monorail 2007

"MONORAIL" --- MONORAIL BEAM ANALYSIS

Program Description:

"MONORAIL" 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 DescriptionDoc 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) 9th Edition Allowable Stress Design (ASD) Manual (1989)

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)

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

British Steel Code B.S. 449: Lbo = 3*Lo (for top flange of monorail beam unrestrained at support)

the contents of that particular "comment box".)

"MONORAIL.xls" ProgramVersion 1.6

3 of 14 04/08/2023 15:49:07

MONORAIL BEAM ANALYSISFor S-shaped Underhung Monorails Analyzed as Simple-Spans with / without Overhang

Per AISC 9th Edition ASD Manual and CMAA Specification No. 74 (2004)Job Name: Modificacion Tubos de carga 412-FC-01 Subject: Cerro Matoso

Job Number: 3448 Originator: Indisa S.A. Checker:

Input:RL(min)=-1.34 RR(max)=15.67

Monorail Size: L=6.5 Lo=1Select: S12x50 x=3.1

Design Parameters: S=0.6

Beam Fy = 36 ksi

Beam Simple-Span, L = 6.5000 ft. S12x50Unbraced Length, Lb = 6.5000 ft.

Bending Coef., Cb = 1.00 Pv=13.95Overhang Length, Lo = 1.0000 ft. Nomenclature

Unbraced Length, Lbo = 7.5000 ft.

Bending Coef., Cbo = 1.00 S12x50 Member Properties:Lifted Load, P = 11.000 kips A = 14.60 in.^2 d/Af = 3.32

Trolley Weight, Wt = 0.100 kips d = 12.000 in. Ix = 303.00 in.^4

Hoist Weight, Wh = 0.100 kips tw = 0.687 in. Sx = 50.60 in.^3

Vert. Impact Factor, Vi = 25 % bf = 5.480 in. Iy = 15.60 in.^4

Horz. Load Factor, HLF = 10 % tf = 0.659 in. Sy = 5.69 in.^3

Total No. Wheels, Nw = 4 k= 1.438 in. J = 2.770 in.^4

Wheel Spacing, S = 0.6000 ft. rt = 1.250 in. Cw = 502.0 in.^6

Distance on Flange, a = 0.5000 in.

Support Reactions: (with overhang)Results: 15.67 = Pv*(L+(Lo-S/2))/L+w/1000/(2*L)*(L+Lo)^2

-1.34 = -Pv*(Lo-S/2)/L+w/1000/(2*L)*(L^2-Lo^2)Parameters and Coefficients:

Pv = 13.950 kips Pv = P*(1+Vi/100)+Wt+Wh (vertical load)Pw = 3.488 kips/wheel Pw = Pv/Nw (load per trolley wheel)Ph = 1.100 kips Ph = HLF*P (horizontal load)ta = 0.514 in. ta = tf-bf/24+a/6 (for S-shape)

0.209Cxo = -0.813Cx1 = 0.687Czo = 0.186Cz1 = 1.783

Bending Moments for Simple-Span:x = 3.100 ft. x = 1/2*(L-S/2) (location of max. moments from left end of simple-span)

Mx = 20.89 ft-kips Mx = (Pv/2)/(2*L)*(L-S/2)^2+w/1000*x/2*(L-x)My = 1.63 ft-kips My = (Ph/2)/(2*L)*(L-S/2)^2

Lateral Flange Bending Moment from Torsion for Simple-Span: (per USS Steel Design Manual, 1981)e = 6.000 in. e = d/2 (assume horiz. load taken at bot. flange)

at = 21.662 at = SQRT(E*Cw/(J*G)) , E=29000 ksi and G=11200 ksiMt = 0.50 ft-kips Mt = Ph*e*at/(2*(d-tf))*TANH(L*12/(2*at))/12

X-axis Stresses for Simple-Span:fbx = 4.95 ksi fbx = Mx/Sx

Lb/rt = 62.40 Lb/rt = Lb*12/rt

RR(max) =RL(min) =

l = l = 2*a/(bf-tw)Cxo = -1.096+1.095*l+0.192*e^(-6.0*l)Cx1 = 3.965-4.835*l-3.965*e^(-2.675*l)Czo = -0.981-1.479*l+1.120*e^(1.322*l)Cz1 = 1.810-1.150*l+1.060*e^(-7.70*l)

C15

From AISC 9th Edition ASD Manual, page 5-47: Cb = 1.75+1.05*(M1/M2)+0.3*(M1/M2) <= 2.3 where: M1/M2 = ratio of smaller to larger bending moment at ends of the unbraced length, 'Lb'.

C17

The unbraced length for the overhang (cantilever) portion, 'Lbo', is often debated. Here are some recommendations from different sources: 1. Per Fluor Enterprises Guideline 000.215.1257 - "Hoisting Facilities" Lbo = Lo+L/2 2. Per Dupont Standard DB1X - "Design and Installation of Monorail Beams" Lbo = 3*Lo 3. Per ANSI MH27.1 - "Underhung Cranes and Monorail Systems" Lbo = 2*Lo 4. Per British Steel Code B.S. 449, pages 42-44 (1959) Lbo = 2*Lo (for top flange of monorail beam restrained at support) Lbo = 3*Lo (for top flange of monorail beam unrestrained at support) 5. Per AISC Journal article by N. Stephen Tanner - "Allowable Bending Stresses for Overhanging Monorails" (3rd Quarter, 1985) Lbo = Lo+L (used with a computed value of 'Cbo' from article)

F28

For a monorail consisting of a simple-span with no overhang, 'RR(max)' and 'RL(min)' are determined by positioning the trolley at the right support. For a monorail consisting of a simple-span with an overhang, 'RR(max)' and 'RL(min)' are determined by positioning the trolley at the end of the overhang. The self-weight of the monorail beam is also included.

F29



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Fbx = 21.60 ksi Fbx = 12000*Cb/(Lb*12*(d/Af)) <= 0.60*Fy fbx <= Fbx, O.K. (continued)


5 of 14 04/08/2023 15:49:07

Y-axis Stresses for Simple-Span:fby = 3.43 ksi fby = My/Sy

fwns = 2.10 ksi fwns = Mt*12/(Sy/2) (warping normal stress)fby(total) = 5.53 ksi fby(total) = fby+fwns

Fby = 27.00 ksi Fby = 0.75*Fy fby <= Fby, O.K.

Combined Stress Ratio for Simple-Span:S.R. = 0.434 S.R. = fbx/Fbx+fby(total)/Fby S.R. <= 1.0, O.K.

Vertical Deflection for Simple-Span:Pv = 11.200 kips Pv = P+Wh+Wt (without vertical impact)

0.0127 in. Pv/2*(L-S)/2/(24*E*I)*(3*L^2-4*((L-S)/2)^2)+5*w/12000*L^4/(384*E*I)L/61540.1733 in. Defl.(max) <= Defl.(allow), O.K.

Bending Moments for Overhang:Mx = 9.79 ft-kips Mx = (Pv/2)*(Lo+(Lo-S))+w/1000*Lo^2/2My = 0.77 ft-kips My = (Ph/2)*(Lo+(Lo-S))

Lateral Flange Bending Moment from Torsion for Overhang: (per USS Steel Design Manual, 1981)e = 6.000 in. e = d/2 (assume horiz. load taken at bot. flange)

at = 21.662 at = SQRT(E*Cw/(J*G)) , E=29000 ksi and G=11200 ksiMt = 1.05 ft-kips Mt = Ph*e*at/(d-tf)*TANH(Lo*12/at)/12

X-axis Stresses for Overhang:fbx = 2.32 ksi fbx = Mx/Sx

Lbo/rt = 72.00 Lbo/rt = Lbo*12/rtFbx = 21.60 ksi Fbx = 12000*Cbo/(Lbo*12*(d/Af)) <= 0.60*Fy fbx <= Fbx, O.K.

Y-axis Stresses for Overhang:fby = 1.62 ksi fby = My/Sy



Combined Stress Ratio for Overhang:S.R. = 0.332 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 = 11.200 kips Pv = P+Wh+Wt (without vertical impact)

0.0054 in. Pv*Lo^2*(L+Lo)/(3*E*I)+w/12000*Lo*(4*Lo^2*L-L^3+3*Lo^3)/(24*E*I)

L/22200.0267 in. Defl.(max) <= Defl.(allow), O.K.

Bottom Flange Bending (simplified):be = 7.200 in. Min. of: be = 12*tf or S*12 (effective flange bending length)tf2 = 0.859 in. tf2 = tf+(bf/2-tw/2)/2*(1/6) (flange thk. at web based on 1:6 slope of flange)

am = 1.818 in. am = (bf/2-tw/2)-(k-tf2) (where: k-tf2 = radius of fillet)Mf = 6.339 in.-kips Mf = Pw*amSf = 0.521 in.^3 Sf = be*tf^2/6fb = 12.16 ksi fb = Mf/Sf

Fb = 27.00 ksi Fb = 0.75*Fy fb <= Fb, O.K.

D(max) = D(max) =D(ratio) = D(ratio) = L*12/D(max)D(allow) = D(allow) = L*12/450

D(max) = D(max) =D(ratio) = D(ratio) = Lo*12/D(max)D(allow) = D(allow) = Lo*12/450


6 of 14 04/08/2023 15:49:07

(continued)


7 of 14 04/08/2023 15:49:07

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

Local Flange Bending Stress @ Point 0: (Sign convention: + = tension, - = compression)-10.73 ksi

2.46 ksi

Local Flange Bending Stress @ Point 1:9.07 ksi

23.53 ksi


-2.46 ksi

Resultant Biaxial Stress @ Point 0:10.23 ksi

-8.05 ksi

0.00 ksi

15.86 ksi <= Fb = 0.66*Fy = 23.76 ksi, O.K.


6.80 ksi

0.00 ksi

23.39 ksi <= Fb = 0.66*Fy = 23.76 ksi, O.K.


8.05 ksi

0.00 ksi

7.41 ksi <= Fb = 0.66*Fy = 23.76 ksi, O.K.

sxo = sxo = Cxo*Pw/ta^2szo = szo = Czo*Pw/ta^2

sx1 = sx1 = Cx1*Pw/ta^2sz1 = sz1 = Cz1*Pw/ta^2

sx2 = sx2 = -sxosz2 = sz2 = -szo

sz = sz = fbx+fby+0.75*szosx = sx = 0.75*sxotxz = txz = 0 (assumed negligible)sto = sto = SQRT(sx^2+sz^2-sx*sz+3*txz^2)

sz = sz = fbx+fby+0.75*sz1sx = sx = 0.75*sx1txz = txz = 0 (assumed negligible)st1 = st1 = SQRT(sx^2+sz^2-sx*sz+3*txz^2)


bf

ta

tw

bf/4

tf

X

Z

Y

PwPw

Pw Pw

Trolley Wheel

S-shape

tw/2

Point 2

Point 0

Point 1


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MONORAIL BEAM ANALYSISFor W-shaped Underhung Monorails Analyzed as Simple-Spans with / without Overhang

Per AISC 9th Edition ASD Manual and CMAA Specification No. 74 (2004)Job Name: Subject:

Job Number: Originator: Checker:

Input:RL(min)=-4.31 RR(max)=21.71

Monorail Size: L=7 Lo=3Select: W44x335 x=3.313

Design Parameters: S=0.75

Beam Fy = 36 ksi

Beam Simple-Span, L = 7.0000 ft. W44x335Unbraced Length, Lb = 7.0000 ft.

Bending Coef., Cb = 1.00 Pv=14.05Overhang Length, Lo = 3.0000 ft. Nomenclature

Unbraced Length, Lbo = 10.0000 ft.

Bending Coef., Cbo = 1.00 W44x335 Member Properties:Lifted Load, P = 11.000 kips A = 98.30 in.^2 d/Af = 1.56

Trolley Weight, Wt = 0.200 kips d = 44.000 in. Ix = 31100.00 in.^4

Hoist Weight, Wh = 0.100 kips tw = 1.020 in. Sx = 1410.00 in.^3

Vert. Impact Factor, Vi = 25 % bf = 16.000 in. Iy = 1200.00 in.^4

Horz. Load Factor, HLF = 10 % tf = 1.770 in. Sy = 151.00 in.^3

Total No. Wheels, Nw = 4 k= 2.560 in. J = 74.400 in.^4

Wheel Spacing, S = 0.7500 ft. rt = 4.120 in. Cw = 536000.0 in.^6

Distance on Flange, a = 0.3750 in.

Support Reactions: (with overhang)Results: 21.71 = Pv*(L+(Lo-S/2))/L+w/1000/(2*L)*(L+Lo)^2

-4.31 = -Pv*(Lo-S/2)/L+w/1000/(2*L)*(L^2-Lo^2)Parameters and Coefficients:

Pv = 14.050 kips Pv = P*(1+Vi/100)+Wt+Wh (vertical load)Pw = 3.512 kips/wheel Pw = Pv/Nw (load per trolley wheel)Ph = 1.100 kips Ph = HLF*P (horizontal load)ta = 1.770 in. ta = tf (for W-shape)

0.050Cxo = -2.000Cx1 = 0.296Czo = 0.193Cz1 = 2.711

Bending Moments for Simple-Span:x = 3.313 ft. x = 1/2*(L-S/2) (location of max. moments from left end of simple-span)

Mx = 24.07 ft-kips Mx = (Pv/2)/(2*L)*(L-S/2)^2+w/1000*x/2*(L-x)My = 1.72 ft-kips My = (Ph/2)/(2*L)*(L-S/2)^2

Lateral Flange Bending Moment from Torsion for Simple-Span: (per USS Steel Design Manual, 1981)e = 22.000 in. e = d/2 (assume horiz. load taken at bot. flange)

at = 136.580 at = SQRT(E*Cw/(J*G)) , E=29000 ksi and G=11200 ksiMt = 0.97 ft-kips Mt = Ph*e*at/(2*(d-tf))*TANH(L*12/(2*at))/12

X-axis Stresses for Simple-Span:fbx = 0.20 ksi fbx = Mx/Sx

Lb/rt = 20.39 Lb/rt = Lb*12/rt

RR(max) =RL(min) =

l = l = 2*a/(bf-tw)Cxo = -2.110+1.977*l+0.0076*e^(6.53*l)Cx1 = 10.108-7.408*l-10.108*e^(-1.364*l)Czo = 0.050-0.580*l+0.148*e^(3.015*l)Cz1 = 2.230-1.490*l+1.390*e^(-18.33*l)

C15

From AISC 9th Edition ASD Manual, page 5-47: Cb = 1.75+1.05*(M1/M2)+0.3*(M1/M2) <= 2.3 where: M1/M2 = ratio of smaller to larger bending moment at ends of the unbraced length, 'Lb'.

C17

The unbraced length for the overhang (cantilever) portion, 'Lbo', is often debated. Here are some recommendations from different sources: 1. Per Fluor Enterprises Guideline 000.215.1257 - "Hoisting Facilities" Lbo = Lo+L/2 2. Per Dupont Standard DB1X - "Design and Installation of Monorail Beams" Lbo = 3*Lo 3. Per ANSI MH27.1 - "Underhung Cranes and Monorail Systems" Lbo = 2*Lo 4. Per British Steel Code B.S. 449, pages 42-44 (1959) Lbo = 2*Lo (for top flange of monorail beam restrained at support) Lbo = 3*Lo (for top flange of monorail beam unrestrained at support) 5. Per AISC Journal article by N. Stephen Tanner - "Allowable Bending Stresses for Overhanging Monorails" (3rd Quarter, 1985) Lbo = Lo+L (used with a computed value of 'Cbo' from article)

F28


F29



10 of 14 04/08/2023 15:49:07

Fbx = 23.76 ksi Fbx = 0.66*Fy fbx <= Fbx, O.K. (continued)


11 of 14 04/08/2023 15:49:07

Y-axis Stresses for Simple-Span:fby = 0.14 ksi fby = My/Sy



Combined Stress Ratio for Simple-Span:S.R. = 0.019 S.R. = fbx/Fbx+fby(total)/Fby S.R. <= 1.0, O.K.

Vertical Deflection for Simple-Span:Pv = 11.300 kips Pv = P+Wh+Wt (without vertical impact)

0.0002 in. Pv/2*(L-S)/2/(24*E*I)*(3*L^2-4*((L-S)/2)^2)+5*w/12000*L^4/(384*E*I)L/487786

0.1867 in. Defl.(max) <= Defl.(allow), O.K.

Bending Moments for Overhang:Mx = 38.39 ft-kips Mx = (Pv/2)*(Lo+(Lo-S))+w/1000*Lo^2/2My = 2.89 ft-kips My = (Ph/2)*(Lo+(Lo-S))

Lateral Flange Bending Moment from Torsion for Overhang: (per USS Steel Design Manual, 1981)e = 22.000 in. e = d/2 (assume horiz. load taken at bot. flange)

at = 136.580 at = SQRT(E*Cw/(J*G)) , E=29000 ksi and G=11200 ksiMt = 3.57 ft-kips Mt = Ph*e*at/(d-tf)*TANH(Lo*12/at)/12

X-axis Stresses for Overhang:fbx = 0.33 ksi fbx = Mx/Sx

Lbo/rt = 29.13 Lbo/rt = Lbo*12/rtFbx = 23.76 ksi Fbx = 0.66*Fy fbx <= Fbx, O.K.

Y-axis Stresses for Overhang:fby = 0.23 ksi fby = My/Sy



Combined Stress Ratio for Overhang:S.R. = 0.043 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 = 11.300 kips Pv = P+Wh+Wt (without vertical impact)

0.0006 in. Pv*Lo^2*(L+Lo)/(3*E*I)+w/12000*Lo*(4*Lo^2*L-L^3+3*Lo^3)/(24*E*I)

L/554950.0800 in. Defl.(max) <= Defl.(allow), O.K.

Bottom Flange Bending (simplified):be = 9.000 in. Min. of: be = 12*tf or S*12 (effective flange bending length)

am = 6.700 in. am = (bf/2-tw/2)-(k-tf) (where: k-tf = radius of fillet)Mf = 23.534 in.-kips Mf = Pw*amSf = 4.699 in.^3 Sf = be*tf^2/6fb = 5.01 ksi fb = Mf/Sf

Fb = 27.00 ksi Fb = 0.75*Fy fb <= Fb, O.K.

D(max) = D(max) =D(ratio) = D(ratio) = L*12/D(max)D(allow) = D(allow) = L*12/450

D(max) = D(max) =D(ratio) = D(ratio) = Lo*12/D(max)D(allow) = D(allow) = Lo*12/450


12 of 14 04/08/2023 15:49:07

(continued)


13 of 14 04/08/2023 15:49:07

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

Local Flange Bending Stress @ Point 0: (Sign convention: + = tension, - = compression)-2.24 ksi

0.22 ksi


3.04 ksi


-0.22 ksi


-1.68 ksi

0.00 ksi

1.98 ksi <= Fb = 0.66*Fy = 23.76 ksi, O.K.


0.25 ksi

0.00 ksi

2.51 ksi <= Fb = 0.66*Fy = 23.76 ksi, O.K.


1.68 ksi

0.00 ksi

1.60 ksi <= Fb = 0.66*Fy = 23.76 ksi, O.K.

sxo = sxo = Cxo*Pw/ta^2szo = szo = Czo*Pw/ta^2

sx1 = sx1 = Cx1*Pw/ta^2sz1 = sz1 = Cz1*Pw/ta^2

sx2 = sx2 = -sxosz2 = sz2 = -szo

sz = sz = fbx+fby+0.75*szosx = sx = 0.75*sxotxz = txz = 0 (assumed negligible)sto = sto = SQRT(sx^2+sz^2-sx*sz+3*txz^2)



tw

Pw Pw

Point 2

Point 1

Point 0

bf

tf

Y

Z

X


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monorail 2007

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