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
[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 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
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