AIAA Design Engineers Guide P2

oO Normal stress, ksi Shear stress, ksi wd Flange flexibility factor, a Angle between neutral axis of beams and direction of diagonal tension, deg* ` fox c Subscripts: “we DT Diagonal

4-8 AIAA DESIGN ENGINEERS GUIDE

Torsion Formulas— Solid and

Tubular Sections—cont

s

u 2

at midpoints of long sides

—

|

3T (approximately bt for narrow rectang/les)

bt? t t v2

J= = [1-083 5 +0.052( `" |

(approximately 3 for narrow rectangles)

.x« T

For sides with thicknesst,, average f, = 2 bd

1

For sides with thickness fo,

tạ average f, = 2t.bd

2

Tụ = 2bở trụn F su (tmin = ty OF to, whichever is smaller)

2b7d?

~ (bit,) (dita)

Thin-walled sections should be checked for buckling

Formulas for C, are not given in this table because for these

cross sections, C, is negligibly small in comparison to J

Torsion Formulas—Thin-Walled

Open Sections

Nomenclature

A = total area of section, in.?

A, = area of one flange in.?

C =torsion-bending constant,*in.®

= moment of inertia of flange no 1 about Y axis, in.4

= moment of inertia of flange no 2 about Y axis, in.‘

~ moment of inertia of section about Y axis, in.4

= radius of gyration of section about Y axis, in

r = radius of gyration of section about X axis, in

1

J= 3 (2ptì + dt3)

d*ly C.=

k 4

J= 3 (b,t} + Dot5 + dt3)

Shear

- Center C.- othe

yi!,— Yale e=

ly

rm

a

mmmmmmmmm*xeẽm1m

STRUCTURAL ELEMENTS Torsion Formulas—Thin-Walled Open Sections—cont

J= — tì ats

C,=0

4-9

1 J= 3 (2b13 + d13)

X X — d“ly x(a — X)

l 2 { y

Shear | j a= — xd?

1

3

‘The torsion constant J is a measure of the stiffness of a mem- ber in pure twisting The torsion-bending constant C, is a mea- sure of the resistance to rotation that arises because of re- straint of warping of the cross section

Position of Flexural Center Q for Different Sections

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Pin S&S coc =

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Position of Flexural Center Q for

Different Sections—cont

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Shear-buckling Curves for Sheet Materials

IV Ky

VK, = Shear-buckling constant

50

+ oO

30 t = Sheet thickness

E = Modulus of Curves are minimum guaranteed and are for room - temperature use only

20

0

tvK,

A AMS 4901 Ti

B 18-8 Type 301 1⁄2 Hard

C 7075—T6 ALC 0.040—0.249

D 7075—T6 ALC 0.012—0.039

E 2024—T81 ALC t2=0.063

F 2024—T81 ALC t20.063

b = Short side of rectangular panel

momen

K

STRUCTURAL ELEMENTS 4-13

Allowable Shear Flow—75S-T6 Clad Web

Excessive wrinkling and oil canning,

6000

5000

4000

Web thickness (t), In

‘ t=.156

125

2000

1500

1000

800

500

400

300

3 4 5 6 7 8910 1.5

Stiffener Spacing (d), in

2253 4

Shear-buckling Constant (VK, )

_- — Cc c S S

3.8

3.6 — a=Long side

c = Clamped

s = Simple support 3.4

3.2

3.0

2.8

2.6

2.4

¬" '

AIAA DESIGN ENGINEERS GUIDE

Beam Diagonal Tension

Nomenclature

4-14

Cross-sectional area, in?

Spacing of uprights, in

Shear flow (shear force per in.), kips per in

Thickness, in (when used without subscript, signifies

thickness of web)

Diagonal-tension factor

depth of beam, in

Length of upright (measured between centroids of upright-

to-flange rivet patterns), in

oO Normal stress, ksi

Shear stress, ksi

wd Flange flexibility factor,

a Angle between neutral axis of beams and direction of

diagonal tension, deg* `

fox c

Subscripts: “we

DT Diagonal tension

IDT Incomplete diagonal tension

PDT Pure diagonal tension

U Upright

e Effective

Maximum Stress to Average Stress in

Web Stiffener

For curved webs: for rings, read abscissa as d/h: for stringers, read

abscissa as h/d

1.8 t ' | i

1 _ | ' ‡ `

`

El > 1.4 4 : -——

1

6 | —

p—1 ._.8 | '

_d

hy

Angle Factor C,

°

tan œ

veka

we

oo

Bek =—= v

STRUCTURAL ELEMENTS 4-15 Stress-concentration Factors C, and C;

(ty + ly)hạ

wd

lo =! of compression flange

I7 =! of tension flange

Diagonal-tension Analysis Chart

9| 0.05.10 15 20'.25.30.35 45 45 Z ff f/f

8 rt —- L + af .- ~ 5056 ff f fo fA

7 — ~ Si 7 f- + * Lf * - - 4 6E + 1 1: 1 90° 2⁄2

c/- 4.20% ;

ọ 5.00 -

9 1.0

4-16 AIAA DESIGN ENGINEERS GUIDE .-

STRUCTURAL ELEMENTS 4-17 Angle of Diagonal Tension Interaction of Column Failure with

The method of analysis of columns subject to loca! failure can

be summarized as follows:

I8

Incomplete diagonal tension

fe)

1.0

45°

a Sections having four cor- b.Sections having two cor

ners, attached to sheets

9

41.99°

L] "LE L along both flanges

cis

c Seclions having only two s-

corners, but restrained E: —” against

column failure | d Sections having three cor

7

34.99° - about axis thru corners ners attached to a sheet

along one unlipped flange

6

30.96°

(Arrow represents direction

0

of column failure) k: - a

For local tailing stress (upper limit of column curve) use Crip-

7

pling stress F (see note)

Pure diagonal tension

e Sections having only twol f Sections having only two

in any direction)

——

Loot FP

of column failure) of column failure)

44

40

For local failing stress (upper limit of column curve) use local buckling stress Fou (see note)

WU

lu

ke 36

e

ờ

32

the allowable ultimate stress, Fo

The calculations for Fre and F should be made by

28

reference to sources not included in this handbook

24

°` h

Ẹ a Comparison of Different Column Curves

20

+ œ œ _ _ = +> rm ' i

Wo Wow

Wo

Ww

W

a -

| Reduced

—L Ÿ modulus

| | | (a)

50,000 Tangent

modulus

= 40,000 line equation

> me IN

2 Pccơ À >

8

S 30,000

(d) Johnson's Parabola 20,000

10,000

: All curves drawn C=10E 10” Ip¿n.2

L

>

Society of Allied Weight Engineers Used with permission

ome

m om

mm

Te

`

RE

`

|

Comparison of Different Column |

coefficient

L

i:

where Pe=criticalcolumnioad C= end fixity coefficient a

ỊP

+

c=1

Uniform column, axially The critical column stress (Øg) is — CHẾE x a L loaded, pinned ends 1,

ve~

where ~ = radius of gyration = 7A

lp b) Reduced Modulus Curve

Ee : =

short column range is

rg

c=4

3 f Uniform column, axially

where

Cc

2

IP

| 4EE, )

= ( ————— E,=~VEE;

E+vET

St tangent modulus

- Uniform column axially c= 2.05 () Tangent Modulus Curve E,

loaded, one end fixed, 1

+

về

sc ek (d) Johnson Parabolic Formula

The Johnson equation gives the critical 2 2 ¬

short column stress (Øee): ƠØc§(L/Ð) Fo

p

(e) Straight Line Equation

L loaded, one end fixed,

—

| where Oo,, k, and VC are chosen to give best agreement with _—

Alloys— Based on Tangent Modulus

Pee Uniform column, distributed

axial load, one end fixed, one 1

—= I’ = INC = equivalent pin-end length E | =

|

‘ON >

- ——

— ô

@

Vo |= total column length

70 r `

{

t

XS, `

| —r

AE C = fixity factor K-

C = 1.87

ø S0Ƒ <3) S3 `@ SA, Curves are based upon “B |

| | axial load, pinned ends a 739

Oo và NON &, So \ values in MIL-HDBK-5 tables —_ - = =O

, i

S OOS `“ \

" = tp

= 50 xà

|

- L

0

~

h- I - Uniform column, distributed

3

PL axial load, fixed ends 1 — 0.365

E 30

c

5

& : m

oO

: ——

tl

©

-— k —

c=3.55

mi Uniform column, distributed (approx )

axial load, one end fixed, 10Ƒ

fe * ) rf one end pinned 1 _ 0.530

0 : _:

2 fe `

p section radius of gyration any

j

é

4-20

AIAA DESIGN ENGINEERS GUIDE

Column Stress for Aluminum Alloy

Columns

80

Johnson-Euler formulas

70

2 L_\?

F2.(—=)

Fc=Fcc- _—_P`>~_—

4x°E

C= restraint coefficient

E = 10,300,000 psi

Fo = crushing stress, psi Euler Formula

Ngư,

30

20

10

0 20 40 60 80 100

L’ip= LpVC

Column Stress for Magnesium Alloy

60

nn oO

W oO

NR Oo

Columns

¬.- (-=)

a cu ee 4r°E pvc

where

Fu = Crushing stress, psi

L = column length, in

\ o = radius of gyration

C = coefficient of restraint

F.=——=—~

E = mod of elas = 6.5 x 108 —

¬

0 „_ 20 40 60 80 100

L'/p=LlprC

120

eae

% Ä

a pepo

OTRUCTURAL ELEMENTS

Column Stress for Steel Columns

4-21

200

Steel columns

Cn L

F.=Fc pve

4r?E where

E = 29,000,000 psi

F = stress, psi

Euier Formula

VC 2 F.= r2E(——)

Applies to corrosion and noncorrosion

20 40 60 80 100 Llp = UpVvC

Column Stress for Titanium Alloy

Columns

70 F =F _ ức) ——— ~—

ˆ C cc 4r?E (=)

where

60 Fu = crushing stress, psi

L = column lengtn, in

p = radius of gyration C= restraint coefficient

50 \ E =mod of

elas = 15.5 x 106

\ E r^E

L’/p= LipVC

4-22

i

dend

moment

Mmac/M,

= max,

bending

moment/applie

AIAA DESIGN ENGINEERS GUIDE

Beam-Column Cures for Centrally

Loaded Columns with End Couples

2

M,

0

Ww

=

seg

O

6 Mạ

4

2

0

Ị I

J

0

P= applied axialioad

Po =F x A=column buckling toad

M,, M2 = applied end moment

Mmax = Max total bending moment including secondaries

Ref: Westergaard, H.M., “Buckling of Elastic Structure,”

Trans ASCE 1922, p 594

Buckling Stress Coefficients for Flat

Plates in Shear

14

Elastic buckling stress Fore, = KE(t/b)2

where t= plate thickness

12

10

e

2

s8

Ỗ

Oo

œ

°

a

6

E

oO

5

2

H

X4

2 ~

Ref TN 1222 1559

Timoshenko, “Theory of Elastic Stabilitv.”

R&S Dsts Snccis “OUessed Skin Structures

0

0.2 0.4 0.6

0.8 1.0

along side

om

STRUCTURAL ELEMENTS 4-23 Buckling Stress Coefficients for Flat

buckling stress

coefficien

K=

0.9 0.8

ectangular Plates Loaded in Compression on the Long Side

Elastic buckling stress F ore = KE(t/b)2 where t = Plate thickness

||

Ỏ

e

>

eet

ae”

f f Ị Ị pà®

Long side clamped b Short side simply Supported” .@/d

1.0 Short side

b

~

a long side

Buckling Stress Coefficients for Flat

NwMWos

buckling stress

coefficient

Ref:

one long side free other side clamped

Rectangular Plates Loaded in Compression on the Short Side

ed

au sides CAE

Long sides clamped and Short sides Simply Supported

Long sid

Elastic buckling stress Fo, = KE(t/b)2 where t = plate thickness

Short sides SiMply supported

ee

0.2 0.4 0.6

0.8 1.0

Đ Short Side

a long side

Timoshenko, “Theory of Elastic Stability,” RAS Data Sheets, “Stressed Skin Structures”

3

“

é my

4

a

3

i

Ba

ý

h

ew

=a

4-27 STRUCTURAL ELEMENTS

Re

Bolt and Screw Strengths—cont

-

ee de

AIAA DESIGN ENGINEERS GUIDE

Bolt and Screw Strengths

4-26

'VPÿL1/8-8-1IW

mm

mm

mm

mm

mm

mm

mm

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nạ

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