Beam Design Formulas with Shear and Moment Diagrams

NATIONAL DESIGN SPECIFICATIONFOR WOOD CONSTRUCTION

American Wood Council

AmericanForest &

PaperAssociation

N D S

2005 EDITION

ANSI/AF&PA NDS-2005Approval Date: JANUARY 6, 2005

WITH COMMENTARY AND

SUPPLEMENT: DESIGN VALUES FOR WOOD CONSTRUCTION

ASD/LRFD

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BEAM DESIGN FBEAM DESIGN FBEAM DESIGN FBEAM DESIGN FBEAM DESIGN FORMULASORMULASORMULASORMULASORMULASWITH SHEAR AND MOMENTWITH SHEAR AND MOMENTWITH SHEAR AND MOMENTWITH SHEAR AND MOMENTWITH SHEAR AND MOMENTDIADIADIADIADIAGRAMSGRAMSGRAMSGRAMSGRAMS

AmericanAmericanAmericanAmericanAmerican

FFFFForest &orest &orest &orest &orest &

PPPPPaperaperaperaperaper

AssociationAssociationAssociationAssociationAssociation

w

R R

V

V

2 2

Shear

Mmax

Moment

x

DESIGN AID NDESIGN AID NDESIGN AID NDESIGN AID NDESIGN AID Nooooo. 6. 6. 6. 6. 6

AMERICAN WOOD COUNCIL

The American Wood Council (AWC) is part of the wood products group of theAmerican Forest & Paper Association (AF&PA). AF&PA is the national tradeassociation of the forest, paper, and wood products industry, representing membercompanies engaged in growing, harvesting, and processing wood and wood fiber,manufacturing pulp, paper, and paperboard products from both virgin and recycledfiber, and producing engineered and traditional wood products. For more informationsee www.afandpa.org.

While every effort has been made to insure the accuracyof the information presented, and special effort has beenmade to assure that the information reflects the state-of-the-art, neither the American Forest & Paper Associationnor its members assume any responsibility for anyparticular design prepared from this publication. Thoseusing this document assume all liability from its use.

Copyright 2007American Forest & Paper Association, Inc.

American Wood Council1111 19th St., NW, Suite 800

Washington, DC 20036202-463-4713

[email protected]

BEAM FBEAM FBEAM FBEAM FBEAM FORMULAS WITHORMULAS WITHORMULAS WITHORMULAS WITHORMULAS WITHSHEAR AND MOMENTSHEAR AND MOMENTSHEAR AND MOMENTSHEAR AND MOMENTSHEAR AND MOMENT

DIADIADIADIADIAGRAMSGRAMSGRAMSGRAMSGRAMS

AMERICAN FOREST & PAPER ASSOCIATION

Figures 1 through 32 provide a series of shearand moment diagrams with accompanying formulasfor design of beams under various static loadingconditions.

Shear and moment diagrams and formulas areexcerpted from the Western Woods Use Book, 4thedition, and are provided herein as a courtesy ofWestern Wood Products Association.

IntrIntrIntrIntrIntroductionoductionoductionoductionoduction Notations Relative to Shear and MomentDiagrams

E = modulus of elasticity, psiI = moment of inertia, in.4L = span length of the bending member, ft.R = span length of the bending member, in.M = maximum bending moment, in.-lbs.P = total concentrated load, lbs.R = reaction load at bearing point, lbs.V = shear force, lbs.W = total uniform load, lbs.w = load per unit length, lbs./in. = deflection or deformation, in.x = horizontal distance from reaction to point

on beam, in.

List of FigurList of FigurList of FigurList of FigurList of Figureseseseses

Figure 1 Simple Beam Uniformly Distributed Load ................................................................................................ 4Figure 2 Simple Beam Uniform Load Partially Distributed..................................................................................... 4Figure 3 Simple Beam Uniform Load Partially Distributed at One End .................................................................. 5Figure 4 Simple Beam Uniform Load Partially Distributed at Each End ................................................................ 5Figure 5 Simple Beam Load Increasing Uniformly to One End .............................................................................. 6Figure 6 Simple Beam Load Increasing Uniformly to Center .................................................................................. 6Figure 7 Simple Beam Concentrated Load at Center ............................................................................................... 7Figure 8 Simple Beam Concentrated Load at Any Point .......................................................................................... 7Figure 9 Simple Beam Two Equal Concentrated Loads Symmetrically Placed....................................................... 8Figure 10 Simple Beam Two Equal Concentrated Loads Unsymmetrically Placed .................................................. 8Figure 11 Simple Beam Two Unequal Concentrated Loads Unsymmetrically Placed .............................................. 9Figure 12 Cantilever Beam Uniformly Distributed Load ........................................................................................... 9Figure 13 Cantilever Beam Concentrated Load at Free End .................................................................................... 10Figure 14 Cantilever Beam Concentrated Load at Any Point .................................................................................. 10Figure 15 Beam Fixed at One End, Supported at Other Uniformly Distributed Load ............................................. 11Figure 16 Beam Fixed at One End, Supported at Other Concentrated Load at Center ........................................... 11Figure 17 Beam Fixed at One End, Supported at Other Concentrated Load at Any Point ..................................... 12Figure 18 Beam Overhanging One Support Uniformly Distributed Load ............................................................... 12Figure 19 Beam Overhanging One Support Uniformly Distributed Load on Overhang ......................................... 13Figure 20 Beam Overhanging One Support Concentrated Load at End of Overhang ............................................. 13Figure 21 Beam Overhanging One Support Concentrated Load at Any Point Between Supports ........................... 14Figure 22 Beam Overhanging Both Supports Unequal Overhangs Uniformly Distributed Load ......................... 14Figure 23 Beam Fixed at Both Ends Uniformly Distributed Load........................................................................... 15Figure 24 Beam Fixed at Both Ends Concentrated Load at Center .......................................................................... 15Figure 25 Beam Fixed at Both Ends Concentrated Load at Any Point .................................................................... 16Figure 26 Continuous Beam Two Equal Spans Uniform Load on One Span ....................................................... 16Figure 27 Continuous Beam Two Equal Spans Concentrated Load at Center of One Span ................................. 17Figure 28 Continuous Beam Two Equal Spans Concentrated Load at Any Point ................................................ 17Figure 29 Continuous Beam Two Equal Spans Uniformly Distributed Load ....................................................... 18Figure 30 Continuous Beam Two Equal Spans Two Equal Concentrated Loads Symmetrically Placed ............. 18Figure 31 Continuous Beam Two Unequal Spans Uniformly Distributed Load ................................................... 19Figure 32 Continuous Beam Two Unequal Spans Concentrated Load on Each Span Symmetrically Placed ..... 19


w

R R

V

V

2 2

Shear

Mmax

Moment

x

7-36 A

a b c

x

R1 R2

V1

V2

Shear

a + R1w

Mmax

Moment

wb

7-36 B

Figure 1 Simple Beam Uniformly Distributed Load

Figure 2 Simple Beam Uniform Load Partially Distributed


R1 R2

V1

V2Shear

x

R1w1

M max

Moment

a b c

w2cw1a

7-37 B

x

R1 R2

V2

V1

Mmax

Moment

Shear

R1w

a

wa

7-37 A

Figure 3 Simple Beam Uniform Load Partially Distributed at One End

Figure 4 Simple Beam Uniform Load Partially Distributed at Each End


xW

2

R R

V

VShear

Mmax

Moment

2

7-38 B

W

R1 R2

V1

V2Shear

Mmax

Moment

x

.57741

7-38 A

Figure 5 Simple Beam Load Increasing Uniformly to One End

Figure 6 Simple Beam Load Increasing Uniformly to Center


x P

a

R1 R2

V1

V2Shear

M max

Moment

b

7-39-b

x P

2

R R

V

VShear

M max

Moment

2

7-39 A

Figure 7 Simple Beam Concentrated Load at Center

Figure 8 Simple Beam Concentrated Load at Any Point

bshowaltRectangle

bshowaltPlaced Image


x P

a

R1 R2

V1

V2 Shear

M1

Moment

P

b

M2

7-40 B

x P

a

R R

V

Shear

Mmax

Moment

P

a

7-40 A

V

Figure 9 Simple Beam Two Equal Concentrated Loads Symmetrically Placed

Figure 10 Simple Beam Two Equal Concentrated Loads UnsymmetricallyPlaced


a

R1 R2

V1

V2 Shear

M 1

Moment

b

M2

7-41-a

xP1 P2

Figure 11 Simple Beam Two Unequal Concentrated Loads UnsymmetricallyPlaced

Figure 12 Cantilever Beam Uniformly Distributed Load

x

R

VShear

Moment

w

Mmax

7-41- B


x

a

V Shear

Moment

b

Mmax

7-42-b

P

R

x

R

V

Shear

Moment

Mmax

P

7-42 A

Figure 13 Cantilever Beam Concentrated Load at Free End

Figure 14 Cantilever Beam Concentrated Load at Any Point


R1

R2

V1

V2Shear

M1

2

M2

311

7-43 B

2

x P

w

R1

7-43 A

R2

V1

V2

4

Shear

M1

Mmax

38

x

Figure 15 Beam Fixed at One End, Supported at Other Uniformly DistributedLoad

Figure 16 Beam Fixed at One End, Supported at Other Concentrated Load atCenter


a b

R1

V2Shear

x

PaR2

M 1

Moment

R2

V 1

P

M2

7-44 A

Figure 17 Beam Fixed at One End, Supported at Other Concentrated Load atAny Point

Figure 18 Beam Overhanging One Support Uniformly Distributed Load

x1

Shear

M2

M1

R1

(1 a2)

Moment

V 3

V 2

(1 a2)2 2

2

x

a

w( + a)

R2

V1

7-44 B


V2

Mmax

x1x

a

Moment

Shear

R1 R2

V 1

P

7-45 B

V2

Mmax

x1x

a

wa

Moment

Shear

R1 R2

V1

7-45 A

Figure 19 Beam Overhanging One Support Uniformly Distributed Load onOverhang

Figure 20 Beam Overhanging One Support Concentrated Load at End ofOverhang


R1 R2

w

a b c

V2V1

V4V3

Mx1

M2M1

M3

X1

X

7-46 B

x

R1 R2

x1

a b

V1

V2

Mmax

Shear

Moment

7-46 A

Figure 21 Beam Overhanging One Support Concentrated Load at Any PointBetween Supports

Figure 22 Beam Overhanging Both Supports Unequal Overhangs UniformlyDistributed Load


x

V

V

2 2

Shear

Moment

R R

Mmax

PPP

Mmax

4

7-47 B

x

V

V

2 2

Shear

M1

Moment

w

R R

Mmax

.2113

7-47 A

Figure 23 Beam Fixed at Both Ends Uniformly Distributed Load

Figure 24 Beam Fixed at Both Ends Concentrated Load at Center


Shear

V1

R3R2

167

M1

V2

Moment

Mmax

x

7-48 B

R1

V3

w

x

Moment

PPP

V2

Ma

M2

R2

V1

M1

7-48 A

R1

Shear

a b

Figure 25 Beam Fixed at Both Ends Concentrated Load at Any Point

Figure 26 Continuous Beam Two Equal Spans Uniform Load on One Span


Shear

V1

R3R2

M1

MomentMmax

PPP

V3V2

R1

ba

7-49 B

Shear

V1

R3R2

M1Moment

PPP

V3V2

R1

2 2

7-49 A

Mmax

Figure 27 Continuous Beam Two Equal Spans Concentrated Load at Centerof One Span

Figure 28 Continuous Beam Two Equal Spans Concentrated Load at AnyPoint


7-50 A

R1 R3R2

3/8

0.4215 0.4215

M1

w w

max

V2V3

M2

3/8

V1V2

V1

R3R2

PPP

V2

R1

PPP

a a a a

Mx

x

V2V3

M1

M2

Figure 29 Continuous Beam Two Equal Spans Uniformly Distributed Load

Figure 30 Continuous Beam Two Equal Spans Two Equal Concentrated LoadsSymmetrically Placed


R1 R3R2

x1

Mx2

M1

V1V4

V3

V2

1 2

1 2w w

x2

Mx1

V1

R3R2

P

V3

R1

P

a a b b

Mm1

V4

M1

Mm2

21

V2

P2P1

Figure 31 Continuous Beam Two Unequal Spans Uniformly Distributed Load

Figure 32 Continuous Beam Two Unequal Spans Concentrated Load on EachSpan Symmetrically Placed

A F & P AAmerican FAmerican FAmerican FAmerican FAmerican Forest & Porest & Porest & Porest & Porest & Paper Associationaper Associationaper Associationaper Associationaper AssociationAmerican WAmerican WAmerican WAmerican WAmerican Wood Councilood Councilood Councilood Councilood Council1111 19th S1111 19th S1111 19th S1111 19th S1111 19th Streetreetreetreetreet, NWt, NWt, NWt, NWt, NWSuitSuitSuitSuitSuite 800e 800e 800e 800e 800WWWWWashingtashingtashingtashingtashington, DC 20036on, DC 20036on, DC 20036on, DC 20036on, DC 20036Phone: 202-463-4Phone: 202-463-4Phone: 202-463-4Phone: 202-463-4Phone: 202-463-4777771111133333FFFFFax: 202-463-2ax: 202-463-2ax: 202-463-2ax: 202-463-2ax: 202-463-2797979797911111aaaaawwwwwcinfcinfcinfcinfcinfo@[email protected]@[email protected]@afandpa.orgwwwwwwwwwwwwwww.a.a.a.a.awwwwwc.orgc.orgc.orgc.orgc.org 11-07

BEAM DESIGN FORMULAS WITH SHEAR AND MOMENT DIAGRAMSCopyrightIntroductionList of FiguresSimple BeamFigure 1 Simple Beam Uniformly Distributed LoadFigure 2 Simple Beam Uniform Load Partially DistributedFigure 3 Simple Beam Uniform Load Partially Distributed at One EndFigure 4 Simple Beam Uniform Load Partially Distributed at Each EndFigure 5 Simple Beam Load Increasing Uniformly to One EndFigure 6 Simple Beam Load Increasing Uniformly to CenterFigure 7 Simple Beam Concentrated Load at CenterFigure 8 Simple Beam Concentrated Load at Any PointFigure 9 Simple Beam Two Equal Concentrated Loads Symmetrically PlacedFigure 10 Simple Beam Two Equal Concentrated Loads Unsymmetrically PlacedFigure 11 Simple Beam Two Unequal Concentrated Loads Unsymmetrically Placed

Cantilever BeamFigure 12 Cantilever Beam Uniformly Distributed LoadFigure 13 Cantilever Beam Concentrated Load at Free EndFigure 14 Cantilever Beam Concentrated Load at Any Point

Beam Fixed at One End, Supported at OtherFigure 15 Beam Fixed at One End, Supported at Other Uniformly Distributed LoadFigure 16 Beam Fixed at One End, Supported at Other Concentrated Load at CenterFigure 17 Beam Fixed at One End, Supported at Other Concentrated Load at Any Point

Beam Overhanging One SupportFigure 18 Beam Overhanging One Support Uniformly Distributed LoadFigure 19 Beam Overhanging One Support Uniformly Distributed Load on OverhangFigure 20 Beam Overhanging One Support Concentrated Load at End of OverhangFigure 21 Beam Overhanging One Support Concentrated Load at Any Point Between Supports

Beam Overhanging Both SupportsFigure 22 Beam Overhanging Both Supports Unequal Overhangs Uniformly Distributed Load

Beam Fixed at Both EndsFigure 23 Beam Fixed at Both Ends Uniformly Distributed LoadFigure 24 Beam Fixed at Both Ends Concentrated Load at CenterFigure 25 Beam Fixed at Both Ends Concentrated Load at Any Point

Continuous BeamFigure 26 Continuous Beam Two Equal Spans Uniform Load on One SpanFigure 27 Continuous Beam Two Equal Spans Concentrated Load at Center of One SpanFigure 28 Continuous Beam Two Equal Spans Concentrated Load at Any PointFigure 29 Continuous Beam Two Equal Spans Uniformly Distributed LoadFigure 30 Continuous Beam Two Equal Spans Two Equal Concentrated Loads Symmetrically PlacedFigure 31 Continuous Beam Two Unequal Spans Uniformly Distributed LoadFigure 32 Continuous Beam Two Unequal Spans Concentrated Load on Each Span Symmetrically Placed

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Beam Design Formulas with Shear and Moment Diagrams