Spring, 2019 ME 323 Mechanics of Materials Lecture 15 Strain and stress in beams Reading assignment: 6.1 6.2 News: Instructor: Prof. Marcial Gonzalez Last modified: 1/6/19 9:42:38 PM
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: Bernoulli-Euler 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
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: Bernoulli-Euler Beam Theory Timoshenko Beam Theory, etc. Longitudinal Axis Longitudinal Plane of Symmetry (Plane of Bending) Pure bending (i.e., and ): - Plane of Bending 4
Bernoulli-Euler 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
Bernoulli-Euler beam theory - Kinematic assumptions: Strain-Displacement Equation y z Neutral Surface Cross Section Undeformed Deformed under pure bending Radius of curvature (+ or -) Curvature (+ or -) 6
Bernoulli-Euler beam theory - Kinematic assumptions: Strain-Displacement 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 strain-displacement eqn., the Bernoulli-Euler beam theory can be used if the beam is slender. 7
Bernoulli-Euler beam theory - Kinematic assumptions: Strain-Displacement 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 strain-displacement eqn., the Bernoulli-Euler beam theory can be used if the beam is slender. 8
Bernoulli-Euler beam theory - Material behavior: isotropic linear elastic material; small deformations positive curvature y Recall our initial assumptions: z Neutral Surface 9
Bernoulli-Euler beam theory - Material behavior: isotropic linear elastic material; small deformations positive curvature z Neutral Surface y Recall our initial assumptions Therefore, the y-coordinate is measured from the centroid!!!! 10
Bernoulli-Euler beam theory - Material behavior: isotropic linear elastic material; small deformations positive curvature y Moment-curvature equation Flexure formula z Neutral Surface 11
Bernoulli-Euler beam theory - Summary positive curvature negative curvature Moment-curvature equation Flexure formula In addition Note: the y-coordinate is measured from the centroid!!!! 12
Example 21: For a T-beam, determine the maximum tensile stress and the maximum compressive stress in the structure. 13
Example 22 (review Statics notes): Determine the location of the centroid. Determine the moment of inertia. 14
Any questions? 15