Lecture 15 – Strain and stress in beams

Instructor: Prof. Marcial Gonzalez

Fall, 2020ME 323 –Mechanics of Materials

Reading assignment: 6.1—6.2

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Last modified: 8/22/20 11:23:31 PM

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Beam theory (@ ME 323)- Geometry of the solid body:

straight, slender member with constant cross sectionthat is design to supporttransverse loads.

- Kinematic assumptions: Bernoulli-Euler Beam TheoryTimoshenko 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)

2) find deformed configuration

Equilibrium of beams

Longitudinal Planeof Symmetry

(Plane of Bending)

Longitudinal Axis

deflection curve

J.L. Lagrange

J. Bernoulli L. EulerS.P. Timoshenko

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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.

Pure bending (i.e., and ):

Equilibrium of beams

Longitudinal Planeof Symmetry

(Plane of Bending)

Longitudinal Axis

- Plane of Bending

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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 .

Equilibrium of beams

- Plane of Bending

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Bernoulli-Euler beam theory- Kinematic assumptions: Strain-Displacement Equation

Equilibrium of beams

y

z

Neutral Surface

Undeformed Deformed underpure bending

Cross Section

Radius of curvature (+ or -) Curvature (+ or -)

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Bernoulli-Euler beam theory- Kinematic assumptions: Strain-Displacement Equation

Equilibrium of beams

y

z

Undeformed Deformed underpure bending

Cross Section

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.

When , the Bernoulli-Euler beam theory can be used if the beam is slender.

Neutral Surface

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Bernoulli-Euler beam theory- Kinematic assumptions: Strain-Displacement Equation

Equilibrium of beams

One half of the cross section is under longitudinal compression,the other half is under longitudinal tension.

When , the Bernoulli-Euler beam theory can be used if the beam is slender.

positivecurvature

negativecurvature

Exist but can be neglected in the derivation of the strain-displacement eqn.

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Bernoulli-Euler beam theory- Material behavior: isotropic linear elastic material; small deformations

Equilibrium of beams

Recall our initial assumptions:

positivecurvature

y

zNeutral Surface

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Bernoulli-Euler beam theory- Material behavior: isotropic linear elastic material; small deformations

Equilibrium of beams

Recall our initial assumptionsTherefore, the y-coordinate is measured from the centroid!!!!

positivecurvature

y

zNeutral Surface

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Bernoulli-Euler beam theory- Material behavior: isotropic linear elastic material; small deformations

Equilibrium of beams

Moment-curvature equation – Flexure formula

positivecurvature

y

zNeutral Surface

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Bernoulli-Euler beam theory - Summary

Equilibrium of beams

Moment-curvature equation – Flexure formula – In addition

Note: the y-coordinate is measured from the centroid!!!!

positivecurvature

negativecurvature

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Equilibrium of beams

Example 21: For a T-beam, determine the maximum tensile stress and the maximum compressive stress in the structure.

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Equilibrium of beams

Example 22 (review Statics notes): Determine the location of the centroid.Determine the moment of inertia.

Any questions?

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Equilibrium of beams