EML 4230 Introduction to Composite Materials

Chapter 2 Macromechanical Analysis of a Lamina

Tsai-Hill Failure Theory

Dr. Autar KawDepartment of Mechanical Engineering

University of South Florida, Tampa, FL 33620

Courtesy of the TextbookMechanics of Composite Materials by Kaw

Strength Failure Theories for an Angle Lamina

The failure theories are generally based on the normal and shear strengths of a unidirectional lamina.

In the case of a unidirectional lamina, the five strength parameters are:

Longitudinal tensile strength Longitudinal compressive strength Transverse tensile strength Transverse compressive strength In-plane shear strength

ultTσ1

ultCσ1

ultTσ 2

ultCσ 2

ult12

Tsai-Hill Failure Theory

Based on the distortion energy theory, Tsai and Hill proposed that a lamina has failed if:

This theory is based on the interaction failure theory.

The components G1 thru G6 of the strength criteria depend on the strengths of a unidirectional lamina.

12222

222126

2135

2234321

3122132321

2231

2132

< τG+τG+τG+σσGσσGσσGσG+G+σG+G+σG+G

Components of Tsai-Hill Failure Theory

Apply to a unidirectional lamina, then the lamina will fail. Hence, Equation reduces to:

,σ=σ ultT11

12132 ultTσG+G

12222

222126

2135

2234321

3122132321

2231

2132

< τG+τG+τG+σσGσσGσσGσG+G+σG+G+σG+G

Components of Tsai-Hill Failure Theory

Apply to a unidirectional lamina, then the lamina will fail. Hence, Equation reduces to:

,σ=σ ultT22

12231 ultTσG+G

12222

222126

2135

2234321

3122132321

2231

2132

< τG+τG+τG+σσGσσGσσGσG+G+σG+G+σG+G

Components of Tsai-Hill Failure Theory

Apply to a unidirectional lamina, and assuming that the normal tensile failure strength is the same in direction (2) and (3), then the lamina will fail. Hence, Equation reduces to:

,σ=σ ultT23

12221 ultTσG+G

12222

222126

2135

2234321

3122132321

2231

2132

< τG+τG+τG+σσGσσGσσGσG+G+σG+G+σG+G

Components of Tsai-Hill Failure Theory

Apply to a unidirectional lamina, then the lamina will fail. Hence, Equation reduces to

ultτ=τ 1212

12 2126 =τG ult

12222

222126

2135

2234321

3122132321

2231

2132

< τG+τG+τG+σσGσσGσσGσG+G+σG+G+σG+G

Components of Tsai-Hill Failure Theory

2

12

2

112

21

)σ()σ(=G

ultT

ultT

)σ(=G

ultT1

221

21

)σ(=G

ultT1

231

21

)τ(=G

ult1226

121

12132 ultTσG+G

12231 ultTσG+G

12221 ultTσG+G

12 2126 =τG ult

Tsai-Hill Failure Theory – Plane Stress

Because the unidirectional lamina is assumed to be under plane stress - that is, , = τ = τ = σ 023313

1)()()()( 12

122

2

22

21

21

1

12

ult

ultT

ultT

ultT

ττ+

σσ+σσ

σσ

12222

222126

2135

2234321

3122132321

2231

2132

< τG+τG+τG+σσGσσGσσGσG+G+σG+G+σG+G

Tsai-Hill Failure Theory

Unlike the Maximum Strain and Maximum Stress Failure Theories, the Tsai-Hill failure theory considers the interaction among the three unidirectional lamina strength parameter.

The Tsai-Hill Failure Theory does not distinguish between the compressive and tensile strengths in its equation. This can result in underestimation of the maximum loads that can be applied when compared to other failure theories.

Tsai-Hill Failure Theory underestimates the failure stress because the transverse strength of a unidirectional lamina is generally much less than its transverse compressive strength.

Example

Find the maximum value of S>0 if a stress of SSS xyyx 4 and,3,2

is applied to a 60o lamina of Graphite/Epoxy. Use Tsai-Hill Failure Theory. Use properties of a unidirectional Graphite/Epoxy lamina given in Table 2.1 of the textbook Mechanics of Composite Materials by Autar Kaw.

FIGURE 2.33Off-axis loading in the x-direction

From Example 2.13,

S,1.714 = 1

S,2.714- = 2

4.165S.- = 12

Using the Tsai-Hill failure theory from Equation (2.150),

1 < 1068

4.165S- + 1040

2.714S- + 101500

2.714S- 101500

1.714S - 101500

1.714S6

2

6

2

666

2

10.94.MPa < S

a) The Tsai-Hill failure theory considers the interaction between the three unidirectional lamina

strength parameters, unlike the Maximum Strain and Maximum Stress failure theories.

b) The Tsai-Hill failure theory does not distinguish between the compressive and tensile strengths in

its equations. This can result in underestimation of the maximum loads that can be applied when

compared to other failure theories. For the load of MPa3- = 2MPa, = yx and MPa4 = xy

as found in Examples 2.15, 2.17 and 2.18, the strength ratios are given by

Example

SR = 10.94 (Tsai-Hill failure theory),

= 16.33 (Maximum Stress failure theory),

= 16.33 (Maximum Strain failure theory).

Tsai-Hill failure theory underestimates the failure stress because the transverse tensile

strength of a unidirectional lamina is generally much less than its transverse compressive

strength. The compressive strengths are not used in the Tsai-Hill failure theory. The

Tsai-Hill failure theory can be modified to use corresponding strengths, tensile or

compressive, in the failure theory as follows

1 < S

+ Y

+ XX

- X

122

22

2

2

2

1

1

12

(2.151)

where

X1 = )( ultT1 if σ1 > 0

= )( ultC1 if σ1 < 0

X2 = )( ultT1 if σ2 > 0

= )( ultC1 if σ2 < 0

Example 2.18

Y = )( ultT2 if σ2 > 0

= )( ultC2 if σ2 < 0

S = .)( ult12

For Example 2.18, the modified Tsai-Hill failure theory given by Equation (2.151) now gives

1 < 1068

4.165- + 10246

2.714- + 101500

2.714- 101500

1.714 - 101500

1.7146

2

6

2

666

2

σ < 16.06 MPa.

which implies that the strength ratio is SR = 16.06 (modified Tsai-Hill failure theory)

This value is closer to the values obtained using Maximum Stress and Maximum Strain failure

theories.

c) The Tsai-Hill failure theory is a unified theory and hence does not give the mode of failure like

the Maximum Stress and Maximum Strain failure theories.

However, you can make a reasonable guess of the failure mode by calculating |,)/(| ultT11

|)/(| ultT22 and |)/(| ult1212 . The maximum of these three values gives the associated mode

of failure. In the modified Tsai-Hill failure theory, calculate the maximum of

|/S| and 12 |/Y| |,X/| 211 for the associated mode of failure.

Example 2.18

Modified Tsai-Hill Failure Theory

1122

22

2

2

2

1

1

12

Sτ

Yσ

Xσ

Xσ

Xσ

0,

0,

11

111

if

ifX

ultC

ultT

0,

0,

21

212

if

ifX

ultC

ultT

0,

0,

22

22

if

ifY

ultC

ultT

ultS 12

END