ALTERNATIVE FIRST PRINCIPLE APPROACH FOR DETERMINATION OF ELEMENTS OF BEAM STIFFNESS MATRIX

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1 Nigerian Journal of Technology (NIJOTECH) Vol.. No.. July, Coyright Faculty of Engineering, University of Nigeria, Nsukka, ISSN: ATERNATIVE FIRST PRINCIPE APPROACH FOR DETERINATION OF EEENTS OF BEA STIFFNESS ATRIX C. A. Chidolue *, N. N. Osadebe + *DEPT OF CIVI ENGINEERING, NNADI AZIKIWE UNIVERSITY, AWKA. +DEPT OF CIVI ENGINEERING, UNIVERSITY OF NIGERIA, NSUKKA. *(corresonding author) chidoluealfred@gmail.com + nkemuamaka@yahoo.com Abstract Stiffness coefficients which in essence are elements of stiffness matri of a uniform beam element are derived in this work from first rinciles using elastic curve equation initial value method. The obtained initial value solution enables eact values of stiffness coefficients, fied end moments shears as well as dislacement (deflection rotation) of any given beam element under arbitrary lateral load to be evaluated. Keywords: First rincile, beam stiffness matri, elastic curve, fied end moments. Introduction It is a common knowledge that the force F generated in an elastic beam is directly roortional to the induced dislacement (deflection or sloe) in the same beam, []. Consequently, F = K () where K is a constant which measures how stiff or resistant the elastic beam is to the induced dislacement. The constant K for an isolated dislacement is called stiffness coefficient for an array or vector of dislacements it is called stiffness matri, []. If in equation () the dislacement is assigned a unit value, we obtain that F = K () Consequently, K is numerically equal to the force necessary to induce a unit dislacement in the structure. Stiffness coefficients are indisensable ingredients for dislacement analysis of redundant beams other redundant assemblages such as continuous beams, indeterminate frames etc [], [4], [5]. They also form an imortant tool for finite element analysis of beam systems [6]. Traditional means used to obtain these coefficients considered a fied ended uniform beam element as a system with two degrees of indeterminacy. By formulating the comatibility equations using fleibility aroach the stiffness coefficients are obtained as fied end forces (moments shears) necessary to induce a unit dislacement i.e., unit deflection or sloe at the beam s fied end. Though this aroach equally gives eact results, it dems evaluation of fleibility influence coefficients before solving the comatibility equations. In this resent work, the equation of the elastic curve of a uniform beam element is solved using initial value method to obtain a set of solutions for dislacement, sloe, bending moment shear force in terms of initial values of these quantities i.e., their values at =, as unknown arameters. With this set of solutions the stiffness coefficients of any given beam element with stiulated end conditions are obtained. The advantage of this resent formulation is that fied-end moments associated shear forces due to any arbitrary loads can be obtained,

2 circumventing the numerical work involved using traditional method.. The Elastic Curve Equation Consider an ordinary beam element under the action of generalized load as shown in Fig.. The elastic curve y(), consequent uon the action of the imosed load, is given by; EIy ''( ) ( ) () where, sagging moments are considered ositive, EI = Fleural rigidity, y '' denotes second derivative of the elastic beam curve with resect to. After two successive differentiations with resect to we obtain, iv EIy ( ) q( ) (4) In the absence of lateral load q(), the homogeneous equation of elastic beam curve is IV y ( ) (5) By successive integration of equation (5) we obtain the following eressions; y '''( ) C (6) y ''( ) C C (7) C y '( ) C C (8) C C C C (9) 4 6 where, C, C, C, C 4 are arbitrary constants of integration which can be determined using initial value methods as follows. y q y() elastic curve Fig. Beam element under generalized load et the initial conditions for determination of the coefficients be stiulated as follows. y() y, y '() () () (), Q() Q () Substituting equations () () into equations (6) to (9), taking note of equation () we obtain that C4 y, C, C, Q C. EI EI Consequently, Q y EI () Q ( ) EI EI () ( ) Q (4) Q( ) Q (5) Equations () to (5) constitute the initial value solution for the elastic curve. They are used as shown below, to obtain the stiffness coefficients of ordinary elastic beams.. Stiffness Coefficients of Elastic Beams In the develoment that follows, the set of initial value solutions is alied to elastic beams with various fied end conditions to obtain their stiffness coefficients.. Case ; Fied ended beam element with induced unit deflection at = Fig. Fied ended beam element with induced unit deflection at = In this case, y, At =; y() =, ( ), ( ) Q( ) Q Using equations () () we obtain that; Q (6) EI Q EI EI (7) Solving equations (6) (7) yields EI ; Q ; ; EI Q The fied end moment diagram is shown in Table. Q NIGERIAN JOURNA OF TECHNOOGY VO., NO., JUY 5

3 . Case : Fied ended beam element with induced unit rotation at =.4 Case 4: Proed cantilever with induced unit rotation at = Q Q Fig. Fied-ended uniform beam with induced unit rotation at one end In this case, y, Using equations () () we obtain that; Q (8) EI Q EI EI (9) Solving equations (8) (9) above yields, 4EI ; Q, EI ; Q The lots of these moments shears are shown in Table. Case : Proed cantilever with induced unit deflection at = Fig.4 Proed cantilever with induced unit deflection at = In this case, y, At = ; y() =, () =, Q( ) Q Substituting these into equations () () we obtain that, Q EI () Q () EI EI Solving gives EI EI ; Q, EI Q The moment diagram is shown in Table. Q Fig. 5: Proosed cantilever with induced unit rotation at = In this case, y, At = ; y( ) y ; ( ) ; Q( ) Q From equations () () we have Q () EI Q EI EI () Solving equations () () gives EI ; EI Q EI Q ; 4. Determination of Fied End oments of aterally oaded Beams Using Initial Value Solutions of the Elastic Curve In the foregoing resentations, we considered only the homogenous solution which enabled us to obtain stiffness coefficients. In the derivations that follow it is shown that the initial value solution can be used, in the face of imosed loads, to obtain fied end moments shears. 4. Case : Fied ended beam with a oint load We consider a uniform beam of length subjected to a lateral oint load P as shown in Fig. 6. Equations () to () constitute the homogenous solution when the imosed lateral load is absent. In order to obtain a articular integral due to the imosed load we consider the additional effect of the imosed load on the beam uniform element. The imosed oint load P, Fig. 6, has the similitude of shear its articular integral on the dislacement y(), sloe ( ), moment (), shear Q(), can be obtained by considering the imosed load P as the arameter Q. However, the origin is seemingly dislaced to the oint of alication so that initial distance which NIGERIAN JOURNA OF TECHNOOGY VO., NO., JUY 54

4 measured in the homogeneous solution will now measure (-a) distance. Again the arameter Q has oosite direction with P, thus by introducing P with a negative sign changing all distances to (-a) the articular integral is obtained as follows. a P b y Fig.6: Fied ended beam with a oint load P a P a, EI (4) P a; Q where the subscrit ( ) indicates the articular integral of the indicated arameters. Using these eressions, a set of general solution is obtained as follows. Q a (5) y EI Q ( a) ( ) EI EI EI (6) ( ) Q P( a) (7) Q( ) Q P (8) In order to obtain fied end moments shears under the action of the oint load, we note that the dislacement sloe at both ends of the beam are all zero. Thus, y(), (), y ( ), ( ) (9) Consequently, eing equation (9) keeing in view of equations (5) (6) we obtain that; Q Pb () EI Q Pb () EI EI EI Solving yields, after simlification; Pb a, Q Pb a ( ) Substituting these values into equations (7) (8) we obtain: Q Pb Q ( a) b 4. Case : Fied ended beam with moment at any arbitrary oint along the beam y o = θ o = Fig.7 Fied ended beam with moment at arbitrary oint on the beam In this eamle the arameter has similitude with o in the homogeneous solution is of the same sign (direction). Therefore by relacing o with changing to (-a) in the initial value solution, equations () to (5), the articular integrals are obtained as follows. y ( a) EI,,, Q The general solution becomes ( a) Q y () EI EI ( a) Q ( ) () EI EI EI ( ) Q (4) Q( ) Q (5) Q As in case 5, the deflection sloe at both ends of the beam are all zero. Therefore eing equation (9) in view of equations (5) (6) we obtain that; b Q 6 (6) Q b (7) a Solving equations (6) (7) gives, after simlification; b ( ) a b, l 6ab Q Substituting into equations (4) (5) gives, after simlification, a ( ) b a, 6ab Q 4. Case : Fied ended beam with UD b Q NIGERIAN JOURNA OF TECHNOOGY VO., NO., JUY 55

5 Fig.8 :Fied ended beam with uniformly distributed load In the case of uniformly distributed load, the articular integrals are obtained as follows; et y,, Q be the articular integrals. qu dy =- du q y u du q 4 / 4EI qu d du, q q u du EI EI d dq y o = θ o = qudu ; qdu ; q q udu Q q du q Suerimosing these articular integrals to the homogeneous solutions gives the general solutions, equations (8) to (4). 4 ( ) Q q y y EI 4EI (8) Q q ( ) EI EI (9) q ( ) Q (4) Q( ) Q q (4) Q. du. qdu The dislacement sloe at both ends of the beam are zero, i.e. y(), (), y( ), ( ) (4) Therefore eing equation (4) using equations (8) (9) we obtain that; 4 Q q (4) 6 4 Q q (44) 6 Solving equations (4) (44) gives; q, q Q Substituting into equation (4) (4) gives, q, q Q. Q The summary of fied end moments shears is given on Table. 5. Stiffness atri of Beam Elements The stiffness coefficients obtained above can be synthesized into a stiffness matri of the considered beam element. Consider the uniform beam element, Fig.9, subjected to clockwise coules,, at its etreme nodal oints together with vertical forces Q Q. et y y be the dislacements in the y-direction at the nodes,, while are clockwise rotations at the same nodes resectively. Using the stiffness coefficients obtained earlier, the bending moment shear forces Q Q can be eressed in terms of dislacements rotations as follows; 4EI EI y y EI Q y y EI 4EI y y In matri notation the above equations take the form; 6 6 Q y EI Q 6 6 y Consequently the stiffness matri is K EI NIGERIAN JOURNA OF TECHNOOGY VO., NO., JUY 56

6 Q Fig.9 Fied ended beam element Q 6. Discussion of Results The stiffness coefficients (Table ) obtained for the various beam fied end conditions can be used to build u the stiffness matri of a beam / or beam-column assemblages as eemlified in section 5.. The fied end moments shears (Table ) obtained for three stard beam loading conditions are the same as for those found in literatures, []. However, the advantages of initial value method which are; simlicity, ease of alication, room for reetitive work, were utilized in this work thus, reducing the comutational time rocedures involved. 7. Conclusion From the foregoing, it can be seen that the obtained stiffness coefficients, fied end moments shears are identical with the ones obtained in literatures [], [6]. The advantage of this method is that the comutation of fleibility influence coefficients before evaluating the comatibility conditions which lead to the desired stiffness coefficients are circumvented. References [] arshall, W.T. Nelson, H.., Structures, second edition. Pitman International tet, 98. [] Reddy, C.S., Basic structural analysis, second edition, cgraw-hill ublishing comany limited, New Delhi, 996. [] Darkov, A. Kuznetsov, V., Structural echanics, Translated from the Russian by B. achinov, ir ublishers, oscow, 985. [4] Chidolue, C.A., Aginam, C.H., Effect of shear on stress distribution in redundant frames, International journal of soft comuting, vol. issue,, 99-6 [5] Osadebe, N.N., Chidolue, C.A., Effect of shear deflections on stiffness matri coefficients, International Journal of Engineering, vol.5 no. 4,, [6] Ross, C.T.F., Finite element method in structural mechanics. Elis Horwood series in engineering science, 985 Table : Summary of Elements of Beam Stiffness Coefficients Case Tye of beam loading Bending moment diagram Beam stiffness coefficients y = Q Constant EI = Q 4EI EI Q Q NIGERIAN JOURNA OF TECHNOOGY VO., NO., JUY 57

7 Case Tye of beam loading Bending moment diagram Beam stiffness coefficients y = Q Constant EI o EI EI Q Q EI Q 4 Constant EI = Q EI EI Q EI Q Case Table : Fied - End oments Shears For aterally oaded Beams Tye of beam Bending moment diagram Fied-end moments loading shears a a l P b l b Q Q Q Pb a, Pba Pb Q a, Pab Q b ( ) a b 6ab Q a b a 6ab Q q, q Q q, q Q NIGERIAN JOURNA OF TECHNOOGY VO., NO., JUY 58

Methods of Analysis. Force or Flexibility Method

Methods of Analysis. Force or Flexibility Method INTRODUCTION: The structural analysis is a mathematical process by which the response of a structure to specified loads is determined. This response is measured by determining the internal forces or stresses