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Unformatted text preview: ME/EMA 508 ‘ ‘
Strength Theories R. E. Rowlands I All Rights Reserved TABLE OF CONTENTS 1. INTRODUCTION 2.ISYNOPSS OF STRENGTH THEORIES
2.1 Relevant'Equations
2.2 Overview
2.3 Brief History
2.4 Classification of Strength Criteria
2.5 Laminate Strength 75‘ TQEN‘GTH pair 751" ' .i .{a‘hﬁbl'iﬂt ‘ 3. ISOTROPIC CRITERIA 3.1 General Comments I 3.2 Maximum Normal Stress Theory
(Rankine) ' 3.3 Maximum Strain Theory (St. Venant)
3.4 Maximum Shear Stress Theory (Tresca) 3.5 Von Mises, .Octahedral Shear Stress or'
Distortional Energy Theory 4. ANISOTROPIC CRITERIA 4.1 General Comments 
4.2 Strength and Ultimate Strain
 . ConventiOn
4.3 Composite Strength Envelope
4.4, Maximum Strain Theory
4.5 Maximum Stress Theory
4.6 Hill Theory 
4.7_ Norris Theory 4. ANISOTROPIC CRITERIA — Cont. 4.1 General Comments 4.2 Strength and Ultimate Strain Convention
4.3 Composite Strength Envelope 4.4 Maximum Strain Theory , 4.5 Maximum Stress Theory 4.6 Hill Theory 4.7 Norris Theory 4.8. TsaiHill Theory 4.9 TsaiWu Theory 4.10 Cowin Theory 4.11 Comparison of Interactive Lamina Criteria
and Influence of F12 4.12 Importance of Sign of Shear Stress 4.13 Modes of Failure 5. LAMINATE STRENGTH
6. LAMINATE CRITERIA
7. CONCLUDING . COMMENTS REFERENCES Lamination
Theory ,
Constitutive
Properties Strength
Properties ' Structufal Environment.
Expectations (loads ,
deformations) Candidate Laminate
(ply construction, materials) Stress Analysis
I Strength Criterion Compare Predicted
Stresses Against:
Allowable Strengths
 Deciae What:
Constitutes Laminate
Failure If unacceptable ,‘
Kodify
Laminate Fig. 1 Laminate Design 2. SYNOPSIS OF STRENGTH TH EORIES 2.1 Relevant Equations 4 The stressstrain relationship for an orthotropic kth lamina in
state of plane stress (03 = 113 = 123 = 0 ) and parallel to its material dire Equation (1) can be inverted to yield Upon transforming Eq. (3) to the orthotropic laminate axes x, y, it
becomes {61? }: [THCw }’ {312 }: [T ]{exy} (5) a and {avianlion}, {em }=[ men} (6) Where is the transformation matrix From claSsical plate theory, the laminate strains sxy at any distance 2
from the middle surface are given by ‘ {a} = 80 + /2,
{M,2vy,my;Mx,My,MW}= fh/2(of,of,rfy;zaf,zoﬁ,zrg)dz (10> lnverting Eq. (8) ,
iv Lat1], [mp—[A413],
[B][A"l] : [D*]=[D][B][A’IIB] , [A'1=[A*]—[B*1[D*llc*1, [B']=[B*1[D*l], ' [A [C *]= [C']%[D*‘IIC*]» ‘ [D']=[D*1]‘. (14) P) Xi!) Y3) Y9!)
Zr, 20 Fig. 2 3D Strength Envelope Described in
Terms of stresses . Table 1 Brief History of Strength Criteria 1850 Maximum normal stress theory proposed 1870 St. Venant’s isotropic maximum strain theory 1920 . Jenkins extended maximum stress theory to wood 1921 Hankinson developed simple theory for wood 1928 Von Mises developed isotropic yield criterion 19391950 Norris’ theory for wood 1950 Hill extended Von Mises’ yield theory to anisotropy , ' 1960 Fischer developed theory for brittle composites 1961 StowellLiu developed theory based on
.micromechanics 1962 GriffithBaldwin developed a distortionalenergy
anisotropic strength criterion Table 1 Brief History of Strength Criteria — cont. 1965 Gol‘denblatKopnov developed a composite theory
independent of coordinate rotation 1965 Azzi—Tsai extended Hill's anisotropic yield criterion into a
composite strength theory V 1967 Hoffman extends TsaiHill theory to account for different tensile
and compressive strengths 1969 Petit—Waddoup developed nonlinear strain criterion 1971 TsaiWu developed criterion containing several mathematical features
1972 . Puppo—Evensen developed laminate criterion
1974 _ Sandhu developed total energy criterion 2.4 Classiﬁcation of Strength
Criteria Strength (failure) criteria can
' be categorized as follows7:  Stress Dominant
rStrain Dominant
 Interactive Examples of stressbased theories include:
0 Maximum Stress
' _TsaiHil  Cowin
 TsaiWu  :Fischer
0 Hoffman a  Norris Strainbased theories are
0 Maximum Strain o PetitWaddoups, while Saudhu’s'criterion is an energybased
theory. ' Examples of interactive theories are  Tsai—Hill  Tennyson
' TsaiWu ' Cowin
 Fischer .  Hoffman 10 > Most composite failure theories are
phenomological. They attempt to predict
through mathematical modeling the load level
to cause failure. A laminate is assumed to be
made up of individual, homogeneoUs and ortho
tropic plies. Discrete fibermatrix aspects are
typically not addressed. The mode of failure is
usually not predicted, although composite
failures tend to be characterized by one or more
of thefollowing: ° FiberDominated Failures
 MatrixDominated Failures
 FiberMatrix Failure 5. LAMINATE STRENGTH Having recognized several criteria for predicting
strength of individual plies, one can consider the failure
(strength) of laminates fabricated of individual, bonded
plies. Strength interrogation Of a laminate usually falls
into one ofthe following categories (Fig. 1):  Design    expected loads, deformations and
environment are known and one
evaluates candidate composites. 0 Analysis   there is an expected or existing ‘ design or component and one desires
to predict allowable loads,
deformations, etc. 11 Structural Environment. ' Expectations (loads.
deformations) Candidate. Laminate
(ply construction, materials) Lamination
If unacceptable, Theory,
Properties  1 Modify
Strength Laminate
Properties  Strength Criterion l
Compare. Predicted ‘
Stresses Against , Allowable Strengths —    Decide What: Constitutes Laminate Paili'zre Fig. 1 Laminate Design Awareness of laminate failure analysis raises the
following considerations:  FirstPly Failure
 Behavior after FirstPly Failure Recognizing that a laminate design based on first
ply failure is excessively conservative, one must
then consider (mathematically in the scheme 'of
Fig. 1) the damage/consequences of any failed
plies. Approaches include:  TotalPly Failure
 Ply Failure Distribution 12 . Isotropic Theories  3.2 Maximum Normal Stress
Theory (Rankine)38 This theOry states that the largest normal
stress remains less than the uniaxial
strength X of the material. This can be
written as c <x (15) _ p
Neither linearity nor elasticity is assumed, no
‘ interaction between stress components is
involved and prediction is independent of
cq or or. 13 P) Kit) YZ) Y3;
' ' Zr, 20 Fig. 2 3D Strength Envelope Described in
Terms of stresses ‘ 3.3. Maximum Strain Theory (St. ,
~ Venant)38 ' Failure predicted to occur under any state of
stress when the maximumstrain at the
point reaches some critical value for the
material as determined from a simple
uniaxial test. Since criterion is usually
uSed to predict allowable stress, the
concept normally assumes‘linearelastic response. 14 9&0 X,’, Y2) Y7:
Zr, 20 b Fig. 2 3D Strength Envelope Described in
Terms of stresses 3.4 Maximum Shear Stress
Theory (Tresca)38 Failure'by yielding is predicted to occur
when the maximum shear stress at the
'point reaches its critical value as 
obtained from a simple uniaxial test. Under
plane stress, the theory states yielding will
occur if the following is violated lap0",] < 8y (16)
where Sy is the yield stress under uniaxial
loading. 15 l 3.5 Von Mises or Distortional
Energy Theory38 Nothing that the hydrostatic state of stress changes only the volume
and not the shape of an isotropic material, Von Mises postulated that
yielding occurs due to the energy of distortion. This criterion can be written
as (op(5‘1)2 + (sq(5,)2 + (a V (17) Under planestress, cr = 0, the above becomes Although energy based, it is written in terms of stress. The theory
‘ provides for Interaction between different stress components. Several
orthotropic failure criteria of this same (general form. Equation (18) describes an envelope of Fig. 2 which egenerates to an ellipse in the op  oq plane and whose center is at the coordinate origin. i 4.2 Strength and Ultimate Strain
' . Conventions Tensile strengths in the directions of
material symmetry are denoted by X, Y and 3,
while positive ultimate strains are .
correspondingly denoted by XE, Y£ and SE.
Normal strengths (X’,Y’) and ultimate strains (X8,
Y8) in compression are denoted as positive
quantities. This is common usage, although
these represent negative states of stress and strain. Uniaxial strengths and ultimate strains are
defined in Figs. 2 and 3. 16 Fig. 3 Sign
Convention for
Strengths and
Ultimate Strains 4.3 Composite Strength
Envelopes Laminates are usually subjected'to plane stress.
Moreover, principal directions of stress and material
symmetry need not coinCide in anisotropic materials.
These reasons make it convenient to describe strength
envelopes of 'orthotropic materials in terms of inplane 1
2 axes of material symmetry, and associated shear
stress r12. Figure 4a illustrates the enclosed strength
envelope for an orthotropic material under plane stress
(as = :13 = 123 = 0), whereas Fig. 4b shows two
_ dimensional projections at representative levels of shear v
stress :12 = 1A, 123 and TC. 17 Fig. 4 Strength Envelope
for Orthotropic Material
under PlaneStress $60) xmb yz’) Fig. 2 3D Strength Envelope Described. in
Terms of stresses 18 4.4 MaximumStrain Theory This is an extension of St. Venant’s maximum principal ’
strain theory (isotropysection 3.3) to orthotropy. For
orthotropic lamina, the strain. components are referred to
the principal material axes; therefore, three Strain
components can exist in the criterion. Linearelasticity is
normally assumed to failure such that the criterion
becomes a strength prediction in. terms of applied
stresses. A ply of a laminate is considered to have
failed when either its longitudinal, transverse or shear
strain reaches a limiting value as determined from
simple onedimensional stress experiments. As with
other laminabasedstrength criteria, the minimum
common envelope of the superposition of the failure
diagrams of all, individual plies related to the global axes
pfllthe laminate represents the initiation of laminate ai ure. Maximum Strain Theory  cont. Under
planestress, the criterion predicts failure occurs when
any one or more of the following is violated: 'X’£<'£1 < XE
“Y’s < 52 < r;
IY12I < Se Where X£(X’E) are the ultimate tensile (compressive)
strains in the 1 direction, Fig. 3 yeti/’8) are the ultimate tensile (compression)
strains in the 2direction, and S8 is the ultimate inplane shear strain relative
to the 12 material axes. 19 Fig. 4 Strength Envelope
for Orthotropic Material
under PlaneStress (—‘(m x,’ , Y2) Y3’)
Zr, 20 Fig. 2 3D Strength Envelope Described in
Terms of stresses ' 20 4.5 Maximum Stress (0mm)
v , Theory Jenkins extended the concept of the maximum normal stress
theory to predict the strength of planar orthotropic materials such as
wood. Stresses are resolved in directions of material symmetry, and
failure is postulated to occur when one or more of these stresses
attains a respective limiting value. This criterion states failure will not
occur as long as the the following all prevail. X’<c1<x,Y’<oz<Y,r12'<S (20) X, X’, Y, Y’ and S are the respective uniaxial tensile and
compressive normal and shear strengths in the inplane directions of
material symmetry, Fig. 3. For this criterion, the failure surface of
Fig. 4' becomes a rectangular box of height 28 and whose ﬂat sides
are parallel to the coordinate axes of material symmetry 01, c .
There are no interactions among stresses or the modes ofgfailure
with this theory, and no stressstrain relationship is assumed. Fig. 4 Strength Envelope
for Orthotropic Material
under PlaneStress 21 6 vs ’8 >
max max
Consider a unidirectionally reinforced material subjected to uniaxial tension 0' at some angle 6 to the ﬁbers and using the transformation of Eqs. (6), the
maximal allowable loading according to this prediction is the smallest of the following:
X ' Y S ‘ (21)
0': 2 , a: _ 2 or 0'=—_————
cos 6 sm 6 511160036 For comparison, if strength were predicted according to the maximum strain
theory of Section 4.4, then the expressions corresponding to Eqs. (21)
' become: a=__.__X a———————Y or 0'=———S (22) . a _ . .
cos2 6 — v12 sm2 6 sm2 6 — v21 cos2 6 3m 6 cos 6 S or a = —,—————
sm6cos6 Y S
. 0'=——————.——,0'=—.————— or a=,———— 22
cos2 6 —v12 sm2 6 sm2 6 —V21 cos2 6 sm 6cos6 ( ) The only difference between these maximum
stress and maximum strain prediction is the
inclusion of the Poisson’s ratio terms in E.q. (22). Like the maximumstrain criterion, the
popularity of this maximumstress theory tends
to be based as much on simplicity of concept
and ease of use as on validity or rationale. 22 The unidirectional strength of omax theory of Eq.: (21) is plotted in Fig. 5 for an offaxis ’
I unidirectional glassepoxy composite. Figure 6 is a similar plot of the smax theory of Eq. (22). Off
axis angle theta is that between the loading
direction and the direction of material symmetry.
These predictions are based on the properties of
Table 2. The lines of Figs. 5 and 6 are the
predictions. The lowest or solid lines govern.
Neither of the maximumstress (Fig. 5) nor
maximumstrain (Fig. 6) criteria adequately
predicts the experimental results represented _
here by solid symbols. 0‘ u an (S ‘0 75 0° 8 Fig. 5 MaximumStress Predicted Strength
of OffAxis Unidirectional GlassEpoxy, 23 Fig. 6 MaximumStrain Predicted Strength of Offaxis Unidirectional GlassEpoxy
Composite Table 2 Mechanical Properties of Unidirectional GlassEpoxy
Composite, Ref. 39 ' E11 = 53.7. GPa
E22 = 17.9 GPa
u12= 0.25 . '
G12 = 9 GPa
X = 1GPa
Y = 0.028 GPa
S = 0.04 GPa
X’ = 1 GPa
Y’ = 0.14 GPa 24 ...
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This note was uploaded on 08/08/2008 for the course ME 508 taught by Professor Rowlands during the Spring '05 term at University of Wisconsin.
 Spring '05
 ROWLANDS
 Composite Materials

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