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
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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