DESIGN OF GANTRY GIRDER
Design a gantry girder to be used in an industrial building carrying a manually operated overhead traveling crane for the following data: Crane capacity - 200kN Self-weight of the crane girder excluding trolley - 200kN Self-weight of the trolley, electric motor, hook, etc. - 40kN Approximate minimum approach of the crane hook to the gantry
girder - 1.20m Wheel base - 3.5m c/c distance between gantry rails - 16m c/c distance between columns (span of gantry girder) - 8m Self weight of rail section - 300N/m Diameter of crane wheels - 150mm Steel is of grade Fe 410. Design also the field welded connection if required. The support bracket connection need to designed.
SOLUTION : For Fe410 grade of steel: fu = 410 Mpa fy = fyw = fyf = 410 MPa For hand operated OT crane: Lateral loads = 5% of maximum static wheel load Longitudinal loads = 5% of weight of crab and weight lifted Maximum permissible deflection = L/500 Partial safety factors עm0 = 1.10 עmw = 1.50 (for site welds) Load factor עm1 = 1.50 ξ = ξw = ξf = √ (250/ fy) = √ (250/ 250) = 1.0
STEP 1: CALCULATION OF DESIGN FORCES Maximum wheel load: Maximum concentrated load on crane = 200 + 40 = 240kN Maximum factored load on crane = 1.5 x 240 = 360kN The crane will carry the self-weight as a UDL = 200/16 = 12.5 kN/m Factored uniform load = 1.5 x 12.5 = 18.75kN/m
For maximum reaction on the gantry girder the loads are placed on the crane girder as shown in the figure below,
Taking moment about B, RA x 16 = 360 x (16 - 1.2) + (18.75 x 16 x 16) / 2 RA = 483kN RB = 177kN
The reaction from the crane girder is distributed equally on the two wheels at the end of the crane girder. Therefore, Maximum wheel load on each = 483/2 = 241.5 kN wheel of the crane
STEP 2: CALCULATION OF MAXIMUM BM & SF: Maximum Bending Moment: It consists of maximum moments caused by the moving wheel loads on the gantry girder and self weight of the gantry girder.
For Maximum Bending Moment, the wheel loads shall be placed as shown in figure.
The calculation of maximum bending moments due to wheel loads and self weight of gantry girder has been done separately because calculation of impact load and bending moment due to it involve live load only,
Assume, Self Weight of the Gantry Girder as 2 kN/m. Total Dead Load, w = 2000 + 300 = 2300 N/m = 2.3 kN/m Factored Dead Load = 1.5 x 2.3 = 3.45kN/m
The position of one wheel load from the mid point of span = Wheel Base/4 = 3.5/4 = 0.875 m Bending moment due to live load only, Taking moment about D,
Rc x 8 = 241.5 x (8 - 1.375) + 241.5 x 3.125 Rc = 294.33kN Taking moment about C, RD x 8 = 241.5 x 1.375 + 241.5 x 4.875 RD = 188.67kN Maximum Bending Moment due to Live Load = 188.67 x 3.125 = 589.6 kNm Bending Moment due to Impact = 0.10 x 589.6 = 58.96 kNm
Total Bending Moment due to live load and impact loads = 589.6 + 58.96 = 648.56 kNm Bending Moment due to Dead Load = wl2/8 = 3.45 x (82/8) = 27.6 kNm
Therefore, Maximum B.M = 648.56 + 27.6 = 676.16 kNm = 676.16 x 106 N_mm Maximum Shear Force: It consists of maximum shear due to moving wheel loads and self
weight of the girder. For maximum shear force the wheel loads shall be placed as shown in figure, i.e one of the wheel loads should at the support.
Taking Moment about D, Rc x 8 = (241.5 x 8) + (241.5 x 4.5) Rc = 377.34 kN Hence,
Maximum Shear Force due to wheel loads = 377.34 kN
STEP 3: CALCULATION OF LATERAL FORCES: Lateral force transverse to the rails = 5% of weight of crab and weight lifted = 0.05 x 240 = 12kN
Factored lateral force = 1.5 x 12 = 18 kN
Maximum horizontal reaction due to lateral force by proportion at C, = (lateral force x Reaction at c due to vertical load) (maximum wheel load due to vertical load) = (9 x 294.33) / 241.5
= 10.97 kN Horizontal reaction due to lateral force at D = 18 - 10.97 = 7.03 kN Maximum bending moment due to lateral load by proportion = (9.0 / 241.5) x 589.6 = 21.97 kNm Maximum shear force due to lateral load by proportion = (377.34 / 241.5) x 9.0 = 14.06 kN
STEP 4: SELECTION OF PRELIMINARY TRIAL SECTION : Approximate depth of section = L / 12 = (8 x 103) / 12 = 666.66mm ≈ 600mm Approximate width of the flange = L / 30 = (8 x 103) / 30 = 266.66mm ≈ 300 mm Approximate section modulus required, Zpz = 1.4 (Mz / fy ) = (1.4 x 676.16 x 106) / 250 = 3786.5 x 103 mm3 Let us try ISWB 600 @ 1311.6 N/m with ISMC300 @ 351.2 N/m on its
top flange as shown in figure,
Properties
Notation
I-Section ISWB 600
Channel Section ISMC 300
Area
A
17038 mm2
4564 mm2
Thickness of Flange
tf
21.3 mm
13.6 mm
Thickness of Web
tw
11.2 mm
7.6 mm
Width of Flange
bf
250 mm
90 mm
Moment of Inertia
Iz
106198.5 x 104 mm4
6362.6 x 104 mm4
Iy
4702.5 x 104 mm4
310.8 x 104 mm4
Depth of Section
h
600 mm
300 mm
Radius at root
Rl
17 mm
Cyy
23.6 mm
STEP 5: MOMENT OF INERTIA OF GANTRY GIRDER : The distance of NA of built up section from the extreme fibre of compression flange, ỹ = (∑AY) / ∑A = {17038 X (300 +7.6)} + {4564 X 23.6} = 247.59 mm (4564 + 17038) Gross Moment of Inertia of the Built-up section Iz gross = Iz beam + Iz channel (The channel is placed over the I-section in such a manner that its yyaxis becomes the zz-axis. Therefore, in calculating Iz of the total section, Iy of the channel will become Iz of the channel.)
Iz gross = [(106198.5 x 104) + (17038 x (307.6 - 247.59)2)] + [(310.8 x 104) + (4564 x (247.59 -23.6)2)] = 135,543x104 mm4
Iy gross
= Iy beam + Iy channel = (4702.5 x 104) + (6362.6 x 104) = 11,065.1x104 mm4 Zez = (Iy /y) = (135,543 x 104) / (600 + 7.6 - 247.59) = 3764.98x103 mm3 Plastic Modulus of Section (Ignoring the Fillets) Equal area axis (Refer to Fig) = 4564 + (250 X 21.3) + (ỹ1 t w) = 250 X 21.3 + (600-2 X 21.3 ỹ1 ) X 11.2 Z = 74.95 mm (From Lower surface of top flange of I – Section)
Plastic section modulus of the section above equal axis, Zpz1 = [300 X 7.6 X (74.95 + 21.3 + (7.6 / 2))]
+ [2 X (90-7.6)x13.6(74.95 + 21.3 – (90-7.6/2))] + [250 X 21.3 x (74.95 + (21.3/2))] + [74.95 X 11.2 X (74.95/2)] Zpz1 = 838.775 X 103 mm3 Plastic section modulus of the section below equal axis, Zpz2 = [250 X 21.3 X ( 600 - 21.3 - 74.95 - ( 21.3/2))] + [((600 – 2 X 21.3 – 74.95)2 / 2)] Zpz2 = 3929.202 X 103 mm3 Zpz = Zpz1 + Zpz2 = (838.775 X 103) + (3929.202 X 103)
Zpz = 4767.977 X 103 mm3
Plastic section modulus of compression flange about yy – axis, Zpfy = [((250 X 21.3 X 250) /4)] + [((2 X (300 – (13.6 X 2))2 X 7.6) / 8)] + [2 x (13.6 x 90 x ( 300 – 13.6) / 2)] = 824.763 X 103 mm3
STEP 6: CLASSIFICATION OF SECTION: Outstand of flange of I-Section, b = bf/2 = 250/2 = 125 mm. b/tf of flange of I-section = 125/21.3 = 5.86 < 8.4 (8.4ε = 8.4 x 1 = 8.4)
Outstand of flange of Channel Section, b = bf - tw = 90 - 7.6 = 82.4 mm. b/tf of flange of Channel section = 82.4/13.6 = 6.05 < 8.4 (8.4 ε = 8.4 x 1 = 8.4)
d/tw of web of I-Section = ((h – 2tf) / tw) = ((600 – 2 x 21.3) / 11.2) = 49.76 < 84 (84 ε = 84 x 1 = 84) Hence the entire section is plastic, (βb = 1.0)
STEP 7: CHECK FOR MOMENT CAPACITY: Local Moment capacity, Mdz = βb Zpz (fy / עm0 ) ≤ 1.2 x Ze (fy / עm0 ) Mdz = 1.0 x 4767.977 x 103 x (250 x 10-6 / 1.10) = 1083.63 kNm ≤ 1.2 x 3764.98 x 103 x (250 x 10-6 / 1.10) = 1026.81 kNm Hence, Moment capacity of the section, Mdz = 1026.81 kNm > 676.16 kNm which is safe.
Moment capacity of compression flange about y-axis, Mdy,f = βb Zpyf (fy / עm0 ) ≤ 1.2 x Zey,f (fy / עm0 ) = 1.0 x 824.76 x 103 x (250 x 10-6 / 1.10) = 187.44 kNm ≤ 1.2 x 580.92 x 103 x (250 x 10-6 / 1.10) = 158.43 kNm Hence moment capacity of flange, Mdy,f = 158.43 kNm Combined check for local moment capacity, (Mz / Mdz ) + (My,f / Mdy,f) ≤ 1.0 (676.16 / 1026.81) + (21.97 / 158.43) = 0.797 < 1.0 Which is safe.
STEP 8: BUCKLING RESISTANCE IN BENDING CHECK: The elastic lateral buckling moment, Mcr = c1(π2EIYhf/2L2LT)[1+1/20((LLT/ry)/(hf/tf))2]0.5 Overall depth of the section, hf = h = 600 + 7.6 = 607.6 mm
Note: hf is the centre to centre distance between flanges. In the calculation, value of hf has been taken equal to overall depth of the section to avoid cumbersome calculations. Further, this approximation will result in conservative design. Effective length, LLT = 8 x 102 mm Thickness of flange, tf = 21.3 + 7.6 = 28.9 mm Radius of gyration, ry = √(IY/A) = √((11065.1x104)/(17038+4564)) = 71.57mm
The coefficient, c1 = 1.132 (From Appendix XIV, assuming uniform load condition) Mcr = [(1.132x2x2x105x11065.1x104x607.6) / (2x(8x103)2)] x [1+1/20((8x103/71.57)/(607.6/28.9))2]0.5 = 1823.24 x 106 Nmm Non dimensional slenderness ratio, λLTz = √(βb Zpz fy / Mcr) = √((1 x 4783.594 x 103 x 250) / (1823.24 x 106)) = 0.809 ΦLTz = 0.5 [1 + αLT (λLTz – 0.2) + λLTz2] αLT = 0.21 (assuming connection of channel with I-section flange by intermittent fillet welds. For uniform weld, αLT = 0.49) ΦLTz = 0.5 x [1+0.21 x (0.809 – 0.2) + 0.8092] = 0.891
χLTz
= [1/( ΦLTz + ΦLTz2 - λLTz2)0.5] = [1/(0.891 + 0.8912 – 0.8092)0.5] = 0.791
Design bending compressive stress, fbd = χLTz (fy / עm0 ) = 0.791 (250/1.10) = 179.77 N/mm2 The design bending strength, Mdz = βb Zpz fbd = 1.0 x 4767.977 x 103 x 179.77 x 10-6 = 857.14 kNm > 676.16 kNm (which is alright)
Hence, the beam is safe in bending under vertical load.
STEP 9: Since lateral forces are also acting, the beam must be checked for biaxial bending. The bending strength about y-axis will be provided by the top flange only as the lateral loads are applied there only. Mdy =Zyt (fy/γm0) Zyt = section modulus of top flange about yy-axis. = (4702.5x104/2 +6362.6x104)/(300/2) = 580.92 x 103 mm3
(Assuming the moment of inertia of top flange to be half of the moment of inertia of I-section) Mdy = 580.92 x 103 x (250/1.1) x (10-6) = 132.02kNm. (Mz/Mdz) + (My/Mdy) < 1.0 = (676.16/859.946) + (21.97/132.02) = 0.952 < 1.0 Hence, the section is safe.
STEP 10: CHECK FOR SHEAR CAPACITY : Maximum shear force due to wheel load = 377.34 kN Impact load = 0.1 x 377.34 = 37.73 kN Design shear force = 377.34 + 37.73 = 415.05 kN Shear capacity = Av fyw / (√3 x γmo)
= [((600x11.2) x 250x10-3) / (√3x1.1) ] = 881.77 kN > 415.07kN, which is safe. Maximum shear, V = 415.07 kN < 529 kN (i.e. 0.6*Vd = 0.6 x 881.77 = 529kN) Since V < 0.6*Vd, the case is of low shear. No reduction will be therefore there in the moment capacity.
STEP 11: WEB-BUCKING CHECK: We should check for buckling under the wheel load. Buckling Resistance = (b1+n1)tw*fcd Where, b1 = bearing length=150mm(diameter of wheel) n1 = (600/2) +2x7.6 = 315.2 mm
Slenderness ratio of the web, λw = 2.45 (d1/tw) = 2.45(600 -2x(21.3 + 17))/11.2 = 114.49 For λw=114.49, fy=250N/mm2 and buckling curve c, the design compressive stress from Table 7.6, fcd = 89.64 N/mm2 Therefore, Buckling Resistance = (150 + 315.2) x 7.6 x 89.64 x 10-3 = 316.9 kN > 241.5kN (which is safe)
STEP 12: DEFLECTION CHECK: δ = wL3x(3a/4L –a3/L3) 6EI W = Maximum Static Wheel Load = (241.5 / 1.5) = 161 kN a = (L - c) /2 = [(8x103) – (3.5x103)] / 2 = 2.25 x 103 mm Therefore, Vertical deflection = (161x103)x(8x103)3x{(3x2.25x103)/(4x8x103)-(2.25x103)3/(8x103)3)
[6 x (2x105) x 135,543x104] = 9.56 mm Permissible maximum deflection, = L / 500 = 8000 / 500 = 16 mm > 9.56 mm, which is safe.
STEP 13: DESIGN OF CONNECTIONS : The required shear capacity of the weld, q = Vaỹ / Iz V = 377.34 kN A = 4564 mm2 (area above the section) ỹ = (247.59 -7.6) = 239.9 mm Iz = 135,543x104 mm4
Therefore, q = [(377.34 x 103) x 4564 x 239.9] / [135,543x104] = 304.82 N/mm Let us provide 3 mm weld to connect channel with flange of I-section. Strength of the weld provided, = (0.7x3x410)/(√3x1.50) = 331.39 N/mm > 304.82 N/mm Hence, provide 3 mm size intermittent fillet welds for making the connection.
DRAWINGS & DETAILING
