BEAM DESIGN FORMULAS WITH SHEAR AND MOMENT DIA DI AGRAMS DIA 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 ᐉ w
American Wood Council
l i c n u o C d o o W n a c i r e m A
R
R ᐉ
ᐉ
2
2
V Shear
V
M max
Moment
American Forest &
DESIGN AID No. 6
Paper Association
BEAM FORMULAS WITH SHEAR AND MOMENT DIA IAGRAMS
The American Wood Council (AWC)
is part of the wood products group of the American Ameri can Forest & Paper Pape r Associati Asso ciation on (AF&PA). AF&PA AF&PA is the national natio nal 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, fiber, and producing engineered and traditional traditional wood products. products. For more more information see www.afandpa.org. 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 & Pa per Association, Inc. American Wood Council 1111 19th St., NW, Suite 800 Washington, DC 20036 202-463-4713
[email protected] 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 .
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 w ᐉ
R
R ᐉ
ᐉ
2
2
V Shear
V
M max
Moment
Figure 2
7-36 A
Simple Beam – Uniform Load Partially Distributed
ᐉ
a
b wb
c
R 2
R 1 x V 1
Shear
V 2
R 1 a + — w
M max Moment
7-36 B AMERICAN WOOD COUNCIL
Figure 3
Simple Beam – Uniform Load Partially Distributed at One End ᐉ
a wa
R 1
R 2 x
V 1
Shear
V 2 R w
—1
M max
Moment
Figure 4
Simple Beam – Uniform Load Partially Distributed at Each End
7-37 A
ᐉ
a
b
w a 1
c w c 2
R 1
R 2 x
V 1 Shear
V 2
R 1 — w 1
M max Moment
7-37 B
AMERICAN FOREST & PAPER ASSOCIATION
Figure 5
Simple Beam – Load Increasing Uniformly to One End ᐉ
x
W
R 1
R 2
.57741
V 1
Shear
V 2
M max
Moment
Figure 6
Simple Beam – Load Increasing Uniformly to Center
7-38 A ᐉ
x W
R
R ᐉ
ᐉ
2
2
V Shear
V
M max
Moment
7-38 B AMERICAN WOOD COUNCIL
Figure 7
Simple Beam – Concentrated Load at Center ᐉ
x
P
R
R ᐉ
ᐉ
2
2
V Shear
V
M max
Moment
Figure 8
Simple Beam – Concentrated Load at Any Point 7-39 A ᐉ
x
P
R 1
R 2 a
b
V 1 V 2 Shear
M max
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
M max
Moment
Figure 10 7-40 A Simple Beam – Two Equal Concentrated Loads Unsymmetrically Placed ᐉ
x
P
P
R 1
R 2 a
b
V 1 Shear
V 2
M 2
M 1 Moment
7-40 B
AMERICAN WOOD COUNCIL
Figure 11
Simple Beam – Two Unequal Concentrated Loads Unsymmetrically Placed ᐉ
P 1
P 2
x
R 1
R 2 a
b
V 1 Shear
V 2
M 2
M 1 Moment
Figure 12
Cantilever Beam – Uniformly Distributed Load
ᐉ
w ᐉ
7-41-a
R
x
V
Shear
M max
Moment
7-41- B AMERICAN FOREST & PAPER ASSOCIATION
Figure 13
Cantilever Beam – Concentrated Load at Free End
ᐉ
P
R
x
V
Shear
M max
Moment
Figure 14
Cantilever Beam – Concentrated Load at Any Point
7-42 A
ᐉ
x
P R
a
b
Shear
V
M max Moment
-
- AMERICAN WOOD COUNCIL
Figure 15
Beam Fixed at One End, Supported at Other – Uniformly Distributed Load ᐉ
w ᐉ
R 2 R 1 x
V 1
Shear
V 2
3 ᐉ 8
ᐉ
—
4
M 1 M max
Figure 16
Beam Fixed at One End, Supported at Other – Concentrated Load at Center 7-43 A ᐉ
P
x
R 2 R 1 ᐉ
ᐉ
2
2
V 1
Shear
V 2
M 1 M 2
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
R 2 R 1 a
b
V 1 Shear
V 2
M 1 Moment
M 2
Pa —
R 2
Figure 18
Beam Overhanging One Support – Uniformly Distributed Load ᐉ 7-44 A
x
w (ᐉ+a )
a x 1
R 2
R 1 ᐉ
2
(1–
2
)
a ᐉ2
V 1
V 2
Shear
V 3
M 1
Moment ᐉ
(1–
M 2
2
)
a ᐉ2
7-44 B
AMERICAN WOOD COUNCIL
Figure 19
Beam Overhanging One Support – Uniformly Distributed Load on Overhang ᐉ
a x 1 wa
x
R 2
R 1
V 2 V 1 Shear
M max Moment
7-45 A
Figure 20
Beam Overhanging One Support – Concentrated Load at End of Overhang ᐉ
a x 1
x
R 1
P
R 2
V 2 V 1 Shear
M max Moment
7-45 B AMERICAN FOREST & PAPER ASSOCIATION
Figure 21
Beam Overhanging One Support – Concentrated Load at Any Point Between Supports ᐉ
x
x 1
R 1
R 2 a
b
V 1 V 2
Shear
M max
Moment
Figure 22
Beam Overhanging Both Supports – Unequal Overhangs – Uniformly Distributed Load
7-46 A
w ᐉ
R 1
R 2 ᐉ
a
b
c
V 4
V 2 V 1
V 3 X X 1 M 3
M x
1
M 1 M 2
7-46 B
AMERICAN WOOD COUNCIL
Figure 23
Beam Fixed at Both Ends – Uniformly Distributed Load ᐉ
x w ᐉ R
R
ᐉ
ᐉ
2
2
V Shear
V
.2113ᐉ
M 1 Moment
Figure 24
M max
7-47 A
Beam Fixed at Both Ends – Concentrated Load at Center
ᐉ
x P R
R
ᐉ
ᐉ
2
2
V Shear
V
ᐉ
4
M max Moment
M max
7-47 B AMERICAN FOREST & PAPER ASSOCIATION
Figure 25
Beam Fixed at Both Ends – Concentrated Load at Any Point ᐉ
x P R 1
R 2
b
a
V 1 Shear
V 2
M a Moment
M 1
M 2
7-48 A
Figure 26
Continuous Beam – Two Equal Spans – Uniform Load on One Span
x ᐉ w
R 1
R 2
ᐉ
R 3
ᐉ
V 1 Shear
V 2
V 3
7ᐉ 16
M max M 1
Moment
7-48 B
AMERICAN WOOD COUNCIL
Figure 27
Continuous Beam – Two Equal Spans – Concentrated Load at Center of One Span
ᐉ
ᐉ
2
2
P P
R 1
R 2
ᐉ
R 3
ᐉ
V 1
V 3 Shear
V 2
M max M 1
Moment
7-49 A
Figure 28
a
Continuous Beam – Two Equal Spans – Concentrated Load at Any Point b P
R 1
R 2 ᐉ
R 3 ᐉ
V 1 Shear
V 2
M max
V 3
Moment
M 1
7-49 B
AMERICAN FOREST & PAPER ASSOCIATION
Figure 29
Continuous Beam – Two Equal Spans – Uniformly Distributed Load w
w
R1
R2
R3
V 1
V 2
V 2
V 3 3 /8
3 /8
M2
M1
Δ max 0.4215
0.4215
7-50 A
Figure 30
Continuous Beam – Two Equal Spans – Two Equal Concentrated Loads Symmetrically Placed P
P
R1 a
R2 a
a
R3 a V 2
V 1
V 3 V 2
x Mx
M2 M1
AMERICAN WOOD COUNCIL
Figure 31 w
Continuous Beam – Two Unequal Spans – Uniformly Distributed Load w
1
R1
2
R2
1
R3
2
V 3
V 1 V 4 x1
V 2
x2
Mx
Mx
1
2
M1
Figure 32
Continuous Beam – Two Unequal Spans – Concentrated Load on Each Span Symmetrically Placed P1
P2
R1
R2 2
1
a V 1
R3
a
b
b V 3 V 4
V 2
Mm
1
Mm
2
M1
AMERICAN FOREST & PAPER ASSOCIATION
American F orest & P aper Association Forest 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
[email protected] [email protected] www .a wc.org www.a .aw
®
AF&PA
11 -07
