Assessment of The Theoretical Methods Used for Calculation ...

JORDAN JOURNAL OF APPLIED SCIENCE “Natural Sciences Series”, Volume 13 2016

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Assessment of The Theoretical Methods Used for Calculation of Distribution Factors of Live Loads for The Slab on Girders Bridges

DR. Eng.Rafat Edlebi Associate professor

Transportation Engineering DepartmentFaculty of Civil Engineering

Damascus [email protected]

Eng. Husam Soleman HaddadMaster student

Transportation Engineering DepartmentFaculty of Civil Engineering

Damascus [email protected]

Accepted: 13/4/2015Received:16/12/2014

معاملات لحساب النظرية الطرائق من العديد هناك الملخص: وإنشاء تصميم كودات من والعديد الحية، للحمولات التوزيع لدراسة اعتمدت طرائق محددة المختلفة العالم بلدان في الجسور توزع الحمولات الحية ، لكن في سوريا وفي معظم البلدان العربية لا يوجد كود معتمد لتصميم وإنشاء الجسور. في هذا البحث تم شرح مجموعة من هذه الطرائق ومقارنة القيم النظرية مع نتائج تجربتي المحسوبة القيم أن أظهرت النتائج سوريا. في لجسرين تحميل كانت OHBDC كود وطريقة ماسونيت غويون طريقة وفق AASH� الـ أن طريقتي التجريبية في حين القيم قريبة جدا من TO وطريقة الضغط اللامركزي كانت أقل دقة. النتائج التجريبية أظهرت أن طريقة غويون ماسونيت هي الطريقة الأفضل من بين الدقة البحث من حيث المدروسة في هذا اليدوية الحساب طرائق ومجال الاستخدام، أيضا فإن البحث أوصى بدراسة تأثير الروابط العرضية والاستمرارية والانحراف في محور الجسر على دقة هذه

الطرائق.

Abstract: There are many methods to calculate live load distribution factors. Certain methods are accepted in international bridge codes in different countries worldwide. In Syria and most of Arabcountries there is no code imposed for bridge de�sign and construction .In this research, many theo� retical methods were discussed. On the other hand, loading tests on two selected bridges in Syria weredemonstrated. Then, the theoretically calculated re� sults were compared with experimental results. The results showed that Guyon –Massonnet method and the method of OHBDC code gave results very close to experimental results. Whereas, Methods of AASHTO and eccentric compression method wereless accurate. Test results showed that Guyon –Massonnet is the best method out of manual calcu� lation methods studied in this research in terms of accuracy and field use. follow-up investigation that takes into consideration diaphragms, continuity or.skew is recommended Key words: distribution factors, slab on girdersbridges, Guyon –Massonnet, eccentric compres�.sion, AASHTO, OHBDC

1-Introduction1-1Background

There are many methods to study the distribution of live loads on the superstructure of the bridge, certain methods are accepted in international bridge codes in different coun�tries worldwide, but In Syria and most of the Arab countries, there is no code imposed for bridge design and construction. This research presents a set of methods to study the distribution of live loads in slab on girder bridges. On the other hand, loading tests im�plemented on two selected bridges constructed in Dummar Suburb – Damascus. Then, theoretically calculated results were compared with experimental results. The methods that were studied are:

1. Finite�Element Method2. Eccentric Compression Method3. Guyon – Massonnet Method4. AASHTO LRFD Bridge Design Specification Method5. AASHTO Standard Specification for Highway Bridges Method6. Ontario Highway Bridge Design Code (OHBDC) Method

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1-2-Previous researchesResearch Program of The Cement and Concrete Association- 1962[8]

This program depended on an series of tests on laboratory small model concrete bridges. The results showed that the agreement between the theoretical values of Guyon�Masson�net method and experimental values was very good.

Research of Lin, C. and Vanhorn, D. (1968) [7]The research compared between the distribution factors based on Guyon-Massonnet

method with those determined with experimental values that fixed up from static live load test in a prestressed concrete girder bridge. This comparison showed that the theoretical values are not consistent enough with the experimental values. The researchers indicated that this disagreement arose from the fact the extra stiffness of the curbs and parapets, created some difficultly in replacing the actual structure by a uniform orthotropic plate.

Research of Barr, Paul., Staton, J. and Eberhard, M. (2000), [4]This research presented an evaluation of flexural live loads distribution factors for a

series of prestressed concrete girder bridges. The evaluation depended on finite element Method. The response of one bridge, measured during a static live load test, was used to evaluate the reliability of a finite element model. Many variations of this model were then used to evaluate the procedures for computing flexural live load distribution factors that embodied in three bridge design codes. Then the values of cods compared with those computed by finite element method. The comparison showed that the discrepancy for values of the AASHTO LRFD Code was 27%, for values of AASHTO Standard Specifi�cation was 30%, and for values of OHBDC was 22%.

2- Definition of Live Loads Distribution Factors All methods mentioned previously aims eventually to find Live Loads Distribution Fac�

tors for each girder. The concept of Live Loads Distribution Factors differ according to each method will be studied. Live Loads Distribution Factors (DFi) can be defined for certain girder (gi) for certain location of load as the ratio between the maximum bending moment on the girder (Mi) to the maximum bending moment on entire cross�section of the bridge (M) as in relationship(1) [3] :

Alternatively, in some other way the maximum bending moment on the girder divided by the sum of maximum bending moments for all girders.

3-Theoritical methods to calculate the live loads distribution factorsHere is review of a set of theoretical methods to calculate live load distribution tors:

3-1 Finite element method: This method depends on dividing the structure to small elements connected together by

nodes, and relationship (2) is applied for each node:

Where u, k and f are displacement, stiffness and force, respectively.

In this research SAP2000 software was used to implement the Finite element method.


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3-2 Eccentric compression Method:This method is simple and easy, but it does not take into consideration all the coefficients

that affect the loads distribution, nevertheless it is still the most common method followed in Syria. The distribution factors for each girder is given depending only on the distances between girders and locations of girders and loads. For load P in Fig (1) the distribution factors of girders are calculated by using relationship (3) [6].

Fig (1) Eccentric compression method

Where the distribution factor of the girder; X: the distance between the location of the force and the middle of cross section; n: the number of girders 3-3 Guyon – Massonnet method:Guyon – Massonnet method take into account material properties (E,G), flexural and torsional rigidities in longitudinal and transverse directions, and the ratio of width to span of the bridge. This method depends on orthotropic plate concept, and it is the most one which take into consideration the coefficients influence on load distribution, but it is long and complicated. Firstly, the parameters α and θ are calculated as follow [3]:

Where: α: the torsion parameter; θ: the flexure parameter; Dx: the longitudinal flexural

rigidity per unit width; Dy: the transverse flexural rigidity per unit length; D

xy: the lon�gitudinal torsional rigidity per unit width; D

yx : the transverse torsional rigidity per unit

length; D1 : the longitudinal coupling rigidity per unit width; D2 : the transverse coupling

rigidity per unit length; 2b: the width of the bridge and L: the span of the bridge.After cal�culating θ, we can get K0 and K1 using diagrams that belong to this method. The slab is di�vided into eight longitudinal strips Fig (2), so we have nine positions in cross-section. The diagrams of K0 are shown in Figs (3), (4), (5), (6), (7) and diagrams of K1 can be found in [9]. The horizontal axis represent the values of θ, the vertical axis represent the values of K0, and the distance written near the curve refers to the location of load. Then it becomes possible to calculate the distribution factor for each girder wherever its positions by using a pro-rate. Some references contain the values of these schemes scheduled [6],[5]. The distribution factor for each girder is calculated according to relationship (6) [9]:

Where n refers to the number of girders.

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This method take into account material properties (E,G), flexural and torsional rigidities in longitudinal and transverse directions, and the ratio of width to span of the bridge.

Fig (2) Division of cross-section according to Guyon – Massonnet method


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e: correction coefficient; de: distance between the center of exterior girder and interior edge of curb (ft) This method, unlike eccentric compression and Guyon - Massonnet methods does not consider the sites of the loads in the cross� section.3-5 AASHTO Standard Specification for Highway Bridges methodOf the methods studied, the AASHTO Standard Specification is the simplest, where dis�tribution factor is calculated according to relationship (10) [2]:

Where S: girder spacing (in), D: a coefficient determined according to the shape of cross section. To calculate the moment on girder the distribution factor is multiplied by the entire mo�ment due to one line load. The AASHTO Standard Specification does not take into ac�count rigidities, sites of loads or girders, or any other coefficients. 3-6 Ontario Highway Bridge Design Code (OHBDC) MethodThis method as Guyon – Massonnet method depends on the orthotropic plate concept. Firstly, values of α and θ are calculated, then the factor D is obtained from the schemes exist in the code in [3]. Figure (8) shows the values of D for external and internal girder and correction factor Cf for two lanes. The factor Dd is ob�tained as in relationship (11):

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To calculate moment on the girder, this factor is multiplied by entire moment due to one load�ed lane. This method also does not take into account the sites of loads in cross section.


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Fig (9) Dimensions of precast girder Cross-section (cm)

Fig (10) Cross-section of small bridge and the location of vehicles (cm)

Fig (11) Longitudinal section of small bridge and location of vehicles (cm)

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Fig (12) Small bridge loading experiment

The total moment on the entire bridge section due to total load (120t) equals after calcu�lation to 591 t.m. Since the relationship between the deflections and bending moments is linear, and all girders have same cross-section, it is possible to calculate the moment for each girder by multiplying total moment by the rate of girder deflection to the sum of all girders deflections. The deflection relationship is indicated as follow [9]:

4-2 The Big BridgeThe superstructure consists of six precast prestressed girders. The same vehicles, which were used in the small bridge, were used in the big bridge. Fig (14) shows the cross�section of big bridge and the sites of the vehicles. The sites of vehicles in longitudinal section of big bridge are the same in longitudinal section of the small bridge Fig(11).

Fig (13) Measurement apparatus fixed under the small bridge to measuring deflections


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Fig (14) Cross-section of the big bridge and the sites of vehicles

The deflections are measured only for girder B1 and girder B2 as shown in table (3), so it isn`t possible to use the same way used with the small bridge to calculate moments. The moments on girders B1 and B2 are obtained by using relationship (12). Note that the both bridges have the same material properties and load case, so the value of ξ is same in both experiments.

Table (3) Deflections due to big bridge experiment

BeamB1B2

)mm( deflection8.589.14

Moment on B1:

Moment on B2:

5- Calculating of Moments for Small Bridge Girders Using Theoretical MethodsThe total calculated maximum moment on entire cross�section due to the total loads (120t) is M=591t.m, and the maximum moment due to one loaded lane is M/2=295.5t.m,and the maximum moment due to one line loads is M/4=147.75t.m.Hereafter we will present calculating moments on small bridge girders using theoretical methods:5-1Finite-Element MethodCalculation according Finite Element method were done using program SAP2000. The table (4) shows the results.

Table (4) Moments due to live loads on small bridge girders using SAP2000

beamB1B2B3B4B5

Moment)(t.m140.85148.41139.64103.6160.70

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Fig (15) Model of small bridge using SAP2000

5-2 Eccentric Compression Method By using relationship (2) we get the values of η 0 , η 1 , η2 , η 1’ , η2’ for p1 , p2 , p3 , p4

Fig (16) Eccentric compression method

For p1: , For p2:

For p3: , For p4:

Therefore, the moment on external girder B1 is:

Moments on other girders which calculated by same manner are shown in table (5)Table (5) Moment on small bridge girders using eccentric compression method

BeamB1B2B3B4B5

mo�)t.m( ment168.14142.7117.993.1568.24


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5-3 Guyon-Massonnet MethodFirstly, we will calculate the values of α and θ:

where are the modulus of elasticity, shear modulus, and Poisson”s ratio, respectively; subscripts g and s refer to precast girder concrete and slab cast in situ concrete; are the girder second moments of area and torsion constant, respectively. Table (6) shows the values of K0 and K1 obtained from Guyon�Massonnet schemes.

Table (6) Values of K0 and K1 according to default sites

sectionb3b/4b/2b/40-b/4-b/2-3b/4-b

Load site

K0

0-0.410.311.051.742.091.741.050.31-0.41

b/40.210.991.702.071.741.110.49-0.04-0.54

b/21.662.022.151.701.050.490.10-0.19-0.43

3b/44.223.222.020.990.31-0.04-0.19-0.23-0.24

B7.744.221.660.21-0.41-0.54-0.43-0.24-0.03

K1

00.570.730.971.271.471.270.970.730.57

b/40.961.141.381.521.270.920.640.460.34

b/21.571.691.711.380.970.640.420.290.20

3b/42.472.221.691.140.730.460.290.190.13

B3.652.471.570.960.570.340.200.130.09

The values in table (6) agree with default locations of girders and loads determined by this method, but this locations don`t identify the locations of girders and loads of the studied bridge. Firstly, we will calculate the values accepting to the locations of the bridge girders by pro-rate, these values are shown in table (7):

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Table (7) Values of K0 and K1 for small bridge girders according to defaults sites of loads

beam

Loads locationsB1B2B3B4B5

K0

100.181.332.091.330.18

b/40.841.851.740.74-0.14

b/21.961.971.050.26-0.23

3b/43.411.600.31-0.13-0.23

b4.881.07-0.41-0.47-0.20

K1

00.701.091.471.090.70

b/41.111.441.270.750.43

b/21.671.580.970.510.27

3b/42.261.470.730.360.18

b2.691.320.570.430.12

Buy sing the relationship (6) we get the distribution factors of each girder according to the default loads location as in table (8):

Table (8) Distribution factors of small bridge girders according to default locations of loads

beam

Loads locations

B1B2B3B4B5

DF

00.090.240.360.240.09

b/40.190.330.300.150.03

b/20.360.360.200.080.00

3b/40.570.310.100.02-0.01

b0.760.240.01-0.01-0.01

Next, we redistribute actual lines loads to correspond the default locations by pro-rate. For example, line loads p1 is distributed to correspond locations: as follow:

where the values (80.7) and (38.6) represent distance between actual lines loads and the locations and the value (119.25) represent the distance between this locations. The lines loads after redistribution are as shown in Fig (17).


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Now we will calculate the moment for entire section due to new lines loads after taking into account that the locations of loads don’t change in longitudinal section as follow:

Where w1: new line load, p1: actual line load, M/4: the moment in entire section due to actual line loads

It is also for the other moments:

And by multiplying the value of DF by total moment for entire section due to the new lines loads, we get the final moment for each girder as follow:

beam B1: M1=

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5-4 AASHTO LRFD Bridge Design Specifications MethodBy using the relationship (8) for internal girders, we get the following results:

Then the moment for internal girder is found by multiplying the value of D by the mo�ment in entire section due to on loaded lane (M/2):

the correction factor of external girder is calculated by relationship (9):

As a result, the moment on external girder is equal to the moment on internal girder: M

2=M

1=180.55 t.m

5-5 AASHTO Standard Specification for Highway Bridge methodAccording to relationship (10) we write: Then we get the moment on the external and internal girders by multiplying DF by mo�ment on entire section due to one loaded lane (M/4):

5-6 Ontario Highway Bridge Design Code (OHBDC) Method

After calculating α and θ in section (5-3) value of D for external and internal girders and value of correction factor Cf are obtained by using scheme in Fig (8):For external girder:

For internal girder:

6-The Moments of Big Bridge Girders According to Theoretical MethodTable (9) shows the theoretically calculated moments for big bridge girders according to theoretical methods, which was calculated similarly for small bridge.

Table (9) Moments on girders of big bridge according to theoretical methods (in t.m )

BeamB1B2B3B4B5B6

Method

Experimental136.78147.19------------

finite-element139.83154.31141.5398.6847.6818.4

Guyon –Massonnet129.429154.842146.568102.83449.05311.820

Eccentric Compression182.48148.89115.381.7048.1114.52

AASHTO LRFD180.85180.85------------

AASHTO Standard167.25167.25------------

OHBDC132.38156.62------------


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7-Comparison of ResultsBecause of the differences in the concept of distribution factors according to each in�

vestigated method, we will calculate the distribution factor as defined previously in rela�tionship (1), by dividing the moment of each girder according to each method by the total moment (M) of entire section. Fig (18) represents the results for the small bridge and Fig (19) for big bridge.

Fig (18) Values of distribution factors for small bridge

Fig (19) Values of distribution factors for big bridge

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8-Discussion of the ResultsFigs (20) and (21) show the values of the difference between theoretical and experimental results divided by experimental results as a percentage

.• This percentage did not exceed 5% for the finite element method• Inclination of the results in Guyon Masonat method did not exceed 10% of

experimental results.• Inclination of the results in eccentric compression method was big and reached

to 30%. Figs (18) and (19) show that the diagram determined for the entire sec�tion of the bridge according to eccentric compression method was linear with constant slope. This method assumes that all the points in cross-section round by the same angel around a longitudinal axis and this behavior agrees with cross-section of the bridge when there are diaphragms and long span. Therefore, the validity of this method is limited.

• Inclination the results of the LRFD code method arrived to 33%• Inclination the results of standard Specification code method arrived to 23%• Inclination the results of OHBDC code method did not exceed 10%.

Fig (20) Percentages of the difference between theoretical results and experimental results for the small bridge


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Fig (21) Percentages of the difference between theoretical results and experimental results for the

small bridge

9-Conclusions and Recommendations• The results of Guyon� Masonnet method were very close to the experimental re�

sults, and it is the best method out of manual calculation methods studied in this research in terms of accuracy and field use, but it is long and complex, which can be treated by using the scheduled values of schemes and computer programs such as Excel program

• The results of OHBDC were very close to the experimental results. This meth�od is easy, but it does not take into account the loads locations in cross-section, which means that the results can incline more than we found if we used vehicles that had other dimensions, or used other locations in the cross-section.

• Methods of AASHTO and eccentric compression are easy, but they are less accurate than the Guyon Masonat method, and they do not take into account all coefficients that influence on loads distribution.

• Methods that do not take loads locations in the cross section into account not capable to determine the lateral distribution of loads when the loads are at the middle of the cross�section (such as abnormal loads). Where they assume that the live loads applied in the external lanes. In this case, the best method is Guy�on� Masonat.

• Guyon�Masonat method can be considered as accurate method and the other studied methods can be considered as simplified methods.

• This research was limited to assessing the accuracy of theoretical methods with�out diaphragms, continuity or skew, therefore it is recommended follow-up to investigate the influence of these factors on the accuracy of the results of these methods.

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10- References:[1] American Association of State Highway and Transportatio Officials (AASHTO). (1994), LRFD Bridge Design Specifications. (1st edition), Washington: Washington, D.C.[2] American Association of State Highway and Transportation Officials (AASHTO). (1996), Standard Specification for Highway Bridges,(16st edition), Washington: Washington, D.C.[3] Bakht, B. and Jaeger, L. G. (1985), Bridge Analysis Simplified, New York: McGraw-Hill.[4] Barr, Paul., Staton, J. and Eberhard, M. (2000), Live Load Distribution Factors for Washington State SR 18/SR516 Overcrossing, Washington: Washington State Department of Transportation. [5] Biliszczul, J. and Onysyk, J. (2003), Lecturers for practical design of concrete bridge 2,3 special IM fourth year, Bridge Institute, Wroclaw: Technical University of Wroclaw. [6] Kaminski, L. (1979), Theory of Engineering Construction, Wroclaw: Polytechnic of Wroclaw.[7] Lin, C. and Vanhorn, D. (1968), The Effect of Midspan Diaphragms on Load Distribution in a Pre-stressed Concrete Box-Beam Bridge-Philadelphia Bridge, Pennsylvania: Lehigh University Bethlehem.[8] Row, R. E. (1962), Concrete Bridge Design, London: C.R. Books Limited.[9] Witold.A.M. (2010), Design of concrete bridges, Warszawa: WKL.

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