strengthparti - ME/EMA 508 ‘ ‘ Strength Theories R E...

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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.ISYNOPS|S 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. Tsai-Hill Theory 4.9 Tsai-Wu 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 stress-strain 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? }: [THC-w }’ {312 }: [T ]{exy} (5) a and {avian-lion}, {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 Lat-1], [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 3-D 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 1939-1950 Norris’ theory for wood 1950 Hill extended Von Mises’ yield theory to anisotropy , ' 1960 Fischer developed theory for brittle composites 1961 Stowell-Liu developed theory based on .micromechanics 1962 Griffith-Baldwin developed a distortional-energy anisotropic strength criterion Table 1 Brief History of Strength Criteria — cont. 1965 Gol‘denblat-Kopnov 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 Tsai-Hill theory to account for different tensile and compressive strengths 1969 Petit—Waddoup developed nonlinear strain criterion 1971 Tsai-Wu 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 stress-based theories include: 0 Maximum Stress ' -_Tsai-Hi|l - Cowin - Tsai-Wu - :Fischer 0 Hoffman a - Norris Strain-based theories are 0 Maximum Strain o Petit-Waddoups, while Saudhu’s'criterion is an energy-based theory. ' Examples of interactive theories are - Tsai—Hill - Tennyson '- Tsai-Wu ' 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 fiber-matrix 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: ° Fiber-Dominated Failures - Matrix-Dominated Failures - Fiber-Matrix 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: - First-Ply Failure - Behavior after First-Ply 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: - Total-Ply 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 3-D 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‘linear-elastic response. 14 9&0 X,’, Y2) Y7: Zr, 20 b Fig. 2 3-D 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 lap-0",] < 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 plane-stress, 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 in-plane 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 Plane-Stress $60) xmb yz’) Fig. 2 3-D Strength Envelope Described. in Terms of stresses 18 4.4 MaximumStrain Theory This is an extension of St. Venant’s maximum principal ’ strain theory (isotropy-section 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. Linear-elasticity 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 one-dimensional stress experiments. As with other lamina-basedstrength 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 plane-stress, 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 2-direction, and S8 is the ultimate in-plane shear strain relative to the 1-2 material axes. 19 Fig. 4 Strength Envelope for Orthotropic Material under Plane-Stress (—‘(m x,’ , Y2) Y3’) Zr, 20 Fig. 2 3-D 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 in-plane 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 stress-strain relationship is assumed. Fig. 4 Strength Envelope for Orthotropic Material under Plane-Stress 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 maximum-strain criterion, the popularity of this maximum-stress 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 off-axis ’ I unidirectional glass-epoxy 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 maximum-stress (Fig. 5) nor maximum-strain (Fig. 6) criteria adequately predicts the experimental results represented _ here by solid symbols. 0‘ u an (S ‘0 75 0° 8 Fig. 5 Maximum-Stress Predicted Strength of Off-Axis Unidirectional Glass-Epoxy, 23 Fig. 6 Maximum-Strain Predicted Strength of Off-axis Unidirectional Glass-Epoxy Composite Table 2 Mechanical Properties of Unidirectional Glass-Epoxy 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.

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strengthparti - ME/EMA 508 ‘ ‘ Strength Theories R E...

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