chapter 2 macromechanical analysis of a...

Chapter 2 Macromechanical Analysis of a Lamina Tsai-Hill Failure Theory

Dr. Autar Kaw

Department of Mechanical Engineering University of South Florida, Tampa, FL 33620

Courtesy of the Textbook

Mechanics of Composite Materials by Kaw

http://autarkaw.com/books/composite/index.html



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

( )ult12τ

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

−

−−

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


−

−−


( ) ,σ=σ ultT22

( )( ) 12231 =ultTσG+G

( ) ( ) ( )12222

222126

2135

2234321

3122132321

2231

2132


−

−−

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


−

−−


( )ultτ=τ 1212

( ) 12 2126 =τG ult

( ) ( ) ( )12222

222126

2135

2234321

3122132321

2231

2132


−

−−

[ ] [ ]

− 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

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

σ

( ) ( ) ( )12222

222126

2135

2234321

3122132321

2231

2132


−

−−

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.

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

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.


From Example 2.13,

S,1.714 = 1σ

S,2.714- = 2σ

4.165S.- = 12τ

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 σσ and MPa4 = xyτ

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

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

τσσσσ (2.151)

where

X1 = )( ultT1σ if σ1 > 0

= )( ultC1σ if σ1 < 0

X2 = )( ultT1σ if σ2 > 0

= )( ultC1σ if σ2 < 0

Y = )( ultT2σ if σ2 > 0

= )( ultC2σ if σ2 < 0

S = .)( ult12τ

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

×

×

×

×

×σσσσσ

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 σσ

|)/(| ultT22 σσ and |)/(| ult1212 ττ . 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 12τσσ |/Y| |,X/| 211 for the associated mode of failure.

1122

22

2

2

2

1

1

12

<

+

+

−

Sτ

Yσ

Xσ

Xσ

Xσ

( )( ) 0,

0,

11

111

<=

>=

σσ

σσ

if

ifX

ultC

ultT

( )( ) 0,

0,

21

212

<=

>=

σσ

σσ

if

ifX

ultC

ultT

( )( ) 0,

0,

22

22

<=

>=

σσ

σσ

if

ifY

ultC

ultT

( )ultS 12τ=

chapter 2 macromechanical analysis of a...

Documents