Beam Deflction Page A1

Using Tables for Formulas to Find Beam Deflection, y (pages 734 – 742)

General machine partKeep ymax below 0.0005 in/in of length Page 459Or check for interference with other machine partsCases not in appendix, use Roark (BB, Web Resources, must be on campus or VPN)

90-1, 9-6- Page 515Round, steel shaft, diameter 32mm, 700mm long3.0 kN loadSimply supported at ends – find deflection at center9-1 load in center9-6 load 175mm from left support

Page 734

9-1 load at center – figure (a)

ymax=−PL3

48 E I

9-6 load 175 mm from left support – figure (b)

y=−Pbx6 EIL

(L2−b2−x2)

Beam Deflction Page A2

Find the deflection at the load G due to beam bending. All other members are rigid.Steel, 20mm tall, 10 mm wide.

Page 735 figure (g) find yG

Find reactions at E and F

Draw free-body diagram

Beam Deflction – Statically Indeterminate using tables Page B1

Must use deformation properties to find reactions

Page 516 9-27Use A-25(f) with P = 35 kN, L = 4.0 m, a = 2.50 mFind reactionsDraw shear and moment diagramGive maximum V with x and maximum M with xGive maximum deflectionSelect material and cross-section shape for static loadingConfirm deflection less than L/360

Page 740

RA=Pb2

L3 (3a+b)

RC=Pa2

L3 (3b+a)

M A=−Pab2

L2 M B=2 Pa2b2

L3

MC=−Pa2bL2

Beam Deflction – Statically Indeterminate using tables Page B2

Check ymax E = 207 kN/mm2

ymax=−2Pa3b2

3EI (3a+b)2

Choose steel, assume structure, building Wshape, try Sy 296 MPA 1020 cold drawnFind S for M

Check V page 437

page 723 701 714

Beam Deflection - Superposition Page C1If loading condition and/or support condition aren’t given in the tables, use superposition.Method: break conditions into separate sketches of know table deflectionsPage 517, 9-54 Find deflection at each loadAlumunim 2014-T4, I = 16960 mm4

Find deflection at B due to B

A B C D

Find deflection at B due to C

Find deflection at C due to C

Find deflection at C due to B

300 900 600 600500

400

Page 734

x x

300 300500

x

600

600

400

x300 300

Deflection at B Deflection at C

Beam Deflection – Superposition – Statically Indeterminate Page D1If loading condition and/or support are statically indeterminate, use superposition.Method: break conditions into separate sketches of know table deflections Solve each sketch. Apply constraint conditions to the y values

Statically Indeterminate

Remove redundant reaction RB

Find y at location B

Remove redundant force PFind RB to return B to y=0 constraint

9-68 page 518- find all reactions – draw shear and moment diagrams.

1.60.8x

A

80kN0.8

B C D

Find y at C due to 80kN Page 734

1.61.6

A B C D

Find C to cause y

Find y at C due to 40kN/m Pg 735

1.61.6x

A

40kN/m

B C D

Page 734

Beam Deflection - Design Page E1

Recommended Deflection Limits (page 459)General machine parts

Moderate precision

High precision

Or use company standards or codesOr check for interference with other machine parts

Absent any recommendation use Where L = length between spans or total length

ymaxL

=0.000 5 ¿0.003ymaxL

=0.00001¿0.0005ymaxL

=0.000 001 ¿0.000 01

Check previous problems for precision

ymax<L

360

Page C1 check stress

Page A1

Page A2

Page C1Page B1 and B2

Beam Deflection – Successive Integration Page G1

∫V 0

V

dV=∫x0

x

−wdx

∫M 0

M

dM=∫x0

x

V dx

θEI=∫x0

x

Mdx

yEI=∫x0

x

θEIdx

θ = slope, y = deflection

9-82 (use E = 69 GPa, I = 1.79x107 mm4 EI = 1235 kNm)Find the maximum deflection in the beam.

1R

=d2 ydx2 =M

EI

∫Mdx=EI∫ d2 ydx2 dx

∫Mdx=EI∫ dydx

=EIθ

∫EI θ dx=EI∫ dydxdx=EIy

Pages 484-486

Find reactions and draw shear and moment diagrams

Beam Deflection – Successive Integration Page G2

AB, x = 0 m– 0.4 m

V=

∫M 0

M

dM=∫x0

x

V dx

M(x) =

θEI=∫Mdx

θEI ( x )=¿

yEI=∫θEIdx

yEI ( x )=¿

BC, x = 0.4 m– 1.2 m

V=

∫M 0

M

dM=∫x0

x

V dx

M(x) =

θEI=∫Mdx

θEI ( x )=¿

yEI=∫θEIdx

yEI ( x )=¿

Write equation and solve Find ymax where θEI=0Assume in BC and confirm

C1 = -10.58 C4= 0.22C3= -12.18 C5= -33.82

C6 = 8.94

CD, x = 1.2 m– 2 m

V=

∫M 0

M

dM=∫x0

x

V dx

M(x) =

θEI=∫Mdx

θEI ( x )=¿

yEI=∫θEIdx

yEI ( x )=¿

AB BC CD

Write equations and solve

Beam Deflection – Moment-area Method Page G1

Use Moment-Area Method for complex loading and/or changing cross sections

Page 498

Page 500

Beam Deflection – Moment-area Method Page G2

9-94

Beam Deflection – Moment-area Method Page G3