Lecture 19 – Deflection of Beams

Fall, 2021 ME 323 – Mechanics of Materials

Lecture 19 – Deflection of beams

Reading assignment: 7.1 – 7.4

Instructor: Prof. Marcial Gonzalez

Last modified: 8/16/21 9:20:26 AM Deflection of beams

Beam theory (@ ME 323) J. Bernoulli L. Euler

- Geometry of the solid body: Longitudinal Plane straight, slender member with of Symmetry constant cross section Longitudinal Axis that is design to support transverse 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

4 Deflection of beams

Why do we study deflection of beams? Lateral deflection of H/500

Design of jumping poles Design of fishing poles Atomic force microscopy (commercially available since 1989)

+ Solving statically indeterminate beams!! 5 Deflection of beams

Moment-curvature equation From Lecture 15:

Relationship between the deflection inclination and the inclination angle (~slope): angle (~slope)

deflection

Relationship between the slope and the radius of curvature :

Moment-curvature equation

Note: second-order, ODE 6 Deflection of beams

Load-deflection equation From Lecture 13:

Using the moment-curvature equation

Shear-deflection equation

Load-deflection equation Note: fourth-order, ODE

(constant cross-section and material properties)

Shear-deflection equation

Load-deflection equation Note: fourth-order, ODE 7 Deflection of beams

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.

(follow sign conventions)

8 Deflection of beams = |

= | Boundary conditions

(follow sign conventions)

9 Deflection of beams = |

= | Continuity conditions

10 Deflection of beams

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

Plus boundary conditions

Q: Maximum deflection?

Q: Slope at free end?

(follow sign conventions)

11 Deflection of beams

Any questions?