Saudi Journal of Engineering and Technology (SJEAT) ISSN 2415-6272 (Print) Scholars Middle East Publishers ISSN 2415-6264 (Online) Dubai, United Arab Emirates Website: http://scholarsmepub.com/ Ritz Method for the Analysis of Statically Indeterminate Euler – Bernoulli Beam Charles Chinwuba Ike*, Edwin Uchechukwu Ikwueze Department of Civil Engineering, Enugu State University of Science and Technology, Enugu State, Nigeria Abstract: The Ritz method was used in this paper for the flexural analysis of a Original Research Article statically indeterminate Euler – Bernoulli beam with a prismatic cross section. The beam considered was a propped cantilever of length, l, fixed at x = 0, and simply *Corresponding author supported at x = l; and carrying a linearly distributed transverse load on the Charles Chinwuba Ike longitudinal axis. Two cases of coordinate (basis) functions were studied. In the first case, the basis functions were constructed to satisfy the deflection boundary Article History conditions, but not the force boundary conditions. In the second case, the basis Received: 02.03.2018 functions were constructed to satisfy all the boundary conditions. It was found that Accepted: 12.03.2018 the stiffness equations formed with the basis functions that satisfied all the boundary Published: 30.03.2018 conditions gave the exact solutions for deflection, bending moments and shear force distributions along the beam’s longitudinal axis. The effectiveness of the Ritz method DOI: for solving statically indeterminate Euler Bernoulli beam flexure problems was thus 10.21276/sjeat.2018.3.3.3 highlighted. Keywords: Ritz method, Euler – Bernoulli beam, stiffness equations, statically indeterminate beam, deflection, bending moment distribution, shears force distribution. INTRODUCTION Beams are structural members that carry loads applied transversely to the longitudinal axis by the development of bending moments. They are used in buildings, bridges, machines, aerospace structures and naval structures [1, 2]. Several theories are used to describe the behaviour of beams. Some of these are: Euler – Bernoulli beam theory [3], Mindlin beam theory, Timoshenko beam theory [4] and Vlasov beam theory. Euler – Bernoulli beam theory has been found to be suitable for slender beams where the flexural behaviour governs the structural behaviour [5,6]. For deep and moderately thick beams, shear deformation plays a significant role in the structural behaviour, and the Timoshenko beam theory is used [4]. The main defect of the Euler – Bernoulli beam theory is that it is inaccurate for deep (thick) beams. Thick beams are ones for which the thickness is not negligible compared to the length (longitudinal dimension). A more accurate beam theory or a complete three dimensional (3D) solid mechanics approach is used to analyse thick beams. Timoshenko beam theory can be considered an extension of the Euler – Bernoulli beam theory in that it considers the effect of shear deformation, which is disregarded in the Euler Bernoulli beam theory. The fundamental assumptions of the Timoshenko beam theory are: The longitudinal axis of the unloaded beam is straight. Loads are applied transversely to the longitudinal axis. The deformations and strains are small. Hooke’s law of linear elasticity can be used for the stress – strain relations. Plane cross – sections which are initially normal to the longitudinal axis will remain plane after bending deformation. This paper adopts the Euler – Bernoulli beam theory. Several methods are used in the analysis of the Euler Bernoulli beam to determine the internal bending moments and shear force distribution. For statically indeterminate beams, the methods found in the literature include: Force methods Flexibility methods Available Online: http://scholarsmepub.com/sjet/ 133 Charles Chinwuba Ike & Edwin Uchechukwu Ikwueze., Saudi J. Eng. Technol., Vol-3, Iss-3 (Mar, 2018): 133-140 Energy methods [7, 8] Fundamental assumptions of the Euler – Bernoulli beam theory The assumptions of the Euler – Bernoulli beam theory are as follows: [9] The beam is prismatic and has a straight centroidal axis, which is defined as the x – axis. The beam’s cross-section has an axis of symmetry. All transverse loadings act in the plane of symmetry. Plane sections perpendicular to the centroidal axis remain plane after bending deformation. The beam material is elastic, isotropic and homogeneous. Transverse deflections are small. METHODOLOGY The statically indeterminate Euler – Bernoulli beam problem shown in Figure 1, was considered. Fig-1: A statically indeterminate Euler – Bernoulli beam under linear load distribution The origin of coordinates is chosen at the fixed support, A. The boundary conditions are: w ()00 (1); w ()00 (2); wl() 0 (3); Ml() 0 (4); wl() 0 (5) The deflection shape functions are constructed from third degree and fourth degree polynomials as follows: x x2 x 3 x x 2 x 3 x 4 wxcaaa()1012 a 3 cbbb 2012 b 3 b 4 lll2 l 3 l 2 l 3 l 4 c1 N 1()() x c 2 N 2 x (7) where c1 and c2 are the unknown generalized deflections, N1 and N2 are the shape functions. w() x c1 N 1 c 2 N 2 (8) 2 2 3 11x x x x x wxcaa()1 1 2 2 3 a 3 cbb 2 1 2 2 3 b 3 4 b 4 (9) lll2 l 3 l 2 l 3 l 4 2 11x x x w() x c12 a 2 6 a 3 c 2 2 b 2 6 b 3 1 2 b 4 (10) l2 l 3 l 2 l 3 l 4 w()0 0 a00 b 0 (11) w()0 0 a11 b 0 (12) w() l00 a23 a (13) b234 b b 0 (14) wl() 0 2 a 6 a 2 3 0 (15) ll22 2bb6 b 1 2 243 0 (16) l2 l 2 l 2 2aa23 6 0 (17) 2b2 6 b 3 1 2 b 4 0 (18) Solving, using w(),(),()0 0 w 0 0 w l 0 Available online: http://scholarsmepub.com/sjet/ 134 Charles Chinwuba Ike & Edwin Uchechukwu Ikwueze., Saudi J. Eng. Technol., Vol-3, Iss-3 (Mar, 2018): 133-140 aa23 1 (19) bb241 b 3 2 (20) x2 x 3 x 2 x 3 x 4 w() x c1 c 2 2 c 1 N 1 c 2 N 2 (21) l2 l 3 l 2 l 3 l 4 2 3 2 3 4 x x x x x N ; 2 (22) l l l l l 2 1 x x x N 2 6 2 1 2 1 2 (23) ll22 ll The stiffness matrix is l T K E I N N d x (24) 0 Where T denotes the transpose matrix operation, E is the Young’s modulus, and I is the moment of inertia. l N 1 K E I() N N d x 12 (25) N 0 2 l 2 ()NNN1 1 2 K E I d x (26) 2 0 NNN2 1() 2 6 x 2 l 2 E I l 6 x 1 2 x 1 2 x K 22 d x (27) 42 2 ll ll0 1 2xx 1 2 2 l l 2 l EI aa1 1 1 2 K d x (28) 4 aa l 0 2 1 2 2 2 2 6x 2 4 x 3 6 x a 11 24 (29) ll l 2 2 6x 1 2 x 1 2 x a 12 22 (30) ll l 2 23 3 6x 9 6 x 7 2 x a 12 4 (31) l ll23 2 1 2x 1 2 x 6 x a 21 22 (32) lll 2 23 3 6x 9 6 x 7 2 x a 21 4 (33) l ll23 2 2 1 2xx 1 2 a 22 2 (34) l l 2 2 3 4 2 48x 48 x 288 x 144 x 144 x a 22 4 (35) l l2 l 3 l 4 l 2 Evaluating the integrals, we obtain: Available online: http://scholarsmepub.com/sjet/ 135 Charles Chinwuba Ike & Edwin Uchechukwu Ikwueze., Saudi J. Eng. Technol., Vol-3, Iss-3 (Mar, 2018): 133-140 E Il4 0 E I 4 0 K (36) 43 ll0 0.. 8 0 0 8 Force (load) matrix The force matrix is l Nx1 () F p() x d x (37) Nx() 0 2 23 xx l x ll F p0 1 d x (38) l 2 3 4 0 x x x 2 l l l 23 x x x l 1 l l l F p0 d x (39) 2 3 4 0 x x x x 12 l l l l 2 3 4 x x x l 2 l l l F p0 d x (40) 2 3 4 5 0 x x x x 33 l l l l 1 / 3 0 F p0 l (41) 1 / 6 0 The structure stiffness matrix is EI 4 0 c1 1 / 3 0 pl (42) 3 0 l 0 0. 8 c2 1 / 6 0 Solving, c 1 /1 2 0 4 1 pl0 (43) c2 1 / 4 8 EI The deflection function becomes: 4 2 3 2 3 4 pl0 11x x x x x wx() 2 (44) E I1 2 0 l l 4 8 l l l Simplifying, the deflection function is: 4 2 3 4 pl0 x x x wx()7 1 2 5 (45) 240E I l l l The bending moments M(x) and shear force Q(x) distributions are found from the bending moment deflection and the shear force – deflection relations as: 2 2 pl0 3 6xx 3 0 M() x E Iw 7 (46) 120 l l 2 pl 15x Q()() x E Iw x 0 8 (47) 30 l We observe that at x = l, Available online: http://scholarsmepub.com/sjet/ 136 Charles Chinwuba Ike & Edwin Uchechukwu Ikwueze., Saudi J.