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Strength and Failure Analysis

10.

x

y

z

Equivalent Volume

of Laminated Structures Using the Zor Model

(To understand this section, please first review Chapter 9.)

Mechanics of Composite Materials- Lecture Notes (pdf and pptx files) / Mehmet Zor

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10.1 Objectives of This Chapter: To determine the tensile, compressive and shear strength limits of the Zor equivalent volume described in Section 9.4, and to perform stress-strain calculations and failure evaluations for different loading conditions.

n layered structure

Zor equivalent volume

1

i

2

n

z

y

x

?

3. While determining the strength limit of the equivalent volume for a loading type, the weakest layer that will fail first under that loading is taken as the basis. The equivalent stress that brings this weakest layer to its own strength limit is determined. This equivalent stress is accepted as the strength-limit value of the equivalent volume.

4. . In the Zor approach, failure evaluation is performed in terms of the equivalent volume of the structure. In this way, rather than local failures that may occur in the layers beforehand, it is determined whether the structure as a whole has failed or not. (In CLT, failure evaluation is performed for each layer.)

10.2 General Framework

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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y

x

i

y

x

i

1

i

2

n

F_x

z

y

x

k

F_x

1

2

i

n

k

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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(9.4.14)

(9.4.15)

;

y

x

i

(10.1.a)

(10.1.b)

(10.1.c)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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Substituting Equations (9.4.14 and 9.4.15) into Equations (10.1 a–c) and rearranging, we obtain:

(10.2.a)

(10.2.b)

(10.2.c)

Layer stress coefficients:

(10.3.a)

(10.3.b)

(10.3.c)

(10.3.d)

(10.3.e)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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Local stresses before failure in terms of the stress coefficients:

(10.4.a)

(10.4.b)

(10.4.c)

(10.5)

(10.6.a)

(10.6.b)

(10.6.c)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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10.3.2 Application of the Failure Criterion to the Layers

(10.7.a)

Substituting the local stresses in Equation (10.6), which are valid at the moment of failure, into Equation (10.7.a), we obtain:

(10.7.b)

(10.7.c)

(10.7.d)

(i =1,2, …n )

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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(10.8.a)

Substituting the local stresses in Equation (10.6), which are valid at the moment of failure, into Equation (10.8.a), we obtain:..>>

(i =1,2, …n )

(10.8.b)

(10.8.c)

(10.8.d)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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(10.9)

From Equation (7.9), the coefficients are:

According to the Tsai–Hill failure criterion: :

According to the Hoffman failure criterion:

According to the Mises-Hencky failure criterion:

(10.10.a)

(10.10.b)

(10.10.c)

(10.10.d)

(10.11.a)

(10.11.b)

(10.11.c)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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(10.13)

(10.12)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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(10.15)

(10.14)

(10.16)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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y

x

i

(10.1.a)

(10.1.b)

(10.1.c)

(10.17.a)

(10.17.b)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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Substituting Equations (10.17.a and 10.17.b) into Equations (10.1 a–c) and rearranging, we obtain:

Layer stress coefficients:

(10.18.a)

(10.18.b)

(10.18.c)

(10.19.a)

(10.19.b)

(10.19.c)

(10.19.d)

(10.19.e)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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Local stresses before failure in terms of the stress coefficients:

(10.20.a)

(10.20.b)

(10.20.c)

(10.21)

(10.22.a)

(10.22.b)

(10.22.c)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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10.4.2.a According to the Tsai–Hill Failure Criterion:

10.4.2.b According to the Modified Tsai–Hill Failure Criterion :

(10.24.a)

(10.24.b)

(10.23)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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(10.25)

(10.26)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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(10.27)

(10.28)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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y

x

(10.29)

(10.30)

(10.31)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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(10.34)

(10.32)

(10.33)

From the transformation Equations 6.3 a–c:

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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(10.35)

(10.7.a)

At the moment of failure

(10.36)

After calculations are performed for all layers, from Equation (10.29) :

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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(10.37)

At the moment of failure

(10.38)

(10.8.a)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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(10.9)

In the case of pure shear, at the moment of failure:

(10.39)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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(10.14)

In the case of pure shear,

at the moment of failure:

(10.40)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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10.8 Stress–Strain Calculations in the Zor Equivalent Volume :

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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Example 10.1

48⁰

0⁰

6mm

F_x

x

y

F_y

F_x

F_y

200mm

(The structure, load, and material properties in Example 8.2 were used.)

E₁	E₂	ν₁₂	G₁₂
(GPa)	(GPa)		(GPa)	(MPa)	(MPa)	(MPa)	(MPa)	(MPa)
81	30	0,35	15	101	180	25	50	12

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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Solution:

Step 2. Calculation of the Elastic Properties of the Zor Equivalent Volume:

First, the elastic properties of each layer with respect to the global axes are determined:

The cross-Poisson ratios are calculated from Equation (9.1):

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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From eqs. (9.4.51):

The cross-Poisson ratios are calculated from Equation (9.1):

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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In the Zor Model, for the case of pure shear, the shear modulus is obtained by the Voigt-type volumetric average.

From eq. (9.4.33):

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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Step 3: The Strength Limits of the Zor Equivalent Volume Are Calculated (According to the Modified Tsai–Hill Criterion):

a. Calculation of the Layer Stress Coefficients:

From eq. (10.3)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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From Eqs. (10.3)

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From Eqs. (10.19)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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From eq. (10.8.c), (i=1 , 3)

The equivalent stress that will cause failure of layers 1 and 3

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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From eq. (10.8.c) , (i=2)

According to Equation (10.8.d), the tensile strength of the equivalent volume in the x direction:

The equivalent stress that will cause failure of layer 2

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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Writing Equation (10.8.c) for the compression case :

The equivalent stress that will cause failure of layers 1 and 3

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

(i=1 , 3)

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the compressive strength of the equivalent volume in the x direction:

The equivalent compressive stress that will cause failure of layer 2:

Writing Equation (10.8.c) for the compression case : (i=2)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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From Eq. (10.24) , (i=1 , 3)

The equivalent tensile stress that will cause failure of layers 1 and 3

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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The equivalent tensile stress that will cause failure of layer 2

the tensile strength of the equivalent volume in the y direction:

From Eq. (10.24) , (i=2)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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The equivalent compressive stress that will cause failure of layers 1 and 3

Writing Equation (10.24.b) for the compression case (i=1 , 3)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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Writing Equation (10.24.b) for the compression case (i=2)

The equivalent compressive stress that will cause failure of layers 2

the compressive strength of the equivalent volume in the y direction

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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The equivalent shear stress that will cause failure of layers 1 and 3

From eq. (10.38)

From eq (10.31):

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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The equivalent shear stress that will cause failure of layer 2.

From eq. (10.38)

From eq. (10.31) :

From eq.(10.29):

Shear strength of the entire structure (equivalent volume)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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Step 4 – Calculation of the Stresses in the Equivalent Volume

6mm

x

y

200mm

6mm

200mm

x

y

Laminated Structure

Equivalent Volume and Forces

200mm

6mm

200mm

x

y

Equivalent Volume and Stresses

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes / Mehmet Zor

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Step 5 – Failure Check of the Equivalent Volume According to the Modified Tsai–Hill Criterion

For a single orthotropic layer, the no-failure condition of this criterion is written in the local 1–2 axes as given in Equations 7.7 or 10.8.a. The equivalent volume of the Zor model exhibits orthotropic behavior in the global axes. Therefore, for the Zor equivalent volume, this condition can be adapted to the global x–y coordinate system as shown in Equation (10.41).

Meaning of the values in the denominator:

(10.41)

Equivalent stresses calculated for this example:

In all cases:

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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6mm

200mm

x

y

200mm

6mm

x

y

b-) In order to compare the local stress results with the CLT results in Example 8.2, we must also calculate the local stresses occurring in the layers in the Zor Model solution. This is because CLT gives results on a layer basis. However, it should also be remembered that, in strength calculations or failure checks in the Zor Model, the equivalent volume is taken as the basis, and there is no need for the layer stress or strain. According to the principle of superposition, we can apply the external forces successively and calculate the local stresses occurring in each layer.

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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Total Local Stresses in the Layers and Comparisons with the CLT Results:

Local normal stress in the 1 direction:

Local normal stress in the 2 direction:

Total local stresses in layers 1 and 3

Shear stress in the 1–2 plane:

Total local stresses in layer 2

Local normal stress in the 1 direction:

Local normal stress in the 2 direction:

Shear stress in the 1–2 plane :

CLT

Zor

The Zor Model and CLT results are very close to each other. This is mainly because the structure in the example is symmetric and therefore the Poisson interactions remain at a low level. In asymmetric stacking sequences, the differences between the results of the methods become somewhat more pronounced because the Poisson interactions are greater. However, it should not be forgotten that the Zor Model and CLT are different approaches. In Example 9.2, the CLT and Zor Model results were also compared for an asymmetric stacking sequence.

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200mm

6mm

200mm

x

y

z

h

b

x

z

At the instant of failure :

(total bending moment that will bring the entire structure to the strength limit)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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Example 10.2-)

x

y

200mm

6mm

Solution)

Stresses occurring in the equivalent volume

200mm

6mm

200mm

x

y

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

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Calculation of the Stiffness Matrix of the Equivalent Volume:

(from Equation 9.4.50):

The material properties calculated in Example 10.1 are substituted into the matrix:

Compliance Matrix of the Equivalent Volume:

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The strains in Equivalent Volume:

The total elongation/shortening occurring in the equivalent volume :

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200mm

6mm

200mm

x

y

z

b

h

From Eq. (10.41)

Failure occurs.

b-)

10. Strength and Failure Analysis of Laminated Structures Using the Zor Model

Mechanics of Composite Materials- Lecture Notes (pdf and pptx files) / Mehmet Zor