Lecture 15 Strain and stress in beams


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1 Spring, 2019 ME 323 Mechanics of Materials Lecture 15 Strain and stress in beams Reading assignment: News: Instructor: Prof. Marcial Gonzalez Last modified: 1/6/19 9:42:38 PM
2 Beam theory ME 323)  Geometry of the solid body: straight, slender member with constant cross section that is design to support transverse loads. J.L. Lagrange Longitudinal Axis Longitudinal Plane of Symmetry (Plane of Bending) S.P. Timoshenko  Kinematic assumptions: BernoulliEuler Beam Theory Timoshenko Beam Theory, etc.  Material behavior: isotropic linear elastic material; small deformations.  Equilibrium: 1) relate stress distribution (normal and shear stress) with internal resultants (only shear and bending moment) J. Bernoulli L. Euler 2) find deformed configuration deflection curve 3
3 Beam theory ME 323)  Geometry of the solid body: straight, slender member with constant cross section that is design to support transverse loads.  Kinematic assumptions: BernoulliEuler Beam Theory Timoshenko Beam Theory, etc. Longitudinal Axis Longitudinal Plane of Symmetry (Plane of Bending) Pure bending (i.e., and ):  Plane of Bending 4
4 BernoulliEuler beam theory  Kinematic assumptions: 1. the beam possesses a longitudinal plane of symmetry or plane of bending, and is loaded and supported symmetrically with respect to this plane; 2. there is a longitudinal plane perpendicular to the plane of bending that remains free of strain (i.e., ) as the beam deforms. This plane is called the neutral surface (NS) the neutral axis (NA) is the intersection of the NS with the cross section; 3. cross sections remain plane and perpendicular to the deflection curve of the deformed beam; 4. transverse strains (i.e., ) may be neglected in deriving an expression for the longitudinal strain.  Plane of Bending 5
5 BernoulliEuler beam theory  Kinematic assumptions: StrainDisplacement Equation y z Neutral Surface Cross Section Undeformed Deformed under pure bending Radius of curvature (+ or ) Curvature (+ or ) 6
6 BernoulliEuler beam theory  Kinematic assumptions: StrainDisplacement Equation y z Neutral Surface Cross Section Undeformed Deformed under pure bending When One half of the cross section is under longitudinal compression, the other half is under longitudinal tension. Exist but can be neglected in the derivation of the straindisplacement eqn., the BernoulliEuler beam theory can be used if the beam is slender. 7
7 BernoulliEuler beam theory  Kinematic assumptions: StrainDisplacement Equation positive curvature negative curvature When One half of the cross section is under longitudinal compression, the other half is under longitudinal tension. Exist but can be neglected in the derivation of the straindisplacement eqn., the BernoulliEuler beam theory can be used if the beam is slender. 8
8 BernoulliEuler beam theory  Material behavior: isotropic linear elastic material; small deformations positive curvature y Recall our initial assumptions: z Neutral Surface 9
9 BernoulliEuler beam theory  Material behavior: isotropic linear elastic material; small deformations positive curvature z Neutral Surface y Recall our initial assumptions Therefore, the ycoordinate is measured from the centroid!!!! 10
10 BernoulliEuler beam theory  Material behavior: isotropic linear elastic material; small deformations positive curvature y Momentcurvature equation Flexure formula z Neutral Surface 11
11 BernoulliEuler beam theory  Summary positive curvature negative curvature Momentcurvature equation Flexure formula In addition Note: the ycoordinate is measured from the centroid!!!! 12
12 Example 21: For a Tbeam, determine the maximum tensile stress and the maximum compressive stress in the structure. 13
13 Example 22 (review Statics notes): Determine the location of the centroid. Determine the moment of inertia. 14
14 Any questions? 15
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