Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler General Plane-Strain Transformation Normal strains are positive if they cause elongation along the x and y axes, respectively the shear strain is positive if the interior angle AOB becomes smaller then 90 o.
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler General Equations of Plane-Stress Transformation Normal & Shear Strains
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler Principal Strains and Maximum In-Plane Shear Strain Principal Strains Maximum In-Plane Shear Strain
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler Mohr’s Circle—Plane Strain Plane strain transformation is able to have a graphical solution that is easy to remember. Above equations represent the equation of Mohr’s circle for strain. It has a center on the ε-axis at point C(ε avg,0) and a radius R.
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler Absolute Maximum Shear Strain If the principal in-plane strains cause elongations If one of the principal in-plane strain is of opposite sign to the other principal in-plane strain, so that ε 1 causes elongation and ε 2 causes contraction
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler Material-Property Relationships Generalized Hooke’s Law If the material at a point is subjected to a state of triaxial stress (Fig.a), associated normal strains will be developed in the material. The stresses can be related to these strains by using the principle of superposition, Poissons ratio,ε lat =-vε long, and Hooke’s law, as it applies in the uniaxial direction, ε= σ/E.
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler Material-Property Relationships (cont...) If we apply shear stress Ƭ xy to the element, (Fig.a), the material will deform only due to a shear strain ɣ xy ; that is Ƭ xy will not cause other strains in the material.Likewise Ƭ yz and Ƭ xz will only cause shear strains ɣ yz and ɣ xz. Hooke’s law for shear stress and shear strain can be written as
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler Relationship Involving E,v and G The modulus of elasticity E is related to the shear modulus G by G= E/2(1+v) By Hooke’s law In case of pure shear (σ x =σ y =σ z =0)
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler Dilatation and Bulk Modulus When an elastic material is subjected to normal stress, its volume will change, therefore δV= (1+ε x ) (1+ε y ) (1+ε z ) dx dy dz – dx dy dz neglecting the products of the strains as they are very small: δV= (ε x +ε y +ε z ) dx dy dz Volumetric strain or the dilatation can be written as Volumemodulus of elasticity or the bulk modulus is
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler
Copyright ©2011 by Pearson Education, Inc. All rights reserved. Mechanics of Materials, Eighth Edition Russell C. Hibbeler