Lecture 21 – Deflection of beams (cont.)

Instructor: Prof. Marcial Gonzalez

Fall, 2021ME 323 – Mechanics of Materials

Reading assignment: 7.1 – 7.4

Last modified: 10/29/21 2:15:51 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 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

Load-deflection equations

(constant cross-section and material properties)

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Deflection of beams

inclinationangle (~slope)

deflection

Shear-deflection eqn.

Load-deflection eqn.

Moment-curvature eqn.

(2nd order) (4nd order)

(follow sign conventions)

Boundary conditions

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Deflection of beams

(follow sign conventions)

= |

= |

Constrainedrotation end

>0

>0

Continuity conditions

7

Deflection of beams

>0

>0

= |

= |

8

Deflection of beams – Indeterminate problems

Example 33 (last lecture):The uniformly loaded beam shown in the figure is completely fixed at ends A and B. Determine an expression for the deflection curve usingthe second-order method.

8

40

2 384LEI

v pL=æ ö

ç ÷è ø 2

' 0v Læ öç ÷

ø=

è

2 2 3 40 0 0

1 1 1 124 12

( )24

p L x p Lx p xE

xI

v é= ù- +ê úë û

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Example 34 (Practice problem):The beam shown in the figure is completely fixed at ends A and B. Determine an expression for the deflection curve using the second-order method.

Deflection of beams – Indeterminate problems

2 3

3 2 2 3

, 0( )

, /

12 20 / 3

16 32 9 15 7

Lx x x LPEI L L x L x

vx x

xL L

- + £ £

- + - £ £

ìï= íïî

2

2

8 20 / 3

54 3

, 0'( )

10 7 , / 3

Lx x x LPEI L L x L

vx

xx L

- + £ £

- - £ £

ìï= í+ïî

Example 35:Determine an expression for the deflection curve using second order method. L

Deflection of beams – Indeterminate problems

Example 35 (cont.):Determine an expression for the deflection curve using second order method.

Deflection of beams – Indeterminate problems

( )2 2 3 40( ) 3 5 248wv x L Lx xEI

x= - + -

( )20 2 3'( ) 6 15 848

xwv x L Lx xEI

= - + -

L

Outline for 2nd order method (determinate or indeterminate):– FBD– Equilibrium for external forces and couples– Find internal moment 𝑀(𝑥) for each section– Integrate moment-curvature equation 𝐸𝐼𝑣!! 𝑥 = 𝑀(𝑥)– Apply boundary and continuity conditions– Solve for unknowns– Check units!

Deflection of beams – Indeterminate problems

Any questions?

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Equilibrium of beams