Beams under load are subject to displacement. In the calculations ofthese, mathematical formulations are used. In this paper, displacement of different beams under different loads has been analysed by using mathematical formulations. Thus, coherent formulations of loading on beams and displacement of beams are obtained. Furthermore to obtain these formulas, boundary conditions have been taken into consideration

Hanifi Binici

Murat Sirinoglu

Sevket Civelek

Epoka University

“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.668Math Applications on the Displacement of Beam CalculationsHanifi BİNİCİ1, Şevket CİVELEK2, Murat ŞİRİNOĞLU31Department of Civil Engineering, Kahramanmaraş Sütçü ImamUniversity, Kahramanmaraş-TURKEY, Email: hbinici@ksu.edu.tr2Department of Computer Engineering, EPOKA University, Tirana-ALBANIA,Email: scivelek@epoka.edu.al3Elbistan Vocatinal School, Kahramanmaras Sutcu Imam University,Kahramanmaraş, TurkeyEmail: muratsirinoglu@ksu.edu.trABSTRACTBeams under load are subject to displacement. In the calculations ofthese, mathematical formulations are used. In this paper, displacement ofdifferent beams under different loads has been analysed by usingmathematical formulations. Thus, coherent formulations of loading on beamsand displacement of beams are obtained. Furthermore to obtain theseformulas, boundary conditions have been taken into consideration.INTRODUCTIONMath has been used in many engeneering deciplins [1]. Among thesecivil engineering plays a crucial role. Civil Engineering is a field of widescope that ranges from the design and construction of structures, roadwaysand pollution control processes to the management of our natural andengineered resources. Width of the field has led to division into a number ofsub disciplines including structural engineering, geotechnical engineering,transportation engineering, environmental engineering, materials,construction management, and water resources engineering. Civilengineering included a number of science disciplines including such aschemistry, physics, ecology, geology, microbiology, material science,economics and mathematics, and statistics [2]. Matematical definacitions“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.669have a big range of applications in strength. Only one of these is mathapplications on the displacement of beam calculations.Some part of these areas for instance displacement of beamcalculations can be formulized. As part of these calculations it should notassess the maximum deflections that will occur in the beams of the structureand make sure that they are not excessive. The application of math on allbranches of engineering will require numerical integration techniques tosolve some problems such as beam deflection. Using techniques form thebasis of the calculations that would be undertaken in real life albeit oftencarried out using sophisticated and powerful computer analysis software [3].BACKGROUND THEORYTo calculate beam deflections a standard fundamental formula is usedto determine deflections base on beam curvature. This is given by theexpression:Where:M = The bending moment at the section, distance x from the originy = The vertical deflection at the section distance x from the origin.The radius of the shape of the curved beam at a distance x from the origin,normally taken at the left or right hand end of the beamE = The elastic or Young’s modulus of the material from which the beam isfabricated.I = The second moment of area of the beam’s cross-section. This valuedepends on the shape of the cross section and is normally obtained fromtables. Its units are m4 or mm4 or cm4.22dxydEIM(1)In the above formula E and I are normally constant values whilst y, x and Mare variables. M can be expressed in terms of distance x and hence doubleintegration techniques can be used to solve the above expression to calculatethe deflection y.“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.670In other words,dxdxEIMy (2)The bending moment is a means of describing mathematically theamount of bending and deflection that will occur in a beam under a givenloading system and is defined as the sum of the moments of all forces to theleft or right of the section under consideration. It doesn’t matter whether theleft or right is taken as the answer will be the same in both cases. Forexample in Figure 1 below the simply supported beam shown carries auniformly distributed load of P KN/m. (note the units). When a load isdescribed as uniformly distributed it means that the load intensity is the samethroughout.Figure 1.Solution 1.At x, a distance x from the left hand support the bending moment, Mx,will be given as:2xxPM  . If this value is put in equations (1),EIxPdxyd.2.22And if the first entegral is calculated,dxxdxydPEI  222= 12)4( cxdxdyPEIL/2PL/2x“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.671First boundary condition is : 0)2( LdxdyIf this value is put in the equation above, constant C1 is obtained.2121 81021)2( LCLC  .If this value (C1) is put in the equations above,22814LxdxdyPEIIf the second entegral of both sides is calculated,dxLxdxdyPEI )814( 22  = 2238.12CxLxdxdyPEISecond boundary condition is: y(0)=0,In other words because the left and right hand ends are both supports thenthey can not deflect downwards.If this values are put in the equations above C2=0 is obtained.If this value is put in the equation again, xLxyPEI.8112.23From here; ).8112( 23xLxEIPy  can be written and it is regulated,Equation of elastic curve in (3) is obtained.334.648 LxLxEIPLy(3)Note: The above expression can now be used to calculate the deflectionat any point on the beam. In practice it is the maximum deflection that is of“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.672interest and common sense would say that for this example this occurs atmid-span and can be calculated by substituting x=L/2 into equation 3 above.If it is not obvious where the maximum deflection occurs this will be wherethere is a change in slope of the beam where dy/dx=0 . Hence equation (3), orits equivalent in a similar but different problem, could be differentiated andequated to find the distance xmax where the rotation, dy/dx, is zero.Substituting this value for xmax into equation (3) will give the maximumdeflection, ymax.From the boundary condition,maksimum deflection can be obtained inthe centre of beam. If the deflections x=L/2 is  written in (3), equations  (4) isobtained.EIPLy483max (4)Example 2.Let`s find the elastic curve equation of Uniform loaded simple beamand the maximum displacement value(figure 2)Figure 2Solution 2.).(2.xLxqM x  is obtained. If this definition is put in the equation(1) ,BLq“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.673)(222xLEIqxdxyd the definition is regulated according to constants ,)(2 22xLxdxydqEI1. Integral is calculated, then it will be obtained;132223).2(2)(2 CxLxdxdyqEIdxxLxdxydqEI First boundary condition is : 0)2( Ldxdyand if this definition iswritten above;21 121 LC  is obtained.If the obtained integral is put in constant place then 2. integral can becalculated;22432321212.62.)123.2()2( CxLxLxqEIydxLxLxdxdxdyqEI Second boundary condition is : y(0)=0and if this boundry condition is written above C2=0 is obtained and if thisvalue is put in the equation (5) then the equation of  the elastic curve isobtained.  )()(2)(24. 344LxLxLxEILqy(5)From boundary conditions, for2Lx  the displacement is maximumand maximum displacement is given in equation (6).“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.674EILqy384..5 4(6)Example 3.In the figure below, the condition of loading in simple beam is givenLet`s find the displacements in x=2m, x=4m . I=4.105cm4, E=6.106kg/cm4Figure 3.Solution 3:The equation of Elastic curve was found with definition (5) in previousproblem.  )()(2)(24. 344LxLxLxEIlqycmyy 036,0)2(6262.26210.410.6)600.(60)2(34564  cmyy 042,0)3(6363.2310.410.6)600.(60)3(34564cmyy 036,0)4(6464.26410.410.6)600.(60)4(345646m6t/m“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.675these results are given in table 1.Table.1 Deflactions of the beamI(cm4) E(kg/cm4) x(m) y(cm)4.105 6.106 2 0,0364.105 6.106 3 0,0424.105 6.106 4 0,036Example 4.(In the figure 4) If the beam which loading condition is given of isconcrete and I=1,6.105cm4 , E=2.106kg/cm2 then let`s find the displacementvalues of x=1m,x=2m and x=3m .Figure 4.Solution 4.The general elastic curve equation of simple beam under the conditionof having conservative load was founded by equation (3).334.3..48.LxLxEILPy365341441.3.10.2.10.6,1.48400.12000)3()1( yy =0,036 cmThe displacement of 2m can be found with definition (4). ..48.3EILPy 65310.2.10.6,1.48400.12000 0,042 cm2m12t2mx0,042“1st International Symposium on Computing in Informatics and Mathematics (ISCIM 2011)”in Collabaration between EPOKA University and “Aleksandër Moisiu” University of Durrëson June  2-4 2011, Tirana-Durres, ALBANIA.676CONCLUSIONNote that five of the problems are based on cantilever beams where thebeam is held rigidly at one end and is unsupported at the other end. Theboundary conditions in this case are that at the built-in end both rotation anddeflection will be zero.The major difference between the needs of current and past civilengineers relates to the techniques available for solving complex problems.For example, we believe that it is important for students to understand thatintegration can be used to determine the beam deflection, and thatdifferentiation can be used to determine the slope of a curve at a particularlocation. Displacement on conservative or uniform loaded simple beam isobtained by simple mathematical equations.REFERENCES[1] Engin, T. and Çengel, Y.A. (2008) Mühendisler İçin DiferensiyelDenklemler, Sakarya.[2]  Katz, L., Report Editor Kenneth Roby and Susan Ganter, (May 4–7),2000Workshop Organizers, Engineering: Civil Engineering, CRAFTYCurriculum Foundations Project, Clemson University.[3] The mathematics of beam deflection, The Royal Academy ofEngineering, Construction skill report.

Math Applications on the Displacement of Beam Calculations

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Math Applications on the Displacement of Beam Calculations

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