BEAM DESIGN FORMULAS
WITH SHEAR AND MOMENT
DIAGRAMS
2005 EDITION
ANSI/AF&PA NDS-2005
Approval Date: JANUARY 6, 2005
ASD/LRFD
N DS
®
NATIONAL DESIGN SPECIFICATION®
FOR WOOD CONSTRUCTION
WITH COMMENTARY AND
SUPPLEMENT: DESIGN VALUES FOR WOOD CONSTRUCTION
American
Forest &
Paper
Association
ᐉ
x
wᐉ
American W
ood Council
Wood
American Wood Council
R
R
ᐉ
2
ᐉ
2
V
Shear
V
Mmax
Moment
American
Forest &
DESIGN AID No. 6
Paper
Association
BEAM FORMULAS WITH
SHEAR AND MOMENT
DIAGRAMS
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-ofthe-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
AMERICAN WOOD COUNCIL
Notations Relative to “Shear and Moment
Diagrams”
Intr
oduction
Introduction
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.
E =
I =
L =
R =
M=
P =
R =
V =
W=
w =
Δ =
x =
modulus of elasticity, psi
moment of inertia, in.4
span length of the bending member, ft.
span length of the bending member, in.
maximum bending moment, in.-lbs.
total concentrated load, lbs.
reaction load at bearing point, lbs.
shear force, lbs.
total uniform load, lbs.
load per unit length, lbs./in.
deflection or deformation, in.
horizontal distance from reaction to point
on beam, in.
List of Figur
es
Figures
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
Figure 9
Figure 10
Figure 11
Figure 12
Figure 13
Figure 14
Figure 15
Figure 16
Figure 17
Figure 18
Figure 19
Figure 20
Figure 21
Figure 22
Figure 23
Figure 24
Figure 25
Figure 26
Figure 27
Figure 28
Figure 29
Figure 30
Figure 31
Figure 32
Simple Beam – Uniformly Distributed Load ................................................................................................ 4
Simple Beam – Uniform Load Partially Distributed..................................................................................... 4
Simple Beam – Uniform Load Partially Distributed at One End .................................................................. 5
Simple Beam – Uniform Load Partially Distributed at Each End ................................................................ 5
Simple Beam – Load Increasing Uniformly to One End .............................................................................. 6
Simple Beam – Load Increasing Uniformly to Center .................................................................................. 6
Simple Beam – Concentrated Load at Center ............................................................................................... 7
Simple Beam – Concentrated Load at Any Point.......................................................................................... 7
Simple Beam – Two Equal Concentrated Loads Symmetrically Placed....................................................... 8
Simple Beam – Two Equal Concentrated Loads Unsymmetrically Placed .................................................. 8
Simple Beam – Two Unequal Concentrated Loads Unsymmetrically Placed .............................................. 9
Cantilever Beam – Uniformly Distributed Load ........................................................................................... 9
Cantilever Beam – Concentrated Load at Free End .................................................................................... 10
Cantilever Beam – Concentrated Load at Any Point .................................................................................. 10
Beam Fixed at One End, Supported at Other – Uniformly Distributed Load ............................................. 11
Beam Fixed at One End, Supported at Other – Concentrated Load at Center ........................................... 11
Beam Fixed at One End, Supported at Other – Concentrated Load at Any Point ..................................... 12
Beam Overhanging One Support – Uniformly Distributed Load ............................................................... 12
Beam Overhanging One Support – Uniformly Distributed Load on Overhang ......................................... 13
Beam Overhanging One Support – Concentrated Load at End of Overhang ............................................. 13
Beam Overhanging One Support – Concentrated Load at Any Point Between Supports ........................... 14
Beam Overhanging Both Supports – Unequal Overhangs – Uniformly Distributed Load ......................... 14
Beam Fixed at Both Ends – Uniformly Distributed Load ........................................................................... 15
Beam Fixed at Both Ends – Concentrated Load at Center .......................................................................... 15
Beam Fixed at Both Ends – Concentrated Load at Any Point .................................................................... 16
Continuous Beam – Two Equal Spans – Uniform Load on One Span ....................................................... 16
Continuous Beam – Two Equal Spans – Concentrated Load at Center of One Span ................................. 17
Continuous Beam – Two Equal Spans – Concentrated Load at Any Point ................................................ 17
Continuous Beam – Two Equal Spans – Uniformly Distributed Load ....................................................... 18
Continuous Beam – Two Equal Spans – Two Equal Concentrated Loads Symmetrically Placed ............. 18
Continuous Beam – Two Unequal Spans – Uniformly Distributed Load ................................................... 19
Continuous Beam – Two Unequal Spans – Concentrated Load on Each Span Symmetrically Placed ..... 19
AMERICAN FOREST & PAPER ASSOCIATION
Figure 1
Simple Beam – Uniformly Distributed Load
ᐉ
x
wᐉ
R
R
ᐉ
2
ᐉ
2
V
Shear
V
Mmax
Moment
Figure 2
7-36 A
Simple Beam – Uniform Load Partially Distributed
ᐉ
a
b
wb
c
R2
R1
x
V1
Shear
V2
R
a + —1
w
Mmax
Moment
7-36 B
AMERICAN WOOD COUNCIL
Figure 3
Simple Beam – Uniform Load Partially Distributed at One End
ᐉ
a
wa
R2
R1
x
V1
Shear
V2
R
—1
w
Mmax
Moment
Figure 4
7-37 ASimple
Beam – Uniform Load Partially Distributed at Each End
ᐉ
a
b
w1a
c
w2c
R1
R2
x
V1
Shear
V2
R1
—
w1
Mmax
Moment
7-37 B
AMERICAN FOREST & PAPER ASSOCIATION
Figure 5
Simple Beam – Load Increasing Uniformly to One End
ᐉ
x
W
R1
R2
.57741
V1
Shear
V2
Mmax
Moment
Figure 6
Simple Beam – Load Increasing Uniformly to Center
7-38 A
ᐉ
x
W
R
R
ᐉ
2
ᐉ
2
V
Shear
V
Mmax
Moment
7-38 B
AMERICAN WOOD COUNCIL
Figure 7
Simple Beam – Concentrated Load at Center
ᐉ
x
P
R
R
ᐉ
2
ᐉ
2
V
Shear
V
Mmax
Moment
Figure 8
Simple Beam – Concentrated Load at Any Point
7-39 A
ᐉ
x
P
R1
R2
a
b
V1
V2
Shear
Mmax
Moment
7-39-b
AMERICAN FOREST & PAPER ASSOCIATION
Figure 9
Simple Beam – Two Equal Concentrated Loads Symmetrically Placed
ᐉ
x
P
P
R
R
a
a
V
Shear
V
Mmax
Moment
Figure 10 7-40 A Simple Beam – Two Equal Concentrated Loads Unsymmetrically
Placed
ᐉ
x
P
P
R1
R2
b
a
V1
Shear
V2
M2
M1
Moment
7-40 B
AMERICAN WOOD COUNCIL
Figure 11
Simple Beam – Two Unequal Concentrated Loads Unsymmetrically
Placed
ᐉ
P2
P1
x
R1
R2
a
b
V1
Shear
V2
M2
M1
Moment
Figure 12
Cantilever Beam – Uniformly Distributed Load
ᐉ
wᐉ
7-41-a
R
x
V
Shear
Mmax
Moment
7-41- B
AMERICAN FOREST & PAPER ASSOCIATION
Figure 13
Cantilever Beam – Concentrated Load at Free End
ᐉ
P
R
x
V
Shear
Mmax
Moment
Figure 14
Cantilever Beam – Concentrated Load at Any Point
7-42 A
ᐉ
P
x
R
a
b
Shear
V
Mmax
Moment
7-42-b
AMERICAN WOOD COUNCIL
Figure 15
Beam Fixed at One End, Supported at Other – Uniformly Distributed
Load
ᐉ
wᐉ
R2
R1
x
V1
Shear
V2
3
ᐉ
8
ᐉ
—
4
M1
Mmax
Figure 16
Beam Fixed at One End, Supported at Other – Concentrated Load at
Center
7-43 A
ᐉ
P
x
R2
R1
ᐉ
2
ᐉ
2
V1
Shear
V2
M1
M2
3
—ᐉ
11
AMERICAN FOREST & PAPER ASSOCIATION
7-43 B
Figure 17
Beam Fixed at One End, Supported at Other – Concentrated Load at
Any Point
P
x
R2
R1
a
b
V1
Shear
V2
M1
Moment
M2
Pa
—
R2
Figure 18
Beam Overhanging One Support – Uniformly Distributed Load
ᐉ 7-44 A
x
w(ᐉ+ a)
a
x1
R2
R1
ᐉ (1 – a 2 )
2
ᐉ2
V1
V2
Shear
V3
M1
Moment
M2
a2 )
ᐉ (1 –
ᐉ2
7-44 B
AMERICAN WOOD COUNCIL
Figure 19
Beam Overhanging One Support – Uniformly Distributed Load on
Overhang
ᐉ
a
x1
wa
x
R2
R1
V2
V1
Shear
Mmax
Moment
7-45 A
Figure 20
Beam Overhanging One Support – Concentrated Load at End of
Overhang
ᐉ
a
x1
x
P
R2
R1
V2
V1
Shear
Mmax
Moment
7-45 B
AMERICAN FOREST & PAPER ASSOCIATION
Figure 21
Beam Overhanging One Support – Concentrated Load at Any Point
Between Supports
ᐉ
x
x1
R1
R2
a
b
V1
V2
Shear
Mmax
Moment
Figure 22
7-46
A
Beam
Overhanging Both Supports – Unequal Overhangs – Uniformly
Distributed Load
wᐉ
R1
R2
ᐉ
a
b
c
V4
V2
V1
V3
X
X1
M3
Mx 1
M1
M2
7-46 B
AMERICAN WOOD COUNCIL
Figure 23
Beam Fixed at Both Ends – Uniformly Distributed Load
ᐉ
x
wᐉ
R
R
ᐉ
2
ᐉ
2
V
Shear
V
.2113 ᐉ
M1
Moment
Figure 24
Mmax
7-47 A
Beam Fixed at Both Ends – Concentrated Load at Center
ᐉ
x
P
R
R
ᐉ
2
ᐉ
2
V
Shear
V
ᐉ
4
Mmax
Moment
Mmax
7-47 B
AMERICAN FOREST & PAPER ASSOCIATION
Figure 25
Beam Fixed at Both Ends – Concentrated Load at Any Point
ᐉ
x
P
R1
R2
b
a
V1
Shear
V2
Ma
Moment
M1
M2
7-48 A
Figure 26
Continuous Beam – Two Equal Spans – Uniform Load on One Span
x
wᐉ
R1
R2
R3
ᐉ
ᐉ
V1
Shear
V2
V3
7ᐉ
16
Mmax
M1
Moment
7-48 B
AMERICAN WOOD COUNCIL
Figure 27
Continuous Beam – Two Equal Spans – Concentrated Load at Center
of One Span
ᐉ
2
ᐉ
2
P
R1
R2
ᐉ
R3
ᐉ
V1
V3
Shear
V2
Mmax
M1
Moment
7-49 A
Figure 28
Continuous Beam – Two Equal Spans – Concentrated Load at Any
Point
a
b
P
R1
R2
ᐉ
R3
ᐉ
V1
Shear
V2
Mmax
V3
Moment
M1
7-49 B
AMERICAN FOREST & PAPER ASSOCIATION
Figure 29
Continuous Beam – Two Equal Spans – Uniformly Distributed Load
wl
wl
R1
R2
R3
l
l
V1
V2
V3
V2
3l/8
3l/8
M2
M1
Δ max
0.4215 l
0.4215 l
7-50 A
Figure 30
Continuous Beam – Two Equal Spans – Two Equal Concentrated Loads
Symmetrically Placed
P
P
R1
R2
a
a
R3
a
a
V2
V3 V2
V1
x
M2
Mx
M1
l
l
AMERICAN WOOD COUNCIL
Figure 31
Continuous Beam – Two Unequal Spans – Uniformly Distributed Load
w l1
w l2
R1
R2
l
R3
l
1
2
V3
V1
V4
x1
V2
x2
Mx2
Mx1
M1
Figure 32
Continuous Beam – Two Unequal Spans – Concentrated Load on Each
Span Symmetrically Placed
P1
P2
R1
R2
l
a
V1
R3
l
1
a
b
2
b
V3
V4
V2
Mm
1
Mm
2
M1
AMERICAN FOREST & PAPER ASSOCIATION
American FForest
orest & P
aper Association
Paper
American W
ood Council
Wood
1111 19th S
tree
t, NW
Stree
treet,
Suit
e 800
Suite
Washingt
on, DC 20036
ashington,
Phone: 202-463-4
713
202-463-47
Fax: 202-463-2
79
1
202-463-279
791
awcinf
o@afandpa.org
cinfo@afandpa.org
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.a
wc.org
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.aw
A F & P A®
11-07