"MONORAIL13" --- MONORAIL BEAM ANALYSIS

Program Description:

"MONORAIL13" is a spreadsheet program written in MS-Excel for the purpose of analysis of either S-shape or
W-shape underhung monorail beams analyzed as simple-spans with or without overhangs (cantilevers).
Specifically, the x-axis and y-axis bending moments as well as any torsion effects are calculated. The actual and
allowable stresses are determined, and the effect of lower flange bending is also addressed by two different
approaches.

This program is a workbook consisting of three (3) worksheets, described as follows:

Worksheet Name Description

Doc This documentation sheet
S-shaped Monorail Beam Monorail beam analysis for S-shaped beams
W-shaped Monorail Beam Monorail beam analysis for W-shaped beams

Program Assumptions and Limitations:

1. The following references were used in the development of this program:

a. Fluor Enterprises, Inc. - Guideline 000.215.1257 - "Hoisting Facilities" (August 22, 2005)
b. Dupont Engineering Design Standard: DB1X - "Design and Installation of Monorail Beams" (May 2000)
c. American National Standards Institute (ANSI): MH27.1 - "Underhung Cranes and Monorail Syatems"
d. American Institute of Steel Construction (AISC) 13th Edition Allowable Stress Design (ASD) Manual (2005)
e. "Allowable Bending Stresses for Overhanging Monorails" - by N. Stephen Tanner -
AISC Engineering Journal (3rd Quarter, 1985)
f. Crane Manufacturers Association of America, Inc. (CMAA) - Publication No. 74 -
"Specifications for Top Running & Under Running Single Girder Electric Traveling Cranes
Utilizing Under Running Trolley Hoist" (2004)
g. "Design of Monorail Systems" - by Thomas H. Orihuela Jr., PE (www.pdhengineer.com)
h. British Steel Code B.S. 449, pages 42-44 (1959)
i. USS Steel Design Manual - Chapter 7 "Torsion" - by R. L. Brockenbrough and B.G. Johnston (1981)
j. AISC Steel Design Guide Series No. 9 - "Torsional Analysis of Structural Steel Members" -
by Paul A. Seaburg, PhD, PE and Charlie J. Carter, PE (1997)
k. "Technical Note: Torsion Analysis of Steel Sections" - by William E. Moore II and Keith M. Mueller -
AISC Engineering Journal (4th Quarter, 2002)
2. The unbraced length for the overhang (cantilever) portion, 'Lbo', of an underhung monorail beam is often debated.
The following are some recommendations from the references cited above:
a. Fluor Guideline 000.215.1257: Lbo = Lo+L/2
b. Dupont Standard DB1X: Lbo = 3*Lo
c. ANSI Standard MH27.1: Lbo = 2*Lo
d. British Steel Code B.S. 449: Lbo = 2*Lo (for top flange of monorail beam restrained at support)
British Steel Code B.S. 449: Lbo = 3*Lo (for top flange of monorail beam unrestrained at support)
e. AISC Eng. Journal Article by Tanner: Lbo = Lo+L (used with a computed value of 'Cbo' from article)
3. This program also determines the calculated value of the bending coefficient, 'Cbo', for the overhang (cantilever)
portion of the monorail beam from reference "e" in note #1 above. This is located off of the main calculation page.
Note: if this computed value of 'Cbo' is used and input, then per this reference the total value of Lo+L should be
used for the unbraced length, 'Lbo', for the overhang portion of the monorail beam.
4. This program ignores effects of axial compressive stress produced by any longitudinal (traction) force which is
usually considered minimal for underhung, hand-operated monorail systems.
5. This program contains “comment boxes” which contain a wide variety of information including explanations of
input or output items, equations used, data tables, etc. (Note: presence of a “comment box” is denoted by a
“red triangle” in the upper right-hand corner of a cell. Merely move the mouse pointer to the desired cell to view
the contents of that particular "comment box".)
 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

MONORAIL BEAM ANALYSIS

For S-shaped Underhung Monorails Analyzed as Simple-Spans with / without Overhang
Per AISC 13th Edition ASD Manual and CMAA Specification No. 74 (2004)
Project Name: Client:
Project No.: Prep. By: Date:

Input:
RL(min)=-0.63 kips RR(max)=4.68 kips
Monorail Size: L=12.5 ft. Lo=3 ft.
Select: S10X35 x=6.25ft.
Design Parameters: S=0 in.
Beam Fy = 50 ksi
Beam Simple-Span, L = 12.5000 ft. S10X35
Unbraced Length, Lb = 12.5000 ft.
Bending Coef., Cb = 1.00 Pv=3.5 kips
Overhang Length, Lo = 3.0000 ft. Nomenclature
Unbraced Length, Lbo = 15.5000 ft. Shape Factor X =
Bending Coef., Cbo = 0.73 S10X35 Member Properties:
Lifted Load, P = 2.000 kips A= 10.30 in.^2 Zx = 35.40
Trolley Weight, Wt = 0.500 kips d= 10.000 in. Iy = 8.30
Hoist Weight, Wh = 0.500 kips tw = 0.594 in. Sy = 3.36
Vert. Impact Factor, Vi = 25 % bf = 4.940 in. ry = 0.899
Horz. Load Factor, HLF = 10 % tf = 0.491 in. Zy = 6.19
Total No. Wheels, Nw = 2 k= 1.130 in. J = 1.290
Wheel Spacing, S = 0.0000 in. Ix = 147.00 in.^4 Cw = 188.0
Distance on Flange, a = 0.3750 in. Sx = 29.40 in.^3 wt / ft. = 35.0

Support Reactions: (with overhang)

Results: RR(max) = 4.68 = Pv*(L+Lo)/L+w/1000/(2*L)*(L+Lo)^2
RL(min) = -0.63 = -Pv*Lo/L+w/1000/(2*L)*(L^2-Lo^2)
Parameters and Coefficients:
Pv = 3.500 kips Pv = P*(1+Vi/100)+Wt+Wh (vertical load)
Pw = 1.750 kips/wheel Pw = Pv/Nw (load per trolley wheel)
Ph = 0.200 kips Ph = HLF*P (horizontal load)
ta = 0.348 in. ta = tf-bf/24+a/6 (for S-shape)
l = 0.173 l = 2*a/(bf-tw)
Cxo = -0.839 Cxo = -1.096+1.095*l+0.192*e^(-6.0*l)
Cx1 = 0.632 Cx1 = 3.965-4.835*l-3.965*e^(-2.675*l)
Czo = 0.171 Czo = -0.981-1.479*l+1.120*e^(1.322*l) Design Chapter
Cz1 = 1.892 Cz1 = 1.810-1.150*l+1.060*e^(-7.70*l)
Bending Moments for Simple-Span:
x = 6.250 ft. x = L/2 (location of max. moments from left end of simple-span)
Mx = 11.62 ft-kips Mx = Pv*L/4+w/1000*L^2/8
My = 0.63 ft-kips My = Ph*L/4

Lateral Flange Bending Moment from Torsion for Simple-Span: (per USS Steel Design Manual, 1981)
e = 5.000 in. e = d/2 (assume horiz. load taken at bot. flange)
at = 19.426 at = SQRT(E*Cw/(J*G)) , E=29000 ksi and G=11200 ksi

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

Mt = 0.09 ft-kips Mt = Ph*e*at/(2*(d-tf))*TANH(L*12/(2*at))/12 Is Lb<=Lp?

X-axis Stresses for Simple-Span:

fbx = 4.74 ksi fbx = Mx/Sx
Lr = 15.76 ft. (Eqn. F2-6, max. value of Lb for inelastic LTB) Use: Fbx =
Fbx = 24.87 ksi Eqn. F2-2 Controls fbx <= Fbx, O.K.

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

Y-axis Stresses for Simple-Span:

fby = 2.23 ksi fby = My/Sy
fwns = 0.61 ksi fwns = Mt*12/(Sy/2) (warping normal stress) fby(total) =
fby(total) = 2.84 ksi fby(total) = fby+fwns
Fby = 47.90 ksi Eqn. F6-1 Controls fby <= Fby, O.K.

Combined Stress Ratio for Simple-Span:

S.R. = 0.250 S.R. = fbx/Fbx+fby(total)/Fby S.R. <= 1.0, O.K.

Vertical Deflection for Simple-Span:

Pv = 3.000 kips Pv = P+Wh+Wt (without vertical impact)
D(max) = 0.0540 in. D(max) = Pv*L^3/(48*E*I)+5*w/12000*L^4/(384*E*I)
D(ratio) = L/2778 D(ratio) = L*12/D(max)
D(allow) = 0.3333 in. D(allow) = L*12/450 Defl.(max) <= Defl.(allow), O.K.

Bending Moments for Overhang:

Mx = 10.66 ft-kips Mx = Pv*Lo+w/1000*Lo^2/2
My = 0.60 ft-kips My = Ph*Lo

Lateral Flange Bending Moment from Torsion for Overhang: (per USS Steel Design Manual, 1981)
e = 5.000 in. e = d/2 (assume horiz. load taken at bot. flange)
at = 19.426 at = SQRT(E*Cw/(J*G)) , E=29000 ksi and G=11200 ksi
Mt = 0.17 ft-kips Mt = Ph*e*at/(d-tf)*TANH(Lo*12/at)/12
Design Chapter
X-axis Stresses for Overhang:
fbx = 4.35 ksi fbx = Mx/Sx
Lr = 15.76 ft. (Eqn. F2-6, max. value of Lb for inelastic LTB)
Fbx = 15.53 ksi Eqn. F2-2 Controls fbx <= Fbx, O.K.

Y-axis Stresses for Overhang:

fby = 2.14 ksi fby = My/Sy
fwns = 1.22 ksi fwns = Mt*12/(Sy/2) (warping normal stress)
fby(total) = 3.36 ksi fby(total) = fby+fwns
Fby = 47.90 ksi Eqn. F6-1 Controls fby <= Fby, O.K.

Combined Stress Ratio for Overhang:

S.R. = 0.350 S.R. = fbx/Fbx+fby(total)/Fby S.R. <= 1.0, O.K.
Use: Fbx =
Vertical Deflection for Overhang: (assuming full design load, Pv without impact, at end of overhang)
Pv = 3.000 kips Pv = P+Wh+Wt (without vertical impact)
D(max) = 0.0540 in. D(max) = Pv*Lo^2*(L+Lo)/(3*E*I)+w/12000*Lo*(4*Lo^2*L-L^3+3*Lo^3)/(24*E*I)
D(ratio) = L/666 D(ratio) = Lo*12/D(max)
D(allow) = 0.0800 in. D(allow) = Lo*12/450 Defl.(max) <= Defl.(allow), O.K.
fby(total) =
Bottom Flange Bending (simplified):
be = 5.892 in. be = 12*tf (effective flange bending length)
tf2 = 0.672 in. tf2 = tf+(bf/2-tw/2)/2*(1/6) (flange thk. at web based on 1:6 slope of flange)

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

am = 1.715 in. am = (bf/2-tw/2)-(k-tf2) (where: k-tf2 = radius of fillet)

Mf = 2.345 in.-kips Mf = Pw*(am-a) Use: Fby =
Sf = 0.237 in.^3 Sf = be*tf^2/6
fb = 9.91 ksi fb = Mf/Sf
Fb = 45.00 ksi Fb = 0.9*Fy fb <= Fb, O.K.

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

Bottom Flange Bending per CMAA Specification No. 74 (2004): (Note: torsion is neglected)

Local Flange Bending Stress @ Point 0: (Sign convention: + = tension, - = compression)

sxo = -12.15 ksi sxo = Cxo*Pw/ta^2
szo = 2.47 ksi szo = Czo*Pw/ta^2
S-shape
Local Flange Bending Stress @ Point 1:
sx1 = 9.15 ksi sx1 = Cx1*Pw/ta^2 Trolley
sz1 = 27.40 ksi sz1 = Cz1*Pw/ta^2 Wheel

Local Flange Bending Stress @ Point 2:

sx2 = 12.15 ksi sx2 = -sxo
sz2 = -2.47 ksi sz2 = -szo Pw Pw

Resultant Biaxial Stress @ Point 0:

fbxo = 4.74 ksi fbxo = Mx*(d/2)/Ix = Mx/Sx
fbyo = 0.27 ksi fbyo = My*(tw/2)/Iy
sz = 6.87 ksi sz = fbxo+fbyo+0.75*szo
sx = -9.11 ksi sx = 0.75*sxo
txz = 0.00 ksi txz = 0 (assumed negligible)
sto = 13.88 ksi sto = SQRT(sx^2+sz^2-sx*sz+3*txz^2) <= Fb = 0.66*Fy = 33 ksi, O.K.

Resultant Biaxial Stress @ Point 1:

fbx1 = 4.74 ksi fbx1 = Mx*(d/2)/Ix = Mx/Sx
fby1 = 1.89 ksi fby1 = My*(bf/2-a)/Iy
sz = 27.18 ksi sz = fbx1+fby1+0.75*sz1
sx = 6.86 ksi sx = 0.75*sx1
txz = 0.00 ksi txz = 0 (assumed negligible)
st1 = 24.49 ksi st1 = SQRT(sx^2+sz^2-sx*sz+3*txz^2) <= Fb = 0.66*Fy = 33 ksi, O.K.

Resultant Biaxial Stress @ Point 2:

fbx2 = 4.28 ksi fbx2 = Mx*(d/2-tf)/Ix
fby2 = 0.27 ksi fby2 = My*(tw/2)/Iy
sz = 2.69 ksi sz = fbx2+fby2+0.75*sz2
sx = 9.11 ksi sx = 0.75*sx2
txz = 0.00 ksi txz = 0 (assumed negligible)
st2 = 8.11 ksi st2 = SQRT(sx^2+sz^2-sx*sz+3*txz^2) <= Fb = 0.66*Fy = 33 ksi, O.K.

tw

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

X
Pw Z Pw

tf

Point 2 Point 1 ta
bf/4
Point 0 tw/2 bf

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

MONORAIL BEAM ANALYSIS

g Monorails Analyzed as Simple-Spans with / without Overhang
ition ASD Manual and CMAA Specification No. 74 (2004)
###
###
###
###
ta =
l=
Cxo =
Cx1 =
Czo =
Cz1 =
Section Ratios and Parameters:
bf/(2*tf) =
h/tw =
Shape Factor X =
S10X35 Member Properties:
in.^3
in.^4
in.^3
in.
in.^3
in.^4
in.^6
plf
For Lo > 0 (with overhang):
RR(max) =
RL(min) =
Bending Moments for Simple-Span:
x=
Mx =
My =
Lateral Flange Bending Moment from Torsion for Simple-Spa
e=
at =
Mt =

Design Chapter
kc =
Lp =
rts =
c=
R1 =
R2 =
(per USS Steel Design Manual, 1981) Lr =
R3 =
fbx =

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

Is Lb<=Lp?
Fbx =
Fbx =
Fbx =
Use: Fbx =
fbx <= Fbx, O.K.
(continued)

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

fby =
fwns =
fby(total) =
Fby =
fby <= Fby, O.K.
Fby =
Fcr =
S.R. <= 1.0, O.K.

S.R. =

Pv =
D(max) =
Defl.(max) <= Defl.(allow), O.K.
D(allow) =

Mx =
My =
Lateral Flange Bending Moment from Torsion for Overhang:
(per USS Steel Design Manual, 1981) e =
at =
Mt =

Design Chapter
kc =
Lp =
rts =
fbx <= Fbx, O.K.
R1 =
R2 =
Lr =
R3 =
fbx =
fby <= Fby, O.K.
Fbx =
Fbx =
S.R. <= 1.0, O.K.
Use: Fbx =
Mrx =
S.R. =
Pv*Lo^2*(L+Lo)/(3*E*I)+w/12000*Lo*(4*Lo^2*L-L^3+3*Lo^3)/(24*E*I)
fby =
Defl.(max) <= Defl.(allow), O.K.
fby(total) =
Fby =
Fby =
tf2 = tf+(bf/2-tw/2)/2*(1/6) (flange thk. at web based on 1:6 slope of flange)
Fby =

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

Fcr =
Use: Fby =

S.R. =
fb <= Fb, O.K.
Pv =
(continued)

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

D(ratio) =
D(allow) =

(Sign convention: + = tension, - = compression)

be =
tf2 =
S-shape am =
Mf =
Trolley Sf =
Wheel fb =
Fb =
Bottom Flange Bending per CMAA Specification No. 74 (200
Local Flange Bending Stress @ Point 0:
sxo =
szo =

sx1 =
sz1 =

sx2 =
sz2 =
<= Fb = 0.66*Fy = 33 ksi, O.K.
fbxo =
fbyo =
sz =
sx =
txz =
sto =

<= Fb = 0.66*Fy = 33 ksi, O.K.

fby1 =
sz =
sx =
txz =
st1 =

fbx2 =
<= Fb = 0.66*Fy = 33 ksi, O.K.
sz =
sx =
txz =
st2 =

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

MONORAIL BEAM ANALYSIS

For W-shaped Underhung Monorails Analyzed as Simple-Spans with / without Overhang
Per AISC 13th Edition ASD Manual and CMAA Specification No. 74 (2004)
Project Name: Client:
Project No.: Prep. By: Date:

Input:
RL(min)=-3.93 kips RR(max)=45.7 kips
Monorail Size: L=25 ft. Lo=4 ft.
Select: W18X130 x=12.313 ft
Design Parameters: S=9 in.
Beam Fy = 50 ksi
Beam Simple-Span, L = 25.0000 ft. W18X130
Unbraced Length, Lb = 25.0000 ft.
Bending Coef., Cb = 1.00 Pv=38 kips
Overhang Length, Lo = 4.0000 ft. Nomenclature
Unbraced Length, Lbo = 29.0000 ft. Shape Factor X =
Bending Coef., Cbo = 1.00 W18X130 Member Properties:
Lifted Load, P = 30.000 kips A= 38.20 in.^2 Zx = 290.00
Trolley Weight, Wt = 0.400 kips d= 19.300 in. Iy = 278.00
Hoist Weight, Wh = 0.100 kips tw = 0.670 in. Sy = 49.90
Vert. Impact Factor, Vi = 25 % bf = 11.200 in. ry = 2.700
Horz. Load Factor, HLF = 10 % tf = 1.200 in. Zy = 76.70
Total No. Wheels, Nw = 4 k= 1.600 in. J = 14.500
Wheel Spacing, S = 9.0000 in. Ix = 2460.00 in.^4 Cw = 22700.0
Distance on Flange, a = 0.3750 in. Sx = 256.00 in.^3 wt / ft. = 130.0

Support Reactions: (with overhang)

Results: RR(max) = 45.70 = Pv*(L+(Lo-(S/12)/2))/L+w/1000/(2*L)*(L+Lo)^2
RL(min) = -3.93 = -Pv*(Lo-(S/12)/2)/L+w/1000/(2*L)*(L^2-Lo^2)
Parameters and Coefficients:
Pv = 38.000 kips Pv = P*(1+Vi/100)+Wt+Wh (vertical load)
Pw = 9.500 kips/wheel Pw = Pv/Nw (load per trolley wheel)
Ph = 3.000 kips Ph = HLF*P (horizontal load)
ta = 1.200 in. ta = tf (for W-shape)
l = 0.071 l = 2*a/(bf-tw)
Cxo = -1.957 Cxo = -2.110+1.977*l+0.0076*e^(6.53*l)
Cx1 = 0.408 Cx1 = 10.108-7.408*l-10.108*e^(-1.364*l)
Czo = 0.192 Czo = 0.050-0.580*l+0.148*e^(3.015*l) Design Chapter
Cz1 = 2.501 Cz1 = 2.230-1.490*l+1.390*e^(-18.33*l)
Bending Moments for Simple-Span:
x = 12.313 ft. x = 1/2*(L-(S/12)/2) (location of max. moments from left end of simple-span)
Mx = 240.58 ft-kips Mx = (Pv/2)/(2*L)*(L-(S/12)/2)^2+w/1000*x/2*(L-x)
My = 18.19 ft-kips My = (Ph/2)/(2*L)*(L-(S/12)/2)^2

Lateral Flange Bending Moment from Torsion for Simple-Span: (per USS Steel Design Manual, 1981)
e = 9.650 in. e = d/2 (assume horiz. load taken at bot. flange)
at = 63.668 at = SQRT(E*Cw/(J*G)) , E=29000 ksi and G=11200 ksi

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

Mt = 4.17 ft-kips Mt = Ph*e*at/(2*(d-tf))*TANH(L*12/(2*at))/12 Is Lb<=Lp?

X-axis Stresses for Simple-Span:

fbx = 11.28 ksi fbx = Mx/Sx
Lr = 36.67 ft. (Eqn. F2-6, max. value of Lb for inelastic LTB)
Fbx = 26.53 ksi Eqn. F2-2 Controls fbx <= Fbx, O.K.

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

Y-axis Stresses for Simple-Span:

fby = 4.37 ksi fby = My/Sy
fwns = 2.00 ksi fwns = Mt*12/(Sy/2) (warping normal stress)
fby(total) = 6.38 ksi fby(total) = fby+fwns
Fby = 46.02 ksi Eqn. F6-1 Controls fby <= Fby, O.K.

Combined Stress Ratio for Simple-Span:

S.R. = 0.564 S.R. = fbx/Fbx+fby(total)/Fby S.R. <= 1.0, O.K.

Vertical Deflection for Simple-Span: Use: Fby =

Pv = 30.500 kips Pv = P+Wh+Wt (without vertical impact)
D(max) = 0.2562 in. D(max) = Pv/2*(L-(S/12))/2/(24*E*I)*(3*L^2-4*((L-(S/12))/2)^2)+5*w/12000*L^4/(384*E*I)
D(ratio) = L/1171 D(ratio) = L*12/D(max)
D(allow) = 0.6667 in. D(allow) = L*12/450 Defl.(max) <= Defl.(allow), O.K.

Bending Moments for Overhang:

Mx = 138.79 ft-kips Mx = (Pv/2)*(Lo+(Lo-(S/12)))+w/1000*Lo^2/2
My = 10.88 ft-kips My = (Ph/2)*(Lo+(Lo-S/12))

Lateral Flange Bending Moment from Torsion for Overhang: (per USS Steel Design Manual, 1981)
e = 9.650 in. e = d/2 (assume horiz. load taken at bot. flange)
at = 63.668 at = SQRT(E*Cw/(J*G)) , E=29000 ksi and G=11200 ksi
Mt = 8.48 ft-kips Mt = Ph*e*at/(d-tf)*TANH(Lo*12/at)/12

X-axis Stresses for Overhang:

fbx = 6.51 ksi fbx = Mx/Sx Design Chapter
Lr = 36.67 ft. (Eqn. F2-6, max. value of Lb for inelastic LTB)
Fbx = 24.62 ksi Eqn. F2-2 Controls fbx <= Fbx, O.K.

Y-axis Stresses for Overhang:

fby = 2.62 ksi fby = My/Sy
fwns = 4.08 ksi fwns = Mt*12/(Sy/2) (warping normal stress)
fby(total) = 6.70 ksi fby(total) = fby+fwns
Fby = 46.02 ksi Eqn. F6-1 Controls fby <= Fby, O.K.

Combined Stress Ratio for Overhang: Is Lb<=Lp?

S.R. = 0.410 S.R. = fbx/Fbx+fby(total)/Fby S.R. <= 1.0, O.K.

Vertical Deflection for Overhang: (assuming full design load, Pv without impact, at end of overhang)
Pv = 30.500 kips Pv = P+Wh+Wt (without vertical impact)
D(max) = 0.1070 in. D(max) = Pv*Lo^2*(L+Lo)/(3*E*I)+w/12000*Lo*(4*Lo^2*L-L^3+3*Lo^3)/(24*E*I)
D(ratio) = L/449 D(ratio) = Lo*12/D(max) Use: Fbx =
D(allow) = 0.1067 in. D(allow) = Lo*12/450 Defl.(max) > Defl.(allow), N.G.

Bottom Flange Bending (simplified):

be = 9.000 in. Min. of: be = 12*tf or S (effective flange bending length)
am = 4.865 in. am = (bf/2-tw/2)-(k-tf) (where: k-tf = radius of fillet)

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

Mf = 42.655 in.-kips Mf = Pw*(am-a) fby(total) =

Sf = 2.160 in.^3 Sf = be*tf^2/6
fb = 19.75 ksi fb = Mf/Sf
Fb = 45.00 ksi Fb = 0.90*Fy fb <= Fb, O.K.

Use: Fby =

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

Bottom Flange Bending per CMAA Specification No. 74 (2004): (Note: torsion is neglected)

Local Flange Bending Stress @ Point 0: (Sign convention: + = tension, - = compression)

sxo = -12.91 ksi sxo = Cxo*Pw/ta^2
szo = 1.27 ksi szo = Czo*Pw/ta^2

Local Flange Bending Stress @ Point 1:

sx1 = 2.69 ksi sx1 = Cx1*Pw/ta^2
sz1 = 16.50 ksi sz1 = Cz1*Pw/ta^2

Local Flange Bending Stress @ Point 2:

sx2 = 12.91 ksi sx2 = -sxo
sz2 = -1.27 ksi sz2 = -szo

Resultant Biaxial Stress @ Point 0:

fbxo = 11.28 ksi fbxo = Mx*(d/2)/Ix = Mx/Sx
fbyo = 0.26 ksi fbyo = My*(tw/2)/Iy
sz = 12.49 ksi sz = fbxo+fbyo+0.75*szo
sx = -9.68 ksi sx = 0.75*sxo
txz = 0.00 ksi txz = 0 (assumed negligible)
sto = 19.25 ksi sto = SQRT(sx^2+sz^2-sx*sz+3*txz^2) <= Fb = 0.66*Fy = 33 ksi, O.K.

Resultant Biaxial Stress @ Point 1:

fbx1 = 11.28 ksi fbx1 = Mx*(d/2)/Ix = Mx/Sx
fby1 = 4.10 ksi fby1 = My*(bf/2-a)/Iy
sz = 27.75 ksi sz = fbx1+fby1+0.75*sz1
sx = 2.02 ksi sx = 0.75*sx1
txz = 0.00 ksi txz = 0 (assumed negligible)
st1 = 26.80 ksi st1 = SQRT(sx^2+sz^2-sx*sz+3*txz^2) <= Fb = 0.66*Fy = 33 ksi, O.K.

Resultant Biaxial Stress @ Point 2:

fbx2 = 9.92 ksi fbx2 = Mx*(d/2-tf)/Ix
fby2 = 0.26 ksi fby2 = My*(tw/2)/Iy
sz = 9.23 ksi sz = fbx2+fby2+0.75*sz2
sx = 9.68 ksi sx = 0.75*sx2
txz = 0.00 ksi txz = 0 (assumed negligible)
st2 = 9.46 ksi st2 = SQRT(sx^2+sz^2-sx*sz+3*txz^2) <= Fb = 0.66*Fy = 33 ksi, O.K.

tw
X
Pw Pw

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

X
Pw Pw
Z

Point 2

tf
Point 0
Point 1
bf

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

MONORAIL BEAM ANALYSIS

g Monorails Analyzed as Simple-Spans with / without Overhang
ition ASD Manual and CMAA Specification No. 74 (2004)
###
###
###
###
ta =
l=
Cxo =
Cx1 =
Czo =
Cz1 =
Section Ratios and Parameters:
bf/(2*tf) =
h/tw =
Shape Factor X =
W18X130 Member Properties:
in.^3
in.^4
in.^3
in.
in.^3
in.^4
in.^6
plf
For Lo > 0 (with overhang):
RR(max) =
RL(min) =
= Pv*(L+(Lo-(S/12)/2))/L+w/1000/(2*L)*(L+Lo)^2
= -Pv*(Lo-(S/12)/2)/L+w/1000/(2*L)*(L^2-Lo^2) Bending Moments for Simple-Span:
x=
Mx =
My =
Lateral Flange Bending Moment from Torsion for Simple-Spa
e=
at =
Mt =
X-axis Stresses for Simple-Span:
Design Chapter
kc =
Lp =
x = 1/2*(L-(S/12)/2) (location of max. moments from left end of simple-span)
rts =
c=
R1 =
R2 =
(per USS Steel Design Manual, 1981) Lr =
R3 =
fbx =

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

Is Lb<=Lp?
Fbx =
Fbx =
Fbx =
Fbx =
fbx <= Fbx, O.K.
(continued)

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

Mrx =
S.R. =
Y-axis Stresses for Simple-Span:
fby =
fwns =
fby <= Fby, O.K.
Fby =
Fby =
S.R. <= 1.0, O.K.
Fcr =
Use: Fby =
Combined Stress Ratio for Simple-Span:
S.R.
Pv/2*(L-(S/12))/2/(24*E*I)*(3*L^2-4*((L-(S/12))/2)^2)+5*w/12000*L^4/(384*E*I) =
Vertical Deflection for Simple-Span:
Defl.(max) <= Defl.(allow), O.K.
D(max) =
D(ratio) =
D(allow) =
Bending Moments for Overhang:
Mx =
(per USS Steel Design Manual, 1981) My =
Lateral Flange Bending Moment from Torsion for Overhang:
e=
at =
Mt =
X-axis Stresses for Overhang:
Design Chapter
kc =
fbx <= Fbx, O.K.
rts =
c=
R1 =
R2 =
Lr =
fby <= Fby, O.K.
fbx =
Is Lb<=Lp?
S.R. <= 1.0, O.K.
Fbx =
Fbx =
Fbx =
Pv*Lo^2*(L+Lo)/(3*E*I)+w/12000*Lo*(4*Lo^2*L-L^3+3*Lo^3)/(24*E*I) Fbx =
Use: Fbx =
Defl.(max) > Defl.(allow), N.G.
S.R. =
Y-axis Stresses for Overhang:
fby =
fwns =

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

fby(total) =
Fby =
Fby =
fb <= Fb, O.K.
Fcr =
Use: Fby =
(continued)

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

S.R. =
Vertical Deflection for Overhang:
Pv =
(Sign convention: + = tension, - = compression)
D(ratio) =
D(allow) =

Bottom Flange Local Bending (simplified):

be =
tf2 =
am =
Mf =
Sf =
fb =
Fb =
Bottom Flange Bending per CMAA Specification No. 74 (200
Local Flange Bending Stress @ Point 0:
sxo =
szo =
Local Flange Bending Stress @ Point 1:
sx1 =
<= Fb = 0.66*Fy = 33 ksi, O.K.
Local Flange Bending Stress @ Point 2:
sx2 =
sz2 =
Resultant Biaxial Stress @ Point 0:
fbxo =
fbyo =
sz =
<= Fb = 0.66*Fy = 33 ksi, O.K.
txz =
sto =
Resultant Biaxial Stress @ Point 1:
fbxo =
fbyo =
sz =
sx =
<= Fb = 0.66*Fy = 33 ksi, O.K.
st1 =
Resultant Biaxial Stress @ Point 2:
fbx2 =
fby2 =
sz =
sx =
txz =
st2 =
W18X55
W18X50

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 "MONORAIL13.xls" Program
Created By: Joel Berg, P.E.
Based on a Spreadsheet By: Alex Tomanovich, P.E.
Version 1.3

W18X46
W18X40
W18X35
W16X100
W16X89
W16X77
W16X67
W16X57
W16X50
W16X45
W16X40
W16X36

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