Lecture 11 – Equilibrium of beams

Instructor: Prof. Marcial Gonzalez

Spring, 2020ME 323 – Mechanics of Materials

Reading assignment: 5.1—5.3

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Last modified: 2/7/20 8:57:25 AM

Exam 1- Wednesday February 12th , 8:00-10:00 p.m., room LYNN 1136

(please arrive 15 minutes before the exam and bring a picture ID)

- Formula sheet will be provided (and uploaded to Blackboard during the weekend)

- Those of you who require extra accommodation please inform me before Friday 7th, 9:30 a.m.

- Start working on the lecture book!

Announcements

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https://www.purdue.edu/freeform/me323/

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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 (Lecture 15)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)

2) find deformed configuration

Equilibrium of beams

Longitudinal Planeof Symmetry

Longitudinal Axis

J. Bernoulli L. EulerS.P. Timoshenko

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Equilibrium – Free body diagrams- Types of beams

Equilibrium of beams

Simply supported beam Cantilever beam

Overhanging beam

All configurations are statically determinate

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Equilibrium – Free body diagrams- Types of beams

Equilibrium of beams

Two-spanned continuous beam

Propped cantilever beam

All configurations are statically indeterminate

Q: How can we solve forequilibrium in this case?

A: We use the displacement at thesupports as additional information!

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Equilibrium – Free body diagrams- Stress & Internal resultants

Equilibrium of beams

Free body diagram

Internal resultants at cross section C

Distribution of stresses and Internal resultants at cross section C

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Equilibrium – Free body diagrams- Stress & Internal resultants

Equilibrium of beams

Distribution of stresses and Internal resultants at cross section C

Internal resultants at cross section C

Transverse shear force:

Bending moment:

Sign Convention

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Example 19:Determine expressions for and

Reactions:

Shear and bending diagrams

Sign Convention

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Equilibrium relationships- Shear force and bending

moments diagrams

Equilibrium of beams

Change in shear force from x1 to x2 is equal to the area under the load curve from 1 to 2

Change in bending moment from x1 to x2 is equal to the area under shear force curve from 1 to 2

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Sign Convention

Example 19 (cont.):Draw shear force and bending moment diagrams

Shear and bending diagrams

Equilibrium relationships- Sign convention!

Equilibrium of beams

These are all positive external loads and couples

Note: become familiar with sign convention for external loads and for internal reactions.11

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Any questions?

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Equilibrium of beams