beam design formulas with shear and moment
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NATIONAL DESIGN SPECIFICATION®
FOR 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
awcinfo@afandpa.orgwww.awc.org
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.4
L = 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.Δ = 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
AMERICAN WOOD COUNCIL
w
R R
V
V
2 2
Shear
Mmax
Moment
x
7-36 A
�
�
� �
a b c
x
R1 R2
V1
V2
Shear
a + —R1
w
Mmax
Moment
wb
7-36 B
�
Figure 1 Simple Beam – Uniformly Distributed Load
Figure 2 Simple Beam – Uniform Load Partially Distributed
AMERICAN FOREST & PAPER ASSOCIATION
R1 R2
V1
V2Shear
x
R1—w1
M max
Moment
a b c
w2cw1a
7-37 B
�
x
R1 R2
V2
V1
Mmax
Moment
Shear
R1—w
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
AMERICAN WOOD COUNCIL
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
AMERICAN FOREST & PAPER ASSOCIATION
x P
a
R1 R2
V1
V2
Shear
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
AMERICAN WOOD COUNCIL
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
AMERICAN FOREST & PAPER ASSOCIATION
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
�
�
AMERICAN WOOD COUNCIL
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
AMERICAN FOREST & PAPER ASSOCIATION
R1
R2
V1
V2Shear
M1
2
M2
3—11
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
AMERICAN WOOD COUNCIL
a b
R1
V2Shear
x
Pa—R2
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
�
�
��
��
AMERICAN FOREST & PAPER ASSOCIATION
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
AMERICAN WOOD COUNCIL
R1 R2
w
a b c
V2V1
V4
V3
Mx1
M2
M1
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
AMERICAN FOREST & PAPER ASSOCIATION
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
AMERICAN WOOD COUNCIL
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
AMERICAN FOREST & PAPER ASSOCIATION
Shear
V1
R3R2
M1
MomentMmax
PPP
V3V2
R1
ba
7-49 B
� �
Shear
V1
R3R2
M1Moment
PPP
V3
V2
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
AMERICAN WOOD COUNCIL
7-50 A
R1 R3R2
3�/8
0.4215 0.4215
M1
w w
maxΔ
V2
V3
� �
��
��
M2
3�/8
V1
V2
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
AMERICAN FOREST & PAPER ASSOCIATION
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
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