Mechanics of Materials Instructor: Dr. Botao Lin & Dr. Wei Liu Page 1 Chapter 10 Strain Transformation 2015 Mechanics of Materials Foil gauge LVDT

Mechanics of Materials Instructor: Dr. Botao Lin & Dr. Wei Liu Page 2 Chapter 10 §10.1 Plane Strain 2015 Mechanics of Materials  Normal strain—produced by changes in length  Shear strain—produced by the relative rotation of two adjacent sides of an element

Mechanics of Materials Instructor: Dr. Botao Lin & Dr. Wei Liu Page 3 Chapter 10 §10.1 Plane Strain 2015 Mechanics of Materials  Question:  What is the physical meaning of γ xy ?

Mechanics of Materials Instructor: Dr. Botao Lin & Dr. Wei Liu Page 4 Chapter 10 §10.1 Plane Strain 2015 Mechanics of Materials Two dimensional geometric deformation of an infinitesimal material element

Mechanics of Materials Instructor: Dr. Botao Lin & Dr. Wei Liu Page 5 Chapter 10 §10.1 Plane Strain 2015 Mechanics of Materials Differences between plane stress and plane strain Plane stress Plane strain σ x, σ y & τ xy ε x, ε y & γ xy

Mechanics of Materials Instructor: Dr. Botao Lin & Dr. Wei Liu Page 6 Chapter 10 §10.2 General Equations of Plane Strain Transformation 2015 Mechanics of Materials  Sign convention of strains  Normal strains ε x, ε y are positive when causing elongation along x and y axes  Shear strain γ xy is positive if the interior angle becomes smaller than 90 º  Similar to the stress transformation, the strains at a specific orientation x’ y’ can be determined as: Compare (10- 5),(10-6),(10- 7) with (9- 1),(9-2) & (9-3)

Mechanics of Materials Instructor: Dr. Botao Lin & Dr. Wei Liu Page 7 Chapter 11 Design of Beams and Shafts 2015 Mechanics of Materials Beams are important structural members that are used to support roof and floor loadings

Mechanics of Materials Instructor: Dr. Botao Lin & Dr. Wei Liu Page 8 Chapter 11 §11.1 Basis for Beam Design 2015 Mechanics of Materials  Design on the basis of strength  A beam is designed such as to resist both shear and bending stresses  Requires the use of the shear and flexure formulas Whenever large shear loads occur on a beam it is important to use stiffeners such as at A, in order to prevent any localized failure such as crimping of the beam flanges Pre-assumptions: 1.Homogeneous 2.Linear-elastic

Mechanics of Materials Instructor: Dr. Botao Lin & Dr. Wei Liu Page 9 Chapter 11 §11.1 Basis for Beam Design 2015 Mechanics of Materials The internal shear V and moment M are developed from a parabolic shear distribution and a linear normal-stress distribution Element 1 and 5 are subjected only to the max. normal stress, Element 3 is subjected only to the max. shear stress Elements 2 and 4--both normal and shear stresses

Mechanics of Materials Instructor: Dr. Botao Lin & Dr. Wei Liu Page 10 Chapter 11 §11.2 Prismatic Beam Design 2015 Mechanics of Materials  Section modulus  The ratio of I and c  Is used for a bending design S>S req,d A handful of wide-flange dimensions are available from a handbook Usually the smallest cross section area is chosen—made of less material and more economical Assuming that the tension and the compression result in the same allowable bending stress— ”Bauschinger effect” Check with

Mechanics of Materials Instructor: Dr. Botao Lin & Dr. Wei Liu Page 11 Chapter 11 §11.2 Prismatic Beam Design 2015 Mechanics of Materials Manufactured steel beam Steel plate girders Wood beams