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# Static analysis of thin beams by interpolation method approach

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### Static analysis of thin beams by interpolation method approach

1. 1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME254STATIC ANALYSIS OF THIN BEAMS BY INTERPOLATIONMETHOD APPROACH TO MATLABPrabhat Kumar Sinha Vijay Kumar*, Piyush Pandey Manas TiwariMechanical Engineering DepartmentSam Higginbottom Institute of Agriculture Technology and sciences, AllahabadABSTRACTEuler-Bernoulli beam theory (also known as Engineer’s beam theory or classicalbeam theory) is a simplification of the linear theory of elasticity which provides a means ofcalculating the load carrying and deflection characteristics of beams. It covers the case forsmall deflection of a beam which is subjected to lateral loads only for a local point inbetween the class-interval in ‫-ݔ‬direction by using the interpolation method, to make the tableof ‫ݔ‬ and ‫,ݕ‬ then ‫ݕ‬ ൌ ݂ሺ‫ݔ‬ሻ, where, y is a deflection of beam and slope ሺௗ௬ௗ௫ሻ at any point in thethin beams, apply the initial and boundary conditions, this can be calculating and plotting thegraph by using the MATLAB is a fast technique method will give results, the result is alsoshown with numerical analytically procedure. The successful demonstrated it quickly becauseengineering and an enabler of the Industrial Revolution.Additional analysis tools have been developed such as plate theory and finite elementanalysis, but the simplicity of beam theory makes it an important tool in the science,especially structural and Mechanical Engineering.Keywords: Static Analysis, Interpolation Method, Flexural Stiffness, Isotropic Materials,MATLAB.INTRODUCTIONWhen a thin beam bends it takes up various shapes [1]. The shapes may besuperimposed on ‫ݔ‬ െ ‫ݕ‬ graph with the origin at the left or right end of the beam (before it isloaded). At any distance x meters from the left or right end, the beam will have a deflection ‫ݕ‬and gradient or slopeሺௗ௬ௗ௫ሻ. The statement ‫ݕ‬ ൌ ݂ሺ‫ݔ‬ሻ, ‫ݔ‬଴ ൑ ‫ݔ‬ ൑ ‫ݔ‬௡ means: corresponding toINTERNATIONAL JOURNAL OF MECHANICAL ENGINEERINGAND TECHNOLOGY (IJMET)ISSN 0976 – 6340 (Print)ISSN 0976 – 6359 (Online)Volume 4, Issue 2, March - April (2013), pp. 254-271© IAEME: www.iaeme.com/ijmet.aspJournal Impact Factor (2013): 5.7731 (Calculated by GISI)www.jifactor.comIJMET© I A E M E
2. 2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME255every value of ‫ݔ‬ in the range ‫ݔ‬଴ ൑ ‫ݔ‬ ൑ ‫ݔ‬௡, there exists one or more values of y. Assumingthat ݂ሺ‫ݔ‬ሻ is a single-valued and continuous and that it is known explicitly, then the values of݂ሺ‫ݔ‬ሻ corresponding to certain given values of ‫,ݔ‬ say ‫ݔ‬଴, ‫ݔ‬ଵ,…, ‫ݔ‬௡, can easily be computed andtabulated. The central problem of numerical analysis is the converse one: Given the set oftabular values ሺ‫ݔ‬଴, ‫ݕ‬଴ሻ, ሺ‫ݔ‬ଵ , ‫ݕ‬ଵሻ, ሺ‫ݔ‬ଶ, ‫ݕ‬ଶሻ, … , ሺ‫ݔ‬௡, ‫ݕ‬௡ሻ satisfying the relation ‫ݕ‬ ൌ ݂ሺ‫ݔ‬ሻ wherethe explicit nature of ݂ሺ‫ݔ‬ሻ is not known, it is required to simpler function, ‫׎‬ሺ‫ݔ‬ሻ such that݂ሺ‫ݔ‬ሻ and ‫׎‬ሺ‫ݔ‬ሻ agree at the set of tabulated points. Such a process is interpolation. If ‫׎‬ሺ‫ݔ‬ሻ isa polynomial, then the process is called polynomial interpolation and ‫׎‬ሺ‫ݔ‬ሻ is called theinterpolating polynomial. As a justification for the approximation of unknown function bymeans of a polynomial, we state that famous theorem due to Weierstrass: If ݂ሺ‫ݔ‬ሻ iscontinuous in ‫ݔ‬଴ ൑ ‫ݔ‬ ൑ ‫ݔ‬௡, then given any ߳ ൐0, there exists a polynomial ܲሺ‫ݔ‬ሻ such that( ) ( )f x P x− <∈, for all in ሺ 0x , nx ).This means that it is possible to find a polynomial ܲሺ‫ݔ‬ሻ whose graph remains within theregion bounded by ‫ݕ‬ ൌ ݂ሺ‫ݔ‬ሻ-߳ and ‫ݕ‬ ൌ ݂ሺ‫ݔ‬ሻ+߳ for all ‫ݔ‬ between ‫ݔ‬଴ and ‫ݔ‬௡, however small ߳may be [2].SLOPE, DEFLECTION AND RADIUS OF CURVATUREWe have already known the equation relating bending moment and radius ofcurvature in a beam, namely,ெ௒ൌாோWhere,M is the bending moment.I is second moment of area about the centroid.E is the Modulus of Elasticity andR is the radius of curvature,Rearranging we have,1/ܴ ൌ ‫ܧ/ܯ‬Figure-1 illustrates the radius of curvature which is defined as the radius of circle that has atangent the same as the point on x-y graph.Figure-1Consider an elemental length ܲܳ ൌ ݀‫ݏ‬ of a curve. Let the tangents at P and Q make angles ߰and ߰+݀߰ with the axis. Let the normal at P and Q meet at C. Then C is called the centre ofcurvature of the curve at any point between P and Q on the curve. The distance CP = CQ = Ris called the radius of curvature at any point between P and Q on the curve.Obviously, ݀‫ݏ‬ ൌ ܴ݀߰
3. 3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME256Or, ܴ ൌ ݀‫߰݀/ݏ‬But we know that if ሺ‫,ݔ‬ ‫ݕ‬ሻ be the coordinate of P,ௗ௬ௗ௫ൌ ‫߰݊ܽݐ‬ௗ௬ௗ௦ൌௗ௦/ௗ௫ௗట/ௗ௫ൌ௦௘௖టௗట/ௗ௫………………………………………………………(1)‫߰݊ܽݐ‬ ൌ݀‫ݕ‬݀‫ݔ‬Differentiating with respect to ‫,ݔ‬ we have‫ܿ݁ݏ‬ଶ‫.ݔ‬݀߰݀‫ݔ‬ൌ ݀ଶ‫ݔ݀/ݕ‬ଶௗటௗ௫ൌௗమ௬ௗ௫మ/‫ܿ݁ݏ‬ଶ߰……………………………………………………………(2)Substituting in equation (1) we have,ܴ ൌ௦௘௖ట೏మ೤೏ೣమ‫ܿ݁ݏ‬ଶ‫ݔ‬ൌ‫ܿ݁ݏ‬ଷ߰݀ଶ‫ݔ݀/ݕ‬ଶTherefore,1ܴൌ݀ଶ‫ݕ‬݀‫ݔ‬ଶ/‫ܿ݁ݏ‬ଷ߰ଵோൌௗమ௬ௗ௫మ / 2 3/2(sec )Ψ orଵோൌௗమ௬ௗ௫మ / 2 3/2(1 tan )+ ΨFor practical member bent due to the bending moment the slope ‫߰݊ܽݐ‬ at any point is a smallquantity, hence ‫݊ܽݐ‬ଶ߰ can be ignored.Therefore,1ܴൌ ݀ଶ‫ݔ݀/ݕ‬ଶIf M be the bending moment which has produced the radius of curvature R, we have,‫ܯ‬‫ܫ‬ൌ‫ܧ‬ܴ1ܴൌ‫ܯ‬‫ܫܧ‬݀ଶ‫ݕ‬݀‫ݔ‬ଶൌ‫ܯ‬‫ܫܧ‬‫ܯ‬ ൌ ‫ܫܧ‬ௗమ௬ௗ௫మ…………………………………………………………..(3)The product EI is called the flexural stiffness of the beam. In order to solve the slope ሺௗ௬ௗ௫ሻ orthe deflection ሺ‫ݕ‬ሻ at any point on the beam, an equation for M in terms of position ‫ݔ‬ must besubstituted into equation (1). We will now examine these cases in the example of cantileverbeam [3].OBJECTIVE OF THE PRESENT WORKThe objective of the present work is to develop a MATLAB program which can workwithout the dependence upon the thin beam materials and the aspect ratio. The input shouldbe geometry dimensions of thin beams for example-plate and circular bar such as length,breadth, thickness and diameter, the materials should be isotropic, materials data such asYoung’s Modulus and Flexural Stiffness and to calculate the slope and deflection at any pointin between 1-class interval by using interpolation method by analytically as well as
5. 5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME258EIௗమ௬ௗ௫మൌ െ‫ݔܨ‬Integrate with respect to ‫ݔ‬, we getEIௗ௬ௗ௫ൌିி௫మଶ൅ ‫ܣ‬………………………………………………………(4)Integrate again and we getEIy=െி௫య଺൅ ‫ݔܣ‬ ൅ ‫ܤ‬……………………………………………………(5)A and B are constants of integration and must be found from the boundary conditions.These are at ‫ݔ‬ ൌ ‫,ܮ‬ ‫ݕ‬ ൌ 0 (no deflection)at ‫ݔ‬ ൌ ‫,ܮ‬ௗ௬ௗ௫ൌ 0 (gradient horizontal)Substitute ‫ݔ‬ ൌ ‫ܮ‬ ܽ݊݀ௗ௬ௗ௫ൌ 0, in equation (4). This gives‫ܫܧ‬ሺ0ሻ ൌ െ‫ܮܨ‬ଶ2൅ ‫ܣ‬ ݄݁݊ܿ݁ ‫ܣ‬ ൌ ‫ܮܨ‬ଶ/2Substitute ‫ܣ‬ ൌி௅మଶ, ‫ݕ‬ ൌ 0 ܽ݊݀ ‫ݔ‬ ൌ ‫ܮ‬ ݅݊‫݋ݐ‬ ݁‫݊݋݅ݐܽݑݍ‬ ሺ5ሻ ܽ݊݀ ‫݁ݓ‬ ݃݁‫ݐ‬‫ܫܧ‬ሺ0ሻ ൌ െ‫ܮܨ‬ଷ6൅‫ܮܨ‬ଷ2൅ ‫ܤ‬ ݄݁݊ܿ݁ ‫ܤ‬ ൌ െ‫ܮܨ‬ଷ3Substitute ‫ܣ‬ ൌி௅మଶܽ݊݀ ‫ܤ‬ ൌ െ‫ܮܨ‬ଷ/3 into equations (4) and (5) and the complete equationsare‫ܫܧ‬ௗ௬ௗ௫ൌ െி௫మଶ൅ி௅మଶ…………………………………………………(6)‫ܫܧ‬ ൌିி௫య଺൅ி௅మ௫ଶെி௅యଷ……………………………………………….(7)The main points of interest is slope and deflection at free end where ‫ݔ‬ ൌ 0.Substituting ‫ݔ‬ ൌ 0 into (6) and (7) gives the standard equations,Slope at free endௗ௬ௗ௫ൌி௅మଶாூ……………………………….(8)Deflection at free end‫ݕ‬ ൌିி௅యଷாூ…………………………………………….(9)Numerical Analysis-1A cantilever thin beam is 4m long and has a point load of 5KN at the free end. Theflexural stiffness is 53.3MN2. Calculate the slope and deflection at the free end.Solution:-Slope equation ‫ݕ‬′ൌ ݂′ሺ‫ݔ‬ሻ݀‫ݕ‬݀‫ݔ‬ൌ ‫ݕ‬′ൌ ሾെ‫ݔܨ‬ଶ2൅‫ܮܨ‬ଶ2ሿ1‫ܫܧ‬‫ݕ‬′ൌ‫ܨ‬2‫ܫܧ‬ሾെ‫ݔ‬ଶ൅ ‫ܮ‬ଶሿ‫ݐݑ݌‬ ‫ݔ‬ ൌ 0݉, ‫ܮ‬ ൌ 6݉‫ݕ‬′ൌ50002 ‫כ‬ 53.3 ‫כ‬ 10଺ሾെ0 ൅ 6ଶሿ‫ݎ݋‬ ‫ݕ‬ଵ′ൌ ሺ1.6885 ‫כ‬ 10ିଷሻ° ሺ݊‫݋‬ ‫ݏݐ݅݊ݑ‬ሻ
6. 6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME259Similarly, ‫ݐݑ݌‬ ‫ݔ‬ ൌ 2݉, ‫ܮ‬ ൌ 6݉‫ݕ‬ଶ′ൌ50002 ‫כ‬ 53.3 ‫כ‬ 10଺ሾെ2ଶ൅ 6ଶሿ‫ݎ݋‬ ‫ݕ‬ଶ′ൌ ሺ1.5 ‫כ‬ 10ିଷሻ°ሺ݊‫݋‬ ‫ݏݐ݅݊ݑ‬ሻAlso, ‫ݐݑ݌‬ ‫ݔ‬ ൌ 4݉, ‫ܮ‬ ൌ 6݉‫ݕ‬ଶ′ൌ50002 ‫כ‬ 53.3 ‫כ‬ 10଺ሾെ4ଶ൅ 6ଶሿ‫ݕ‬ଷ′ൌ ሺ9.38086 ‫כ‬ 10ିସሻ° ሺ݊‫݋‬ ‫ݏݐ݅݊ݑ‬ሻAnd, ‫ݐݑ݌‬ ‫ݔ‬ ൌ 6݉, ‫ܮ‬ ൌ 6݉‫ݕ‬ସ′ൌ50002 ‫כ‬ 53.3 ‫כ‬ 10଺ሾെ6ଶ൅ 6ଶሿ‫ݕ‬ସ′ൌ ሺ0ሻ° ሺ݊‫݋‬ ‫ݏݐ݅݊ݑ‬ሻTable-1‫ݔ‬ሺ݉ሻ 0 2 4 6‫ݕ‬′ሺ°ሻ ‫݁݌݋݈ݏ‬ 1.6885*10-31.5*10-39.38086*10-40.000MATLAB PROGRAM-USING INTERPOLATION METHOD% calculate the slope of beam at any point in between 1-class interval% cantilever beam% point load at free endx=[0 2 4 6];slope=[1.66885*10.^-3 1.5*10.^-3 9.38086*10.^-4 0.0];xi=1;yilin=interp1(x,slope,xi,linear)yilin = 0.0016° (Answer)Plot the graph of slope of beam% plot the graph of slope of beam% cantilever thin beam% point load at free endF=5000;x=[0:1:6];L=6;EI=53.3*10.^6;slope=(F/2)*[-x.^2+L.^2]/(EI);plot(x,slope,--r*,linewidth,2,markersize,12)xlabel(position along the axis (x),Fontsize,12)ylabel(position along the axis (y),Fontsize,12)title(slope of cantilever beam with point load at free end,Fontsize,12)
7. 7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME260Figure-3Deflection equation ‫ݕ‬ ൌ ݂ሺ‫ݔ‬ሻ[15]‫ݕ‬ ൌ‫ܨ‬‫ܫܧ‬ሾെ‫ݔ‬ଷ6൅‫ܮ‬ଶ‫ݔ‬2െ‫ܮ‬ଷ3ሿ‫ݐݑ݌‬ ‫ݔ‬ ൌ 0݉, ‫ܮ‬ ൌ 6݉‫ݕ‬ଵ ൌ500053.3 ‫כ‬ 10଺ቈ0 ൅ 0െ6ଷ3቉ ൌ െ6.7542 ‫כ‬ 10ିଷ݉Similarly, ‫ݐݑ݌‬ ‫ݔ‬ ൌ 2݉, ‫ܮ‬ ൌ 6݉‫ݕ‬ଶ ൌ500053.3 ‫כ‬ 10଺ቈെ2ଷ6൅6ଶ‫כ‬ 22െ6ଷ3቉ ൌ െ3.5021 ‫כ‬ 10ିଷ݉‫ݐݑ݌‬ ‫ݔ‬ ൌ 4݉, ‫ܮ‬ ൌ 6݉‫ݕ‬ଷ ൌ500053.3 ‫כ‬ 10଺ቈെ4ଷ6൅6ଶ‫כ‬ 42െ6ଷ3቉ ൌ െ1.0 ‫כ‬ 10ିଷ݉‫ݐݑ݌‬ ‫ݔ‬ ൌ 6݉, ‫ܮ‬ ൌ 6݉‫ݕ‬ସ ൌ500053.3 ‫כ‬ 10଺ቈെ6ଷ6൅6ଶ‫כ‬ 62െ6ଷ3቉ ൌ െ0.0݉Table-2x(m) 0 2 4 6y(m) -6.7542*10-3-3.5021*10-3-1.0*10-30.0MATLAB PROGRAM- USING INTERPOLATION METHOD% calculate the deflection of cantilever thin beam at any point in between% any 1-class interval% point load at free endx=[0 2 4 6];y=[-6.7542e-3 -3.5021e-3 -1.0e-3 0.0];xi=1;yilin=interp1(x,y,xi,linear)yilin =-0.0051(Answer)0 1 2 3 4 5 600.20.40.60.811.21.41.61.8x 10-3position along the axis (x)positionalongtheaxis(y)slope of cantilever beam with point load at free end
8. 8. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME261Plot the graph of deflection% plot the graph of deflection of beam% cantilever beam% point load at free endF=5000;x=[0:1:6];EI=53.3*10.^6;L=6;y=(F/EI)*[((-x.^3)/6)+((L.^2*x)/2)-((L.^3)/3)];plot(x,y,--r*,linewidth,2,Markersize,12)xlabel(position along the axis (x),Fontsize,12)ylabel(position along the axis (y),Fontsize,12)title(deflection of cantilever beam with point load at freeend,fontsize,12)Figure-4Case 2- Cantilever Thin Beam with Uniformly Distributed Load (U.D.L.)-Figure-5The bending moment at position ‫ݔ‬ is given by ‫ܯ‬ ൌି௪௫మଶ. Substituting this equation (3) wehave,‫ܫܧ‬݀ଶ‫ݕ‬݀‫ݔ‬ଶൌെ‫ݔݓ‬ଶ2Integrate wrt ‫ݔ‬ and we get,0 1 2 3 4 5 6-7-6-5-4-3-2-10x 10-3position along the axis (x)positionalongtheaxis(y)deflection of cantilever beam with point load at free end
9. 9. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME262‫ܫܧ‬݀‫ݕ‬݀‫ݔ‬ൌെ‫ݔݓ‬ଷ6൅ ‫ܣ‬ … … … … … … … … … … … … … … … … … … … … ሺ10ሻIntegrate again we get,‫ݕܫܧ‬ ൌെ‫ݔݓ‬ସ24൅ ‫ݔܣ‬ ൅ ‫ܤ‬ … … … … … … … … … … … … … … … … … … ሺ11ሻA and B are constants of integration and must be found from the boundary conditions. Theseare,ܽ‫ݐ‬ ‫ݔ‬ ൌ ‫,ܮ‬ ‫ݕ‬ ൌ 0 ሺ݊‫݋‬ ݂݈݀݁݁ܿ‫݊݋݅ݐ‬ሻܽ‫ݐ‬ ‫ݔ‬ ൌ ‫,ܮ‬݀‫ݕ‬݀‫ݔ‬ൌ 0 ሺ݄‫݈ܽݐ݊݋ݖ݅ݎ݋‬ሻSubstitute ‫ݔ‬ ൌ ‫ܮ‬ ܽ݊݀ௗ௬ௗ௫ൌ 0 ݅݊ ݁‫݊݋݅ݐܽݑݍ‬ ሺ10ሻ and we get,‫ܫܧ‬ሺ0ሻ ൌെ‫ܮݓ‬ଷ6൅ ‫ܣ‬ ݄݁݊ܿ݁ ‫ܣ‬ ൌ‫ܮݓ‬ଷ6Substitute this into (11) with the known solution‫ݕ‬ ൌ 0 ܽ݊݀ ‫ݔ‬ ൌ ‫ܮ‬ ‫ݏݐ݈ݑݏ݁ݎ‬ ݅݊‫ܫܧ‬ሺ0ሻ ൌെ‫ܮݓ‬ସ24൅‫ܮݓ‬ଷ6൅ ‫ܤ‬ ݄݁݊ܿ݁ ‫ܤ‬ ൌെ‫ܮݓ‬ସ8Putting the results for A and B into equations in (10) and (11) yields the complete equations.‫ܫܧ‬݀‫ݕ‬݀‫ݔ‬ൌെ‫ݔݓ‬ଷ6൅ ‫ݓ‬‫ܮ‬ଷ6… … … … … … … … … … … … … … … … … … … … … ሺ12ሻ‫ݕܫܧ‬ ൌെ‫ݔݓ‬ସ24൅‫ܮݓ‬ଷ‫ݔ‬6െ‫ܮݓ‬ସ8… … … … … … … … … … … … … … … … … … … … ሺ13ሻThe main points of interest is the slope and deflection at free end where‫ݔ‬ ൌ 0, ‫݃݊݅ݐݑݐ݅ݐݏܾݑݏ‬ ‫ݔ‬ ൌ 0 ݅݊‫݋ݐ‬ ሺ12ሻ ܽ݊݀ ሺ13ሻ gives the standard equations.Slope at free endௗ௬ௗ௫ൌ௪௅య଺ாூ… … … … … … … … … … … … … … … … … … … … … ሺ14ሻDeflection at free end ‫ݕ‬ ൌି௪௅ర଼ாூ… … … … … … … … … … … … … … … … … … ሺ15ሻNumerical Analysis-2A cantilever thin beam is 6m long and has a U.D.L. of 300N/m. The flexural stiffnessis 60MN2. Calculate the slope and deflection at the free end. [16]Solution:-Given data:- ‫ݓ‬ ൌଷ଴଴ே௠, ‫ܫܧ‬ ൌ 60‫ܰܯ‬ଶ, ‫ܮ‬ ൌ 6݉Slope equation ‫ݕ‬′ൌ ݉ ൌௗ௬ௗ௫ൌ௪଺ாூሾെ‫ݔ‬ଷ൅ ‫ܮ‬ଷሿ‫ݐݑ݌‬ ‫ݔ‬ ൌ 0݉, ‫ܮ‬ ൌ 6݉݉ଵ ൌଷ଴଴଺‫଺כ‬଴‫כ‬ଵ଴లሾെ0ଷ൅ 6ଷሿ ൌ 1.7999 ‫כ‬ 10ିସሺ݊‫݋‬ ‫ݏݐ݅݊ݑ‬ሻ‫ݐݑ݌‬ ‫ݔ‬ ൌ 2݉, ‫ܮ‬ ൌ 6݉݉ଶ ൌଷ଴଴଺‫଺כ‬଴‫כ‬ଵ଴లሾെ2ଷ൅ 6ଷሿ ൌ 1.7332 ‫כ‬ 10ିସሺ݊‫݋‬ ‫ݏݐ݅݊ݑ‬ሻ,‫ݐݑ݌‬ ‫ݔ‬ ൌ 4݉, ‫ܮ‬ ൌ 6݉݉ଷ ൌଷ଴଴଺‫଺כ‬଴‫כ‬ଵ଴లሾെ4ଷ൅ 6ଷሿ ൌ 1.2667 ‫כ‬ 10ିସሺ݊‫݋‬ ‫ݏݐ݅݊ݑ‬ሻ and‫ݐݑ݌‬ ‫ݔ‬ ൌ 6݉, ‫ܮ‬ ൌ 6݉݉ସ ൌ3006 ‫כ‬ 60 ‫כ‬ 10଺ሾെ6ଷ൅ 6ଷሿ ൌ 0.00 ሺ݊‫݋‬ ‫ݏݐ݅݊ݑ‬ሻ
10. 10. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME263Table-30 2 4 6Slope(m) 1.7999*10.^-41.7332*10.^-41.2667*10.^-40.00MATLAB PROGRAM-USING INTERPOLATION METHOD% calculate the slope of thin beam at any point in between 1-class interval% cantilever UDLx=[0 2 4 6];m=[1.7999e-4 1.7332e-4 1.2666e-4 0.00];xi=1;yilin=interp1(x,m,xi,linear)yilin =(1.7666e-004) (Answer)Plot the graph of slope% plot the graph of slope of beam% cantilever udl thin beamw=300;x=[0:1:6];EI=60*10.^6;L=6;m=(w/6)*(-x.^3+L.^3)/(EI);plot(x,m,--r*,linewidth,2,markersize,12)xlabel(position along the axis (x),fontsize,12)ylabel(position along the axis (y),fontsize,12)title(slope of cantilever udl thin beam,fontsize,12)Figure-60 1 2 3 4 5 601x 10-4position along the axis (x)positionalongtheaxis(y)slope of cantilever udl thin beam
11. 11. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME264Deflection equation:-‫ݕ‬ ൌ‫ݓ‬‫ܫܧ‬ቈെ‫ݔ‬ସ24൅‫ܮ‬ଷ‫ݔ‬6െ‫ܮ‬ସ8቉‫ݐݑ݌‬ ‫ݔ‬ ൌ 0݉, ‫ܮ‬ ൌ 6݉‫ݕ‬ଵ ൌ30060 ‫כ‬ 10଺ቈെ0 ൅ 0 െ6ସ8቉ ൌ െ8.1 ‫כ‬ 10ିସ݉,‫ݐݑ݌‬ ‫ݔ‬ ൌ 2݉, ‫ܮ‬ ൌ 6݉‫ݕ‬ଶ ൌ30060 ‫כ‬ 10଺ቈെ2ସ24൅6ଷ‫כ‬ 26െ6ସ8቉ ൌ െ4.533 ‫כ‬ 10ିସ݉,‫ݐݑ݌‬ ‫ݔ‬ ൌ 4݉, ‫ܮ‬ ൌ 6݉‫ݕ‬ଷ ൌ30060 ‫כ‬ 10଺ቈെ4ସ24൅6ଷ‫כ‬ 46െ6ସ8቉ ൌ െ1.433 ‫כ‬ 10ିସ݉And ‫ݐݑ݌‬ ‫ݔ‬ ൌ 6݉, ‫ܮ‬ ൌ 6݉‫ݕ‬ସ ൌ30060 ‫כ‬ 10଺ቈെ6ସ24൅6ଷ‫כ‬ 66െ6ସ8቉ ൌ 0.00݉Table-4‫ݔ‬ሺ݉ሻ 0 2 4 6‫ݕ‬ሺ݉ሻ -8.1*10.^-4 -4.533*10.^-4 -1.433*10.^-4 0.00MATLAB PROGRAM-USING INTERPOLATION METHOD% calculate the deflection of thin beam at any point in between 1-class% interval% cantilever udl thin beamx=[0 2 4 6];y=[-8.1*10.^-4 -4.533*10.^-4 -1.433*10.^-4 0.00];xi=1;yilin=interp1(x,y,xi,linear)yilin = -6.3165e-004 (Answer)Plot the graph of deflection% plot the graph of deflection of beam% cantilever thin beam% cantilever udlw=300;x=[0:1:6];EI=60*10.^6;L=6;y=(w/EI)*[(-x.^4/24)+(L.^3*x/6)-(L.^4/8)];plot(x,y,--r*,linewidth,2,markersize,12)xlabel(position along the axis (x),fontsize,12)ylabel(position along the axis (y),fontsize,12)title(deflection of cantilever udl thin beam,fontsize,12)
12. 12. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME265Figure-7Case 3- Cantilever Thin Beam with Uniformly Varying Load (U.V.L)-Figure-8Consider a section ‫ݔ‬ െ ‫ݔ‬ at a distance ‫ݔ‬ the fixed end AB intensity of loading at ‫ݔ‬ െ ‫ݔ‬.[17]ൌሺ‫ܮ‬ െ ‫ݔ‬ሻ‫ݓ‬‫ܮ‬‫ݎ݁݌‬ ‫ݐ݅݊ݑ‬ ‫݊ݑݎ‬The bending moment at section ‫ݔ‬ െ ‫ݔ‬ is given by‫ܫܧ‬݀ଶ‫ݕ‬݀‫ݔ‬ଶൌ െ12ሺ‫ܮ‬ െ ‫ݔ‬ሻ‫ݓ‬‫ܮ‬ሺ‫ܮ‬ െ ‫ݔ‬ሻ.ሺ‫ܮ‬ െ ‫ݔ‬ሻ3ൌെ‫ݓ‬ሺ‫ܮ‬ െ ‫ݔ‬ሻଷ6‫ܮ‬Integrating we get,‫ܫܧ‬݀‫ݕ‬݀‫ݔ‬ൌ‫ݓ‬ሺ‫ܮ‬ െ ‫ݔ‬ሻସ24‫ܮ‬൅ ‫ܥ‬ଵAt B the slope is 0,0 1 2 3 4 5 6-9-8-7-6-5-4-3-2-101x10-4position along the axis (x)positionalongtheaxis(y)deflectionof cantilever udlthinbeam
13. 13. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME266Therefore,At ‫ݔ‬ ൌ 0݀‫ݕ‬݀‫ݔ‬ൌ 00 ൌ‫ܮݓ‬ଷ24൅ ‫ܥ‬ଵThen ܿଵ ൌ െ௪௅యଶସ‫ܫܧ‬ௗ௬ௗ௫ൌ௪ሺ௅ି௫ሻరଶସ௅െ௪௅యଶସ… … … … … … … … … … … … … … . . ሺ16ሻ Slope equationIntegrating again we get,‫ݕܫܧ‬ ൌ െ‫ݓ‬ሺ‫ܮ‬ െ ‫ݔ‬ሻହ120‫ܮ‬െ‫ܮݓ‬ଷ24‫ݔ‬ ൅ ܿଶThe deflection at B is 0‫ݔ‬ ൌ 0, ‫ݕ‬ ൌ 00 ൌ െ‫ܮݓ‬ସ120൅ ܿଶܿଶ ൌ‫ܮݓ‬ସ120Therefore,‫ݕܫܧ‬ ൌ െ௪ሺ௅ି௫ሻఱଵଶ଴௅െ௪௅య௫ଶସ൅௪௅రଵଶ଴… … … … … … … … … … ሺ17ሻ Deflection equationTo find the slope at C at the free end putting the value ‫ݔ‬ ൌ ‫ܮ‬ in the slope equation weget,[18]‫ܫܧ‬݀‫ݕ‬݀‫ݔ‬ൌ െ‫ܮݓ‬ଷ24Therefore,݀‫ݕ‬݀‫ݔ‬ൌ െ‫ܮݓ‬ଷ24‫ܫܧ‬To find the deflection at C putting ‫ݔ‬ ൌ ‫ܮ‬ in the deflection equation we get,‫ݕܫܧ‬ ൌ െ‫ܮݓ‬ସ24൅‫ܮݓ‬ସ120ൌ െ‫ܮݓ‬ସ120ሺ5 െ 1ሻ ൌ െ‫ܮݓ‬ସ30Therefore,‫ݕ‬ ൌ െ‫ܮݓ‬ସ30‫ܫܧ‬Download deflection of ܿ ൌ௪௅రଷ଴ாூNumerical Analysis-3Cantilever of length ‫ܮ‬ ൌ 6݉ carrying a distributed load whose intensity variesuniformly from zero at the free end to 800N/m at fixed end, flexural stiffness ሺ‫ܫܧ‬ ൌ643300ܰ݉ଶሻ.[19]Solution:-Given data:-Span lengthሺ‫ܮ‬ሻ ൌ 6݉, ‫ݓ‬ ൌ଼଴଴ே௠, ‫ܫܧ‬ ൌ 643300ܰ݉ଶ,Slope equationௗ௬ௗ௫ൌ ‫ݕ‬′ൌ ݉ ൌ௪ଶସாூቂሺ௅ି௫ሻర௅െ ‫ܮ‬ଷቃ
14. 14. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME267‫ݐݑ݌‬ ‫ݔ‬ ൌ 0, ‫ܮ‬ ൌ 6݉‫′ݕ‬ଵൌ ݉ଵ ൌ଼଴଴ଶସ‫଺כ‬ସଷଷ଴଴ቂሺ଺ି଴ሻర଺െ 6ଷቃ ൌ 0.00° ሺ݊‫݋‬ ‫ݏݐ݅݊ݑ‬ሻ,‫ݐݑ݌‬ ‫ݔ‬ ൌ 2, ‫ܮ‬ ൌ 6݉‫′ݕ‬ଶൌ ݉ଶ ൌ଼଴଴ଶସ‫଺כ‬ସଷଷ଴଴ቂሺ଺ିଶሻర଺െ 6ଷቃ ൌ െ0.0090° ሺ݊‫݋‬ ‫ݏݐ݅݊ݑ‬ሻ,‫ݐݑ݌‬ ‫ݔ‬ ൌ 4, ‫ܮ‬ ൌ 6݉‫ݕ‬Ԣଷ ൌ ݉ଷ ൌ80024 ‫כ‬ 643300ቈሺ6 െ 4ሻସ6െ 6ଷ቉ ൌ െ0.0111° ሺ݊‫݋‬ ‫ݏݐ݅݊ݑ‬ሻ‫ݐݑ݌‬ ‫ݔ‬ൌ 6, ‫ܮ‬ൌ 6݉‫′ݕ‬4ൌ ݉4 ൌ80024 ‫כ‬ 643300ቈሺ6 െ 6ሻ46െ 63቉ ൌ െ0.0112° ሺ݊‫݋‬ ‫ݏݐ݅݊ݑ‬ሻTable-4‫ݔ‬ሺ݉݁‫݁ݎݐ‬ሻ 0 2 4 6‫݁݌݋݈ݏ‬ሺ݉ሻ° 0 -0.0090 -0.0111 -0.0112MATLAB PROGRAM-USING INTERPOLATION METHOD% calculate the slope of thin beam at any point in between 1-class interval% cantilever U.V.L.x=[0 2 4 6];slope=[0 -0.0090 -0.0111 -0.0112];xi=1;yilin=interp1(x,slope,xi,linear)yilin = -0.0045°Plot the graph of slope of cantilever U.V.L.MATLAB PROGRAM-% plot the graph of slope of thin beam% cantilever U.V.L.w=800;x=[0:1:6];EI=643300;L=6;m=(1/24)*(w/EI)*[((L-x).^4/L)-L.^3];plot(x,m,--r*,linewidth,2,markersize,12)xlabel(position along the axis (x),fontsize,12)ylabel(position along the axis (y),fontsize,12)title(slope of cantilever U.V.L. thin beam,fontsize,12)
15. 15. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME268Figure-9Deflection equation W - ሺL-xሻ5 L4y ൌ - L3 x ൅ ሾ20ሿ24 EI 5L 4Put x = 0m, L = 6m800 - (6-0)564y 1 = - 63* 0 + = 0.00m24*70*109*9.19*10- 65*6 5x = 2m, L = 6m800 - (6-2)564y 2 = - 63* 2 + = -0.011m,24*70*109*9.19*10- 65*6 5x = 4m, L = 6m800 - (6-4)564y 3 = - 63* 4 + = -0.031m,24*70*109*9.19*10- 65*6 5x = 4m, L = 6m800 - (6-6)564y 4 = - 63* 6 + = -0.054m,24*70*109*9.19*10- 65*6 5Table-5x (metre) 0 2 4 6y(metre) 0.0000 -0.0110 -0.0310 -0.05400 1 2 3 4 5 6-0.012-0.01-0.008-0.006-0.004-0.0020position along the axis (x)positionalongtheaxis(y)slope of cantilever U.V.L. thin beam
16. 16. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME269MATLAB PROGRAM-USING INTERPOLATION METHOD% calculate the deflection of beam at any point in between 1-class interval% cantilever U.V.L. thin beamx=[0 2 4 6];y=[0.0000 -0.0110 -0.0310 -0.0540];xi=1;yilin=interp1(x,y,xi,linear)yilin =-0.0055Answer is -0.0055mPlot the graph of deflection of cantilever U.V.L.-MATLAB PROGRAM-% plot the graph of deflection of thin beam% cantilever U.V.L.w=800;x=[0 2 4 6];EI=643300;L=6;y=(1/24)*(w/EI)*[(-(L-x).^5/(5*L))-L.^3*x+L.^4/5];plot(x,y,--r*,linewidth,2,markersize,12)xlabel(position along the axis (x),fontsize,12)ylabel(position along the axis (y),fontsize,12)title(deflection of cantilver U.V.L.thin beam,fontsize,12)Figure-10FUTURE SCOPE1- It can be extending as apply in simply supported thin beam (point load at mid, uniformlydistributed load and uniformly varying load).2- It can be extending in case of composite materials of beam which are non-isotropic.3- It can be extending that is used in trusses like perfect, deficient and redundant.4- It can be extending that is used in tapered and triangular beam.5- It can be extending in Aeronautics, Aerodynamics and Space Engineering which is consistingof fixed vanes and crossed moving fixed vanes in rotor.6- It can extending in Orthopaedics in Medical Sciences which is applicable in replace orsupport to the bones.0 1 2 3 4 5 6-0.06-0.05-0.04-0.03-0.02-0.010position along the axis (x)positionalongtheaxis(y)deflection of cantilver U.V.L. thin beam
17. 17. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME270DISCUSSION AND CONCLUSIONIt was observed that in case of Cantilever thin beams (point load at free end, u.d.l. andu.v.l.) are carried out by the numerical analysis and MATLAB programming, made the tableof ‫ݔ‬ verse slope and ‫ݔ‬ verse deflection after that taken at any one point in between any 1-class-interval in thin beam and then calculated value at same point by using InterpolationMethod through the MATLAB programming to analysed the value at that point is slope anddeflection, we have analyzed by plotted the graph of static Slopes and Deflections of thinbeams through the MATLAB programming.REFERENCES[1] Zhang, G.Y., 2010, “A Thin Beam Formulation Based on Interpolation Method”,International Journal of numerical methods in engineering, volume 85, pp. 7-35.[2] Wang, Hu, Guang, Li Yao, 2007, “Successively Point Interpolation for OneDimensional Formulation”, Engineering Analysis with Boundary Elements, volume31, pp. 122-143.[3] Ballarini, Roberto, S, 2003 “Euler-Bernoulli Beam Theory”, Mechanical EngineeringMagazine Online.[4] Liu, Wing Kam, 2010, “Meshless Method for Linear One-Dimensional InterpolationMethod”, International Journal of Computer Methods in Applied Mechanics andEngineering, volume 152, pp. 55-71.[5] Park, S.K. and Gao, X.L., 2007, “Bernoulli-Euler Beam Theory Model Based on aModified Coupled Stress Theory”, International of Journal of Micro-mechanics andMicro- engineering, volume 19, pp. 12-67.[6] Paul, Bourke, 2010, “Interpolation Method”, International Journal of NumericalMethods in Engineering, volume 88, pp. 45-78.[7] Launder, B.E. and Spading D.B., 2010, “The Numerical Computation of ThinBeams”, International Journal of Computer Methods in Applied Mechanics andEngineering, volume 3, pp. 296-289.[8] Ballarini and Roberto, 2009, “Euler-Bernoulli Beam Theory Numerical Study of ThinBeams”, International of Computer in Applied Mechanics and Engineering, volume178, pp. 323-341.[9] Thomson, J.F., Warsi Z.U.A. and Mastin C.W., 1982 “Boundary Fitted Co-ordinatesystem for Numerical Solution of Partial Differential Equations”, Journal ofComputational Physics, volume 47, pp. 1-108.[10] Gilat, Amos, January 2003, “MATLAB An Introduction with Application,Publication- John Wiley and Sons.[11] Hashin, Z and Shtrikman, S., 1963 “A Variation Approach to the Theory of ElasticBehaviour of Multiphase Materials”, Journal of Mechanics and Physics of Solids,volume 11, pp. 127-140.[12] Liu, G.R. and Gu, Y.T., 2001, “A Point Interpolation Method for One-DimensionalSolids”, International Journal of Numerical Methods Engineering, pp. 1081-1106.[13] Katsikade, J.T. and Tsiatas, G.C., 2001, “Large Deflection Analysis of Beams withVariable Stiffness”, International Journal of Numerical Methods Engineering, volume33, pp. 172-177.
18. 18. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 –6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 2, March - April (2013) © IAEME271[14] Atluri, S.N. and Zhu, T., 1988, “A new meshless local Petrov-Galerkin (MLPG)approach in computational mechanics”, Computational Mechanics volume 22, pp.117-127.[15] Atluri, S. N. and Zhu, T., 2000, “New concepts in meshless methods”, InternationalJournal of Numerical Methods Engineering, volume 47, pp. 537-556.[16] Newmark, N.M., 2009, “A Method of Computation for Structural Statics”, Journal ofEngineering Mechanics Division, ASCE, volume 85, pp. 67-94.[17] Bickley, W.G., 1968 “Piecewise Cubic Interpolation and Two-point Boundary ValueProblem”, Computer Journal, volume 11, pp. 200-206.[18] Sastry, S.S., 1976, “Finite Difference Approximations to One Dimensional ParabolicEquation” Journal Computer and Applied Maths., volume 2, pp. 20- 23.[19] Liu, G. R. and Gu,Y. T., 2001, “A Point Interpolation Method for two-dimensionalsolids”, International Journal for Numerical Methods in Engineering, volume 50,pp. 55-60.[20] Timoshenko, Stephen P. and Gere, James M., 1962, “Theory of Elastic Stability”,International Student Edition by McGraw-Hill Book Company, New York.[21] Prabhat Kumar Sinha and Rohit, “Analysis of Complex Composite Beam by usingTimoshenko Beam Theory & Finite Element Method”, International Journal ofDesign and Manufacturing Technology (IJDMT), Volume 4, Issue 1, 2013,pp. 43 - 50, ISSN Print: 0976 – 6995, ISSN Online: 0976 – 7002.[21] Mehdi Zamani, “An Applied Two-Dimensional B-Spline Model for Interpolation ofData”, International Journal of Advanced Research in Engineering & Technology(IJARET), Volume 3, Issue 2, 2012, pp. 322 - 336, ISSN Print: 0976-6480,ISSN Online: 0976-6499.