Design Aid 6 Beam Design Formulas with Shear and Moment Diagrams

BEAMBEAM DESIGNDESIGN FFORMULASORMULAS WITHWITH SHEARSHEAR ANDAND MOMENTMOMENT DIADIAGRAMSGRAMS

2005 EDITION

ANSI/AF&PA NDS-2005 Approval Date: JANUARY 6, 2005

ASD/LRFD NDS® NATIONAL DESIGN SPECIFICATION® FOR WOOD CONSTRUCTION

WITH COMMENTARY AND SUPPLEMENT: DESIGN VALUES FOR WOOD CONSTRUCTION

American ᐉ Forest & x Paper Association wᐉ American Wood Council

R R ᐉᐉ 2 2 V

Shear V ood Council ood Council ood Council ood Council ood Council

Mmax

Moment American W American W American W American W American W American Forest & DESIGNDESIGN AIDAID NNoo.. 66 Paper Association BEAMBEAM FFORMULASORMULAS WITHWITH SHEARSHEAR ANDAND MOMENTMOMENT DIADIAGRAMSGRAMS

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.

American Wood Council 1111 19th St., NW, Suite 800 Washington, DC 20036 202-463-4713 [email protected] www.awc.org

AMERICAN WOOD COUNCIL Introduction Notations Relative to “Shear and Moment Diagrams” Figures 1 through 32 provide a series of shear and moment diagrams with accompanying formulas E = modulus of elasticity, psi for design of beams under various static loading I = moment of inertia, in.4 conditions. L = span length of the bending member, ft. Shear and moment diagrams and formulas are R = span length of the bending member, in. excerpted from the Western Woods Use Book, 4th M = maximum bending moment, in.-lbs. edition, and are provided herein as a courtesy of P = total concentrated load, lbs. Western Wood Products Association. 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 Figures 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

AMERICAN FOREST & PAPER ASSOCIATION Figure 1 Simple Beam – Uniformly Distributed Load

ᐉ x wᐉ

R R ᐉᐉ 2 2 V

Shear V

Mmax

Moment

7-36 A Figure 2 Simple Beam – Uniform Load Partially Distributed

ᐉ ab c wb

R1 R2

V 1 Shear

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

R1 R2

x V 1 Shear

R1 —w

Mmax

Moment

Figure 47-37 A Simple Beam – Uniform Load Partially Distributed at Each End

ᐉ abc

w1a w2c

R1 R2

V Shear 2

R —1 w1

M max

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

RR ᐉᐉ 2 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 ᐉ 2 2

V Shear V

M max

Moment

Figure 8 Simple Beam – Concentrated Load at Any Point 7-39 A

ᐉ

x P

R1 R2

a b

V2 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

Shear V

Mmax

Moment

Figure 107-40 A Simple Beam – Two Equal Concentrated Loads Unsymmetrically Placed

ᐉ

x P P

R1 R2

a b

Shear V2

M2 M1

Moment

7-40 B AMERICAN WOOD COUNCIL Figure 11 Simple Beam – Two Unequal Concentrated Loads Unsymmetrically Placed

ᐉ

P1 P2 x

R1 R2

a b

V1 Shear V2

M2 M 1

Moment

Figure 12 Cantilever Beam – Uniformly Distributed Load

ᐉ

7-41-a wᐉ

V Shear

Mmax Moment

7-41- B

AMERICAN FOREST & PAPER ASSOCIATION Figure 13 Cantilever Beam – Concentrated Load at Free End

ᐉ

Shear

Mmax Moment

Figure 147-42 A Cantilever Beam – Concentrated Load at Any Point

ᐉ

x P

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

V1 Shear V2

3 ᐉ 8 ᐉ — 4

Mmax

Figure 16 Beam Fixed at One End, Supported at Other – Concentrated Load at

7-43Center A

ᐉ

x P

R1 ᐉᐉ 2 2

Shear V2

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

x P

V 1 Shear V2

M 1

Moment M2 Pa — R2

Figure 18 Beam Overhanging One Support – Uniformly Distributed Load

ᐉ 7-44 A a

x x1 w( ᐉ + a )

R1 R2 ᐉ a2 (1– ᐉ ) 2 2

V 1 V 2

V Shear 3

M1 Moment M2 2 ᐉ a (1– ᐉ ) 2

7-44 B AMERICAN WOOD COUNCIL Figure 19 Beam Overhanging One Support – Uniformly Distributed Load on Overhang

ᐉ a

x x1 wa

R1 R2

V1 Shear

Mmax Moment

7-45 A Figure 20 Beam Overhanging One Support – Concentrated Load at End of Overhang

ᐉ a x x 1 P

R1 R2

V 1 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

V2 Shear

Mmax

Moment

Figure 227-46 Beam A Overhanging Both Supports – Unequal Overhangs – Uniformly Distributed Load

wᐉ

R1 R2 ᐉ

ab c

V2 V4 V1 V3 X

M3 M x1 M1 M2

7-46 B

AMERICAN WOOD COUNCIL Figure 23 Beam Fixed at Both Ends – Uniformly Distributed Load

ᐉ x wᐉ

R R

ᐉᐉ 2 2

V Shear V

.2113 ᐉ

M1 Moment Mmax

Figure 247-47 Beam A Fixed at Both Ends – Concentrated Load at Center

ᐉ x P

R R

ᐉᐉ 2 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

a b

V1 Shear V2

M Moment 1 M2

7-48 A Figure 26 Continuous Beam – Two Equal Spans – Uniform Load on One Span

wᐉ

R1 R2 R3

ᐉᐉ

V1 Shear V2 V3 7ᐉ 16

Mmax

M 1 Moment

7-48 B

AMERICAN WOOD COUNCIL Figure 27 Continuous Beam – Two Equal Spans – Concentrated Load at Center of One Span

ᐉᐉ 2 2 P

R1 R2 R3

ᐉ ᐉ

V1 V3 V2 Shear

Mmax

M1 Moment

7-49 A

Figure 28 Continuous Beam – Two Equal Spans – Concentrated Load at Any Point

a b

R1 R2 R3

ᐉᐉ

V1 V3 V2 Shear

Mmax Moment

7-49 B

AMERICAN FOREST & PAPER ASSOCIATION Figure 29 Continuous Beam – Two Equal Spans – Uniformly Distributed Load

wwll

R R R 1 l 2 l 3

V1 V2

V2 V3 3l/8 3l/8

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 R3

aaaa

V2 V1 V 3 V2

M2 Mx

M1 l l

AMERICAN WOOD COUNCIL Figure 31 Continuous Beam – Two Unequal Spans – Uniformly Distributed Load

l l ww12

R1 R2 R3

l l 12

V1 V3 V4 V2

x1 x2

M M x2 x1

Figure 32 Continuous Beam – Two Unequal Spans – Concentrated Load on Each Span Symmetrically Placed

P1 P2

R1 R2 R3 l l 1 2 aa bb

V3 V1 V2 V4

M m1 M m2

AMERICAN FOREST & PAPER ASSOCIATION American Forest & Paper Association A F & P A® American Wood Council 1111 19th Street, NW Suite 800 Washington, DC 20036 Phone: 202-463-4713 Fax: 202-463-2791 [email protected] www.awc.org 11-07