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CLASSİCAL LAMİNATİON THEORY (CLT)

8.

(tvids: 8a and 8b)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

STRESS–STRAİN CALCULATİONS İN LAMİNATED COMPOSİTES

(Note: The Zor Equivalent Volume Model, which is an alternative to CLT, will be explained in Sections 9.4 and Chapter 10.)

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8.1 Our aim in this chapter is to develop equations by which the stresses and strains occurring at any point in a laminated structure obtained by combining orthotropic layers reinforced with continuous fibers can be calculated theoretically.

We call the structures formed as a result of gluing more than one composite layer on top of each other layered or laminated composite structure.

8. Classical Lamination Theory (CLT)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

• The equations will be developed within the scope of the Classical Lamination Theory (CLT) and will also be valid for structures consisting of layers with different fiber orientation angles, different thicknesses, and different material properties.

• In the examples to be solved, damage assessments will also be made according to the criteria explained in section 7.

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Meaning of Internal Force and internal moments in element Q:

N_x: Normal (tension or compression) internal force per unit length in the x direction,

N_y: Normal (tension or compression) internal force per unit length in the y direction,

N_xy = N_yx : Internal shearing forces per unit length,

M_x: Bending internal moment per unit length in a section whose normal is x

M_y: Bending internal moment per unit length in a section with normal y

M_xy: Torsional internal moment per unit length in a section whose normal is x

M_yx: Torsional moment per unit length in a section whose normal is y

• Now we consider that all internal forces and moments in element Q have been previously calculated from static equilibrium and each of them is known.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

8.2 Internal forces and internal moments occurring per unit length:

We are considering a laminated composite structure that is subject to the

influence of external loads (P1, P2,.. M1, M2:..) and is in static equilibrium.

From this structure, we imaginary separate a Q element with the same

thickness as the lamina (layer) and side lengths of 1 mm.

According to the separation principle, this Q element is still in static equilibrium,

8. Classical Lamination Theory (CLT)

• We want to calculate the stresses and deformations at a point such as "b" belonging to the Q element.
• We accept that there is a plane stress state in element "b".

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1. Each layer is elastic, orthotropic and homogeneous within itself.
2. It is accepted that the layers adhere to each other very well.
3. A vertical and straight line (for examp: line CA) drawn before loading

on the midplane passing through the middle of the thickness of the

layered composite plate remains vertical and straight at the end of the

loading (after deformation) (line C'A’). That is, out-of-plane shear deformations do not ocur (γ_xz = γ _yz =0)

6. The Cartesian coordinate system used is placed on the middle plane of the layered composite and the z axis is necessarily

selected in a downward direction.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

8.3 Classical Lamination Theory and Assumptions:

We consider the deformed state of the Q element of the layered (or laminated) structure. When we look at this element from the thickness side and from the x-z plane, we will make the following assumptions:

In the following stages, we will try to obtain the stresses and strains at any point of the layered structure in terms of internal forces and internal moments, which are known values…>>

8. Classical Lamination Theory (CLT)

5. Displacements remain very small compared to the plate thickness.

4. The plate (laminated structure) is assumed to be thin and only subjected to plane stress (σ_z = τ_xz = τ _yz =0)

(center of curvature)

mid-plane

7. Although a layer tends to deform freely in the lateral direction due to the Poisson effect, this lateral deformation is constrained because

of the perfect bonding with the other layers. However, this lateral interaction between the layers (Poisson interaction) is neglected.

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: Vertical (z) displacement of point C, that is, the amount of collapse caused by bending (the index '0' is used for the middle plane)

: Horizontal (x) displacement of point C (total displacement in the x direction due to deformation)

u

: Horizontal (x) displacement of point A

α

: The angle made by the tangent passing through point C' with the horizontal

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

The slope of point C in the midplane is:

(8.1)

8.4 Calculation of Global Stress and Strains According to Mid-plane Values

8. Classical Lamination Theory (CLT)

C

A

z

C'

z

u

α

α

dx

dw₀

.

.

A'

mid-plane

z

x

Now we enlarge the part within the oval frame in the figure and examine it in more detail:

C: a point on the midplane

A: a point on any layer k, at a distance z from C.

C' and A': are the final positions of the points C and A after changing shape.

According to the 3rd assumption, CA=C'A'=z can be written.

: The final deformed shape of the midplane due to bending

α

mid-plane

(center of curvature)

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Similarly, when we look at it from the y-z plane, the displacement of point A in the y direction is:

For small angles;

If we look at the displacement (u) in the x direction of a point A on any layer such as k;

Here, the unit elongation (strain) value in the x direction:

and here the unit elongation (strain) value in the y direction:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(8.2)

(8.3)

(8.4)

(8.5)

(8.6)

(8.7)

8. Classical Lamination Theory (CLT)

mid-plane

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Shear strain angle in x-y plane :

is found

If Equations 8.4 and 8.6 we obtained before are substituted into equation 8.8 above:

(8.8)

(8.9)

8. Classical Lamination Theory (CLT)

(8.4)

(8.6)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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Meanings of the terms on the right side of equation 8.10:

Curvatures of the midplane:

If we write equations 8.5, 8.7 and 8.9 in matrix format:

(8.10)

=

Strains in the midplane:

(8.11)

(8.12)

8. Classical Lamination Theory (CLT)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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Global Strains:

(8.13)

In this case, at a point on layer k, at a distance z from the midplane:

(If equations 8.11 and 8.12 are substituted into 8.10,)

(If we remember equation 6.1)

Global Stresses:

(8.14)

From now on, we will obtain the strains and curvatures in the mid-plane in terms of internal forces and internal moments... >>

8. Classical Lamination Theory (CLT)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

8.5 Coding of a layered structure

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• When solving problems related to layered composites, the layers must be numbered correctly and the interfaces must be coded. (The reason for this will become clearer later.)

• The figure on the side shows a composite structure consisting of n layers, shown in the x-z plane.

The following points should be taken into consideration when coding:

1. First, the layers are numbered. The top one is the 1st layer, the other layer numbers are given downwards.
2. The layer interfaces are then coded. The outer surface of the top layer is h0 (or z0), and as you go down, each interface is given code numbers as h1, h2, ...
3. h values are actually the z coordinates of that interface and can be negative. The symbol z can also be used instead of h.
4. The midplane may not correspond to an interface. In this case, the middle plane is not given a separate code number.

Example 2: Coded version of a three-layer structure

Example 1: Coded version of a two-layer structure

8. Classical Lamination Theory (CLT)

• The origin of the cartesian coordinate system is placed at any point in the midplane passing through the middle of the thickness. The middle plane is coincident with the x-y plane. The z axis is selected in the direction of the thickness and downwards.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

8.6 Calculation of Mid-Plane Strains

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We examine the internal forces in a Q element with a side length of 1 mm in a layered structure:

Stress and internal force in x direction in layer k:

Internal force in the x direction in the area of a 1mm wide, dz thick dA differential strip:

(8.15)

Internal normal force in x direction in

layered Q element:

Internal normal force in y direction in layered Q element:

Internal shear force in layered Q element:

Similarly, if the same operations are performed for the y direction;

(8.16)

(8.17)

Nx, Ny, Nxy : These are the internal forces per unit length. Its units are N/mm.

Internal force in x direction in layer k:

If we write equations 8.15, 8.16 and 8.17 in a matrix format:

(8.18)

k

8. Classical Lamination Theory (CLT)

(midplane)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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Now we examine the internal moments in the Q element.

Stresses in x direction and internal bending moment (Mx-k) in layer k.

• Moment of the stresses 𝜎_𝑥 on the strip area dA about the midplane:

Internal force x

perpendicular distance

• Moment of 𝜎_𝑥 stresses in layer k relative to the mid-plane:

• Internal bending moment in the section with normal x in the layered Q element :

Similarly, if the same operations are performed for the y direction;

• Total bending moment in the section with normal y in the layered Q element:

• Torsional internal moments in the layered Q element:

• If we write equations 8.19, 8.20 and 8.21 in a matrix format:

(8.19)

(8.20)

(8.21)

(8.22)

Internal bending moments per unit length are M_x, M_y; Internal torsional moment: M_xy. Units are Nmm/mm.

8. Classical Lamination Theory (CLT)

(midplane)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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In equation 8.14, we expressed the global stresses in a k layer in terms of the strains of the middle plane as follows:

Now if we substitute Equation 8.14 into Equations 8.18 and 8.22;

The deformations and curvatures of the middle plane do not depend on z.

(8.23)

(8.24)

(8.25)

Stifness Matrices

of the Whole Layered Structure

Since these stiffness matrices belong to the entire layered structure, there is 1 of each.

8. Classical Lamination Theory (CLT)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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(8.26)

(8.27)

Each of the Stifness Matrices is a 3x3 matrix and has 9 terms.

If the stiffness matrices are replaced:

Internal Forces:

Internal Moments:

(8.28)

(8.29)

8. Classical Lamination Theory (CLT)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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Strains and curvatures of the middle plane from equation 8.31:

If we combine Equations 8.28 and 8.29 into a matrix:

(8.30)

(8.31)

Symbolically :

The explicit expression of the matrix is:

(8.32)

8. Classical Lamination Theory (CLT)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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8.7 Special Cases

1st Special Case: In structures that are symmetrical with respect to the midplane in terms of material, load, geometry and fiber arrangement, [B] = 0.

2nd Special Case: In symmetrical structures, if all internal moments are zero, the curvatures of the middle plane are also zero.

In this case, In this case, there is no need to calculate the [D] matrix since it will not be involved in the operations.

3rd Special Case:

In this case, there is no need to calculate the [D] matrix again.

However, in the following special cases we do not need to calculate some of these matrices:

That is, if

then

since z = 0

In symmetrical or non-symmetrical structures, If the stresses or deformations in the mid-plane are asked;

In the stress or strain calculations to be made from Equations 8.13 and 8.14,

For this reason, before proceeding with the calculations of a layered structure, it should be checked whether it falls into the special conditions above.

8. Classical Lamination Theory (CLT)

4th Special Case: In symmetrical structures, if all internal forces are zero, the strains of the middle plane are also zero.

That is, if

then

In this case, there is no need to calculate the [D] matrix since it will not enter into the calculations.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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Example 8.1:

The material properties found through experimental measurements for a newly produced composite layer (lamina) with dimensions of 400mm x 400mm x 8mm are given in the table below.

A laminated (layered) structure was created by gluing two of these layers on top of each other with 45^o and 0^o fiber orientation arrangements, respectively. The internal forces occurring in an element with dimensions of 1 mm x 1 mm x 8 mm at the Q point of the laminated structure are shown in the figure.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

c-) Check whether damage will occur on the upper surface of the top layer according to the Hoffman criterion.

a-) Calculate the local stresses and local strains occurring on the upper and lower surfaces of the upper layer and on the upper, middle and lower surfaces of the lower layer.b-) Determine the proportions in which the Nx force is distributed among the layers

Accordingly, at point Q,

8. Classical Lamination Theory (CLT)

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Solution:

Step 3: Does the Structure Enter Special Cases?

Does this structure fall into any of the special cases described in Article 8.7 ? First of all, this is detected.

1st Special Case?

Since there is no symmetry with respect to the middle plane in terms of fiber arrangement, It does not fall into the 1st special case. Then the [B] matrix will be non-zero.

Since the structure is not symmetrical, it does not fall into the 2nd Special Case.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Since only the stresses or strains in the mid-plane are not asked, it does not fall into the 4th Special Case.

a-)

Step 1: Internal forces and internal moments are calculated.

Internal forces (N_x , N_y , N_xy) and internal moments (M_x , M_y , M_xy) per unit length from static equilibrium are calculated. But in this example, these values are given directly.

Step 2:

The layered structure is coded.

By following the steps below, we will obtain local stresses and local strains at the asked points.

8. Classical Lamination Theory (CLT)

• The midplane (the middle of the total thickness) is the interface of the two layers. The x-y plane coincides with the middle plane. z is selected in the downward direction.

• The layers are numbered from top to bottom(1 and 2).
• As explained in article 8.5, h values are given sequentially to

each external surface and interface from top to bottom.

• In fact, the h value is the z coordinate of that interface.

2nd Special Case?

3rd Special Case?

4th Special Case?

Since the structure is not symmetric, it does not fall into the 3rd Special Case.

As a results, since the structure does not fall into any special cases, all matrices [A], [B] and [D] must be calculated..>>

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Step 4:

The [Q] matrices of each layer are calculated:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Since the materials of both layers are the same, their [Q] matrices are equal:

[Q] matrices depend only on material properties.

(They do not depend on the fiber routing angleθ. )

First, the minor Poisson ratio is calculated;

[Q] matrix of layer 1

[Q] matrix of layer 2

From Equation 2.7:

8. Classical Lamination Theory (CLT)

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Step 5:

The reduced matrices of each layer [𝑸 ̅ ] are calculated:

From equations 6.14;

For 1st layer : θ =45^o

For 2nd layer: θ =0^o

8. Classical Lamination Theory (CLT)

(Don't forget this observation... It brings practicality when solving problems..)

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Step 6:

The necessary ones are calculated from the [A], [B] and [D] matrices belonging to the entire structure.

Since the 2-layer structure in this example does not enter any special states, all 3 matrices will be calculated:

8. Classical Lamination Theory (CLT)

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Let's calculate other stiffness matrices :

8. Classical Lamination Theory (CLT)

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The strains and curvatures of the midplane are calculated:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Step 7:

From Equation 8.32,

https://www.youtube.com/watch?v=AjTBunqakDA

https://www.youtube.com/watch?v=SOOFvGa81nA

Helpful Videos for Matrix Inversion:

In Excell:

In Matlab:

If we take the inverse of the 6x6 matrix on the right with the help of a program and substitute it;

8. Classical Lamination Theory (CLT)

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Similar calculations were made for other asked points and the results are given in the table below.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Global strains are calculated:

Step 8:

From 8.13 :

For the upper surface of the 1st (Top) layer; (z=-4)

8. Classical Lamination Theory (CLT)

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Calculations have been made for other asked points and all global stresses are shown in the table below.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Global stresses are calculated :

Step 9:

If equation 6.1 is applied;

8. Classical Lamination Theory (CLT)

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c:cos45^o , s:sin45^o

As can be noticed, the global stresses (in the table on the previous page) and local stresses occurring in the 2nd layer (layer with 0^o fiber orientation) are equal.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Local stresses are calculated:

Step 10:

From Equation 6.7:

Calculations have been made for other asked points and all local stresses are shown in the table below.

8. Classical Lamination Theory (CLT)

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When calculations are made for other desired points;

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Local strains are calculated:

Step 11:

If equation 6.11 is applied;

c:cos45^o , s:sin45^o

8. Classical Lamination Theory (CLT)

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Alternative solution for option a :

Step 9-) Local strains are obtained from transformation equations:

Step 10-) Afterwards, local stresses are calculated from Hooke's relations :

or Global stresses could be calculated from local stresses:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

By doing the first 8 steps exactly the same, global strains are obtained. After this, the following steps are followed.

8. Classical Lamination Theory (CLT)

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On average, we can calculate the intensity and proportions in which the Nx force is distributed to the layers as follows:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Since

The ratio at which the 1st Layer carries the Nx force:

b )

8. Classical Lamination Theory (CLT)

The ratio at which the 2nd Layer carries the Nx force:

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The local stresses on the upper surface of the upper layer were calculated in step 10 of a as follows:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

c )

Material strength properties given in the question

According to Hoffman criterion;

If the numerical values are substituted into the equation:

(No Damage Condition)

Equation 7.11

… no damage occurs.

8. Classical Lamination Theory (CLT)

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The material properties found through experimental measurements for a newly produced composite lamina with dimensions of 200mm x 200mm x 2mm are given in the table below.

Example 8.2:

8. Classical Lamination Theory (CLT)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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We will obtain local stresses and local strains by following the steps below. We'll check for damage later.

Calculations can be made on a Q element with unit edge lengths.

For this reason, first the total internal forces and then the internal forces per unit length are calculated from the separation principle and static equilubrium.

We cut from the K1 plane and examine the equilbrium of the right side in the x direction:

Step 1: Internal forces and internal moments are calculated.

Solution:

8. Classical Lamination Theory (CLT)

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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Similarly, for the y direction;

Equilibrium in the y direction of the back side of the cut made from the K2 plane:

200 mm

200 mm

K2

Normal internal force in y direction per unit length:

Internal forces in the unit element Q:

• For different solid systems and different boundary conditions, the internal forces at the most critical Q point must first be found from static calculations.

1mm

1mm

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

8. Classical Lamination Theory (CLT)

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Step 3: Control os special cases

Does this structure fall into any of the special cases described in clause 8.7?

Does it fall under Special Case 1?

Does it fall under Special Case 2?

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Step 2:

The layered structure is coded.

If the structure is symmetrical, it enters the 1st Special Case.

• According to the figure above,
• Since a single material is used in our construction, it is symmetrical with respect to the middle plane in terms of material.
• Geometrically, it is symmetrical with respect to the middle plane.
• Since the forces acting on the structure are distributed homogeneously, they are symmetrical with respect to the middle plane.
• In terms of fiber arrangement, our material is again symmetrical with respect to the middle plane.
• Therefore, this structure is a symmetrical structure. The 1st Special Case is satisfied and [B] = 0

Yes

Yes,

except option d

If the structure is symmetrical and internal moments are zero, it enters Special Case 2.

No, except option d

It will be explained while solving option d.

As explained in article 8.5

8. Classical Lamination Theory (CLT)

Does it fall under Special Case 4?

Does it fall under Special Case 3?

No

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Step 4:

The [Q] matrices of each layer are calculated:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Since their materials are the same, the [Q] matrices of all layers are the same.

From equation 2.7:

8. Classical Lamination Theory (CLT)

(Recall that the [Q] and [S] matrices do not depend on the fiber orientation angle.)

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Step 5:

From equatios 6.14;

For layers 1 and 3 : θ =48^o

For the 2nd layer: Since θ =0^o

8. Classical Lamination Theory (CLT)

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Step 6:

The necessary ones are calculated from the A], [B] and [D] matrices belonging to the entire structure.

For the reasons explained in Step 3: [B] = 0. The matrix [D] is non-zero, but there is no need to calculate it since it will not enter into operations. Then it is sufficient to just calculate the [A] matrix. (For option d, a separate explanation will be made.)

It is found as

8. Classical Lamination Theory (CLT)

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The strains and curvatures of the midplane are calculated:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Step 7:

From Equation 8.32

Or, since [B] = 0 and [D] will not enter into operations, the strains of the middle plane can be reduced to a 3x3 matrix multiplication as follows:

(It can also be found using a program such as Matlab, Excel etc.)

As explained in Step 3, the structure enters the 2nd special case and the curvatures of the middle plane become zero:

8. Classical Lamination Theory (CLT)

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Global strains are calculated:

Step 8:

From equation 8.13

Step 9:

Since they are independent of z, the global strains of all points are the same and equal to the strains of the midplane

Local strains are calculated:

The local stresses requested in the question on the lower surface of the upper layer are:

c:cos48^o , s:sin48^o

8. Classical Lamination Theory (CLT)

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Local stresses are calculated:

Step 10:

Since we have obtained local stresses and local deformations, we can move on to damage control..>>

If we apply Equation 6.1 for the bottom surface of the top layer;

As an alternative solution,

After doing the first 8 steps exactly ,

we could find,

global stresses in step 9,

local stresses in step 10,

the local strains n Step 11.

8. Classical Lamination Theory (CLT)

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No Damage Conditions :

a-) According to the Maximum Stress Criterion:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Control for the bottom surface of the top layer (z=-1):

X

Note: Even if the shear stress (𝛕_𝟏𝟐) is negative, it is taken as positive (+) in the equations. Because the direction does not matter in the shear strength of the material..

Previously calculated values

Material Properties Given in the Question

Since all inequalities cannot be provided

on the lower surface of the upper layer.

according to this criterion damage occurs

8. Classical Lamination Theory (CLT)

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b-) According to the Maximum Strains Criterion,

No Damage Conditions :

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Previously calculated values

Additionally, the strength limits for strains are:

Control for the bottom surface of the top layer (z=-1):

X

Since all inequalities cannot be provided

Note: Even if the shear deformation angle (𝛄_𝟏𝟐) is negative, it is taken as positive (+) in the equations. Because the direction does not matter in the shear strength of the material.

on the lower surface of the upper layer.

according to this criterion damage occurs

8. Classical Lamination Theory (CLT)

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No Damage Condition:

c-) According to the Tsai-Hill Criterion

Control for the bottom surface of the top layer (z=-1):

Previously calculated values

8. Classical Lamination Theory (CLT)

on the lower surface of the upper layer.

according to this criterion damage occurs

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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First of all, local stresses in the middle of the middle layer in the Q element must be determined.

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

• Since the structure is symmetrical, it is provided in the first special case and [B] = 0.
• Although there are internal moments in the Q element, since only the stresses

in the middle plane (z = 0) are asked, it enters the 4th special case and

there is no need to calculate the [D] matrix. (See article 8.7)

• Since the internal moments will not be taken into account only for the middle plane,

the intensities of the moments are not important. (We can see this situation by examining Equation 8.32.)

Which of the special cases does option d fall into?

In this case, the first 8 steps done before are also valid for option d.

Otherr Steps..>>

8. Classical Lamination Theory (CLT)

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Alternative to Step 9 :

Local Strains are calculated:

Local strains in the middle of the middle layer:

c:cos0^o , s:sin0^o

Local Stresses are calculated:

Alternative to Step 10 :

Local stresses in the middle of the middle layer:

8. Classical Lamination Theory (CLT)

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According to the Modified Tsai-Hill Criterion (damage control in the middle of the middle layer):

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

No Damage Condition :

Previously calculated values

No damage occurs on the middle surface (midplane) of the Middle Layer.

8. Classical Lamination Theory (CLT)

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Example 8.3: A layered composite elbow was obtained by bending 3 of the continuous fiber reinforced composite layers, the properties of which are given in the table, in an L shape and gluing them on top of each other with a 45⁰ / 0⁰ / 45⁰ fiber arrangement. The upper end of this elbow was connected to a fixed wall, a horizontal tensile force of F = 320kN was applied from a distance of h/2 from the upper surface and a torsional moment of T = 0.96kNm was applied to the lower free surface. Each of the layers is t = 2cm thick and h = 8cm wide. Accordingly, check whether damage will occur on the upper and lower surfaces of the middle layer under these loading conditions in this layered structure, according to the Tsai-Hill criterion.

8. Classical Lamination Theory (CLT)

On the lower surface of the Middle Layer:

On the upper surface of the Middle Layer:

Answers:

1.46 > 1

(according to the Tsai-Hill criterion.)

1 = 1

Damage occur

Damage occur

inequality 7.6

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

Example 8.4:

BCDE beam will be manufactured by placing a bidirectional woven-fabric material no. 2 between two unidirectional and continuous fiber reinforced layers no. 1 and 3. The beam will be connected to a bar AB as shown in the figure and supported by the H element. The beam will also be subjected to uniformly distributed load in its middle region (between C-D). Material properties are given in the table below. Layer 1 has a fiber orientation angle of 60⁰, layer 3 has a fiber orientation angle of 45⁰ and the thickness of both is t₁=t₃=10mm. Material number 2 will have a thickness of t₂ = 12mm and a fiber pair orientation of 0⁰ / 90⁰. In order to avoid damage to the structure under these conditions, determine the minimum value of b width according to the Tsa-Hill criterion.

1 and 3

2

8. Classical Lamination Theory (CLT)

Answer:

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• Sandwich composites are actually a special case of laminated structures, consisting of a total of three layers, and all calculations applicable to laminated composites can also be directly applied to sandwich composites.
• Sandwich composites are a type of laminated composite formed by bonding two stronger face sheets to the top and bottom surfaces of a core material.
• The components of a sandwich composite — the core and the reinforcing top and bottom layers — can each be made of isotropic materials, or they may themselves be different types of composite structures.�The core can be a solid block or have partially hollowed-out regions, such as honeycomb configurations.�For this reason, sandwich composites offer a wide range of possible configurations.

8.8 Sandwich Composites and Calculation Methods

• In sandwich composites, stress-strain calculations can be performed using the [A], [B], and [D] matrices explained in Topic 8, or by employing alternative approaches developed for laminated composites.

8. Classical Lamination Theory (CLT)

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8.9 Thermal Loading in Layered (laminated) Composites

(8.33)

We think that a layered composite structure is placed in a cavity that will prevent deformation in both x and y directions, and its temperature is increased by the amount ΔT.

Eq. (6.18.b)

Eq. (8.14)

Eq. (8.18)

Internal forces per unit length:

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Internal Forces per Unit Length due to temperature difference:

Internal Forces Per Unit Length Resulting from Structural Loads:

Total Internal Forces Per Unit Length Resulting from

Temperature Difference + Structural Loads:

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(8.34)

(8.36)

(8.37)

(8.35)

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Similarly, let's calculate the total internal moments :

Eq. (8.22)..>>

Internal moments per unit length:

If we substitute Equation (8.33) into Equation 8.22, we get

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(8.38)

Internal Moments per Unit Length due to temperature difference:

Internal Moments Per Unit Length Resulting from Structural Loads:

Total Internal Moments Per Unit Length Resulting from

Temperature Difference + Structural Loads:

(8.40)

(8.41)

(8.39)

(It is zero in symmetrical structures.)

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If Equations 8.34 and 8.41 are combined;

(8.42)

(8.43)

In this case, the strains and deformations of the middle plane are:

Equations 8.42 and 8.43 are general equations for layered composites, including the temperature effect.

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8.9.1 Steps for calculating Stresses and Strains (including temperature effect):

Other Steps are the same as described in topic 8. Only instead of equation 8.32, the operations start with equation 8.43.

The only difference with the calculations in topic 8 is that instead of structural internal loads, total internal loads (including temperature) will be used.

Now we will try to understand the subject better by solving an example..>>

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Example 8.6

3 of the layers in Example 7.3 are glued on top of each other with a 30⁰ / 0⁰ / 30⁰ fiber arrangement and placed in the same mold. Each layer is 200mm x 200mm x 2mm in size. By tightening the bolts a little, a compression force of -20kN was created on the lateral surfaces. Additionally, the temperature of the system was increased by 50 ⁰C. Calculate the local stresses arising on the lower surface of the middle layer, ignoring all friction.

Solution:

Material properties will be taken from example 7.3

a-) First of all, if we calculate the structural loads per unit length:

N_x = N_y = -20x10³ N / 200mm = -100N/mm

N_xy = 0 (Because frictions are neglected)

Step 1-) Total internal forces and moments per unit length are calculated.

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b-) Calculation of internal forces and moments due to temperature difference:

For the 1st and 3rd layers with θ =30⁰

[Q] matrix terms for all layers:

(It was calculated in Example 7.3)

Since θ =0⁰ for the middle layer:

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Let's code the structure..>>

Calculation of global thermal expansion coefficients for each layer :

Eq. 6.19.b

Eq. 6.9

for the 2nd (middle) layer, θ =0⁰

for the 1st and 3rd layer, θ =30⁰

Values given in Example 7.3 :

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We will calculate Thermal Stifness Matrices:

From eq. 8.35

Because the structure is symmetrical :

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From eq. 8.38

Internal Forces

per Unit Length

due to temperature difference:

Internal Moments

per Unit Length

due to temperature difference:

From eq. 8.34

Total Internal Forces Per Unit Length Resulting from

Temperature Difference and Structural Loads:

Total Internal Moments Per Unit Length Resulting from

Temperature Difference and Structural Loads:

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• Since the structure is symmetrical, it enters the 1st special case, therefore [B] =0

• Does the structure fall into the special situations in article 8.7?

Other Steps:

• The matrix [D] is non-zero. However, since the total internal moments per unit length are zero, the 2nd special case is achieved. For this reason, there is no need to calculate the [D] matrix as it will not enter into the calculations.

Then only matrix [A] will be calculated.

• Since the stresses in the mid-plane are not asked, the structure does not enter the 3rd special case.

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- We will calculate the strains and curvatures in the midplane from equation 8.43

Or, since [B] = 0 and [D] have no effect in the calculations, the deformations of the middle plane can be reduced to a 3x3 matrix multiplication as follows:

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- Global strains are calculated:

From eq. 8.13

Since the global strains of all points are independent of z, they are the same and equal to the strains of the middle plane.

- Local strains are calculated:

The local stresses on the lower surface of the middle layer, requested in the question are:

c:cos0^o , s:sin0^o

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- Local stresses are calculated :

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

8. Classical Lamination Theory (CLT)