Copy of Monorail-Design

"MONORAIL.xls" ProgramVersion 1.3

1 of 12 04/08/2023 05:59:52

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: Subject:

Job Number: Originator: Checker:

Input:RL(min)=-5.57 RR(max)=22.22

Monorail Size: L=10 Lo=4Select: S15x50 x=4.813

Design Parameters: S=0.75Beam Fy = 50 ksi

Beam Simple-Span, L = 10.0000 ft. S15x50Unbraced Length, Lb = 10.0000 ft.

Bending Coef., Cb = 1.00 Pv=15.95Overhang Length, Lo = 4.0000 ft. Nomenclature

Unbraced Length, Lbo = 4.0000 ft.

Bending Coef., Cbo = 1.00 S15x50 Member Properties:Lifted Load, P = 13.500 kips A = 14.70 in.^2 d/Af = 4.28

Trolley Weight, Wt = 1.000 kips d = 15.000 in. Ix = 485.00 in.^4

Hoist Weight, Wh = 0.100 kips tw = 0.550 in. Sx = 64.70 in.^3

Vert. Impact Factor, Vi = 10 % bf = 5.640 in. Iy = 15.60 in.^4

Horz. Load Factor, HLF = 20 % tf = 0.622 in. Sy = 5.53 in.^3

Total No. Wheels, Nw = 4 k= 1.375 in. J = 2.120 in.^4

Wheel Spacing, S = 0.7500 ft. rt = 1.260 in. Cw = 806.0 in.^6

Distance on Flange, a = 0.3750 in.

Support Reactions: (with overhang)Results: 22.22 = Pv*(L+(Lo-S/2))/L+w/1000/(2*L)*(L+Lo)^2

-5.57 = -Pv*(Lo-S/2)/L+w/1000/(2*L)*(L^2-Lo^2)Parameters and Coefficients:

Pv = 15.950 kips Pv = P*(1+Vi/100)+Wt+Wh (vertical load)Pw = 3.988 kips/wheel Pw = Pv/Nw (load per trolley wheel)Ph = 2.700 kips Ph = HLF*P (horizontal load)ta = 0.450 in. ta = tf-bf/24+a/6 (for S-shape)

0.147Cxo = -0.855Cx1 = 0.579Czo = 0.162Cz1 = 1.981

Bending Moments for Simple-Span:x = 4.813 ft. x = 1/2*(L-S/2) (location of max. moments from left end of simple-span)

Mx = 36.94 ft-kips Mx = (Pv/2)/(2*L)*(L-S/2)^2+w/1000*x/2*(L-x)My = 6.25 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 = 7.500 in. e = d/2 (assume horiz. load taken at bot. flange)

at = 31.375 at = SQRT(E*Cw/(J*G)) , E=29000 ksi and G=11200 ksiMt = 1.76 ft-kips Mt = Ph*e*at/(2*(d-tf))*TANH(L*12/(2*at))/12

X-axis Stresses for Simple-Span:fbx = 6.85 ksi fbx = Mx/Sx

Lb/rt = 95.24 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 = 23.36 ksi Fbx = 12000*Cb/(Lb*12/(d/Af)) <= 0.60*Fy fbx <= Fbx, O.K. (continued)


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Y-axis Stresses for Simple-Span:fby = 13.57 ksi fby = My/Sy

fwns = 7.65 ksi fwns = Mt*12/(Sy/2) (warping normal stress)fby(total) = 21.22 ksi fby(total) = fby+fwns

Fby = 37.50 ksi Fby = 0.75*Fy fby <= Fby, O.K.

Combined Stress Ratio for Simple-Span:S.R. = 0.859 S.R. = fbx/Fbx+fby(total)/Fby S.R. <= 1.0, O.K.

Vertical Deflection for Simple-Span:Pv = 14.600 kips Pv = P+Wh+Wt (without vertical impact)

0.0379 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/31690.2667 in. Defl.(max) <= Defl.(allow), O.K.

Bending Moments for Overhang:Mx = 58.22 ft-kips Mx = (Pv/2)*(Lo+(Lo-S))+w/1000*Lo^2/2My = 9.79 ft-kips My = (Ph/2)*(Lo+(Lo-S))

Lateral Flange Bending Moment from Torsion for Overhang: (per USS Steel Design Manual, 1981)e = 7.500 in. e = d/2 (assume horiz. load taken at bot. flange)

at = 31.375 at = SQRT(E*Cw/(J*G)) , E=29000 ksi and G=11200 ksiMt = 3.68 ft-kips Mt = Ph*e*at/(d-tf)*TANH(Lo*12/at)/12

X-axis Stresses for Overhang:fbx = 10.80 ksi fbx = Mx/Sx

Lbo/rt = 38.10 Lbo/rt = Lbo*12/rtFbx = 33.00 ksi Fbx = 0.66*Fy fbx <= Fbx, O.K.

Y-axis Stresses for Overhang:fby = 21.24 ksi fby = My/Sy



Combined Stress Ratio for Overhang:S.R. = 1.319 S.R. = fbx/Fbx+fby(total)/Fby S.R. > 1.0

Vertical Deflection for Overhang: (assuming full design load, Pv without impact, at end of overhang)Pv = 14.600 kips Pv = P+Wh+Wt (without vertical impact)

0.1338 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/3590.1600 in. Defl.(max) <= Defl.(allow), O.K.

Bottom Flange Bending (simplified):be = 7.464 in. Min. of: be = 12*tf or S*12 (effective flange bending length)tf2 = 0.834 in. tf2 = tf+(bf/2-tw/2)/2*(1/6) (flange thk. at web based on 1:6 slope of flange)

am = 2.004 in. am = (bf/2-tw/2)-(k-tf2) (where: k-tf2 = radius of fillet)Mf = 7.991 in.-kips Mf = Pw*amSf = 0.481 in.^3 Sf = be*tf^2/6fb = 16.60 ksi fb = Mf/Sf

Fb = 37.50 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*12/300


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(continued)


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

Local Flange Bending Stress @ Point 0: (Sign convention: + = tension, - = compression)-16.88 ksi

3.20 ksi

Local Flange Bending Stress @ Point 1:11.43 ksi

39.10 ksi


-3.20 ksi

Resultant Biaxial Stress @ Point 0:22.82 ksi

-12.66 ksi

0.00 ksi

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


8.57 ksi

0.00 ksi

46.06 ksi > Fb = 0.66*Fy = 33 ksi


12.66 ksi

0.00 ksi

16.03 ksi <= Fb = 0.66*Fy = 33 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 = sy = 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)

sz = sz = fbx+fby+0.75*sz2sx = sx = 0.75*sx2txz = txz = 0 (assumed negligible)st2 = st2 = SQRT(sx^2+sz^2-sx*sz+3*txz^2)

bf

ta

tw

bf/4

tf

X

Z

Y

tw/2

PwPw

Pw Pw

Trolley Wheel

S-shape

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)=-8.01 RR(max)=24.93

Monorail Size: L=10 Lo=5.5Select: W12x58 x=4.813

Design Parameters: S=0.75Beam Fy = 36 ksi

Beam Simple-Span, L = 10.0000 ft. W12x58Unbraced Length, Lb = 10.0000 ft.

Bending Coef., Cb = 1.00 Pv=16.025Overhang Length, Lo = 5.5000 ft. Nomenclature

Unbraced Length, Lbo = 5.5000 ft.

Bending Coef., Cbo = 1.00 W12x58 Member Properties:Lifted Load, P = 13.500 kips A = 17.00 in.^2 d/Af = 1.90

Trolley Weight, Wt = 0.400 kips d = 12.200 in. Ix = 475.00 in.^4

Hoist Weight, Wh = 0.100 kips tw = 0.360 in. Sx = 78.00 in.^3

Vert. Impact Factor, Vi = 15 % bf = 10.000 in. Iy = 107.00 in.^4

Horz. Load Factor, HLF = 10 % tf = 0.640 in. Sy = 21.40 in.^3

Total No. Wheels, Nw = 4 k= 1.240 in. J = 2.100 in.^4

Wheel Spacing, S = 0.7500 ft. rt = 2.720 in. Cw = 3570.0 in.^6

Distance on Flange, a = 0.3750 in.

Support Reactions: (with overhang)Results: 24.93 = Pv*(L+(Lo-S/2))/L+w/1000/(2*L)*(L+Lo)^2

-8.01 = -Pv*(Lo-S/2)/L+w/1000/(2*L)*(L^2-Lo^2)Parameters and Coefficients:

Pv = 16.025 kips Pv = P*(1+Vi/100)+Wt+Wh (vertical load)Pw = 4.006 kips/wheel Pw = Pv/Nw (load per trolley wheel)Ph = 1.350 kips Ph = HLF*P (horizontal load)ta = 0.640 in. ta = tf (for W-shape)

0.078Cxo = -1.944Cx1 = 0.441Czo = 0.192Cz1 = 2.448

Bending Moments for Simple-Span:x = 4.813 ft. x = 1/2*(L-S/2) (location of max. moments from left end of simple-span)

Mx = 37.11 ft-kips Mx = (Pv/2)/(2*L)*(L-S/2)^2+w/1000*x/2*(L-x)My = 3.13 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 = 6.100 in. e = d/2 (assume horiz. load taken at bot. flange)

at = 66.346 at = SQRT(E*Cw/(J*G)) , E=29000 ksi and G=11200 ksiMt = 1.41 ft-kips Mt = Ph*e*at/(2*(d-tf))*TANH(L*12/(2*at))/12

X-axis Stresses for Simple-Span:fbx = 5.71 ksi fbx = Mx/Sx

Lb/rt = 44.12 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



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


9 of 12 04/08/2023 05:59:53

Y-axis Stresses for Simple-Span:fby = 1.75 ksi fby = My/Sy



Combined Stress Ratio for Simple-Span:S.R. = 0.364 S.R. = fbx/Fbx+fby(total)/Fby S.R. <= 1.0, O.K.

Vertical Deflection for Simple-Span:Pv = 14.000 kips Pv = P+Wh+Wt (without vertical impact)

0.0372 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/32230.2667 in. Defl.(max) <= Defl.(allow), O.K.

Bending Moments for Overhang:Mx = 83.01 ft-kips Mx = (Pv/2)*(Lo+(Lo-S))+w/1000*Lo^2/2My = 6.92 ft-kips My = (Ph/2)*(Lo+(Lo-S))

Lateral Flange Bending Moment from Torsion for Overhang: (per USS Steel Design Manual, 1981)e = 6.100 in. e = d/2 (assume horiz. load taken at bot. flange)

at = 66.346 at = SQRT(E*Cw/(J*G)) , E=29000 ksi and G=11200 ksiMt = 3.73 ft-kips Mt = Ph*e*at/(d-tf)*TANH(Lo*12/at)/12

X-axis Stresses for Overhang:fbx = 12.77 ksi fbx = Mx/Sx

Lbo/rt = 24.26 Lbo/rt = Lbo*12/rtFbx = 23.76 ksi Fbx = 0.66*Fy fbx <= Fbx, O.K.

Y-axis Stresses for Overhang:fby = 3.88 ksi fby = My/Sy



Combined Stress Ratio for Overhang:S.R. = 0.836 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 = 14.000 kips Pv = P+Wh+Wt (without vertical impact)

0.2757 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/2390.1467 in. Defl.(max) > Defl.(allow)

Bottom Flange Bending (simplified):be = 7.680 in. Min. of: be = 12*tf or S*12 (effective flange bending length)

am = 4.220 in. am = (bf/2-tw/2)-(k-tf) (where: k-tf = radius of fillet)Mf = 16.906 in.-kips Mf = Pw*amSf = 0.524 in.^3 Sf = be*tf^2/6fb = 32.25 ksi fb = Mf/Sf

Fb = 27.00 ksi Fb = 0.75*Fy fb > Fb

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


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(continued)


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

Local Flange Bending Stress @ Point 0: (Sign convention: + = tension, - = compression)-19.01 ksi

1.88 ksi


23.94 ksi


-1.88 ksi


-14.26 ksi

0.00 ksi

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


3.24 ksi

0.00 ksi

23.97 ksi > Fb = 0.66*Fy = 23.76 ksi


14.26 ksi

0.00 ksi

12.39 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 = sy = 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)

sz = sz = fbx+fby+0.75*sz2sx = sx = 0.75*sx2txz = txz = 0 (assumed negligible)st2 = st2 = 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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