Lecture 18 – Deflection of beams

Instructor: Prof. Marcial Gonzalez

Spring, 2020ME 323 – Mechanics of Materials

Reading assignment: 7.1-7.4

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Last modified: 2/7/20 8:57:26 AM

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

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

- Kinematic assumptions: Bernoulli-Euler Beam Theory

- 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

Deflection of beams

Longitudinal Planeof Symmetry

Longitudinal Axis

J. Bernoulli L. Euler

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Why do we study deflection of beams?

Deflection of beams

Atomic force microscopy (commercially available since 1989)

Design of jumping poles Design of fishing poles

Lateral deflection of H/500

+ Solving statically indeterminate beams!!

Moment-curvature equationFrom Lecture 15:

Relationship between the deflectionand the inclination angle (~slope):

Relationship between the slopeand the radius of curvature :

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Deflection of beams

inclinationangle (~slope)

deflection

Moment-curvature equation

Note: second-order, ODE

Load-deflection equationFrom Lecture 13:

Using the moment-curvature equation

(constant cross-section and material properties)

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Deflection of beams

Shear-deflection equation

Note: fourth-order, ODELoad-deflection equation

Shear-deflection equation

Load-deflection equation Note: fourth-order, ODE

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Example 29:The uniformly loaded beam shown in the figure is completely fixed at end B. Determine an expression for the deflection curve .(a) Use the second-order method.

Deflection of beams

(follow sign conventions)

Boundary conditions

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Deflection of beams

(follow sign conventions)

= |

= |

Continuity conditions

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Deflection of beams

>0

>0

= |

= |

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Example 29 (cont.):The uniformly loaded beam shown in the figure is completely fixed at end B. Determine an expression for the deflection curve .

Deflection of beams

Plus boundary conditions

Q: Maximum deflection?

Q: Slope at free end?(follow sign conventions)

Any questions?

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Deflection of beams