Journal of Science & Technology 100 (2014) 006-010 Optimizing Triangular Cross Section for Increasing Load Capability of I-Beam Trinh Dong Tinh*, Vuong Van Thanh Hanoi University ofScience and Technology, No 1, Dai Co Viet Str., Hai Ba Trung, Ha Noi, Viet Nam Received: December 27, 2013; accepted- April 22, 2014 Abstract Results of analysis on load capability of I-beam using as a railway of hoist in single girder crane show that the standard I-beam is applicable only for the crane with shoii span and tight capacity This paper presents a combined beam, make of I-beam and steel plate to change the cross section to triangular type with goal increasing the load capability of the beam, loaded in both vertical and horizontal directions, and improves the torsion resistance as well The dimensions of the combined beam are determined by establishing and solving the structure optimizing problem with the goal to minimize the beam's weight in the terms of strength, stiffness and technology of the structure The globalized reduced gradient method, integrated in Excels as Solver tool is used to solve this nonlinear optimization problem Keywords: Crane metal stnjcture, I-beam, Cross-section optimization Introduction The steel I-beam is widely applied as the main beam in the cranes and the portal bridge crane Besides, it is also used in the monorail systems for mechanical handling of materials in workshops and storages For these machines, the electtic hoists are usually used as equipment for lifting, lowering the load and transporting it along the I-beam as shown in Fig,I [i] The rated load (load capacity) and other parameters of electric hoist are standardized by the hoist manufacture and for each series of rated load the hoist manufacture also specified which size of standard I-beam to use with For example, with the V-series electric hoist, lifting height ranges from to m, the main parameters of hoist and I-beam are listed in Table I When the mechanisms work, the loads acting on the beam include the lifting load, the weight of hoist, and the dynamic loads These loads cause the stiess in the beam and deform it In order to guarantee the work ability of stmcture, the maximum stress and the deformation must be less than the allowable values The sttess and deformation of the main beam of overhead travelling crane could be calculated by using the diagram shown in Fig.2 [2], in which: S is considered as concenttated load, including the lifting load SL, the weight of electric hoist Sec and the 'Corresponding author, Tel, (+84) 904.274.984 Email tinh.ttinhdong@hust.edu dynamic vertical loads; dts the disttibuted load caused of the beam's weight; Sfi and dH are the horizontal loads by the inertial force on the main beam when the crane starts or stops In the case of common cranes, horizontal load is taken by 10% of vertical loads; L is the span of the main beam, and X is the location of electric hoist on the beam, and varies from to L The sttess and deflection will be maximum at the center of beam when the hoist is at this place (JC L/2) The effect of beam's weight is not large [3, 4], and it can be ignored in preliminary calculation In order to satisfy the requirement on the strength, the maximum sttess should satisfy: (T = ^ z + -^y b b b b U b L- b % Fig Forces acting on a beam These results indicate that using single I-beam with the long span and heavy lifting load is impossible For example, when the load is one ton, the 1-250 beam can be used only for cranes with the span to 18m by the sttength criteria and to 7m by the deflection critena When the load is 20 toimes, the I600 beam can be used for very short span (approximately 3m) Since the demand to increase the span and load, using I-beam as the railway for electtic hoist, evaluation of the beam stmcture is a necessary task Some studies have proposed the solutions to enhance load capacity of I-beam [3-5] Fig.4a shows the method to increase the load capacity of I-beam by cutting I-beam into two zigzag parts and assembling them to a large beam, leaving the hexagonal holes at he parameters of electric hoists I-beam height mm 11 Hoist weight kg 145 11 175 200/250/300 8,4 280 200/250/300 7,5 385 250/300/450 6,7 685 300/450 7,5 10 930 450/600 1230 450/600 15 2340 450/600 20 4,2 2940 450/600 Rated load, t Lifting speed fii/ph 0.5 150/200/250 middle of beam On the other hand, this method can be only used when fabricating new beam In the case of updating and repairing, the beams need to be dismantled all stmctures and reproduced Another solution is welding the thick steel plates or U shaped steel on the top side of I-beam (Fig.4b) also mentioned However, these solutions only increase the bending resistance about the y axis The bending resistance about the z axis changes insignificantly and this open section has less torsion strength Fig.4c shows another method to increase the bending and torsion strength [6], but the free height for the railway is reduced that affects to installation and moving of the electric hoist To avoid the disadvantage of this solution, the beam with triangular cross-section shown in Fig.5 can be used On the other hand, many researchers have investigated the optimization of main beam crosssection such as T.V.Chien [3], Koiarov el al [4], Cho and Kwak [7] However, these studies have not mentioned to the cross-section as shown in Fig.5, Determining the Optimal Cross Section of the Beam The weight of beam is approximately proportional to the area of its metal cross-section By this reason, the weight optimization can be changed to solve the problem of determining the optimal cross section of beam [6], For the section in Fig.5, it is a non-linear consttained optimization problem, with discrete variables and can form as the follows Given: input parameters, such as loads on beam (vertical and horizontal), crane span L, variables to be determined are the sizes h, c, t of steel plate and angle a of the section Goal: the metal cross-section to be minimal Journal ofScience & Technology 100 (2014) 006-010 -A-U-I2?0 - * — l 1,-1500 MPa 150 ^/ fy p ô-5t-M50 -ã-lOt-1600 ã151-1600 201-1600 100 30 T „ _ Span,m 10 IS 20 25 Fig Using ability of I-beam in strength and deflection conditions Fig Solutions for increasing the load capacity of I-beam Constrains: the requirements of beam strength and deflection to fulfill, and the plate sizes are in the standard set Fig Triangular cross-section of the beam Where, y, and z, are coordinates of C, in the O(yo,zo) coordinate system, defined as: To find out the sttess and deflection by Eq,(I) and Eq,(3) there have to detenmne the geometinc characteristics of the section The moments of inertia of beam cross-section (Fig,5) are given by: = 0,5ff -~(A4-0,50 (7) (4) (5) Where, /=I,2,3,4 are the parts in the combined section; A^ is the cross sectional area, /yi and /z, are the moment of inertia to the y and z axis, respectively, yc\ and Zcp are the distance from the center of gravity C of combined cross section to the center of gravity d of parts to the y and z axis The coordinates of the center of gravity C in the 0(yo,Zo) coordinate system are: "HA- "E4 (6) h can be calculated by the following relationship: yc Zc, in Eq (4) and (5) are evaluated i y,i=y,-yci ^„=z,-z, (*) The maximum stresses of cross-section are calculated with Eq (1) in the boundary points as Journal of Science & Technology 100 (2014) 006-010 y = Q,5b; z^h-ht-hz y = 0,5B; z = H-z (9) The size of I-beam is selected based on the rated load of the hoist, and the dimensions 5, H and other parameters of beam will refer to the standard The above mentioned optinuzation problems are exammed by Globalized Reduced Gradient method (GRG2), integrated m the MS Excel Solver tool The I-beam is according to the JIS standard [8] as specifying in the Hitachi catalog, Results and Discusions In cases of span less than 7m, the single beam with larger size can be used For example, 1-250 and 1-300 beams can use for the loads of ton, 1-600 - for 10 tonnes (Fig.3) When the span is large, the single beam cannot be used due to insufficient the conditions of the sttength and reflection, and it need to change to the combined beam In this paper, the optunization of combined cross sectional area is examined for the loads of I and 10 tonnes wath the span L of 7m to 25m by using the smallest I-beam specified in Hitachi catalog, Fig.6a and 6b show the survey results of the optimal parameters of the combined beam used 1-200 and the steel plate for the load of ton and 1-450 for the load of 10 tonnes The vertical axes show the optimal combined section area and some characteristics of the section such as total height, maximal sttess in the section and maximal deflection of the beam The obtained results indicate that if the load is small, the deflection of the optimal beam is approximately equal to the allowable value and the sttess is less than the allowable sttess In this case, the cross-section of the main beam is optimized based on the condition of the deflection If the loads are large and the span of beam is small, the sttess on the optimal beam is approximately equal to the allowable sttess, and the cross-section of beam is optimized in the condition of the strength WTien the span increases, the crosssectional area is optimized in the condition of the deflection Thus, m the general case, the cross-section needs to be determined with both two criteria (strength and deflection) The optimization of crosssection only based on one condition as proposed in some studies is not satisfactory Comparing the single I-beam and the combined beam, it can be easily seen that the weight of the combined beam is smaller than that of the single one With the load of I ton, the single 1-300 beam with the cross-section area 97,8Sc[n^ can be used for span to 12m (Fig.3) The triangular combined beam using I200 with the same range cross-section area (90,56cm-) can be used for I9m span (Fig.6a), or with the same span 12m, the area of combined section IS only 65,50cm-, reduced to 2/3 of the single 1-300 beam The weight of combined beam is less than single I-beam significanfly In the case of 10 tonnes load, the single 1-600 (with cross-section I69,40cm^) can use only for span to 10m, and by other hand, for smaller span (7 to 9m), this 1-600 must be used because the 1-450 is not applicable due to deflection condition (Fig.3) For the span of 7, and 9m, the cross-section area of combined beam is 149,3; 156,8 and !64,3cm^ respectively, less than 1-600 (I69,4cm^) Actually, for these cases, the combined beam is not effective because of technological reason, but using the combined triangular section (with I450) can solve the problem with larger span (Fig.6b) When the span of beam is over the given value (about 18-20m for two rated load of electric hoist mentioned above), the cross-sectional area increases rapidly, thus using this type section is not effective, and then other stinictures of beam should be used AOO 350 300 250 150 100 - - - j ^ ^^ — - A, cm2 —*— Stress, MPa ^^ r^^Y^ ^- -7—_ ^ ^—^i^*^ "; i i * •-»'*.^J J — • 10 ,, (im -t-h.^ml 15 !0 —•—Deflection (xO,lmml - Deflection (xClmrn) Allowable detlKtion —Allowable deflection 25 Crane span, m a) Load of ton, 1-200 beam b) Load of 10 tonnes, 1-450 beam Fig Optittuzed results for combined beam Journal ofScience & Technology 100 (2014) 006-010 Conclusions The load capability of the I-beam, using as the railway for thc hoist m the single girder overhead ttavelling cranes is examined The results indicate that in the case of heavy load or large span, the single standard I-beam should not be applicable The load capability of the I-beam can be increased by using steel plate to change the section to ttiangular type Weight optimization problem of this type beam is also examined at all This method for determining load capability and optima! dimensions of the tiiangular crosssection can be used for the hoists and I-beams by other standards The optimal resuhs and method could be applied to design the metal stmcture of highcapacity and long-span single girder cranes It is also applicable for increasing the load capability of monorail systems, using hoist as equipment to move the load up/down and away Acknow ledgments This paper was supported by the Vietnam's National Foundation for Science and Technology Development (NAFOSTED) with project No 107.02.2012.20 References [I] Hitachi Hoists; http://www.hitachi-ies.co.jp/ englislT/cataIog_library/pd£'SH-E090W.pdf [2] Tinh T.D., Thanh V.V Mo phong^ qua trinh thay doi ung suat va tinh toan ben moi cua dam cau true Tap chi Khoa hoc va Cong nghe Vien K H C N V N ! 2A (2010) 765-772 [3] Chian T V (2005) KSt edu thep may nang chuyen Nxb Hai Phong [4] Koiarov I (1988), Metal Stmcture of Material Handling Machines Technica, Sofia, Bulgaria [5] Ray S (2008) Introduction to Handling, New Age International Publishers, New Delhi, India [6] Thuong D.T., Tinh T.D,; Tinh toan toi uu ket can dam chinh cau true mot dam Tap chi khoa hpc va cong nghe 344-35 (2002) 104-109 [7] Cho, S.W., Kwak, B.M (1984) Optimal Design of Overhead Electric Crane Girders ASME Journal of Mechanisms, Transmissions, and Automation in Design, Vol.106, pp 203-208 [S] Japanese Industtial Standard - JIS G3192:2008: Dimensions, mass and permissible variations of hot rolled steel sections Materials (P) Ltd ... determining the optimal cross section of beam [6], For the section in Fig.5, it is a non-linear consttained optimization problem, with discrete variables and can form as the follows Given: input... using steel plate to change the section to ttiangular type Weight optimization problem of this type beam is also examined at all This method for determining load capability and optima! dimensions... sttess, and the cross-section of beam is optimized in the condition of the strength WTien the span increases, the crosssectional area is optimized in the condition of the deflection Thus, m the general