x

2 f h ( f + h )

  ;    

σ

x x

c

≈ 0

τ

x z

f

Q

x

4 f h ( f + h )

[

( h + f

)

2

z

2

]

  ;    

τ

x z

c

Q

x

( f + 2 h )

4 h ( f + h )

Q

x

2 h

displaystyle begin aligned sigma _ xx ^ mathrm f &approx cfrac zM_ x 2fh(f+h) ~;~~&sigma _ xx ^ mathrm c &approx 0\tau _ xz ^ mathrm f &approx cfrac Q_ x 4fh(f+h) left[(h+f)^ 2 -z^ 2 right]~;~~&tau _ xz ^ mathrm c &approx cfrac Q_ x (f+2h) 4h(f+h) approx cfrac Q_ x 2h end aligned

If we assume that the facesheets are thin enough that the stresses may be assumed to be constant through the thickness, we have the approximation

σ

x x

f

≈ ±

M

x

2 f h

  ;    

σ

x x

c

≈ 0

τ

x z

f

≈ 0   ;    

τ

x z

c

Q

x

2 h

displaystyle begin aligned sigma _ xx ^ mathrm f &approx pm cfrac M_ x 2fh ~;~~&sigma _ xx ^ mathrm c &approx 0\tau _ xz ^ mathrm f &approx 0~;~~&tau _ xz ^ mathrm c &approx cfrac Q_ x 2h end aligned

Hence the problem can be split into two parts, one involving only core shear and the other involving only bending stresses in the facesheets. Linear
Linear
sandwich theory[edit] Bending
Bending
of a sandwich beam with thin facesheets[edit]

Bending
Bending
of a sandwich beam after incorporating shear of the core into the deformation.

The main assumptions of linear sandwich theories of beams with thin facesheets are:

the transverse normal stiffness of the core is infinite, i.e., the core thickness in the z-direction does not change during bending the in-plane normal stiffness of the core is small compared to that of the facesheets, i.e., the core does not lengthen or compress in the x-direction the facesheets behave according to the Euler-Bernoulli assumptions, i.e., there is no xz-shear in the facesheets and the z-direction thickness of the facesheets does not change

However, the xz shear-stresses in the core are not neglected. Constitutive assumptions[edit] The constitutive relations for two-dimensional orthotropic linear elastic materials are

[

σ

x x

σ

z z

σ

z x

]

=

[

C

11

C

13

0

C

13

C

33

0

0

0

C

55

]

[

ε

x x

ε

z z

ε

z x

]

displaystyle begin bmatrix sigma _ xx \sigma _ zz \sigma _ zx end bmatrix = begin bmatrix C_ 11 &C_ 13 &0\C_ 13 &C_ 33 &0\0&0&C_ 55 end bmatrix begin bmatrix varepsilon _ xx \varepsilon _ zz \varepsilon _ zx end bmatrix

The assumptions of sandwich theory lead to the simplified relations

σ

x x

f a c e

=

C

11

f a c e

 

ε

x x

f a c e

  ;    

σ

z x

c o r e

=

C

55

c o r e

 

ε

z x

c o r e

  ;    

σ

z z

f a c e

=

σ

x z

f a c e

= 0   ;    

σ

z z

c o r e

=

σ

x x

c o r e

= 0

displaystyle sigma _ xx ^ mathrm face =C_ 11 ^ mathrm face ~varepsilon _ xx ^ mathrm face ~;~~sigma _ zx ^ mathrm core =C_ 55 ^ mathrm core ~varepsilon _ zx ^ mathrm core ~;~~sigma _ zz ^ mathrm face =sigma _ xz ^ mathrm face =0~;~~sigma _ zz ^ mathrm core =sigma _ xx ^ mathrm core =0

and

ε

z z

f a c e

=

ε

x z

f a c e

= 0   ;    

ε

z z

c o r e

=

ε

x x

c o r e

= 0

displaystyle varepsilon _ zz ^ mathrm face =varepsilon _ xz ^ mathrm face =0~;~~varepsilon _ zz ^ mathrm core =varepsilon _ xx ^ mathrm core =0

The equilibrium equations in two dimensions are

σ

x x

∂ x

+

σ

z x

∂ z

= 0   ;    

σ

z x

∂ x

+

σ

z z

∂ z

= 0

displaystyle cfrac partial sigma _ xx partial x + cfrac partial sigma _ zx partial z =0~;~~ cfrac partial sigma _ zx partial x + cfrac partial sigma _ zz partial z =0

The assumptions for a sandwich beam and the equilibrium equation imply that

σ

x x

f a c e

σ

x x

f a c e

( z )   ;    

σ

z x

c o r e

=

c o n s t a n t

displaystyle sigma _ xx ^ mathrm face equiv sigma _ xx ^ mathrm face (z)~;~~sigma _ zx ^ mathrm core =mathrm constant

Therefore, for homogeneous facesheets and core, the strains also have the form

ε

x x

f a c e

ε

x x

f a c e

( z )   ;    

ε

z x

c o r e

=

c o n s t a n t

displaystyle varepsilon _ xx ^ mathrm face equiv varepsilon _ xx ^ mathrm face (z)~;~~varepsilon _ zx ^ mathrm core =mathrm constant

Kinematics[edit]

Bending
Bending
of a sandwich beam. The total deflection is the sum of a bending part wb and a shear part ws

Shear strains during the bending of a sandwich beam.

Let the sandwich beam be subjected to a bending moment

M

displaystyle M

and a shear force

Q

displaystyle Q

. Let the total deflection of the beam due to these loads be

w

displaystyle w

. The adjacent figure shows that, for small displacements, the total deflection of the mid-surface of the beam can be expressed as the sum of two deflections, a pure bending deflection

w

b

displaystyle w_ b

and a pure shear deflection

w

s

displaystyle w_ s

, i.e.,

w ( x ) =

w

b

( x ) +

w

s

( x )

displaystyle w(x)=w_ b (x)+w_ s (x)

From the geometry of the deformation we observe that the engineering shear strain (

γ

displaystyle gamma

) in the core is related the effective shear strain in the composite by the relation

γ

z x

c o r e

=

2 h + f

2 h

 

γ

z x

b e a m

displaystyle gamma _ zx ^ mathrm core = tfrac 2h+f 2h ~gamma _ zx ^ mathrm beam

Note the shear strain in the core is larger than the effective shear strain in the composite and that small deformations (

tan ⁡ γ = γ

displaystyle tan gamma =gamma

) are assumed in deriving the above relation. The effective shear strain in the beam is related to the shear displacement by the relation

γ

z x

b e a m

=

d

w

s

d

x

displaystyle gamma _ zx ^ mathrm beam = cfrac mathrm d w_ s mathrm d x

The facesheets are assumed to deform in accordance with the assumptions of Euler-Bernoulli beam theory. The total deflection of the facesheets is assumed to be the superposition of the deflections due to bending and that due to core shear. The

x

displaystyle x

-direction displacements of the facesheets due to bending are given by

u

b

f a c e

( x , z ) = − z  

d

w

b

d

x

displaystyle u_ b ^ mathrm face (x,z)=-z~ cfrac mathrm d w_ b mathrm d x

The displacement of the top facesheet due to shear in the core is

u

s

t o p f a c e

( x , z ) = −

(

z − h −

f 2

)

 

d

w

s

d

x

displaystyle u_ s ^ mathrm topface (x,z)=-left(z-h- tfrac f 2 right)~ cfrac mathrm d w_ s mathrm d x

and that of the bottom facesheet is

u

s

b o t f a c e

( x , z ) = −

(

z + h +

f 2

)

 

d

w

s

d

x

displaystyle u_ s ^ mathrm botface (x,z)=-left(z+h+ tfrac f 2 right)~ cfrac mathrm d w_ s mathrm d x

The normal strains in the two facesheets are given by

ε

x x

=

u

b

∂ x

+

u

s

∂ x

displaystyle varepsilon _ xx = cfrac partial u_ b partial x + cfrac partial u_ s partial x

Therefore,

ε

x x

t o p f a c e

= − z  

d

2

w

b

d

x

2

(

z − h −

f 2

)

 

d

2

w

s

d

x

2

  ;    

ε

x x

b o t f a c e

= − z  

d

2

w

b

d

x

2

(

z + h +

f 2

)

 

d

2

w

s

d

x

2

displaystyle varepsilon _ xx ^ mathrm topface =-z~ cfrac mathrm d ^ 2 w_ b mathrm d x^ 2 -left(z-h- tfrac f 2 right)~ cfrac mathrm d ^ 2 w_ s mathrm d x^ 2 ~;~~varepsilon _ xx ^ mathrm botface =-z~ cfrac mathrm d ^ 2 w_ b mathrm d x^ 2 -left(z+h+ tfrac f 2 right)~ cfrac mathrm d ^ 2 w_ s mathrm d x^ 2

Stress-displacement relations[edit] The shear stress in the core is given by

σ

z x

c o r e

=

C

55

c o r e

 

ε

z x

c o r e

=

C

55

c o r e

2

 

γ

z x

c o r e

=

2 h + f

4 h

 

C

55

c o r e

 

γ

z x

b e a m

displaystyle sigma _ zx ^ mathrm core =C_ 55 ^ mathrm core ~varepsilon _ zx ^ mathrm core = cfrac C_ 55 ^ mathrm core 2 ~gamma _ zx ^ mathrm core = tfrac 2h+f 4h ~C_ 55 ^ mathrm core ~gamma _ zx ^ mathrm beam

or,

σ

z x

c o r e

=

2 h + f

4 h

 

C

55

c o r e

 

d

w

s

d

x

displaystyle sigma _ zx ^ mathrm core = tfrac 2h+f 4h ~C_ 55 ^ mathrm core ~ cfrac mathrm d w_ s mathrm d x

The normal stresses in the facesheets are given by

σ

x x

f a c e

=

C

11

f a c e

 

ε

x x

f a c e

displaystyle sigma _ xx ^ mathrm face =C_ 11 ^ mathrm face ~varepsilon _ xx ^ mathrm face

Hence,

σ

x x

t o p f a c e

= − z  

C

11

f a c e

 

d

2

w

b

d

x

2

(

z − h −

f 2

)

 

C

11

f a c e

 

d

2

w

s

d

x

2

=

− z  

C

11

f a c e

 

d

2

w

d

x

2

+

(

2 h + f

2

)

 

C

11

f a c e

 

d

2

w

s

d

x

2

σ

x x

b o t f a c e

= − z  

C

11

f a c e

 

d

2

w

b

d

x

2

(

z + h +

f 2

)

 

C

11

f a c e

 

d

2

w

s

d

x

2

=

− z  

C

11

f a c e

 

d

2

w

d

x

2

(

2 h + f

2

)

 

C

11

f a c e

 

d

2

w

s

d

x

2

displaystyle begin aligned sigma _ xx ^ mathrm topface &=-z~C_ 11 ^ mathrm face ~ cfrac mathrm d ^ 2 w_ b mathrm d x^ 2 -left(z-h- tfrac f 2 right)~C_ 11 ^ mathrm face ~ cfrac mathrm d ^ 2 w_ s mathrm d x^ 2 &=&-z~C_ 11 ^ mathrm face ~ cfrac mathrm d ^ 2 w mathrm d x^ 2 +left( tfrac 2h+f 2 right)~C_ 11 ^ mathrm face ~ cfrac mathrm d ^ 2 w_ s mathrm d x^ 2 \sigma _ xx ^ mathrm botface &=-z~C_ 11 ^ mathrm face ~ cfrac mathrm d ^ 2 w_ b mathrm d x^ 2 -left(z+h+ tfrac f 2 right)~C_ 11 ^ mathrm face ~ cfrac mathrm d ^ 2 w_ s mathrm d x^ 2 &=&-z~C_ 11 ^ mathrm face ~ cfrac mathrm d ^ 2 w mathrm d x^ 2 -left( tfrac 2h+f 2 right)~C_ 11 ^ mathrm face ~ cfrac mathrm d ^ 2 w_ s mathrm d x^ 2 end aligned

Resultant forces and moments[edit] The resultant normal force in a face sheet is defined as

N

x x

f a c e

:=

− f

/

2

f

/

2

σ

x x

f a c e

 

d

z

f

displaystyle N_ xx ^ mathrm face :=int _ -f/2 ^ f/2 sigma _ xx ^ mathrm face ~mathrm d z_ f

and the resultant moments are defined as

M

x x

f a c e

:=

− f

/

2

f

/

2

z

f

 

σ

x x

f a c e

 

d

z

f

displaystyle M_ xx ^ mathrm face :=int _ -f/2 ^ f/2 z_ f ~sigma _ xx ^ mathrm face ~mathrm d z_ f

where

z

f

t o p f a c e

:= z − h −

f 2

  ;    

z

f

b o t f a c e

:= z + h +

f 2

displaystyle z_ f ^ mathrm topface :=z-h- tfrac f 2 ~;~~z_ f ^ mathrm botface :=z+h+ tfrac f 2

Using the expressions for the normal stress in the two facesheets gives

N

x x

t o p f a c e

= − f

(

h +

f 2

)

 

C

11

f a c e

 

d

2

w

b

d

x

2

= −

N

x x

b o t f a c e

M

x x

t o p f a c e

= −

f

3

 

C

11

f a c e

12

(

d

2

w

b

d

x

2

+

d

2

w

s

d

x

2

)

= −

f

3

 

C

11

f a c e

12

 

d

2

w

d

x

2

=

M

x x

b o t f a c e

displaystyle begin aligned N_ xx ^ mathrm topface &=-fleft(h+ tfrac f 2 right)~C_ 11 ^ mathrm face ~ cfrac mathrm d ^ 2 w_ b mathrm d x^ 2 =-N_ xx ^ mathrm botface \M_ xx ^ mathrm topface &=- cfrac f^ 3 ~C_ 11 ^ mathrm face 12 left( cfrac mathrm d ^ 2 w_ b mathrm d x^ 2 + cfrac mathrm d ^ 2 w_ s mathrm d x^ 2 right)=- cfrac f^ 3 ~C_ 11 ^ mathrm face 12 ~ cfrac mathrm d ^ 2 w mathrm d x^ 2 =M_ xx ^ mathrm botface end aligned

In the core, the resultant moment is

M

x x

c o r e

:=

− h

h

z  

σ

x x

c o r e

 

d

z = 0

displaystyle M_ xx ^ mathrm core :=int _ -h ^ h z~sigma _ xx ^ mathrm core ~mathrm d z=0

The total bending moment in the beam is

M =

N

x x

t o p f a c e

  ( 2 h + f ) + 2  

M

x x

t o p f a c e

displaystyle M=N_ xx ^ mathrm topface ~(2h+f)+2~M_ xx ^ mathrm topface

or,

M = −

f ( 2 h + f

)

2

2

 

C

11

f a c e

 

d

2

w

b

d

x

2

f

3

6

 

C

11

f a c e

 

d

2

w

d

x

2

displaystyle M=- cfrac f(2h+f)^ 2 2 ~C_ 11 ^ mathrm face ~ cfrac mathrm d ^ 2 w_ b mathrm d x^ 2 - cfrac f^ 3 6 ~C_ 11 ^ mathrm face ~ cfrac mathrm d ^ 2 w mathrm d x^ 2

The shear force

Q

x

displaystyle Q_ x

in the core is defined as

Q

x

c o r e

= κ

− h

h

σ

x z

  d z =

κ ( 2 h + f )

2

 

C

55

c o r e

 

d

w

s

d

x

displaystyle Q_ x ^ mathrm core =kappa int _ -h ^ h sigma _ xz ~dz= tfrac kappa (2h+f) 2 ~C_ 55 ^ mathrm core ~ cfrac mathrm d w_ s mathrm d x

where

κ

displaystyle kappa

is a shear correction coefficient. The shear force in the facesheets can be computed from the bending moments using the relation

Q

x

f a c e

=

d

M

x x

f a c e

d

x

displaystyle Q_ x ^ mathrm face = cfrac mathrm d M_ xx ^ mathrm face mathrm d x

or,

Q

x

f a c e

= −

f

3

 

C

11

f a c e

12

 

d

3

w

d

x

3

displaystyle Q_ x ^ mathrm face =- cfrac f^ 3 ~C_ 11 ^ mathrm face 12 ~ cfrac mathrm d ^ 3 w mathrm d x^ 3

For thin facesheets, the shear force in the facesheets is usually ignored.[2] Bending
Bending
and shear stiffness[edit] The bending stiffness of the sandwich beam is given by

D

b e a m

= − M

/

d

2

w

d

x

2

displaystyle D^ mathrm beam =-M/ tfrac mathrm d ^ 2 w mathrm d x^ 2

From the expression for the total bending moment in the beam, we have

M = −

f ( 2 h + f

)

2

2

 

C

11

f a c e

 

d

2

w

b

d

x

2

f

3

6

 

C

11

f a c e

 

d

2

w

d

x

2

displaystyle M=- cfrac f(2h+f)^ 2 2 ~C_ 11 ^ mathrm face ~ cfrac mathrm d ^ 2 w_ b mathrm d x^ 2 - cfrac f^ 3 6 ~C_ 11 ^ mathrm face ~ cfrac mathrm d ^ 2 w mathrm d x^ 2

For small shear deformations, the above expression can be written as

M ≈ −

f [ 3 ( 2 h + f

)

2

+

f

2

]

6

 

C

11

f a c e

 

d

2

w

d

x

2

displaystyle Mapprox - cfrac f[3(2h+f)^ 2 +f^ 2 ] 6 ~C_ 11 ^ mathrm face ~ cfrac mathrm d ^ 2 w mathrm d x^ 2

Therefore, the bending stiffness of the sandwich beam (with

f ≪ 2 h

displaystyle fll 2h

) is given by

D

b e a m

f [ 3 ( 2 h + f

)

2

+

f

2

]

6

 

C

11

f a c e

f ( 2 h + f

)

2

2

 

C

11

f a c e

displaystyle D^ mathrm beam approx cfrac f[3(2h+f)^ 2 +f^ 2 ] 6 ~C_ 11 ^ mathrm face approx cfrac f(2h+f)^ 2 2 ~C_ 11 ^ mathrm face

and that of the facesheets is

D

f a c e

=

f

3

12

 

C

11

f a c e

displaystyle D^ mathrm face = cfrac f^ 3 12 ~C_ 11 ^ mathrm face

The shear stiffness of the beam is given by

S

b e a m

=

Q

x

/

d

w

s

d

x

displaystyle S^ mathrm beam =Q_ x / tfrac mathrm d w_ s mathrm d x

Therefore, the shear stiffness of the beam, which is equal to the shear stiffness of the core, is

S

b e a m

=

S

c o r e

=

κ ( 2 h + f )

2

 

C

55

c o r e

displaystyle S^ mathrm beam =S^ mathrm core = cfrac kappa (2h+f) 2 ~C_ 55 ^ mathrm core

Relation between bending and shear deflections[edit] A relation can be obtained between the bending and shear deflections by using the continuity of tractions between the core and the facesheets. If we equate the tractions directly we get

n

x

 

σ

x x

f a c e

=

n

z

 

σ

z x

c o r e

displaystyle n_ x ~sigma _ xx ^ mathrm face =n_ z ~sigma _ zx ^ mathrm core

At both the facesheet-core interfaces

n

x

= 1

displaystyle n_ x =1

but at the top of the core

n

z

= 1

displaystyle n_ z =1

and at the bottom of the core

n

z

= − 1

displaystyle n_ z =-1

. Therefore, traction continuity at

z = ± h

displaystyle z=pm h

leads to

2 f h  

C

11

f a c e

 

d

2

w

s

d

x

2

− ( 2 h + f )  

C

55

c o r e

 

d

w

s

d

x

= 4

h

2

 

C

11

f a c e

 

d

2

w

b

d

x

2

displaystyle 2fh~C_ 11 ^ mathrm face ~ cfrac mathrm d ^ 2 w_ s mathrm d x^ 2 -(2h+f)~C_ 55 ^ mathrm core ~ cfrac mathrm d w_ s mathrm d x =4h^ 2 ~C_ 11 ^ mathrm face ~ cfrac mathrm d ^ 2 w_ b mathrm d x^ 2

The above relation is rarely used because of the presence of second derivatives of the shear deflection. Instead it is assumed that

n

z

 

σ

z x

c o r e

=

d

N

x x

f a c e

d

x

displaystyle n_ z ~sigma _ zx ^ mathrm core = cfrac mathrm d N_ xx ^ mathrm face mathrm d x

which implies that

d

w

s

d

x

= − 2 f h  

(

C

11

f a c e

C

55

c o r e

)

 

d

3

w

b

d

x

3

displaystyle cfrac mathrm d w_ s mathrm d x =-2fh~left( cfrac C_ 11 ^ mathrm face C_ 55 ^ mathrm core right)~ cfrac mathrm d ^ 3 w_ b mathrm d x^ 3

Governing equations[edit] Using the above definitions, the governing balance equations for the bending moment and shear force are

M

=

D

b e a m

 

d

2

w

s

d

x

2

(

D

b e a m

+ 2

D

f a c e

)

 

d

2

w

d

x

2

Q

=

S

c o r e

 

d

w

s

d

x

− 2

D

f a c e

 

d

3

w

d

x

3

displaystyle begin aligned M&=D^ mathrm beam ~ cfrac mathrm d ^ 2 w_ s mathrm d x^ 2 -left(D^ mathrm beam +2D^ mathrm face right)~ cfrac mathrm d ^ 2 w mathrm d x^ 2 \Q&=S^ mathrm core ~ cfrac mathrm d w_ s mathrm d x -2D^ mathrm face ~ cfrac mathrm d ^ 3 w mathrm d x^ 3 end aligned

We can alternatively express the above as two equations that can be solved for

w

displaystyle w

and

w

s

displaystyle w_ s

as

(

2

D

f a c e

S

c o r e

)

d

4

w

d

x

4

(

1 +

2

D

f a c e

D

b e a m

)

d

2

w

d

x

2

+

(

1

S

c o r e

)

 

d

Q

d

x

=

M

D

b e a m

(

D

b e a m

S

c o r e

)

d

3

w

s

d

x

3

(

1 +

D

b e a m

2

D

f a c e

)

d

w

s

d

x

1

S

c o r e

 

d

M

d

x

= −

(

1 +

D

b e a m

2

D

f a c e

)

Q

S

c o r e

displaystyle begin aligned &left( frac 2D^ mathrm face S^ mathrm core right) cfrac mathrm d ^ 4 w mathrm d x^ 4 -left(1+ frac 2D^ mathrm face D^ mathrm beam right) cfrac mathrm d ^ 2 w mathrm d x^ 2 +left( cfrac 1 S^ mathrm core right)~ cfrac mathrm d Q mathrm d x = frac M D^ mathrm beam \&left( frac D^ mathrm beam S^ mathrm core right) cfrac mathrm d ^ 3 w_ s mathrm d x^ 3 -left(1+ frac D^ mathrm beam 2D^ mathrm face right) cfrac mathrm d w_ s mathrm d x - cfrac 1 S^ mathrm core ~ cfrac mathrm d M mathrm d x =-left(1+ cfrac D^ mathrm beam 2D^ mathrm face right) frac Q S^ mathrm core ,end aligned

Using the approximations

Q ≈

d

M

d

x

  ;     q ≈

d

Q

d

x

displaystyle Qapprox cfrac mathrm d M mathrm d x ~;~~qapprox cfrac mathrm d Q mathrm d x

where

q

displaystyle q

is the intensity of the applied load on the beam, we have

(

2

D

f a c e

S

c o r e

)

d

4

w

d

x

4

(

1 +

2

D

f a c e

D

b e a m

)

d

2

w

d

x

2

=

M

D

b e a m

q

S

c o r e

(

D

b e a m

S

c o r e

)

d

3

w

s

d

x

3

(

1 +

D

b e a m

2

D

f a c e

)

d

w

s

d

x

= −

(

D

b e a m

2

D

f a c e

)

Q

S

c o r e

displaystyle begin aligned &left( frac 2D^ mathrm face S^ mathrm core right) cfrac mathrm d ^ 4 w mathrm d x^ 4 -left(1+ frac 2D^ mathrm face D^ mathrm beam right) cfrac mathrm d ^ 2 w mathrm d x^ 2 = frac M D^ mathrm beam - cfrac q S^ mathrm core \&left( frac D^ mathrm beam S^ mathrm core right) cfrac mathrm d ^ 3 w_ s mathrm d x^ 3 -left(1+ frac D^ mathrm beam 2D^ mathrm face right) cfrac mathrm d w_ s mathrm d x =-left( cfrac D^ mathrm beam 2D^ mathrm face right) frac Q S^ mathrm core ,end aligned

Several techniques may be used to solve this system of two coupled ordinary differential equations given the applied load and the applied bending moment and displacement boundary conditions. Temperature dependent alternative form of governing equations[edit] Assuming that each partial cross section fulfills Bernoulli's hypothesis, the balance of forces and moments on the deformed sandwich beam element can be used to deduce the bending equation for the sandwich beam.

Figure 1 - Equilibration of a deflected sandwich beam under temperature load and burden in comparison with the undeflected cross section

The stress resultants and the corresponding deformations of the beam and of the cross section can be seen in Figure 1. The following relationships can be derived using the theory of linear elasticity:[3][4]

M

c o r e

=

D

b e a m

(

d

γ

2

d

x

+ ϑ

)

=

D

b e a m

(

d

γ

d

x

d

2

w

d

x

2

+ ϑ

)

M

f a c e

= −

D

f a c e

d

2

w

d

x

2

Q

c o r e

=

S

c o r e

γ

Q

f a c e

= −

D

f a c e

d

3

w

d

x

3

displaystyle begin aligned M^ mathrm core &=D^ mathrm beam left( cfrac mathrm d gamma _ 2 mathrm d x +vartheta right)=D^ mathrm beam left( cfrac mathrm d gamma mathrm d x - cfrac mathrm d ^ 2 w mathrm d x^ 2 +vartheta right)\M^ mathrm face &=-D^ mathrm face cfrac mathrm d ^ 2 w mathrm d x^ 2 \Q^ mathrm core &=S^ mathrm core gamma \Q^ mathrm face &=-D^ mathrm face cfrac mathrm d ^ 3 w mathrm d x^ 3 end aligned ,

where

w

displaystyle w,

transverse displacement of the beam

γ

displaystyle gamma ,

Average shear strain in the sandwich

γ =

γ

1

+

γ

2

displaystyle gamma =gamma _ 1 +gamma _ 2 ,

γ

1

displaystyle gamma _ 1 ,

Rotation of the facesheets

γ

1

=

d

w

d

x

displaystyle gamma _ 1 = cfrac mathrm d w mathrm d x ,

γ

2

displaystyle gamma _ 2 ,

Shear strain in the core

M

c o r e

displaystyle M^ mathrm core ,

Bending
Bending
moment in the core

D

b e a m

displaystyle D^ mathrm beam ,

Bending
Bending
stiffness of the sandwich beam

M

f a c e

displaystyle M^ mathrm face ,

Bending
Bending
moment in the facesheets

D

f a c e

displaystyle D^ mathrm face ,

Bending
Bending
stiffness of the facesheets

Q

c o r e

displaystyle Q^ mathrm core ,

Shear force
Shear force
in the core

Q

f a c e

displaystyle Q^ mathrm face ,

Shear force
Shear force
in the facesheets

S

c o r e

displaystyle S^ mathrm core ,

Shear stiffness of the core

ϑ

displaystyle vartheta ,

Additional bending as a consequence of a temperature drop

ϑ =

α (

T

o

T

u

)

e

displaystyle vartheta = frac alpha (T_ o -T_ u ) e ,

α

displaystyle alpha ,

Temperature coefficient of expansion of the converings

Superposition of the equations for the facesheets and the core leads to the following equations for the total shear force

Q

displaystyle Q

and the total bending moment

M

displaystyle M

:

S

c o r e

γ −

D

f a c e

d

3

w

d

x

3

= Q

( 1 )

D

b e a m

(

d

γ

d

x

+ ϑ

)

(

D

b e a m

+

D

f a c e

)

d

2

w

d

x

2

= M

( 2 )

displaystyle begin alignedat 3 &S^ mathrm core gamma -D^ mathrm face cfrac mathrm d ^ 3 w mathrm d x^ 3 =Q&quad quad &(1)\&D^ mathrm beam left( cfrac mathrm d gamma mathrm d x +vartheta right)-left(D^ mathrm beam +D^ mathrm face right) cfrac mathrm d ^ 2 w mathrm d x^ 2 =M&quad quad &(2),end alignedat

We can alternatively express the above as two equations that can be solved for

w

displaystyle w

and

γ

displaystyle gamma

, i.e.,

(

D

f a c e

S

c o r e

)

d

4

w

d

x

4

(

1 +

D

f a c e

D

b e a m

)

d

2

w

d

x

2

=

M

D

b e a m

q

S

c o r e

− ϑ

(

D

b e a m

S

c o r e

)

d

2

γ

d

x

2

(

1 +

D

b e a m

D

f a c e

)

γ = −

(

D

b e a m

D

f a c e

)

Q

S

c o r e

displaystyle begin aligned &left( frac D^ mathrm face S^ mathrm core right) cfrac mathrm d ^ 4 w mathrm d x^ 4 -left(1+ frac D^ mathrm face D^ mathrm beam right) cfrac mathrm d ^ 2 w mathrm d x^ 2 = frac M D^ mathrm beam - cfrac q S^ mathrm core -vartheta \&left( frac D^ mathrm beam S^ mathrm core right) cfrac mathrm d ^ 2 gamma mathrm d x^ 2 -left(1+ frac D^ mathrm beam D^ mathrm face right)gamma =-left( cfrac D^ mathrm beam D^ mathrm face right) frac Q S^ mathrm core ,end aligned

Solution approaches[edit]

Shear and bending deformation of a sandwich composite beam.

The bending behavior and stresses in a continuous sandwich beam can be computed by solving the two governing differential equations. Analytical approach[edit] For simple geometries such as double span beams under uniformly distributed loads, the governing equations can be solved by using appropriate boundary conditions and using the superposition principle. Such results are listed in the standard DIN EN 14509:2006[5](Table E10.1). Energy methods may also be used to compute solutions directly. Numerical approach[edit] The differential equation of sandwich continuous beams can be solved by the use of numerical methods such as finite differences and finite elements. For finite differences Berner[6] recommends a two-stage approach. After solving the differential equation for the normal forces in the cover sheets for a single span beam under a given load, the energy method can be used to expand the approach for the calculation of multi-span beams. Sandwich continuous beam with flexible cover sheets can also be laid on top of each other when using this technique. However, the cross-section of the beam has to be constant across the spans. A more specialized approach recommended by Schwarze[4] involves solving for the homogeneous part of the governing equation exactly and for the particular part approximately. Recall that the governing equation for a sandwich beam is

(

2

D

f a c e

S

c o r e

)

d

4

w

d

x

4

(

1 +

2

D

f a c e

D

b e a m

)

d

2

w

d

x

2

=

M

D

b e a m

q

S

c o r e

displaystyle left( frac 2D^ mathrm face S^ mathrm core right) cfrac mathrm d ^ 4 w mathrm d x^ 4 -left(1+ frac 2D^ mathrm face D^ mathrm beam right) cfrac mathrm d ^ 2 w mathrm d x^ 2 = frac M D^ mathrm beam - cfrac q S^ mathrm core

If we define

α :=

2

D

f a c e

D

b e a m

  ;     β :=

2

D

f a c e

S

c o r e

  ;     W ( x ) :=

d

2

w

d

x

2

displaystyle alpha := cfrac 2D^ mathrm face D^ mathrm beam ~;~~beta := cfrac 2D^ mathrm face S^ mathrm core ~;~~W(x):= cfrac mathrm d ^ 2 w mathrm d x^ 2

we get

d

2

W

d

x

2

(

1 + α

β

)

  W =

M

β

D

b e a m

q

D

f a c e

displaystyle cfrac mathrm d ^ 2 W mathrm d x^ 2 -left( cfrac 1+alpha beta right)~W= frac M beta D^ mathrm beam - cfrac q D^ mathrm face

Schwarze uses the general solution for the homogeneous part of the above equation and a polynomial approximation for the particular solution for sections of a sandwich beam. Interfaces between sections are tied together by matching boundary conditions. This approach has been used in the open source code swe2. Practical Importance[edit] Results predicted by linear sandwich theory correlate well with the experimentally determined results. The theory is used as a basis for the structural report which is needed for the construction of large industrial and commercial buildings which are clad with sandwich panels . Its use is explicitly demanded for approvals and in the relevant engineering standards.[5] Mohammed Rahif Hakmi and others conducted researches into numerical, experimental behavior of materials and fire and blast behavior of Composite material. He published multiple research articles:

Local buckling of Sandwich Panels.[7][8] Face buckling stress in Sandwich Panels.[9] Post-buckling behaviour of foam-filled thin-walled steel beams.[10] Fire resistance of composite floor slabs using a model fire test facility[11] Fire Resistant Sandwich Panels for Offshore structuresSandwich Panels.[12] Numerical Temperature Analysis of Hygroscopic
Hygroscopic
Panels Exposed to Fire.[13] Cost Effective Use of Fibre Reinforced Composites Offshore.[14]

Hakmi developed a design method, which had been recommended by the CIB Working Commission W056 Sandwich Panels, ECCS/CIB Joint Committee and has been used in the European recommendations for the design of sandwich panels (CIB, 2000).[15][16][17] See also[edit]

Bending Beam theory Composite material Hill yield criteria Sandwich structured composite Sandwich plate system Composite honeycomb Timoshenko beam theory Plate theory Sandwich Panel

References[edit]

^ Plantema, F, J., 1966, Sandwich Construction: The Bending
Bending
and Buckling of Sandwich Beams, Plates, and Shells, Jon Wiley and Sons, New York. ^ a b c Zenkert, D., 1995, An Introduction to Sandwich Construction, Engineering Materials Advisory Services Ltd, UK. ^ K. Stamm, H. Witte: Sandwichkonstruktionen - Berechnung, Fertigung, Ausführung. Springer-Verlag, Wien - New York 1974. ^ a b Knut Schwarze: „Numerische Methoden zur Berechnung von Sandwichelementen“. In Stahlbau. 12/1984, ISSN 0038-9145. ^ a b EN 14509 (D):Self-supporting double skin metal faced insulating panels. November 2006. ^ Klaus Berner: Erarbeitung vollständiger Bemessungsgrundlagen im Rahmen bautechnischer Zulassungen für Sandwichbauteile.Fraunhofer IRB Verlag, Stuttgart 2000 (Teil 1). ^ " Mohammed Rahif Hakmi Research".  ^ [1] Local buckling of Sandwich Panels ^ Davies M J and Hakmi M R (1991) "Face buckling stress in sandwich panels", Nordic Conference Steel Colloquium, pp. 99-110. ^ Davies, J.M., Hakmi, M.R. and Hassinen, P. (1991), "Postbuckling behaviour of foam-filled thin-walled steel beams" Journal of Constructional Steel Research 20: 75 - 83. ^ " Fire resistance of composite floor slabs using a model fire test facility, Author(s) ABDEL-HALIM M. A. H. (1) ; HAKMI M. R. (2) ; O'LEARY D. C. (2) ;Affiliation(s) du ou des auteurs / Author(s) Affiliation(s),(1) Department of Civil Engineering, Jordan University of Science and Technology, PO Box 3030., Irbid, JORDANIE(2) Department of Civil Engineering, University of Salford, Salford, M5 4WT, ROYAUME-UNI. ^ Davies, J.M., Dr. Hakmi R. and McNicholas J.B.: Fire Resistant SandwichPanels for Offshore Structures, Cost Effective Use of Fibre ReinforcedComposites Offshore, CP07 Research Report, Marinetech North WestProgramme, Phase 1, 1991. ^ Davies,J.M., Hakmi, R. and Wang,H.B.: Numerical TemperatureAnalysis of Hygroscopic
Hygroscopic
Panels Exposed to Fire, p1624-1635,Numerical Methods in Thermal Problems, Vol. VIII Part 2,Proceedings of the Eighth International Conference Held inSwansea, July 12-16th, 1993. Pineridge Press, UK. ^ [2] HSE,The cost effective use of fibre reinforced composites offshore CP07,Fire Resistant Sandwich Panels for Offshore Structures Professor J.M.Davies, Dr. R. Hakim, Dr. J.B. McNicholas, University of Salford 45 pages ^ "European Recommendations for Sandwich Panels".  ^ Davies, J.M. & Hakmi, M.R. 1990. Local Buckling of Profiled Sandwich Plates. Proc. IABSE Symposium, Mixed Structures including New Materials, Brussels, September, pp. 533-538 ^ "Local Buckling of Profiled Sandwich Plates". 

Bibliography[edit]

Mohammed Rahif Hakmi Klaus Berner, Oliver Raabe: Bemessung von Sandwichbauteilen. IFBS-Schrift 5.08, IFBS e.V., Düsseldorf 2006. Ralf Möller, Hans Pöter, Knut Schwarze: Planen und Bauen mit Trapezprofilen und Sandwichelementen. Band 1, Ernst & Sohn, Berlin 2004, ISBN 3-433-01595-3.

External links[edit]

Mohammed Rahif Hakmi Research for Sandwich Panels Institute for Sandwich Technology http://www.diabgroup.com/europe/literature/e_pdf_files/man_pdf/sandwich_hb.pdf DIAB Sandwich Handbook http://www.swe1.com Programm zur Ermittlung der Schnittgrössen und Spannungen von Sandwich-Wandplatten mit biegeweichen Deckschichten (Open Source) http://www.swe2.com Computation of sandwich beams with corrugated f

.