Practical Beam Formulas

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

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 [email protected] [email protected] www .a wc.org www.a .aw

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