apxb lts 5trev

B-i

APPENDIX B: DESIGN AIDS

TABLE OF CONTENTS B

B1—SECTIONAL PROPERTIES FOR TUBULAR SHAPES .......................................................................................... B-1

B2—STRESSES FOR TUBULAR SECTIONS.................................................................................................................. B-2

B3—DEFLECTION EQUATIONS FOR TAPERED TUBULAR CANTILEVERED BEAMS ....................................... B-4

B4—DEFLECTION EQUATIONS FOR TAPERED CANTILEVERED BEAMS ........................................................... B-5

B5—EFFECTIVE LENGTH FACTORS FOR COLUMNS............................................................................................... B-6

B6—ALLOWABLE STRESSES FOR SELECTED ALUMINUM ALLOYS................................................................... B-7

B7—REFERENCES.......................................................................................................................................................... B-42

B-1

APPENDIX B

DESIGN AIDS B1—SECTIONAL PROPERTIES FOR TUBULAR SHAPES

Table B-1 provides approximate equations to compute sectional properties of tubular shapes.

Table B-1—Estimated Sectional Properties for Common Tubular Shapes

Property Round Tube Hexdecagonal Tube Dodecagonal Tube Octagonal Tube Square Tube Square Tube

(Axis on Diagonal) Moment of inertia, I 3.14R3t 3.22R3t 3.29R3t 3.50R3t 5.33R3t 5.33R3t Section modulus, S 3.14R2t 3.22R2t 3.29R2t 3.50R2t 5.33R2t 3.77R2t Area, A 6.28Rt 6.37Rt 6.43Rt 6.63Rt 8.00Rt 8.00Rt Shape factor, Kp 1.27 1.27 1.26 1.24 1.12 — Radius of gyration, r 0.707R 0.711R 0.715R 0.727R 0.816R 0.816R Cross-sectional constant, C 3.14 3.22 3.29 3.50 5.33 — Pictorial representation

Notation:

C = cross-sectional constant used in Table B-3

R = radius measured to the mid-thickness of the wall

t = wall thickness

B-2 STANDARD SPECIFICATIONS FOR STRUCTURAL SUPPORTS FOR HIGHWAY SIGNS, LUMINAIRES, AND TRAFFIC SIGNALS

B2—STRESSES FOR TUBULAR SECTIONS The following equations calculate stresses for round and multisided tubular shapes. If more than one equation is provided, then the greatest value should be used.

For round tubes in biaxial bending, the maximum bending stress occurs at the resulting moment. For multisided tubular shapes in biaxial bending, the maximum bending stress occurs at one of the corners. The maximum transverse shear stress for tubular shape occurs in the tube walls that are intersected by the neutral axis, which is normal to the resultantant shear force.

Table B-2—Formulas for Maximum Stresses in Common Tubular Shapes

Stress Round Tube Hexdecagonal Tube Dodecagonal Tube Octagonal Tube Square Tube Square Tube

(Axis on Diagonal) Maximum bending stress, fb 2 2

x yf f+ 0.199 +

or0.567 +0.848

or0.848 +0.567

or +0.199

x y

x y

x y

x y

f f

f f

f f

f f

0.732( + )

or+0.268

or0.268 +

x y

x y

x y

f f

f f

f f

+0.414

or0.414 +

x y

x y

f f

f f

+ x yf f fx

or

fy

Maximum shear stress due to transverse loads, fvb

2.0 sVA

2.02 sV

A

2.025 sVA

2.05 sV

A

2.25 sVA

2.12 sV

A

Maximum shear stress due to torsion, fvt 26.28

zM

R t 26.37

z tM k

R t 26.43

z tM k

R t 26.63

z tM k

R t 28.00

z tM k

R t 28.00

z tM k

R t

Values of kt, stress concentration factor See Figure B-1 Pictorial Representation

Notation: A = area fvb = shear stress due to transverse loads fvt = shear stress due to torsion fx = stress due to bending about the x-axis

fy = stress due to bending about the y-axis kt = stress concentration factor for multisided tubular

shapes Mz = torsional moment

R = radius measured to the mid-thickness of the wall t = wall thickness Vs = applied shear

APPENDIX B: DESIGN AIDS B-3

Torsional shear stresses are present on a transverse cross-section due to torsional moments, Mz, and are uniform around the periphery of thin-walled tubular section, except for stress concentrations at the corners of polygonal tubes. The stress concentration factor for a polygonal tube is obtained from Timoshenko’s Strength of Materials. The stress concentration factor may be found by using Figure B-1 or by using the following equation:

1 1

12 1.0

1ln

R nt n tn t nk

nR Rn

t

′ +− + ′′ ′

= + ≥′ +

′

⎡ ⎛ ⎞ ⎤⎜ ⎟⎢ ⎥⎝ ⎠⎢ ⎥

⎛ ⎞⎢ ⎥⎜ ⎟⎣ ⎦⎝ ⎠

where:

kt = stress concentration factor

t = wall thickness

R = radius measured to the mid-thickness of the wall

n′ = ratio of the inside-corner radius to wall thickness

Figure B-1—Stress-Concentration Factors for Polygonal Tubes in Torsion (Source: Timoshenko, S. 1957. Strength of Materials—Part II. D. Van Nostrand Company, Princeton, NJ.)


B3—DEFLECTION EQUATIONS FOR TAPERED TUBULAR CANTILEVERED BEAMS

Equations to compute deflections of prismatic beams are not applicable for tapered beams. Table B-3 provides equations to compute deflection and slope at the tip of tubular cantilever tapered beams. Equations provided in Table B-3 are valid only when the wall thickness t is very small in comparison with the mean radius R of the cross-section. Table B-3—Estimated Maximum Deflection and Slope of Tapered Hollow Beams

Transverse Point Load at Tip of Beam

( )

3

max 3 2 ln 32

B B A A

A B BB A

R R R RWLyR R RECt R R

⎡ ⎤⎛ ⎞ ⎛ ⎞⎛ ⎞−= − −⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟

⎢ ⎥− ⎝ ⎠ ⎝ ⎠⎝ ⎠⎣ ⎦

2

max 2θ2 A B

WLECtR R

=

End Moment Applied at Tip of Beam

( )2

max 22 A B

MLyECt R R

=

( )

max 2 2θ2

A B

A B

ML R RECtR R

+=

Uniform Load

( )

23

max 4 213 ln262

B A AA B

A B BB A

R R RWLy R RR R RECt R R

⎡ ⎤⎛ ⎞⎛ ⎞= − − + + +⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟− ⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦

( )( )

2

max 3 23θ ln 4222

B AB A

A BB A

R RWL R RR RECt R R

⎡ ⎤⎛ ⎞= + − −⎢ ⎥⎜ ⎟

⎢ ⎥− ⎝ ⎠⎣ ⎦

W wL= Triangular Load

( )

2 332 2

max 5 28

12 ln 86

B A AA A B B

A B BB A

R R RWLy R R R RR R RECt R R

⎡ ⎤⎛ ⎞⎛ ⎞= + − − +⎢ ⎥⎜ ⎟⎜ ⎟ ⎜ ⎟− ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

( )

22

max 4 21θ ln2 36

B A A BA

A B BB A

R R R RWL RR R RECt R R

⎡ ⎤⎛ ⎞⎛ ⎞= − − + − +⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟− ⎢ ⎥⎝ ⎠⎝ ⎠⎣ ⎦

2wLW =

Notation:

C = cross-sectional constant defined per Table B-1

E = modulus of elasticity

L = length of the beam (from ground line to tip)

M = applied moment

RA = radius measured to mid-thickness of wall at free end (see Table B-1)

RB = radius measured to mid-thickness of wall at fixed end (see Table B-1)

t = wall thickness

W = load

w = load per unit length

ymax = maximum horizontal deflection at free end of beam

θmax = maximum slope at free end of beam (rad)

Note: Equations are based on the small deflection theory; therefore tan θ = θ.

Adapted from: Hopkins, R. B. 1970. Design Analysis of Shafts and Beams. McGraw–Hill, New York, NY.


B4—DEFLECTION EQUATIONS FOR TAPERED CANTILEVERED BEAMS Table B-4 provides equations to compute tip deflections and slope for solid round cantilevered beams, such as wood poles.

Table B-4—Maximum Deflection and Slope of Solid Round Tapered Beams

Slope Deflection Transverse Point Load at Free End

2

max 2.497θ

2.068 ba

a

PL

dEId

=⎛ ⎞⎜ ⎟⎝ ⎠

3

max 3

3 ba

a

PLydEId

=⎛ ⎞⎜ ⎟⎝ ⎠

End Moment Applied at Free End

max 1 587θ

1.075 ba

a

ML

dEId

=⎛ ⎞⎜ ⎟⎝ ⎠

. 2

max 2.497

2.068 ba

a

MLydEId

=⎛ ⎞⎜ ⎟⎝ ⎠

Uniform Load Applied Along Length

3

max 3θ

6 ba

a

wL

dEId

=⎛ ⎞⎜ ⎟⎝ ⎠

4

max 3.282

7.872 ba

a

wLydEId

=⎛ ⎞⎜ ⎟⎝ ⎠

Notation:

da = diameter at free end

db = diameter at fixed end

E = modulus of elasticity

Ia = moment of inertia of cross-section at free end of beam

L = length

M = moment applied at free end of beam

P = horizontal load applied at free end of beam

w = uniformly distributed load applied along the beam length

ymax = deflection at free end of beam

θmax = slope at free end of beam (rad)

Source: Author, Initials. 1985. “Graphs to Determine Structure Deflections,” Transmission and Distribution. July 1985, pp. X–Y.


B5—EFFECTIVE LENGTH FACTORS FOR COLUMNS Table B-5 provides effective length factors for centrally loaded columns with various end conditions.

Table B-5—Effective Length Factors for Centrally Loaded Columns

Buckled Shape of Column Shown by Dashed Line

Theoretical K Value 0.50 0.70 1.00 1.00 2.00 2.00 Recommended Design Values when Ideal Conditions are Approximated 0.65 0.80 1.00 1.20 2.10 2.00

Rotation fixed, Translation fixed

Rotation free, Translation fixed

Rotation fixed, Translation free

End Conditions Code

Rotation free, Translation free


B6—ALLOWABLE STRESSES FOR SELECTED ALUMINUM ALLOYS Tables B-6 through B-11 provide allowable stresses for welded and nonwelded members for A6005-T5, A6061-T6, and A6063-T6.

Table B-6—Allowable Stresses for Aluminum 6005—T5 Alloy (MPa)

Type of Stress Type of Member or Component

Eq. Set Allowable Stress

Tension, axial, net section

Any tension member

6-1 131 72.4 6005—T5 Extrusions Thickness Up Through 25 mm

Rectangular tubes, structural shapes bent around strong axis

6-2 131 72.4 White bars apply to nonwelded members and to welded members at locations farther than 25.4 mm from a weld.

Round or oval tubes

6-3 165 82.7 Shaded bars apply to within 25.4 mm of a weld.

Tension in Beams, extreme fiber, net section

Shapes bent about weak axis, bars, plates

6-4 193 93.1 Notes:

See Articles 6.4.1, 6.4.2.1, and 6.4.4.1 for additional provisions regarding allowable stresses.

For tubes with circumferential welds, R/t and Rb/t, as applicable, shall be ≤20, except when the design meets the details and post-weld heat treatment requirements of Article 6.5.

On bolts 6-5 234 124 See Article 6.5 for additional provisions regarding allowable stresses in welded members.

The article numbers and equations referenced in this table refer to Section 6, “Aluminum Design.”

Bearing

On flat surfaces and on bolts in slotted holes

6-6 159 82.7

Continued on next page



Table B-6—Allowable Stresses for Aluminum 6005—T5 Alloy (MPa)—Continued


Eq. Set

(a) Allowable Stress Slenderness ≤ S1

(b) Slenderness Limit S1

(c) Allowable Stress Slenderness Between S1 and S2

(d) Slenderness Limit S2

(e) Allowable Stress Slenderness ≥ S2

131 9.5kL

r= 139.3 0.869 kL

r⎛ ⎞− ⎜ ⎟⎝ ⎠

66kLr

= 351600

2kLr

⎛ ⎞⎜ ⎟⎝ ⎠

Compression in Columns, axial, gross section

All columns

6-7

72.4 — 72.4 70kL

r=

3516002kL

r⎛ ⎞⎜ ⎟⎝ ⎠

131 5.2b

t= 159.3 5.45 b

t⎛ ⎞⎜ ⎟⎝ ⎠

− 10bt

= 1062

bt

⎛ ⎞⎜ ⎟⎝ ⎠

Flat plates supported along one edge—columns buckling about a symmetry axis

6-8

72.4 — 72.4 15b

t=

1062bt

⎛ ⎞⎜ ⎟⎝ ⎠

131 5.2b

t= 159.3 5.45 b

t⎛ ⎞⎜ ⎟⎝ ⎠

− 12bt

= 13580

2bt

⎛ ⎞⎜ ⎟⎝ ⎠

Compression in Components of Columns, gross section

Flat plates supported along one edge—columns not buckling about a symmetry axis

6-9

72.4 — 72.4 14b

t=

135802b

t⎛ ⎞⎜ ⎟⎝ ⎠


ibn




Eq. Set

(a) Allowable Stress Slenderness ≤ S1


(c) Allowable Stress Slenderness Between S1 and S2

(d) Slenderness Limit S2


131 16b

t= 159.3 1.72 b

t⎛ ⎞− ⎜ ⎟⎝ ⎠

33bt

= 3378

bt

⎛ ⎞⎜ ⎟⎝ ⎠

Flat plates with both edges supported

6-10

72.4 — 72.4 47b

t=

3378bt

⎛ ⎞⎜ ⎟⎝ ⎠

131 16R

t= 153.1 5.52 R

t− 141R

t=

220602

135

RR tt

⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟ + ⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠


Curved plates supported on both edges, walls of round or oval tubesb

6-11

72.4 5.1R

t= 178.6 2.76 sin θR

t−− 340R

t=

220602

135

RR tt

⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟ + ⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠




Eq. Set

(a) Allowable Stress

Slenderness ≤ S1 (b) Slenderness Limit S1

(c) Allowable Stress Slenderness Between

S1 and S2 (d) Slenderness Limit S2


145 23b

y

Lr = 164.8 0.855 Lb

r y− 79Lb

r y= 2

600000

by

Lr

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

Single web beams bent about strong axis

6-12

72.4 —

72.4 91Lb

r y=

2600000

by

Lr

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

172 28Rb

t= 271.0 18.62 Rb

t− 81Rb

t=

same as Eq. 6-11 with R = Rb Round or oval tubesb

6-13

82.7 53Rb

t= 128.2 6.89 Rb

t− 134Rb

t=

same as Eq. 6-11 with R = Rb

193 13bLd

t d= 279.2 6.41 bLd

t d− 29d Lb

t d= 2

78600

bLdt d

⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Solid rectangular and round section beams

6-14

93.1 — 93.1 29d Lb

t d= 2

78600

bLdt d

⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

145 146

0.5L Sb c

I Jy= 164.8 1.66

0.5b c

y

L SL J

− 17000.5

b c

y

L SL J

= 165500

0.5b c

y

L SI J

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

Compression in Beams, extreme fiber, gross section

Rectangular tubes and box sections

6-15

72.4 — 72.4 2290

0.5b c

y

L SI J

= 165500

0.5b c

y

L SI J

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠


t




Eq. Set




S1 and S2 (d) Slenderness Limit S2 (e) Allowable Stress

Slenderness ≥ S2 145

6.8bt

= 188.2 6.41bt

− 10bt

= 1262

bt

⎛ ⎞⎜ ⎟⎝ ⎠

Flat plates supported on one edge

6-16

72.4 — 72.4 17b

t=

1262bt

⎛ ⎞⎜ ⎟⎝ ⎠

145 22b

t= 188.2 2.0 b

t− 33b

t=

4000bt

⎛ ⎞⎜ ⎟⎝ ⎠

Compression in Components of Beams, component under uniform compression, gross section


6-17

93.1 — 72.4 55b

t=

4000bt

⎛ ⎞⎜ ⎟⎝ ⎠

172 1.6bR

t= 180.6 6.48 bR

t− 141bR

t=

26200

135

bb

RR tt

⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟+⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠

Curved plates supported on

both edgesb

6-18

82.7 1.3bR

t= 85.5 02.41 bR

t− 340bR

t=

Same as nonwelded members





Eq. Set






193 8.9b

t= 279.2 9.72 b

t− 19b

t= 2

33780

bt

⎛ ⎞⎜ ⎟⎝ ⎠

Flat plates with compression edge free, tension edge supported

6-19

93.1 — 93.1 19b

t= 2

33780

bt

⎛ ⎞⎜ ⎟⎝ ⎠

193 46h

t= 279.2 1.86 h

t− 75h

t=

10480ht

⎛ ⎞⎜ ⎟⎝ ⎠

Compression in Components of Beams, component under bending in own plane, gross section

Flat plate with both edges supported

6-20

93.1 — 93.1 113h

t=

10480ht

⎛ ⎞⎜ ⎟⎝ ⎠

83 36h

t= 107.6 0.683 h

t− 65h

t= 2

268900

ht

⎛ ⎞⎜ ⎟⎝ ⎠

Shear in Webs, gross section

Unstiffened flat webs

6-21

41.4 — 41.4 81h

t= 2

268900

ht

⎛ ⎞⎜ ⎟⎝ ⎠


Table B-7—Allowable Stresses for Aluminum 6005—T5 Alloy (ksi)




Any tension member

6-1 19 10.5 6005—T5 Extrusions Thickness ≤1.0 in


6-2 19 10.5 White bars apply to nonwelded members and to welded members at locations farther than 1.0 in from a weld.

Round or oval tubes

6-3 24 12 Shaded bars apply to within 1.0 in of a weld.



6-4 28 13.5

On bolts

6-5 34 18 Bearing


6-6 23 12

Notes: a See Articles 6.4.1, 6.4.2.1, and 6.4.4.1 for additional provisions regarding

allowable stresses. b For tubes with circumferential welds, R/t and Rb/t, as applicable, shall be ≤20,

except when the design meets the details and post-weld heat treatment requirements of Article 6.5.

c See Article 6.5 for additional provisions regarding allowable stresses in welded members.

d The article numbers and equations referenced in this table refer to Section 6, “Aluminum Design.”



Table B-7—Allowable Stresses for Aluminum 6005—T5 Alloy (ksi)—Continued


Eq. Set






9.5kLr

= 20.2 0.126 kLr

⎛ ⎞− ⎜ ⎟⎝ ⎠

66kLr

= 251000

kLr

⎛ ⎞⎜ ⎟⎝ ⎠


All columns

6-7

10.5 — 10.5 70kL

r= 2

51000

kLr

⎛ ⎞⎜ ⎟⎝ ⎠

19 5.2b

t= 23.1 0.79 b

t⎛ ⎞− ⎜ ⎟⎝ ⎠

10bt

= 154

bt

⎛ ⎞⎜ ⎟⎝ ⎠


6-8

10.5 — 10.5 15b

t=

154bt

⎛ ⎞⎜ ⎟⎝ ⎠

19 5.2b

t= 23.1 0.79 b

t⎛ ⎞− ⎜ ⎟⎝ ⎠

12bt

= 21970

bt

⎛ ⎞⎜ ⎟⎝ ⎠


6-9

10.5 — 10.5

14bt

= 21970

bt

⎛ ⎞⎜ ⎟⎝ ⎠

19 16b

t= 23.1 0.25 b

t⎛ ⎞− ⎜ ⎟⎝ ⎠

33bt

= 490bt

⎛ ⎞⎜ ⎟⎝ ⎠



6-10

10.5 — 10.5 47b

t=

490bt

⎛ ⎞⎜ ⎟⎝ ⎠





Eq. Set






16Rt

= 22.2 0.80 Rt

− 141Rt

= 23200

135

RR tt

⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟+⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠



6-11

10.5 5.1R

t= 11.4 0.40 R

t− 340R

t= 2

3200

135

RR tt

⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟+⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠





Eq. Set






21 23b

y

Lr

= 23.9 0.124 by

Lr

− 79by

Lr

= 287000

by

Lr

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠


6-12

10.5 — 10.5 91b

y

Lr

= 2

87000

by

Lr

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

25 28bR

t= 39.3 2.70 bR

t− 81bR

t=

Same as Eq. 6-11 with R = Rb

Round or oval tubesb

6-13

12 53bR

t= 18.6 1.00 bR

t− 134bR

t=


28 13bLd

t d= 40.5 0.93 bLd

t d− 29bLd

t d= 2

11400

bLdt d

⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠


6-14

13.5 — 13.5 29bLd

t d= 2

11400

bLdt d

⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

21 146

0.5b c

y

L SI J

= 23.9 0.240.5

b c

y

L SI J

− 17000.5

b c

y

L Sl J

= 24000

0.5b c

y

L Sl J

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠



6-15

10.5 — 10.5 2290

0.5b c

y

L Sl J

= 24000

0.5b c

y

L Sl J

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠


t




Eq. Set



(c) Allowable Stress Slenderness Between S1 and

S2 (d) Slenderness Limit S2 (e) Allowable Stress


6.8bt

= 27.3 0.93 bt

− 10bt

= 183

bt

⎛ ⎞⎜ ⎟⎝ ⎠


6-16

10.5 — 10.5 17b

t=

183bt

⎛ ⎞⎜ ⎟⎝ ⎠

21 22b

t= 27.3 0.29 b

t− 33b

t=

580bt

⎛ ⎞⎜ ⎟⎝ ⎠


6-17

13.5 — 10.5 55b

t=

580bt

⎛ ⎞⎜ ⎟⎝ ⎠

25 1.6bR

t= 26.2 0.94 bR

t− 141bR

t= 2

3800

135

bb

RR tt

⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟+⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠


Curved plates supported on both edgesb

6-18

12 1.3bR

t= 12.4 0.35 bR

t− 340bR

t=






Eq. Set






8.9bt

= 40.5 1.41bt

− 19bt

= 24900

bt

⎛ ⎞⎜ ⎟⎝ ⎠


6-19

13.5 — 13.5 19b

t= 2

4900

bt

⎛ ⎞⎜ ⎟⎝ ⎠

28 46h

t= 40.5 0.27 h

t− 75h

t=

1520ht

⎛ ⎞⎜ ⎟⎝ ⎠



6-20

13.5 — 13.5 113h

t=

1520ht

⎛ ⎞⎜ ⎟⎝ ⎠

12 36h

t= 15.6 0.099 h

t− 65h

t= 2

39000

ht

⎛ ⎞⎜ ⎟⎝ ⎠



6-21

6 — 6 81h

t= 2

39000

ht

⎛ ⎞⎜ ⎟⎝ ⎠






Any tension member

6-1 131 75.8† 6061—T6 Extrusions Up Through 25 mm, Sheet and Plate, Standard Structural Shapes, Drawn Tube, Rolled Rod and Bar


6-2 131 75.8† White bars apply to nonwelded members and to welded members at locations farther than 25 mm from a weld

Round or oval tubes

6-3 165 93† Shaded bars apply to within 25 mm of a weld



6-4 193 110†

On bolts

6-5 234 124† Bearing


6-6 159 82.7†


allowable stresses. b For tubes with circumferential welds, R/t and Rb/t, as applicable, shall be ≤20,

except when the design meets the details and post-weld heat treatment requirements of Article 6.5

c See Article 6.5 for additional provision regarding allowable stresses in welded members.


† Values when welded with 5183, 5356, or 5556 alloy filler wire, regardless of thickness. Values also apply to thicknesses ≤9.5 mm when welded with 4043 alloy filler wire; for greater thicknesses multiply allowable tensile stresses by 0.8 and allowable compressive and shear stresses by 0.75.

‡ For all thicknesses with filler alloys 5183, 5356 or 5556; and for thicknesses ≤9.5 mm for 4043 filler alloy.






Eq. Set


Slenderness ≤ S1 (b) Slenderness

Limit S1


S1 and S2 (d) Slenderness Limit

S2 (e) Allowable Stress

Slenderness ≥ S2 6-7 131

9.5kLr

= 139.3 0.869 kLr

⎛ ⎞− ⎜ ⎟⎝ ⎠

66kLr

= 2351600

kLr

⎛ ⎞⎜ ⎟⎝ ⎠


All columns

82.7† — 82.7† 65kLr

= ‡ 2351600

kLr

⎛ ⎞⎜ ⎟⎝ ⎠

6-8 131 5.2b

t= 159.3 5.45 b

t⎛ ⎞− ⎜ ⎟⎝ ⎠

10bt

= 1062

bt

⎛ ⎞⎜ ⎟⎝ ⎠


82.7† — 82.7†

13bt

= ‡ 1062

bt

⎛ ⎞⎜ ⎟⎝ ⎠

6-9 131 5.2b

t= 159.3 5.45 b

t⎛ ⎞− ⎜ ⎟⎝ ⎠

12bt

= 213580

bt

⎛ ⎞⎜ ⎟⎝ ⎠


82.7† — 82.7† 13b

t= ‡ 2

13580

bt

⎛ ⎞⎜ ⎟⎝ ⎠

6-10 131 16b

t= 159.3 1.72 b

t⎛ ⎞− ⎜ ⎟⎝ ⎠

33bt

= 3378

bt

⎛ ⎞⎜ ⎟⎝ ⎠



82.7† — 82.7† 41b

t= ‡

3378bt

⎛ ⎞⎜ ⎟⎝ ⎠




Eq. Set



Limit S1





16Rt

= 153.1 5.52 Rt

− 141Rt

= 222060

135

RR tt

⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟+⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠



6-11

82.7† 9.0R

t= 93.1 3.45 R

t− 290R

t= 2

22060

135

RR tt

⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟+⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠

145 23b

y

Lr

= 164.8 0.855 by

Lr

− 79by

Lr

= 2

600000

by

Lr

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠


6-12

82.7† — 82.7† 85by

Lr

= ‡ 2600000

by

Lr

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

172 28bR

t= 270.6 18.61 bR

t− 81bR

t=




6-13

96.5† 51bR

t= 164.8 9.58 bR

t− 137bR

t=






Eq. Set



Limit S1




6-12 145 23b

y

Lr

= 164.8 0.855 by

Lr

− 79by

Lr

= 2600000

by

Lr

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠


82.7† — 82.7†

85by

Lr

= ‡ 2600000

by

Lr

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

6-13 172 28bR

t= 270.6 18.61 bR

t− 81bR

t=



96.5† 51bR

t= 164.8 9.58 bR

t− 137bR

t=


6-14 193 13bLd

t d= 279.2 6.41 bLd

t d− 29bLd

t d= 2

78600

bLdt d

⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠


110† 26bLd

t d= 279.2 6.41 bLd

t d− 29bLd

t d= 2

78600

bLdt d

⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

6-15 145 146

0.5b c

y

L SI J

= 164.8 1.660.5

b c

y

L SI J

− 17000.5

b c

y

L Si J

= 165500

0.5b c

y

L SI J

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠



82.7† — 82.7† 2000

0.5b c

y

L Si J

= ‡ 165500

0.5b c

y

L SI J

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠


t




Eq. Set



Limit S1




6-16 145 6.8b

t= 188.2 6.41b

t− 10b

t=

1255bt

⎛ ⎞⎜ ⎟⎝ ⎠


82.7† — 82.7†

15bt

= ‡ 1255

bt

⎛ ⎞⎜ ⎟⎝ ⎠

6-17 145 22b

t= 188.2 2.0 b

t− 33b

t=

4000bt

⎛ ⎞⎜ ⎟⎝ ⎠


82.7† — 82.7† 48b

t= ‡

4000bt

⎛ ⎞⎜ ⎟⎝ ⎠

6-18 172 1.6bR

t= 180.6 6.48 bR

t− 141bR

t= 2

26200

135

bb

RR tt

⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟+⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠



96.5† 2.5bR

t= 101.4 3.03 bR

t− 290bR

t=






Eq. Set



Limit S1




6-19 193 8.9b

t= 279.2 9.72 b

t− 19b

t= 2

33780

bt

⎛ ⎞⎜ ⎟⎝ ⎠


110† 17b

t= 279.2 9.72 b

t− 19b

t= 2

33780

bt

⎛ ⎞⎜ ⎟⎝ ⎠

6-20 193 46h

t= 279.2 1.86 b

t− 75h

t=

10480ht

⎛ ⎞⎜ ⎟⎝ ⎠



110† — 110† 95h

t= ‡

10480ht

⎛ ⎞⎜ ⎟⎝ ⎠

6-21 83 36h

t= 107.6 0.683 h

t− 65h

t= 2

268900

ht

⎛ ⎞⎜ ⎟⎝ ⎠



51.7† — 51.7†

72ht

= ‡ 2268900

ht

⎛ ⎞⎜ ⎟⎝ ⎠


Table B-9—Allowable Stresses for Aluminum 6061—T6 Alloy (ksi)




Any tension member

6-1 19 11†


6-2 19 11†

Round or oval tubes

6-3 24 13.5†

6061—T6 Extrusions Up Through 1.0 in, Sheet and Plate, Standard Structural Shapes, Drawn Tube, Rolled Rod and Bar White bars apply to nonwelded members and to welded members at locations farther than 1.0 in from a weld Shaded bars apply to within 1.0 in of a weld



6-4 28 16†

On bolts

6-5 34 18† Bearing


6-6 23 12†


allowable stresses. b For tubes with circumferential welds, R/t and Rb/t, as applicable, shall be

≤20, except when the design meets the details and post-weld heat treatment requirements of Article 6.5.



e Values when welded with 5183, 5356, or 5556 alloy filler wire, regardless of thickness. Values also apply to thicknesses ≤0.375 in. when welded with 4043 alloy filler wire; for greater thicknesses multiply allowable tensile stresses by 0.8 and allowable compressive and shear stresses by 0.75.

f For all thicknesses with filler alloys 5183, 5356 or 5556 and for thicknesses ≤0.375 in. for 4043 filler alloy.





Eq. Set






9.5kLr

= 20.2 0.126 kLr

⎛ ⎞− ⎜ ⎟⎝ ⎠

66kLr

= 251000

kLr

⎛ ⎞⎜ ⎟⎝ ⎠


All columns

6-7

12† — 12† 65kL

r= ‡ 2

51000

kLr

⎛ ⎞⎜ ⎟⎝ ⎠

19 5.2b

t= 23.1 0.79 b

t⎛ ⎞− ⎜ ⎟⎝ ⎠

10bt

= 154

bt

⎛ ⎞⎜ ⎟⎝ ⎠


6-8

12† — 12† 13b

t= ‡

154bt

⎛ ⎞⎜ ⎟⎝ ⎠

19 5.2b

t= 23.1 1.39 b

t⎛ ⎞− ⎜ ⎟⎝ ⎠

12bt

= 21970

bt

⎛ ⎞⎜ ⎟⎝ ⎠


6-9

12† — 12† 13b

t= ‡ 2

1970

bt

⎛ ⎞⎜ ⎟⎝ ⎠

19 16b

t= 23.1 0.25 b

t⎛ ⎞− ⎜ ⎟⎝ ⎠

33bt

= 490bt

⎛ ⎞⎜ ⎟⎝ ⎠



6-10

12† — 12† 41b

t= ‡

490bt

⎛ ⎞⎜ ⎟⎝ ⎠





Eq. Set






16Rt

= 22.2 0.80 Rt

− 141Rt

= 23200

135

RR tt

⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟+⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠



6-11

12† 90R

t= 13.5 0.50 R

t− 290R

t= 2

3200

135

RR tt

⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟+⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠





Eq. Set





Slenderness ≥ S2

21 23by

Lr

= 23.9 0.124 by

Lr

− 79by

Lr

= 2

87000

by

Lr

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠


6-12

12† — 12† 85by

Lr

= ‡ 287000

by

Lr

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

25 28bRt

= 39.3 2.70 bRt

− 81bRt

= Same as Eq. 6-11 with

R = Rb Round or oval tubesb

6-13

14† 51bRt

= 23.9 1.39 bRt

− 137bRt

= Same as Eq. 6-11 with

R = Rb

28 13bLd

t d= 40.5 0.93 bLd

t d− 29bLd

t d= 2

11400

bLdt d

⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠


6-14

16† 26bLd

t d= 40.5 0.93 bLd

t d− 29bLd

t d= 2

11400

bLdt d

⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

21 146

0.5b c

y

L SI J

= 23.9 0.240.5

b c

y

L SI J

− 17000.5

b c

y

L SI J

= 24000

0.5b c

y

L SI J

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠



6-15

12† — 12† 20000.5

b c

y

L SI J

= ‡ 24000

0.5b c

y

L SI J

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠


t




Eq. Set


Slenderness ≤S1 (b) Slenderness Limit S1


S1and S2 (d) Slenderness Limit S2


21 6.8b

t= 27.3 0.93 b

t− 10b

t=

182bt

⎛ ⎞⎜ ⎟⎝ ⎠


6-16

12† — 12† 15b

t= ‡

182bt

⎛ ⎞⎜ ⎟⎝ ⎠

21 22b

t= 27.3 0.29 b

t− 33b

t=

580bt

⎛ ⎞⎜ ⎟⎝ ⎠


6-17

12† — 12† 48b

t= ‡

580bt

⎛ ⎞⎜ ⎟⎝ ⎠

25 1.6bR

t= 26.2 0.94 bR

t− 141bR

t=

3800

135

bb

RR tt

⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟+⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠



6-18

14† 2.5bR

t= 1.47 0.44 bR

t− 290bR

t=






Eq. Set


Slenderness ≤S1 (b) Slenderness Limit S1


S1and S2 (d) Slenderness Limit S2


28 8.9bt

= 40.5 1.41bt

− 19bt

= 24900

bt

⎛ ⎞⎜ ⎟⎝ ⎠


6-19

16† 17bt

= 40.5 1.41bt

− 19bt

= 24900

bt

⎛ ⎞⎜ ⎟⎝ ⎠

28 46h

t= 40.5 0.27 h

t− 75h

t=

1520ht

⎛ ⎞⎜ ⎟⎝ ⎠



6-20

16† — 16† 95ht

= ‡ 1520

ht

⎛ ⎞⎜ ⎟⎝ ⎠

12 36ht

= 15.6 0.099 ht

− 65ht

= 239000

ht

⎛ ⎞⎜ ⎟⎝ ⎠



6-21

7.5† — 7.5† 72ht

= ‡ 239000

ht

⎛ ⎞⎜ ⎟⎝ ⎠






Any tension member

6-1 103 44.8 6063—T6 Extrusions, Pipe


6-2 103 44.8 White bars apply to nonwelded members and to welded members at locations farther than 25.4 mm from a weld

Round or oval tubes

6-3 124 55.2 Shaded bars apply to within 25.4 mm of a weld



6-4 138 58.6

Bearing On bolts 6-5 165 93.1 On flat surfaces and on bolts in

slotted holes

6-6 110 62.1



≤20, except when the design meets the details and post-weld heat treatment requirements of Article 6.5







Eq. Set


Slenderness ≤ S1





Slenderness ≥ S2 93.1

9.5kLr

= 97.9 0.510 kLr

⎛ ⎞− ⎜ ⎟⎝ ⎠

78kLr

= 2351600

kLr

⎛ ⎞⎜ ⎟⎝ ⎠


All columns

6-7

44.8 — 44.8 89kL

r= 2

351600

kLr

⎛ ⎞⎜ ⎟⎝ ⎠

93.1 5.7b

t= 111.0 3.17 b

t⎛ ⎞− ⎜ ⎟⎝ ⎠

12bt

= 889bt

⎛ ⎞⎜ ⎟⎝ ⎠


6-8

44.8 — 44.8 20b

t=

889bt

⎛ ⎞⎜ ⎟⎝ ⎠

93.1 5.7b

t= 111.0 3.17 b

t⎛ ⎞− ⎜ ⎟⎝ ⎠

15bt

= 213580

bt

⎛ ⎞⎜ ⎟⎝ ⎠


6-9

44.8 — 44.8 17b

t= 2

13580

bt

⎛ ⎞⎜ ⎟⎝ ⎠

6-10 93.1 18b

t= 111.0 1.0 b

t⎛ ⎞− ⎜ ⎟⎝ ⎠

39bt

= 2827

bt

⎛ ⎞⎜ ⎟⎝ ⎠



44.8 — 44.8 63b

t=

2827bt

⎛ ⎞⎜ ⎟⎝ ⎠





Eq. Set


Slenderness ≤ S1





Slenderness ≥ S2 6-11 93.1

18Rt

= 107.6 3.45 Rt

− 188Rt

= 222060

135

RR tt

⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟+⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠



44.8 10R

t= 49.6 1.52 R

t− 510R

t= 2

22060

135

RR tt

⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟+⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠

103 23b

y

L

r= 151.1 0.503 b

y

L

r− 94b

y

Lr

= 2

600000

by

Lr

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠


6-12

44.8 — 44.8 116b

y

Lr

= 2600000

by

Lr

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

124 33bR

t= 191.0 11.72 bR

t− 102bR

t=




6-13

55.2 62bR

t= 88.3 4.21 bR

t− 206bR

t=






Eq. Set






15bLdt d

= 192.4 3.65 bLdr d

− 35bLdt d

= 278600

bLdt d

⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠


6-14

58.6 — 58.6 37bRd

t t= 2

78600

bLdt d

⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

103 145

0.5b c

y

L SI J

= 115.1 0.9720.5

SLb cJI y

−

23800.5

b c

y

L SI J

= 165500

0.5b c

y

L SI J

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠



6-15

44.8 — 44.8 3690

0.5b c

y

L SI J

= 165500

0.5b c

y

L SI J

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠


t




Eq. Set






7.4bt

= 131.0 3.72 bt

− 12bt

= 1048

bt

⎛ ⎞⎜ ⎟⎝ ⎠


6-16

44.8 — 44.8 23b

t=

1048bt

⎛ ⎞⎜ ⎟⎝ ⎠

103 24b

t= 131.0 1.172 b

t− 39b

t=

3309bt

⎛ ⎞⎜ ⎟⎝ ⎠


6-17

44.8 — 44.8 74b

t=

3309bt

⎛ ⎞⎜ ⎟⎝ ⎠

124 0.7bR

t= 127.6 4.07 bR

t− 188bR

t= 2

26200

135

bb

RR tt

⎛ ⎞⎜ ⎟⎛ ⎞ ⎜ ⎟+⎜ ⎟ ⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠



6-18

55.2 0.3bR

t= 5.45 1.31 bR

t− 510bR

t=






Eq. Set






9.8bt

= 192.4 5.59 bt

− 23bt

= 233780

bt

⎛ ⎞⎜ ⎟⎝ ⎠


6-19

58.6 — 58.6 24b

t= 2

33780

bt

⎛ ⎞⎜ ⎟⎝ ⎠

138 51h

t= 192.4 1.07 h

t− 90h

t=

8687ht

⎛ ⎞⎜ ⎟⎝ ⎠



6-20

58.6 — 58.6 148h

t=

8687ht

⎛ ⎞⎜ ⎟⎝ ⎠

58.6 39h

t= 73.8 0.386 h

t− 78h

t= 2

268900

ht

⎛ ⎞⎜ ⎟⎝ ⎠



6-21

26.9 — 26.9 100h

t= 2

268900

ht

⎛ ⎞⎜ ⎟⎝ ⎠


Table B-11—Allowable Stresses for Aluminum 6063—T6 Alloy




Any tension member

6-1 15 6.5 6063—T6 Extrusions, Pipe


6-2 15 6.5 White bars apply to nonwelded members and to welded members at locations farther than 1.0 in from a weld

Round or oval tubes

6-3 18 8.0 Shaded bars apply to within 1.0 in of a weld



6-4 20 8.5

On bolts 6-5 24 13.5 Bearing On flat surfaces and on bolts in slotted holes

6-6 16 9



≤20, except when the design meets the details and post-weld heat treatment requirements of Article 6.5







Eq. Set






9.5kLr

= 14.2 0.074 kLr

⎛ ⎞− ⎜ ⎟⎝ ⎠

78kLr

= 251000

kLr

⎛ ⎞⎜ ⎟⎝ ⎠


All columns

6-7

6.5 — 6.5 89kL

r= 2

51000

kLr

⎛ ⎞⎜ ⎟⎝ ⎠

13.5 5.7b

t= 16.1 0.46 b

t⎛ ⎞− ⎜ ⎟⎝ ⎠

12bt

= 129

bt

⎛ ⎞⎜ ⎟⎝ ⎠


6-8

6.5 — 6.5 20b

t=

129bt

⎛ ⎞⎜ ⎟⎝ ⎠

13.5 5.7b

t= 16.1 0.46 b

t⎛ ⎞− ⎜ ⎟⎝ ⎠

15bt

= 21970

bt

⎛ ⎞⎜ ⎟⎝ ⎠



6-9

6.5 — 6.5 17b

t= 2

1970

bt

⎛ ⎞⎜ ⎟⎝ ⎠





Eq. Set






18bt

= 16.1 0.144 bt

⎛ ⎞− ⎜ ⎟⎝ ⎠

39bt

= 410bt

⎛ ⎞⎜ ⎟⎝ ⎠


6-10

6.5 — 6.5 63b

t=

410bt

⎛ ⎞⎜ ⎟⎝ ⎠

13.5 18R

t= 15.6 0.50 R

t− 188R

t= 2

3200

135

RR tt

⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟+⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠



6-11

6.5 10R

t= 7.2 0.22 R

t− 510R

t= 2

3200

135

RR tt

⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟+⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠





Eq. Set






15 206bR

t= 16.7 0.073 b

y

Lr

− 94by

Lr

= 2

87000

by

Lr

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠


6-12

6.5 — 6.5 116b

y

Lr

= 2

87000

by

Lr

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

18 33bR

t= 27.7 1.70 bR

t− 102bR

t=



6-13

8 62bR

t= 12.8 0.61 bR

t− 206bR

t=


20 15bLd

t d= 27.9 0.53 bLd

t d− 35bLd

t d= 2

11400

bLdt d

⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠


6-14

8.5 — 8.5 37bLd

t d= 2

11400

bLdt d

⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

15 145

0.5b c

y

L SI J

= 16.7 0.1410.5

b c

y

L SI J

− 23800.5

b c

y

L SI J

= 24000

0.5b c

y

L SI J

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠



6-15

6.5 — 6.5 3690

0.5b c

y

L SI J

= 24000

0.5b c

y

L SI J

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠


t




Eq. Set


Slenderness ≤ S1






7.4bt

= 19.0 0.54 bt

− 12bt

= 152

bt

⎛ ⎞⎜ ⎟⎝ ⎠


6-16

6.5 — 6.5 23b

t=

152bt

⎛ ⎞⎜ ⎟⎝ ⎠

15 24b

t= 19.0 0.170 b

t− 39b

t=

480bt

⎛ ⎞⎜ ⎟⎝ ⎠


6-17

6.5 — 6.5 74b

t=

480bt

⎛ ⎞⎜ ⎟⎝ ⎠

18 0.7bR

t= 18.5 0.59 bR

t− 188bR

t= 2

3800

135

bb

RR tt

⎛ ⎞⎜ ⎟⎛ ⎞⎜ ⎟+⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎝ ⎠



6-18

8 0.3bR

t= 7.9 0.19 bR

t− 510bR

t=






Eq. Set






20 9.8b

t= 27.9 0.81b

t− 23b

t= 2

4900

bt

⎛ ⎞⎜ ⎟⎝ ⎠


6-19

8.5 — 8.5 24b

t= 2

4900

bt

⎛ ⎞⎜ ⎟⎝ ⎠

20 51h

t= 27.9 0.155 h

t− 90h

t=

1260ht

⎛ ⎞⎜ ⎟⎝ ⎠



6-20

8.5 — 8.5 148h

t=

1260ht

⎛ ⎞⎜ ⎟⎝ ⎠

8.5 39h

t= 10.7 0.056 h

t− 78h

t= 2

39000

ht

⎛ ⎞⎜ ⎟⎝ ⎠



6-21

3.9 — 3.9 100h

t= 2

39000

ht

⎛ ⎞⎜ ⎟⎝ ⎠

B7—REFERENCES

Author, Initials. 1985. “Graphs to Determine Structure Deflections,” Transmission and Distribution. July 1985, pp. X–Y.

Timoshenko, S. 1957. Strength of Materials—Part II. D. Van Nostrand Company, Princeton, NJ.

apxb lts 5trev

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