Advanced Strength of Materials
Advanced Strength of Materials CE 715
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This 17 page Class Notes was uploaded by Jermain Lindgren on Thursday October 15, 2015. The Class Notes belongs to CE 715 at North Carolina State University taught by Staff in Fall. Since its upload, it has received 19 views. For similar materials see /class/223741/ce-715-north-carolina-state-university in Civil Engineering at North Carolina State University.
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Date Created: 10/15/15
Advanced Strength of Materials 0 Lecture 6 000 no Review of Last Class Maximum shear stresses and directions not in text Static boundary conditions not in text 0 Octahedral normal and shear stresses Art 244 Mean and deviatoric stresses Art 245 Generalized Green strain de nition Art 27 1D and 2D physical interpretation of Green strains not in text Small deformation strain theoryengineering strain Art 28 Engineering Normal and z O Shearing Strains for 3D case 39 s 8y 2 sz 2 Ya sz y y These equations are applicable to small deformations small strains only 3 O O Generalized Definition of Strain 23 ds 2 ds2 28 de Ziayyaly2 262261122 2yxydxdy Zy dxdz Zyyzdydz Generalized Definition of Strain g Strains obtained from this generalized de nition are called Green strains Strain Tensor gig 8xx 8xy x2 8 gxy 8yy gyz 8x2 yz 22 a 1 a 2 6v 2 6w 2 s 77 7 7 7 a 6x 26x au 0v auau 0v0v 0w8w 777777 yxyii 0y 0x 0x0 0x0 0x0 Transformation of Strain Strain tensortransform the same way as stress tensor 2 2 2 9 re ll syym1 82an 29le 28 mn 28 nl yll yz 11 x211 VXY em smlllz syymlm2 822nan 8 11m2 2m1 8yzm1n2 mznl 8 lln2 lzn1 Principal Strains and Directions Determining principal strains and directions is an eigenvalue problem Principal Strains and Directions 53539 O O 0 O For nontrivial solution 8m M 8W 8W 8W 8W M 8 0 8x2 8yz 822 0 IO 0 Strain Invariants 55 0 i1 i2 7 81m 810 xz I3 81y 8 W 8 8 8 Strain Compatibility Relations Undeformed Body Deformation unacceptable Deform ation unacceptable Deform ation acceptable Strain Compatibility Equations In strength of materials theory line elements must deform in a manner which doesn t create voids cracks crossing of points etc This means there has to be an one to one mapping from undeformed to the deformed state This requirement can be expressed mathematically through the equations of compatibility Strain Compatibility Equations Consider 2D case an 6v au 0v E 1 8M 3 Eyy 6y yxy i q Ifu and v are known Eq 1 can be used to calculate s 2W and 74y Strain Compatibility Equations Say three strains 8 8 Jxy are known We have to determine displacements u and v Three known values But two unknowns and three equations We need to establish an additional condition to be imposed on strain components in order to ensure the existence of singlevalued continuous solution Strain Compatibility Equations 2D Case 61 626 6314 m 0x 9 0y2 0x0y2 6v 628 63v g 7 W 6y 6x2 6x26y au 0v 62 63 6311 77 y y 0x Bx y Bx yz 6x26y See text for strain compatibility equations for 3D case Eq 283 Strain Measurement Strain Rosettes 0 Strain gages measure normal strains along a direction on a surface 0 Todetermine the state of strain is syy yxy at a pomt we need three normal strain measurements These three gages measure ea 2 so during a test Strain Rosettes g 0 8b 8xx112 8yym12 7xyllm1 llCosa mlSina 85 12 9 m2 Solve for a 2W and 74y Steps in Stress Analysis Consider deformation produced by loads conduct experiments or make assumption based on intuition or experience Determine straindeformation relationship Determine stress distribution using stressstrain relation ofthe material Relate stresses to loads or stress resultants Relate loadsto displacements StressStrain Relations g Constitutive Equations 3 A structural analysis problem involves the kinetic variables forces and stresses equilibrium material independent the kinematic variables displacements and strains deformation geometry material independent For solving structural problems we need relationship between kinetic and kinematic variables constitutive relations or equations I StressStrain Relations 5 0 Constitutive Equations Constitutive equation of a material is dependent on the constituents ofthe material and hence referred to as constitutive equation This is basically the stressstrain relations Constitutive equations are developed based on the I StressStrain Relations 5352 I I O Constitutive Equations 6W 2 6U Work done Increase in by external internal forces energy Reading assignment Arts 31 and 311 33 StressStrain Relations O 0 Constitutive Equations It can show that there exist a strainenergy density function U0 U0 U0 gmggyywu yz an an m a W a etc 8 eyy 6U 1an 77 etc O O StressStrain Relations 535quot O Constitutive Equations 39 Physical meaning of strain energy density a 80 8 0 39 39 I StressStrain Relations 52 0 Elastic Plastic Inelastic 0390 quotquotquotquot quot Elastic 8390 8 Elastic relation for 0 s 00 StressStrain Relations 3332 sas Power Law 8 U Z a S 60 Elastic e 60 8 U a gt a 80 60 0 ElasticPlastic O 332 Power Law 33 0 O 0 0 quot 80 00 00 i E 80 60 0 8 StressStrain Relations RambergOsgood Fit 8 2 1 1 E 7 0y StressStrain Relations Constitutive law depends on the type of material Material ElasticAnisotropic Material 1 U 0 ECijsisj Engineering I shearing M W SEQ strain Cns C128 s C1 1 1 1 EC21 syy EC228y2y EC268yy7 1 1 1 EC318MEZ EC3zsyyszz C3sszyyz 1 1 1 EC613M7yz EC628yy7yz EC667Z ElasticAnisotropic Material 1 U0 ECijsiej aZlYO 68168 1 C1 C1 aZUO J J 68168 139 ElasticAnisotropic Material Most generalized case 6U 610 as 0 C118 C12 yy C13 zz C147xy C157xz C167yz 6U 0y g C128 C22 yy C23 zz C247xy C257xz C267yz yy we 622 g C138 C238 C3382 63934ny C35 722 36sz 66 axy C145 C24 yy C34 zz CAAyxy C45 yxz C457yz I1 we 6x2 W C158 C15 yy C3582 C4572 C557xz C567yz we a 7 Cl gxx C26 yy C36 zz C46yxy C56yxz C66yyz 2 6792 ElasticAnisotropic Material 0 C11 C12 C13 C14 C15 C16 8 0 C12 C22 C23 C24 C25 C26 SW 0 2 C13 C23 C33 C34 C35 036 an a C14 C24 C34 C44 C45 C46 yxy an C15 C25 C35 C45 C55 C56 y gyz C16 C26 C36 C46 C56 C66 yyz i 1 26 21 elastic constants 0 CHIS J 126 due to symmetry ElasticAnisotropic Material Example of symmetry in material properties Ci 2 C ji Symmetry Igt C12 2 C21