Beam With Torsion

Daniel PROJECT : PAGE :

Tian Li CLIENT : DESIGN BY : JOB NO. : DATE : REVIEW BY :

Simply Supported Beam Design with Torsional Loading Based on AISC 13th (AISC 360-05)

INPUT DATA & DESIGN SUMMARY

BEAM SECTION = > W10X54 = > A dGRAVITY DISTRIBUTED LOAD w = 1.15 kips / ft 15.8 10.1 4.38 2.55 303 60

LATERAL POINT LOAD AT MID F = 5 kips lTORSION AT MID SPAN T = 5.1 ft-kips 103 20.6 0.01744 0.37 10.00 0.62

AXIAL LOAD P = 96 kips

BEAM LENGTH 15 ft

BEAM YIELD STRESS 50 ksi

VERTICAL BENDING UNBRACED LENGTH 15 ft

AXIAL VERTICAL UNBRACED LENGTH 15 ft

AXIAL HORIZONTAL UNBRACED LENGTH 7.5 ft

ANALYSIS

DETERMINE GOVERNING MOMENTS AT MIDDLE OF SPAN

32.3 ft-kips

18.8 ft-kips

22.7 ft-kips THE BEAM DESIGN IS ADEQUATE.

13.3 ft-kips

0.584 ,(Philip page 101)

CHECK TORSIONAL CAPACITY (AISC 360-05 H3.3 & Philip page 100)

0.73 < 1.00 [Satisfactory]

Where 21.93 ksi

29.94 ksi

CHECK COMBINED COMPRESSION AND BENDING CAPACITY (AISC 360-05, H1)

1.08 < 4/3 [Satisfactory]

Where 96 kips

109.7 ft-kips, (Sim. from Philip page 100)

18.8 ft-kips

721 / 1.67 = 431.971 kips, (AISC 360-05 Chapter E)

> [Satisfactory]

252.623 / 1.67 = 151.271 ft-kips, (AISC 360-05 Chapter F)

> [Satisfactory]

130.417 / 1.67 = 78.0938 ft-kips, (AISC 360-05 Chapter F)

> [Satisfactory]

DETERMINE DEFLECTIONS

0.22127

Where 11200 ksi 29000 ksi

rx ry Ix Sx

Iy Sy tw bf tf

L =

Fy =

Lb =

Lx =

Ly =

Mx = w L2/ 8 =

My = F L / 4 =

M0 = T L / (4d) =

MT = bM0 =

fbx / Fnx =

fbx = Mx / Sx + 2MT / Sy =

Fnx = Fy / WT = Fy / 1.67 =

Pr =

Mrx = (Mx / Sx + 2MT / Sy) Sx =

Mry =

Pc = Pn / Wc =

3/4 Pr

Mcx = Mn / Wb =

Mrx

Mcy = Mn / Wb =

3/4 Mry

o , max twist angle at middle (Philip page 100)

G = Es =

24 sinh

2sinh

L

L L

l

bl l

2sinh

2 sinh2 22 sinh

LT LL

GJ L

lll

l l

8, 0.2

9

, 0.22

ryr rx r

c cx cy c

ryr rx r

c cx cy c

MP M PforP M M P

MP M PforP M M P

J = 1.82 in4

(cont'd)

0.15 in = L / 1207 , vertical deflection at middle

Where 303

0.20 in = L / 885 , horizontal deflection at middle

Where 103

Technical References: 1. AISC: "Steel Construction Manual 13th Edition", American Institute of Steel Construction, 2005. 2. Philip H. Lin: "Simplified Design for Torsional Loading of Rolled Steel Members", Engineering Journal, AISC, 1977.

I3 = Ix sin2(90-) + Iy cos2(90-) = in4 , (AISC 13th Page 17-42)

I4 = Ix cos2(90-) + Iy sin2(90-) = in4 , (AISC 13th Page 17-42)

4

3

5384

wLvert E I

3

448

F Lhoriz E I

Daniel ROJECT : PAGE :

Tian LiCLIENT : DESIGN BY :

JOB NO. : DATE : REVIEW BY :

Plate Girder Design Based on AISC Manual 13th Edition (AISC 360-05)


STEEL YIELD STRESS 50 ksi

SIMPLY SUPPORTED SPAN 15 ft

SUPERIMPOSED UNIFORM DEAD LOAD kips / ft

UNIFORM LIVE LOAD kips / ft

POINT DEAD LOAD kips

POINT LIVE LOAD kips

DISTANCE POINT LOAD TO END ft

TOP FLANGE WIDTH 10.00 in

TOP FLANGE THICKNESS 0.62 in

BOTTOM FLANGE WIDTH 10 in

BOTTOM FLANGE THICKNESS 0.615 in

WEB THICKNESS 0.37 inBEAM DEPTH 10.1 in Err:502

UNBRACED LENGTH 15 ft FLANGE TO WEB WELDING USE 1/4 in - 24 in @ 2293 in o.c.

THE GIRDER DESIGN IS ADEQUATE.

ANALYSIS

CHECK LIMITING WIDTH-THICKNESS RATIOS FOR WEB (AISC 365-05 Table B4.1)

23.97 < 137.27

< 90.55

Compact Web

where E = 29000 ksi

137.27

87.03

90.55

8.87 in 9.46 in

277.5 ft-kips 250.0 ft-kips

CHECK LIMITING WIDTH-THICKNESS RATIOS FOR FLANGES (AISC 365-05 Table B4.1)

8.13 < 25.09

< 9.15

Compact Flanges

where 25.09

9.15

0.76

60 60

35 ksi, (AISC 360-05 Table note B4.1 & Eq F4-6)

DETERMINE CRITERIA FOR ALLOWABLE FLEXURAL STRENGTH (AISC 365-05 Table F1.1)

Required Conditions

Chapter F Sections

F2 F3 F4 F5

Double Symmetric x xCompact Web x x

xNoncompact Web

Slender Web 151.3 ft-kipsCompact Flanges x ( from following analysis)

Noncompact FlangesSlender Flanges

Applicable Section ok

9.02 ft

33.66 ft

where 2.55 in 60

Fy =

S =

DL =

LL =

PDL =

PLL =

c =

bf,top =

tf,top =

bf,bot =

tf,bot =

tw = d =

Lb =

hc / tw = lr =

lp =

lr = 5.7 (E / Fy)0.5 =

lp = (hc / hp) (E / Fy)0.5 / (0.54 Mp / My -0.09)2 = ,for Af,top ≠ Af,bot

lp = 3.76 (E / Fy)0.5 = ,for Af,top = Af,bot

hc = hp =

Mp = My =

0.5 bf,top / tf,top = lr =

lp =

lr = 1.0 (kc E / FL)0.5 =

lp = 0.38 (E / Fy)0.5 =

kc = Min [0.76 , Max (0.35 , 4 / (h / hw)0.5 )] =

Sxt = in3 Sxc = in3

FL =

Mallowable = Mn / Wb =

DETERMINE ALLOWABLE FLEXURAL STRENGTH , Mn / Wb , BASED ON AISC 365-05 Chapter F2

ry = Sx = in3

1.76p yy

EL r

F

2

0

0.7 01.95 1 1 6.760.7 0.7

r ts tsy x y

FE Jc Ey S hxL r rE JcS h FF

9.49 in 103h0 = Iy = in4

(cont'd)

2316.6 , (AISC 365-05 F2.2)

31.8102

(Use J = 1.82

2.85 in

1.0 1.0 , (AISC Manual 13th Table 3-1, page 3-10)

101.524 ksi

252.6 ft-kips

151.3 ft-kips where 1.67 , (AISC 365-05 F1)

<== Not Applicable.

284.1 ft-kips

where 8.13

9.15 25.09

151.3 ft-kips

<== Not Applicable.

6.33 ft

33.85 ft

where 0.53

2.87 in

277.5 ft-kips

250 ft-kips 250 ft-kips

25.64

87.03 137.27

1.11

102.385 0.50 > 0.23, AISC 360-05 F4-5 )

245.2238 ft-kips

Cw = Iy h02 / 4 =

J = [tw d (tw2 + d2)] / 12 = in4, (not applicable if taken web only, EIT Manual page 26)

in4 )

rts =[( Iy Cw)0.5/ Sx)]0.5 =

c = Cb =

Mallowable, F2 =Mn / Wb = Wb =


l = bf / (2 tf) =

lpf = lp = lrf = lr =

Mallowable, F3 =Min(Mn,F2 , Mn,F3) / Wb =


aw =hc tw/ (bfc tfc) =

Mp = Min [Zx Fy , 1.6Sxc Fy ] =

Myc = Sxc Fy = Myt = Sxt Fy =

l = hc / tw =

lpw = lp = lrw = lr =

ksi, (for Iyc / Iy =

22

20

1 0.078b bcr

tsxb

ts

E JcC LF

rS hL

r

, 2

,

0.7 , ,

, ,

p b p

b pp p y p p b rb xn F

r p

cr p r bx

forM L L

L LMin forC SM M F M L L LM

L L

Min forSF M L L

, 3

2

0.7 ,

0.9,

pfp p y x

rf pfn F

c x

for Noncompact FlangesSM M F

MEk S for Slender Flanges

l ll l

l

1.1p ty

EL r

F

2

0

01.95 1 1 6.76 Lr t

L xc

E J S hF xcL rE JS hF

, 4.2

,

, ,

, ,

pc yc b p

b ppc yc pc yc L pc yc p b rb xcn F

r p

cr pc yc r bxc

forR M L L

L LMin forC SR M R M F R M L L LM

L L

Min forSF R M L L

20

0

112

6

fct

w

br

h ha

d dh

, /

1 , , /

pc w pw

yc

pcp p pw p

c w pwyc yc ycrw pw

Mfor h t

MR

M M MMin for h t

M M M

l

l ll

l l

22

20

1 0.078b bcr

txcb

t

E JC LF

rS hLr

(cont'd)

277.5 ft-kips

1.11

146.8 ft-kips

<== Not Applicable.

21.62 ft

41.4981 ksi

50 ksi

1

124.2 ft-kips

DETERMINE ALLOWABLE SHEAR STRENGTH , Vn / Wv , BASED ON AISC 365-05 Chapter G2

9 24 , 3.74

14.7 ft

5.00 1.000

112.11 kips

67.1317 kips 1.67 , (AISC 365-05 G1)

TOTAL SUPERIMPOSED GRAVITY LOAD

w = DL + LL = 0.000 kips / ft , 0.00 kips

CHECK EACH SECTION CAPACITIES

Section Left 0.00 S 0.00 S 0.00 S 0.00 S 0.00 S Point 0.17 S 0.33 S 0.50 S 0.67 S 0.83 S RightDistance 0 0.00 0.00 0.00 0.00 0.00 0.00 2.50 5.00 7.50 10.00 12.50 15.00

10 10 10 10 10 10 10 10 10 10 10 10 105 5 5 5 5 5 5 5 5 5 5 5 5

299 299 299 299 299 299 299 299 299 299 299 299 29953.0 53.0 53.0 53.0 53.0 53.0 53.0 53.0 53.0 53.0 53.0 53.0 53.00.40 0.40 0.40 0.40 0.40 0.40 0.00 -0.27 -0.13 0.00 0.13 0.27 0.40

0 0 0 0 0 0 0 1 1 1 1 1 0

Mallowable, F4 = Min( Mn,F4.2 , Mn,F4.3, RptMyt) / Wb =


Mallowable, F5 = Min( RpgFySxc , RpgFcr,F5.2Sxc , RpgFcr,F5.3Sxc , FySxt) / Wb =

h = d - tf,top - tf,bot = in , h / tw = Aw = in2 ,a =

Vn = 0.6 FyAwCv =

Vallowable = Vn / Wv = Wv =

P = PDL + PLL =

d (in)y (in)

I (in4)Wt (plf)V (kips)M (ft-k)

25

5 , / 3/

5 , / 3v

for a ha h

for a h

k

2

1.0 , / 1.10

1.10, 1.10 / 1.37

/

1.51, 1.37 /

/

vw

y

v v vwv

y y yw

v vw

y yw

Ekfor h tF

E E Ek k kfor h th t F F F

E Ek kfor h tF Fh t

C

0.7r t

y

EL r

F

, 5.2

2

2

,

0.3 , ,

, ,

y b p

b py y y p b rb

r p

cr F

by r b

forF L L

L LMin forC F F F L L L

L L

F

ECMin forF L LLbrt

, 5.3

2

,

0.3 ,

0.9,

2

y

pfy y

rf pfcr F

c

f

f

for Compact FlangesF

for Noncompact FlangesF F

FEk for Slender Flanges

b

t

l ll l

, 10

1 5.7 , 1.01200 300 , 10

w cpg

yw w

Min Ea hMinRMin a t F

, 5.3

2

,

,

0.9,

pc yc

pfpc yc pc yc L xc

n F rf pf

c xc

for Compact FlangesR M

for Noncompact FlangesSR M R M FM

Ek S for Slender Flanges

l ll l

l

, /

1 , , /

pc w pw

yt

ptp p pw p

c w pwyt yt ytrw pw

Mfor h t

MR

M M MMin for h t

M M M

l

l ll

l l

(cont'd)

1.491237 ft-kips @ 7.50 ft, from heel.

< 151.271 ft-kips[Satisfactory]

0.40 kips @ 15.00 ft, from heel.

< 67.1317 kips [Satisfactory]

DETERMINE DEFLECTION AT MID SPAN

0.01 in ( L / 25804 ) (for camber, self Wt included.)

where E = 29000 ksi w = 0.053 kips / ft

I = 299 P = 0 kipsb = 0.3 ft L = 15.0 ft

0.00 #DIV/0!

where P = 0 kips w = 0.000 kips / ft

DETERMINE FLANGE TO WEB WELDING (AISC 360-05 J2.4 )

1/4 in

3/16

4/162.0

0.40 kips

29

0.04 kips / in

A = 24 in 2293 in. o.c.

USE 1/4 in - 24 in @ 2293 in o.c.

DESIGN STIFFENERS

2. CHECK LOCAL WEB YIELDING FOR THE CONCENTRATED LOAD. (AISC 360-05, J10.2)R = P = 0.00 kipsN = 0 in, bearing length, point.

0.87 in1.5

0.00 < [Satisfactory]

Mmax =

Mallowable =

Vmax =

Vallowable =

in4

w =

wmin = in, < w

wmax = in, > wW =

Vmax =

Q = Af(d - y - 0.5 tf,top) = in3

vmax = Vmax Q / I =

1. BEARING STIFFENERS ARE REQUIRED AT EACH END SUPPORT. (AISC 360-05, J10.8)

k = tf,top + w =W =

Fy / W

-160-140-120-100

-80-60-40-20

0

BENDING LOADS & CAPACITY

Length

Mo

me

nts

-68.00

-67.50

-67.00

-66.50

-66.00

-65.50

SHEAR LOADS & CAPACITY

Length

Sh

ea

r F

orc

es

4

3/ 2225 0.06415

384DL

w PbLbL

EI EIL

4

3/ 2225 0.06415

384LL

w PbLbL

EI EIL

0.6 0.707

max

EXX w AFB

v

W

,5

,2.5

Rfor c d

N kt wR

for c dN kt w

(cont'd)3. CHECK WEB CRIPPLING FOR THE CONCENTRATED LOAD. (AISC 360-05, J10.3)

2.0

85.01 > P [Satisfactory]

(Note : If item 2, local web yielding is Satisfactory, this item does not need to be checked.)

4. CHECK SIDESWAY WEB BUCKING FOR THE CONCENTRATED LOAD. (AISC 360-05, J10.4)

8.37 in

960000 ksi

1.261.76

171.48 > P [Satisfactory]

(Note : If item 2, local web yielding is Satisfactory, this item does not need to be checked.)

5. DETERMINE STIFFENER SIZE.

1/3 4 in

12.80[Satisfactory]

4.14 15

R = 0.4 kips

1.67 , (AISC 365-05 E1)

3.5

113

23867.77 ksi

123.9 kips, (AISC 360-05 E2)

> R [Satisfactory]

Techincal Reference: 1. AISC: "Steel Construction Manual 13th Edition", American Institute of Steel Construction, 2005.

W =

dc = d - 2k =

Cr =

(dc / tw ) / (l / bf) =W =

tw = in , bst =

bst / tw = < 0.56 (E / Fy)0.5 , (AISC 360-05 Table B4.1)

Ag = in2 , I = in4

Wc =

K l / r = 0.75 h / ( I / Aeff)0.5 =

Cc = 4.71 (E/ Fy)0.5 =

1.5

2

1.5

2

0.80 1 3 , 0.5

/ 1 /

0.40 1 3 , 0.5

yw fww

f w

yw fww

f w

tEFN t for c dtd t t

RntEFN t for c dt

d t t

W W

33

2

33

2

/ /0.4 , 1.7

/ /

/ // 1 / 1 0.4 , 1.7 2.3

/ /

/, 2.3

/

r w f c w c w

f f

r w f c w c w

f f

c w

f

C t t d t d tforl lb bh

C t t d t d tforRnl lb bh

d tP forl b

W W W

/ /

,0.658

0.877 ,

F yy cFe

R An c g c

e c

klfor CF

r

klfor CF

r

W W

2

2/

FeE

kl r

Daniel PROJECT : PAGE :

Tian Li CLIENT : DESIGN BY : JOB NO. : DATE : REVIEW BY :

Simply Supported Beam Design with Torsional Loading Based on AISC Manual 9th


BEAM SECTION = > W10X54 = > A dGRAVITY DISTRIBUTED LOAD w = 1.15 kips / ft 15.8 10.1 4.38 2.55 303 60

LATERAL POINT LOAD AT MID F = 5 kips lTORSION AT MID SPAN T = 5.1 ft-kips 103 20.6 0.01744 0.37 10.00 0.62

AXIAL LOAD P = 96 kips

BEAM LENGTH 15 ft

BEAM YIELD STRESS 50 ksi

VERTICAL BENDING UNBRACED LENGTH 15 ft

AXIAL VERTICAL UNBRACED LENGTH 15 ft

AXIAL HORIZONTAL UNBRACED LENGTH 7.5 ft

ANALYSIS

CHECK LOCAL BUCKLING (AISC-ASD Tab. B5.1)

8.13 < 9.19

[Satisfactory]

27.30 < 90.51 THE BEAM DESIGN IS ADEQUATE.[Satisfactory]

DETERMINE GOVERNING MOMENTS AT MIDDLE OF SPAN

32.3 ft-kips

18.8 ft-kips

22.7 ft-kips 0.584 ,(Philip page 101)

13.3 ft-kips

DETERMINE GOVERNING UNBALANCED SEGMENT LENGTH (AISC-ASD F1)

8.96 ft

20.30 ft

22.39 ft

Where 1.64

2.66

1.00

DETERMINE ALLOWABLE BENDING STRESSES (AISC-ASD F1)

= = N/A

= = 30.00

= = N/A

= = N/A

Where 25.85 ksi

16.67 ksi

30.00 ksi

CHECK VERTICAL FLEXURAL CAPACITY (AISC-ASD F & Philip page 100)


Where 21.93 ksi

rx ry Ix Sx

Iy Sy tw bf tf

L =

Fy =

Lb =

Lx =

Ly =

bf / (2tf ) = 65 / (Fy)0.5 =

d / tw = 640 / (Fy)0.5 =

Mx = w L2/ 8 =

My = F L / 4 =

M0 = T L / (4d) =

MT = bM0 =

Lc = MIN[76bf/(Fy)0.5 , 20000/(d/Af)Fy] =

Lu = MAX[rT(102000Cb/Fy)0.5 , 12000Cb/(d/Af)0.6Fy] =

L3 = rT(510000Cb/Fy)0.5 =

(d/Af) = in-1

rT =

Cb =

Fbx = {0.66Fy ksi, for Lb @ [0, Lc]

0.60Fy ksi, for Lb @ (Lc, Lu]

MAX(Fb1, Fb3) ksi, for Lb @ (Lu, L3]

MAX(Fb2, Fb3) ksi, for Lb @ (L3, Larger)

Fb1 = MIN{[2/3 - Fy(L/rT)2/(1530000Cb)]Fy , 0.6Fy} =

Fb2 = MIN[170000Cb/(L/rT)2, Fy/3] =

Fb3 = MIN[12000Cb/(Ld/Af), 0.6Fy] =

fbx / Fbx =

fbx = Mx / Sx + 2MT / Sy =

24 sinh

2sinh

L

L L

l

bl l

(cont'd)CHECK COMPRESSION CAPACITY (AISC-ASD E2)


Where 6.08 ksi

K = 1.0

29000 ksi 25.68

107 N/A

41.10 < 200 [Satisfactory]

0.38

CHECK COMBINED STRESS (AISC-ASD H1)

0.24 > 0.15

1.33 < 1.33

Where 1.00

10.92 ksi

37.50 ksi

88.39 ksi 120.18 ksi

1.22 < 1.33

1.26 < 1.33 <== Not applicable.

[Satisfactory]

DETERMINE DEFLECTIONS

0.22127

Where 11200 ksi

J = 1.82

0.15 in = L / 1207 , vertical deflection at middle

Where 303

0.20 in = L / 885 , horizontal deflection at middle

Where 103

Technical References: 1. AISC: "Manual of Steel construction 9th", American Institute of Steel Construction, 1990. 2. Philip H. Lin: "Simplified Design for Torsional Loading of Rolled Steel Members", Engineering Journal, AISC, 1977.

fa / Fa =

fa = P / A =

Es =Fa = {

(1-F2/2)Fy / (5/3+3F/8-F3/8) = ksi, for Cc > (Kl/r)

Cc = (22Es/Fy)0.5 = 122Es/[23(KL/r)2] = ksi, for Cc < (Kl/r)

KL/r = MAX(KLx/rx, KLy/ry) =

F = (KL/ r) / Cc =

fa / Fa =

Cm =

fby = My / Sy =

Fby = 0.75 Fy =

o , max twist angle at middle (Philip page 100)

G =

in4

I3 = Ix sin2(90-) + Iy cos2(90-) = in4 , (AISC-ASD Page 6-23)

I4 = Ix cos2(90-) + Iy sin2(90-) = in4 , (AISC-ASD Page 6-23)

1 1' '

fCf f myC bymxa bxfF a fa aFbx FbyFex Fey

212'2

23

EFex

K l xr x

212'

223

EFey

K l yr y

0.6

ff f bya bxF F Fbx byy

ff f bya bxF F Fa bx by

2sinh

2 sinh2 22 sinh

LT LL

GJ L

lll

l l

4

3

5384

wLvert E I

3

448

F Lhoriz E I

Beam With Torsion

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