"MONORAIL.xls" "MONORAIL.xls" Program Version 1.6
"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 a 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 S-shaped Monorail Beam W-shaped Monorail Beam
This documentation sheet Monorail beam analysis for S-shaped beams Monorail beam analysis for W-shaped beams
Program Assumptions and Limitations: 1. The following references were used in the development of of this program: a. Fluor Enterprises, Inc. - Guideline 000.215.1257 - "Hoisting Facilities" (August 22, 2005) b. Dupont Engineering Design Standard: Standard: DB1X - "Design and Installation of Monorail Beams" Beams" (May 20 c. American National Standards Institute (ANSI): MH27.1 - "Underhung Cranes Cranes and Monorail Syatems d. American Institute of Steel Construction (AISC) 9th Edition Allowable Stress Design (ASD) Manual e. "Allowable Bending Stresses for Overhanging Monorails" - by N. Stephen Tanner AISC Engineering Journal (3rd Quarter, 1985) f. Crane Manufacturers Association of America, 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, 449, pages 42-44 (1959) i. USS Steel Design Manual - Chapter 7 "Torsion" - by R. L. Brockenbrough and B.G. Johnston (1981 (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 overhang (cantilever) portion, 'Lbo', of an underhung underhung monorail beam is often de The following are some recommendations from the references cited above: a. Fluor Guideline Guideline 000.215.1257: Lbo = Lo+L/2 b. Dupont Standard DB1X: Lbo = 3*Lo 3*Lo c. ANSI Standar Standard d MH27.1: MH27.1: Lbo = 2*Lo d. British Steel Code B.S. 449: Lbo = 2*Lo (for top flange flange of monorail monorail beam restrained restrained at support) 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 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 bending coefficient, 'Cbo', for the overhang (cantil portion of the monorail beam from reference reference "e" in note #1 above. This is located off of the main calculatio Note: if this computed value of 'Cbo' is used and input, then per this reference the total value of Lo+L shoul 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 usually considered minimal for underhung, hand-operated monorail systems. 5. This program contains “comment boxes” which contain a wide variety of information including explanations input or output items, equations equations used, data tables, etc. (Note: presence of a “comment box” is denoted by “red triangle” in the upper upper right-hand corner of a cell. Merely move the mouse pointer to the desired cell to the contents of that particular "comment box".)
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"MONORAIL.xls" Program Version 1.6
MONORAIL BEAM ANALYSIS For 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 L=6.5 x=3.1
Monorail Size: Select: Design Parameters: Beam Fy eam Simple-Span, L = Unbraced Length, Lb = Bending Coef., Cb = Overhang Length, Lo = nbraced Length, Lbo = Bending Coef., Cbo = Lifted Load, P = Trolley Weight, Wt = Hoist Weight, Wh = ert. Impact Factor, Vi = rz. Load Factor, HLF = otal No. Wheels, Nw = Wheel Spacing, S = istance on Flange, a =
S12x50
RR(max)=15.67 Lo=1
S=0.6
36 ksi 6.5000 ft. 6.5000 ft. 1.00 1.0000 ft. 7.5000 ft. 1.00 11.000 kips 0.100 kips 0.100 kips 25 % 10 % 4 0.6000 ft. 0.5000 in.
Results: Parameters and Coefficients: Pv = 13.950 kips Pw = 3.488 kips/wheel Ph = 1.100 kips ta = 0.514 in. λ = 0.209 Cxo = -0.813 Cx1 = 0.687 Czo = 0.186 Cz1 = 1.783
S12x50 Pv=13.95
Nomenclature
A= d= tw = bf = tf = k= rt =
S12x50 Member Properties: 14.60 in.^2 d/Af = 3.32 12.000 in. Ix = 303.00 0.687 in. Sx = 50.60 5.480 in. Iy = 15.60 0.659 in. Sy = 5.69 1.438 in. J = 2.770 1.250 in. Cw = 502.0
in.^4 in.^3 in.^4 in.^3 in.^4 in.^6
Support Reactions(with overhang) RR(max) = 15.67 = Pv *(L+(Lo-S/2))/L+w/1000/(2*L)*(L+Lo)^ RL(min) = -1.34 = -Pv* (Lo-S/2)/L+w/1000/(2*L)*(L^2-Lo^2) Pv = P*(1+Vi/100)+Wt+Wh (vertical load) Pw = Pv/Nw (load per trolley wheel) Ph = HLF*P (horizontal load) ta = tf-bf/24+a/6 (for S-shape) λ = 2*a/(bf-tw) Cxo = -1.096+1.095* λ+0.192*e^(-6.0*λ) Cx1 = 3.965-4.835*λ-3.965*e^(-2.675*λ) Czo = -0.981-1.479*λ+1.120*e^(1.322*λ) Cz1 = 1.810-1.150*λ+1.060*e^(-7.70*λ)
Bending Moments for Simple-Span: x = 3.100 ft. x = 1/2*(L-S/2) (location of max. moments from left end of simple-spa 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-S (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 ksi Mt = 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 Fbx = 21.60 ksi Fbx = 12000*Cb/(Lb*12*(d/Af)) <= 0.60*Fy
fbx <= Fbx; O.K. (continued)
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"MONORAIL.xls" Program Version 1.6 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 Combined Stress Ratio for Simple-Span: S.R. = 0.434 S.R. = fbx/Fbx+fby(total)/Fby
fby <= Fby; O.K.
S.R. <= 1.0; O.K.
Vertical Deflection for Simple-Span: Pv = 11.200 kips Pv = P+Wh+Wt (without vertical impact) ∆(max) = 0.0127 in. ∆(max) = Pv/2*(L-S)/2/(24*E*I)*(3*L^2-4*((L-S)/2)^2)+5*w/12000*L^4/(384 ∆(ratio) = L/6154 ∆(ratio) = L*12/∆(max) ∆(allow) = 0.1733 in. ∆(allow) = L*12/450 Defl.(max) <= Defl.(allow); O.K. Bending Moments for Overhang: Mx = 9.79 ft-kips Mx = (Pv/2)*(Lo+(Lo-S))+w/1000*Lo^2/2 My = 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 ksi Mt = 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/rt Fbx = 21.60 ksi Fbx = 12000*Cbo/(Lbo*12*(d/Af)) <= 0.60*Fy Y-axis Stresses for Overhang: fby = 1.62 ksi fwns = 4.42 ksi fby(total) = 6.05 ksi Fby = 27.00 ksi
fby = My/Sy fwns = Mt*12/(Sy/2) (warping normal stress) fby(total) = fby+fwns Fby = 0.75*Fy
Combined Stress Ratio for Overhang: S.R. = 0.332 S.R. = fbx/Fbx+fby(total)/Fby
fbx <= Fbx; O.K.
fby <= Fby; O.K.
S.R. <= 1.0; O.K.
Vertical Deflection for Overhan (assuming full design load, Pv without impact, at end of overhang) Pv = 11.200 kips Pv = P+Wh+Wt (without vertical impact) ∆(max) = 0.0054 in. ∆(max) = Pv*Lo^2*(L+Lo)/(3*E*I)+w/12000*Lo*(4*Lo^2*L-L^3+3*Lo^3)/(24*E*I) ∆(ratio) = L/2220 ∆(ratio) = Lo*12/∆(max) ∆(allow) = 0.0267 in. ∆(allow) = Lo*12/450 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 fl am = 1.818 in. am = (bf/2-tw/2)-(k-tf2) (where: k-tf2 = radius of fillet) Mf = 6.339 in.-kips Mf = Pw*am Sf = 0.521 in.^3 Sf = be*tf^2/6 fb = 12.16 ksi fb = Mf/Sf Fb = 27.00 ksi Fb = 0.75*Fy fb <= Fb; O.K. (continued)
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"MONORAIL.xls" Program Version 1.6 Bottom Flange Bending per CMAA Specification No. 74 (20 (Note: torsion is neglected) Local Flange Bending Stress @ Point 0: (Sign convention: + = tension, - = compression) σxo = -10.73 ksi σxo = Cxo*Pw/ta^2 σzo = σzo = Czo*Pw/ta^2 2.46 ksi Local Flange Bending Stress @ Point 1: σx1 = σx1 = Cx1*Pw/ta^2 9.07 ksi σz1 = 23.53 ksi σz1 = Cz1*Pw/ta^2 Local Flange Bending Stress @ Point 2: σx2 = 10.73 ksi σx2 = - σxo σz2 = σz2 = - σzo -2.46 ksi Resultant Biaxial Stress @ Point 0: σz = 10.23 ksi σz = fbx+fby+0.75* σzo σx = σx = 0.75* σxo -8.05 ksi τxz = τxz = 0 (assumed negligible) 0.00 ksi σto = 15.86 ksi σto = SQRT( σx^2+σz^2-σx*σz+3*<= τxz^2) Fb = 0.66*Fy = 23.76 ksi; O.K. Resultant Biaxial Stress @ Point 1: σz = 26.03 ksi σz = fbx+fby+0.75* σz1 σx = σx = 0.75* σx1 6.80 ksi τxz = τxz = 0 (assumed negligible) 0.00 ksi σt1 = 23.39 ksi σt1 = SQRT( σx^2+σz^2-σx*σz+3*<= τxz^2) Fb = 0.66*Fy = 23.76 ksi; O.K. Resultant Biaxial Stress @ Point 2: σz = σz = fbx+fby+0.75* σz2 6.54 ksi σx = σx = 0.75* σx2 8.05 ksi τxz = τxz = 0 (assumed negligible) 0.00 ksi σt2 = σt2 = SQRT( σx^2+σz^2-σx*σz+3*<= τxz^2) 7.41 ksi Fb = 0.66*Fy = 23.76 ksi; O.K.
Point 1
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"MONORAIL.xls" Program Version 1.6
MONORAIL BEAM ANALYSIS For 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: Monorail Size: Select: W44x335 Design Parameters: Beam Fy 36 ksi eam Simple-Span, L = 7.0000 ft. Unbraced Length, Lb = 7.0000 ft. Bending Coef., Cb = 1.00 Overhang Length, Lo = 3.0000 ft. nbraced Length, Lbo = 10.0000 ft. Bending Coef., Cbo = 1.00 Lifted Load, P = 11.000 kips Trolley Weight, Wt = 0.200 kips Hoist Weight, Wh = 0.100 kips ert. Impact Factor, Vi = 25 % rz. Load Factor, HLF = 10 % otal No. Wheels, Nw = 4 Wheel Spacing, S = 0.7500 ft. istance on Flange, a = 0.3750 in. Results: Parameters and Coefficients: Pv = 14.050 kips Pw = 3.513 kips/wheel Ph = 1.100 kips ta = 1.770 in. λ = 0.050 Cxo = -2.000 Cx1 = 0.296 Czo = 0.193 Cz1 = 2.711
RL(min)=-4.31 L=7 x=3.313
RR(max)=21.71 Lo=3
S=0.75 W44x335 Pv=14.05
Nomenclature
A= d= tw = bf = tf = k= rt =
W44x335 Member 98.30 in.^2 44.000 in. 1.020 in. 16.000 in. 1.770 in. 2.560 in. 4.120 in.
Properties: d/Af = 1.56 Ix = 31100.00 in.^4 Sx = 1410.00 in.^3 Iy = 1200.00 in.^4 Sy = 151.00 in.^3 J = 74.400 in.^4 Cw = 536000.0 in.^6
Support Reactions(with overhang) RR(max) = 21.71 = Pv *(L+(Lo-S/2))/L+w/1000/(2*L)*(L+Lo)^ RL(min) = -4.31 = -Pv* (Lo-S/2)/L+w/1000/(2*L)*(L^2-Lo^2) Pv = P*(1+Vi/100)+Wt+Wh (vertical load) Pw = Pv/Nw (load per trolley wheel) Ph = HLF*P (horizontal load) ta = tf (for W-shape) λ = 2*a/(bf-tw) Cxo = -2.110+1.977* λ+0.0076*e^(6.53*λ) Cx1 = 10.108-7.408*λ-10.108*e^(-1.364*λ) Czo = 0.050-0.580*λ+0.148*e^(3.015*λ) Cz1 = 2.230-1.490*λ+1.390*e^(-18.33*λ)
Bending Moments for Simple-Span: x = 3.313 ft. x = 1/2*(L-S/2) (location of max. moments from left end of simple-spa 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-S (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 ksi Mt = 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 Fbx = 23.76 ksi Fbx = 0.66*Fy
fbx <= Fbx; O.K. (continued)
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"MONORAIL.xls" Program Version 1.6 Y-axis Stresses for Simple-Span: fby = 0.14 ksi fby = My/Sy fwns = 0.15 ksi fwns = Mt*12/(Sy/2) (warping normal stress) fby(total) = 0.29 ksi fby(total) = fby+fwns Fby = 27.00 ksi Fby = 0.75*Fy Combined Stress Ratio for Simple-Span: S.R. = 0.019 S.R. = fbx/Fbx+fby(total)/Fby
fby <= Fby; O.K.
S.R. <= 1.0; O.K.
Vertical Deflection for Simple-Span: Pv = 11.300 kips Pv = P+Wh+Wt (without vertical impact) ∆(max) = 0.0002 in. ∆(max) = Pv/2*(L-S)/2/(24*E*I)*(3*L^2-4*((L-S)/2)^2)+5*w/12000*L^4/(384 ∆(ratio) = L/487786 ∆(ratio) = L*12/∆(max) ∆(allow) = 0.1867 in. ∆(allow) = L*12/450 Defl.(max) <= Defl.(allow); O.K. Bending Moments for Overhang: Mx = 38.39 ft-kips Mx = (Pv/2)*(Lo+(Lo-S))+w/1000*Lo^2/2 My = 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 ksi Mt = 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/rt Fbx = 23.76 ksi Fbx = 0.66*Fy Y-axis Stresses for Overhang: fby = 0.23 ksi fwns = 0.57 ksi fby(total) = 0.80 ksi Fby = 27.00 ksi
fbx <= Fbx; O.K.
fby = My/Sy fwns = Mt*12/(Sy/2) (warping normal stress) fby(total) = fby+fwns Fby = 0.75*Fy
Combined Stress Ratio for Overhang: S.R. = 0.043 S.R. = fbx/Fbx+fby(total)/Fby
fby <= Fby; O.K.
S.R. <= 1.0; O.K.
Vertical Deflection for Overhan (assuming full design load, Pv without impact, at end of overhang) Pv = 11.300 kips Pv = P+Wh+Wt (without vertical impact) ∆(max) = 0.0006 in. ∆(max) = Pv*Lo^2*(L+Lo)/(3*E*I)+w/12000*Lo*(4*Lo^2*L-L^3+3*Lo^3)/(24*E*I) ∆(ratio) = L/55495 ∆(ratio) = Lo*12/∆(max) ∆(allow) = 0.0800 in. ∆(allow) = Lo*12/450 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*am Sf = 4.699 in.^3 Sf = be*tf^2/6 fb = 5.01 ksi fb = Mf/Sf Fb = 27.00 ksi Fb = 0.75*Fy fb <= Fb; O.K.
(continued)
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"MONORAIL.xls" Program Version 1.6 Bottom Flange Bending per CMAA Specification No. 74 (20 (Note: torsion is neglected) Local Flange Bending Stress @ Point 0: (Sign convention: + = tension, - = compression) σxo = σ xo = Cxo*Pw/ta^2 -2.24 ksi σzo = σzo = Czo*Pw/ta^2 0.22 ksi Local Flange Bending Stress @ Point 1: σx1 = σx1 = Cx1*Pw/ta^2 0.33 ksi σz1 = σz1 = Cz1*Pw/ta^2 3.04 ksi Local Flange Bending Stress @ Point 2: σx2 = σx2 = - σxo 2.24 ksi σz2 = σz2 = - σzo -0.22 ksi Resultant Biaxial Stress @ Point 0: σz = σz = fbx+fby+0.75* σzo 0.50 ksi σx = σx = 0.75* σxo -1.68 ksi τxz = τxz = 0 (assumed negligible) 0.00 ksi σto = σto = SQRT( σx^2+σz^2-σx*σz+3*<= τxz^2) 1.98 ksi Fb = 0.66*Fy = 23.76 ksi; O.K. Resultant Biaxial Stress @ Point 1: σz = σz = fbx+fby+0.75* σz1 2.62 ksi σx = σx = 0.75* σx1 0.25 ksi τxz = τxz = 0 (assumed negligible) 0.00 ksi σt1 = σt1 = SQRT( σx^2+σz^2-σx*σz+3*<= τxz^2) 2.51 ksi Fb = 0.66*Fy = 23.76 ksi; O.K. Resultant Biaxial Stress @ Point 2: σz = σz = fbx+fby+0.75* σz2 0.18 ksi σx = σx = 0.75* σx2 1.68 ksi τxz = τxz = 0 (assumed negligible) 0.00 ksi σt2 = σt2 = SQRT( σx^2+σz^2-σx*σz+3*<= τxz^2) 1.60 ksi Fb = 0.66*Fy = 23.76 ksi; O.K.
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