Mechanical Beam Analysis

1. NATIONAL DESIGN SPECIFICATION®
FOR WOOD CONSTRUCTION
American Wood Council
American
Forest &
Paper
Association
N D S
2005 EDITION
®
ANSI/AF&PA NDS-2005
Approval Date: JANUARY 6, 2005
WITH COMMENTARY AND
SUPPLEMENT: DESIGN VALUES FOR WOOD CONSTRUCTION
ASD/LRFD
AmericanWAmericanWAmericanWAmericanWAmericanWoodCounciloodCounciloodCounciloodCounciloodCouncil
BEAM DESIGN FBEAM DESIGN FBEAM DESIGN FBEAM DESIGN FBEAM DESIGN FORMULASORMULASORMULASORMULASORMULAS
WITH SHEAR AND MOMENTWITH SHEAR AND MOMENTWITH SHEAR AND MOMENTWITH SHEAR AND MOMENTWITH SHEAR AND MOMENT
DIADIADIADIADIAGRAMSGRAMSGRAMSGRAMSGRAMS
AmericanAmericanAmericanAmericanAmerican
FFFFForest &orest &orest &orest &orest &
PPPPPaperaperaperaperaper
AssociationAssociationAssociationAssociationAssociation
w
R R
V
V
2 2
Shear
Mmax
Moment
x
ᐉ
ᐉ
ᐉ ᐉ
DESIGN AID NDESIGN AID NDESIGN AID NDESIGN AID NDESIGN AID Nooooo. 6. 6. 6. 6. 6

2. AMERICAN WOOD COUNCIL
The American Wood Council (AWC) is part of the wood products group of the
American Forest & Paper Association (AF&PA). AF&PA is the national trade
association of the forest, paper, and wood products industry, representing member
companies engaged in growing, harvesting, and processing wood and wood fiber,
manufacturing pulp, paper, and paperboard products from both virgin and recycled
fiber, and producing engineered and traditional wood products. For more information
see www.afandpa.org.
While every effort has been made to insure the accuracy
of the information presented, and special effort has been
made to assure that the information reflects the state-of-
the-art, neither the American Forest & Paper Association
nor its members assume any responsibility for any
particular design prepared from this publication. Those
using this document assume all liability from its use.
Copyright © 2007
American Forest & Paper Association, Inc.
American Wood Council
1111 19th
St., NW, Suite 800
Washington, DC 20036
202-463-4713
awcinfo@afandpa.org
www.awc.org
BEAM FBEAM FBEAM FBEAM FBEAM FORMULAS WITHORMULAS WITHORMULAS WITHORMULAS WITHORMULAS WITH
SHEAR AND MOMENTSHEAR AND MOMENTSHEAR AND MOMENTSHEAR AND MOMENTSHEAR AND MOMENT
DIADIADIADIADIAGRAMSGRAMSGRAMSGRAMSGRAMS

3. AMERICAN FOREST & PAPER ASSOCIATION
Figures 1 through 32 provide a series of shear
and moment diagrams with accompanying formulas
for design of beams under various static loading
conditions.
Shear and moment diagrams and formulas are
excerpted from the Western Woods Use Book, 4th
edition, and are provided herein as a courtesy of
Western Wood Products Association.
IntrIntrIntrIntrIntroductionoductionoductionoductionoduction Notations Relative to “Shear and Moment
Diagrams”
E = modulus of elasticity, psi
I = 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 ................................................................................................ 4
Figure 2 Simple Beam – Uniform Load Partially Distributed..................................................................................... 4
Figure 3 Simple Beam – Uniform Load Partially Distributed at One End.................................................................. 5
Figure 4 Simple Beam – Uniform Load Partially Distributed at Each End ................................................................ 5
Figure 5 Simple Beam – Load Increasing Uniformly to One End .............................................................................. 6
Figure 6 Simple Beam – Load Increasing Uniformly to Center.................................................................................. 6
Figure 7 Simple Beam – Concentrated Load at Center ............................................................................................... 7
Figure 8 Simple Beam – Concentrated Load at Any Point.......................................................................................... 7
Figure 9 Simple Beam – Two Equal Concentrated Loads Symmetrically Placed....................................................... 8
Figure 10 Simple Beam – Two Equal Concentrated Loads Unsymmetrically Placed .................................................. 8
Figure 11 Simple Beam – Two Unequal Concentrated Loads Unsymmetrically Placed .............................................. 9
Figure 12 Cantilever Beam – Uniformly Distributed Load........................................................................................... 9
Figure 13 Cantilever Beam – Concentrated Load at Free End .................................................................................... 10
Figure 14 Cantilever Beam – Concentrated Load at Any Point .................................................................................. 10
Figure 15 Beam Fixed at One End, Supported at Other – Uniformly Distributed Load ............................................. 11
Figure 16 Beam Fixed at One End, Supported at Other – Concentrated Load at Center ........................................... 11
Figure 17 Beam Fixed at One End, Supported at Other – Concentrated Load at Any Point ..................................... 12
Figure 18 Beam Overhanging One Support – Uniformly Distributed Load ............................................................... 12
Figure 19 Beam Overhanging One Support – Uniformly Distributed Load on Overhang ......................................... 13
Figure 20 Beam Overhanging One Support – Concentrated Load at End of Overhang ............................................. 13
Figure 21 Beam Overhanging One Support – Concentrated Load at Any Point Between Supports........................... 14
Figure 22 Beam Overhanging Both Supports – Unequal Overhangs – Uniformly Distributed Load......................... 14
Figure 23 Beam Fixed at Both Ends – Uniformly Distributed Load........................................................................... 15
Figure 24 Beam Fixed at Both Ends – Concentrated Load at Center.......................................................................... 15
Figure 25 Beam Fixed at Both Ends – Concentrated Load at Any Point .................................................................... 16
Figure 26 Continuous Beam – Two Equal Spans – Uniform Load on One Span ....................................................... 16
Figure 27 Continuous Beam – Two Equal Spans – Concentrated Load at Center of One Span ................................. 17
Figure 28 Continuous Beam – Two Equal Spans – Concentrated Load at Any Point ................................................ 17
Figure 29 Continuous Beam – Two Equal Spans – Uniformly Distributed Load ....................................................... 18
Figure 30 Continuous Beam – Two Equal Spans – Two Equal Concentrated Loads Symmetrically Placed ............. 18
Figure 31 Continuous Beam – Two Unequal Spans – Uniformly Distributed Load ................................................... 19
Figure 32 Continuous Beam – Two Unequal Spans – Concentrated Load on Each Span Symmetrically Placed ..... 19

4. AMERICAN WOOD COUNCIL
w
R R
V
V
2 2
Shear
Mmax
Moment
x
7-36 A
ᐉ
ᐉ
ᐉ ᐉ
a b c
x
R1 R2
V1
V2
Shear
a + —
R1
w
Mmax
Moment
wb
7-36 B
ᐉ
Figure 1 Simple Beam – Uniformly Distributed Load
Figure 2 Simple Beam – Uniform Load Partially Distributed

5. AMERICAN FOREST & PAPER ASSOCIATION
R1 R2
V1
V2
Shear
x
R1
—w1
Mmax
Moment
a b c
w2c
w1a
7-37 B
ᐉ
x
R1 R2
V2
V1
Mmax
Moment
Shear
R1
—
w
a
wa
7-37 A
ᐉ
Figure 3 Simple Beam – Uniform Load Partially Distributed at One End
Figure 4 Simple Beam – Uniform Load Partially Distributed at Each End

6. AMERICAN WOOD COUNCIL
x
W
2
R R
V
V
Shear
Mmax
Moment
2
7-38 B
ᐉ
ᐉ ᐉ
W
R1 R2
V1
V2
Shear
Mmax
Moment
x
.57741
7-38 A
ᐉ
Figure 5 Simple Beam – Load Increasing Uniformly to One End
Figure 6 Simple Beam – Load Increasing Uniformly to Center

7. AMERICAN FOREST & PAPER ASSOCIATION
x P
a
R1 R2
V1
V2
Shear
Mmax
Moment
b
7-39-b
ᐉ
x P
2
R R
V
V
Shear
Mmax
Moment
2
7-39 A
ᐉᐉ
ᐉ
Figure 7 Simple Beam – Concentrated Load at Center
Figure 8 Simple Beam – Concentrated Load at Any Point

8. AMERICAN WOOD COUNCIL
x P
a
R1 R2
V1
V2
Shear
M1
Moment
P
b
M2
7-40 B
ᐉ
x P
a
R R
V
Shear
Mmax
Moment
P
a
7-40 A
V
ᐉ
Figure 9 Simple Beam – Two Equal Concentrated Loads Symmetrically Placed
Figure 10 Simple Beam – Two Equal Concentrated Loads Unsymmetrically
Placed

9. AMERICAN FOREST & PAPER ASSOCIATION
a
R1 R2
V1
V2
Shear
M1
Moment
b
M2
7-41-a
x
P1 P2
ᐉ
Figure 11 Simple Beam – Two Unequal Concentrated Loads Unsymmetrically
Placed
Figure 12 Cantilever Beam – Uniformly Distributed Load
x
R
V
Shear
Moment
w
Mmax
7-41- B
ᐉ
ᐉ

10. AMERICAN WOOD COUNCIL
x
a
VShear
Moment
b
Mmax
7-42-b
P
R
ᐉ
x
R
V
Shear
Moment
Mmax
P
7-42 A
ᐉ
Figure 13 Cantilever Beam – Concentrated Load at Free End
Figure 14 Cantilever Beam – Concentrated Load at Any Point

11. AMERICAN FOREST & PAPER ASSOCIATION
R1
R2
V1
V2
Shear
M1
2
M2
3
—
11
7-43 B
2
x P
ᐉ
ᐉ ᐉ
ᐉ
w
R1
7-43 A
R2
V1
V2
—
4
Shear
M1
Mmax
3
8
x
ᐉ
ᐉ
ᐉ
ᐉ
Figure 15 Beam Fixed at One End, Supported at Other – Uniformly Distributed
Load
Figure 16 Beam Fixed at One End, Supported at Other – Concentrated Load at
Center

12. AMERICAN WOOD COUNCIL
a b
R1
V2
Shear
x
Pa
—
R2
M1
Moment
R2
V1
P
M2
7-44 A
Figure 17 Beam Fixed at One End, Supported at Other – Concentrated Load at
Any Point
Figure 18 Beam Overhanging One Support – Uniformly Distributed Load
x1
Shear
M2
M1
R1
(1– a2
)
Moment
V3
V2
(1– a2
)2 2
2
x
a
w( +a)
R2
V1
7-44 B
ᐉ
ᐉ
ᐉ
ᐉ
ᐉ
ᐉ

13. AMERICAN FOREST & PAPER ASSOCIATION
V2
Mmax
x1x
a
Moment
Shear
R1 R2
V1
P
7-45 B
ᐉ
V2
Mmax
x1x
a
wa
Moment
Shear
R1 R2
V1
7-45 A
ᐉ
Figure 19 Beam Overhanging One Support – Uniformly Distributed Load on
Overhang
Figure 20 Beam Overhanging One Support – Concentrated Load at End of
Overhang

14. AMERICAN WOOD COUNCIL
R1 R2
w
a b c
V2
V1
V4
V3
Mx1
M2
M1
M3
X1
X
7-46 B
ᐉ
ᐉ
x
R1 R2
x1
a b
V1
V2
Mmax
Shear
Moment
7-46 A
ᐉ
Figure 21 Beam Overhanging One Support – Concentrated Load at Any Point
Between Supports
Figure 22 Beam Overhanging Both Supports – Unequal Overhangs – Uniformly
Distributed Load

15. AMERICAN FOREST & PAPER ASSOCIATION
x
V
V
2 2
Shear
Moment
R R
Mmax
PPP
Mmax
4
7-47 B
ᐉ
ᐉ ᐉ
ᐉ
x
V
V
2 2
Shear
M1
Moment
w
R R
Mmax
.2113
7-47 A
ᐉ
ᐉ
ᐉ ᐉ
ᐉ
Figure 23 Beam Fixed at Both Ends – Uniformly Distributed Load
Figure 24 Beam Fixed at Both Ends – Concentrated Load at Center

16. AMERICAN WOOD COUNCIL
Shear
V1
R3R2
16
7
M1
V2
Moment
Mmax
x
7-48 B
R1
V3
w
ᐉ ᐉ
ᐉ
ᐉ
x
Moment
PPP
V2
Ma
M2
R2
V1
M1
7-48 A
R1
Shear
a b
ᐉ
Figure 25 Beam Fixed at Both Ends – Concentrated Load at Any Point
Figure 26 Continuous Beam – Two Equal Spans – Uniform Load on One Span

17. AMERICAN FOREST & PAPER ASSOCIATION
Shear
V1
R3R2
M1
Moment
Mmax
PPP
V3
V2
R1
ba
7-49 B
ᐉ ᐉ
Shear
V1
R3R2
M1
Moment
PPP
V3
V2
R1
2 2
7-49 A
Mmax
ᐉ ᐉ
ᐉ ᐉ
Figure 27 Continuous Beam – Two Equal Spans – Concentrated Load at Center
of One Span
Figure 28 Continuous Beam – Two Equal Spans – Concentrated Load at Any
Point

18. AMERICAN WOOD COUNCIL
7-50 A
R1 R3R2
3l/8
0.4215 0.4215
M1
w w
maxΔ
V2
V3
l l
ll
ll
M2
3l/8
V1
V2
V1
R3R2
PPP
V2
R1
PPP
a a a a
Mx
x
V2
V3
M1
M2
l l
Figure 29 Continuous Beam – Two Equal Spans – Uniformly Distributed Load
Figure 30 Continuous Beam – Two Equal Spans – Two Equal Concentrated Loads
Symmetrically Placed

19. AMERICAN FOREST & PAPER ASSOCIATION
R1 R3R2
x1
Mx2
M1
V1
V4
V3
V2
1 2
1 2
w w
x2
Mx1
ll
ll
V1
R3R2
P
V3
R1
P
a a b b
Mm1
V4
M1
Mm2
21
V2
P2P1
ll
Figure 31 Continuous Beam – Two Unequal Spans – Uniformly Distributed Load
Figure 32 Continuous Beam – Two Unequal Spans – Concentrated Load on Each
Span Symmetrically Placed

20. A F & P A®
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