Tối ưu hóa các mặt cắt ngang hình tam giác để tăng khả năng chịu tải của thanh dầm chữ i

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

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

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

1 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

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

standard I-beam to use with For example, with the

V-series electric hoist, lifting height ranges from 6 to

8 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

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

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 0 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<{a\ (I) Where My and Mz are the bending moment fo the y and z axis, respectively

When ignoring beam's weight, these values are calculated by the following equations:

M ^ ^ ^ ; M.=^^ (2)

^ 4 ' 4 ^

ly and h are the moment of inertia of the cross section with respect to the y, z axis, respectively

y and z are the coordinates of the points on the cross

section,

[a] is the allowable sttess of beam material, equal to

Table 1 he parameters of electric hoists

by Hitachi

Fig 1 Steel I-beam and electric hoist

To ensure the static stiffness of beam, the

maximum deflection should satisfy:

SL'

Here, [y] is allowable deflection of beam,

Fig.3 illusttates the stuvey results for the using

ability of single I-beam with the different span length

of the crane and load conditions in the constrained

conditions such as in Eq.(I) and Eq.(3) In this paper,

the allowable stress [CT], and the allowable deflection

\y\, are set to ISO MPa a n d i / 7 0 0

k

b b b b U b L- b

%

Fig 2 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

I-600 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

Rated load,

t 0.5

1

2

3

5 7,5

10

15

20

Lifting speed

fii/ph

11

11 8,4 7,5 6,7

6

5

5 4,2

Hoist weight

kg

145

175

280

385

685

930

1230

2340

2940

I-beam height

mm

150/200/250 200/250/300 200/250/300 250/300/450 300/450 450/600 450/600 450/600 450/600 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

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

cross-section 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,

2 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

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

M P a

150

100

^/

y

f p

30 T „ _

1

5 10

—

I S

-A-U-I2?0

- * — l 1,-1500

—«-5t-M50 -•-lOt-1600

—•—151-1600 201-1600

Span,m

20 25 Fig 3 Using ability of I-beam in strength and deflection conditions

Fig 4 Solutions for increasing the load capacity of I-beam Fig 5 Triangular cross-section of the beam

Where,

Constrains: the requirements of beam strength

and deflection to fulfill, and the plate sizes are in the

standard set

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:

y, and z, are coordinates of C, in the O(yo,zo)

coordinate system, defined as:

(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:

h can

relationship:

= 0,5ff -~(A4-0,50

(7)

be calculated by the following

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

y = Q,5b; z^h-ht-hz

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,

3 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 1 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 1 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

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

AOO

350

300

250

150

100

-^^

r^^Y^ ^

10

- j ^

.^^ —

- -7—_ ^—^i^*^

"; i i 1 * • - » ' * ^

15 !0

Crane span, m

,, (im

-t-h.^ml

—*— Stress,

^ —•—Deflection (xO,lmml

J J — • Allowable detlKtion

25

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 cross-sectional 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 cross-section 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

I-200 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 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, 8 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 but using the combined triangular section (with I-450) 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 rapidly, thus using this type section is not effective, and then other stinictures of beam should be used

- A, cm2

MPa

- Deflection (xClmrn)

—Allowable deflection

a) Load of 1 ton, 1-200 beam b) Load of 10 tonnes, 1-450 beam

4 Conclusions

The load capability of the I-beam, using as the

railway for thc hoist m the single girder overhead

that in the case of heavy load or large span, the single

standard I-beam should not be applicable The load

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

cross-section 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

high-capacity 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 do 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 Materials

Handling, New Age International (P) Ltd

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