A new efficient method to evaluate exact stiffness and

Arch Appl Mech DOI 10.1007/s00419-014-0820-7

O R I G I NA L

George C. Tsiatas

A new efficient method to evaluate exact stiffness and mass matrices of non-uniform beams resting on an elastic foundation

Received: 15 July 2013 / Accepted: 10 January 2014 © Springer-Verlag Berlin Heidelberg 2014

Abstract In this paper, a new efficient method to evaluate the exact stiffness and mass matrices of a nonuniform Bernoulli–Euler beam resting on an elastic Winkler foundation is presented. The non-uniformity may result from variable cross-section and/or from inhomogeneous linearly elastic material. It is assumed that there is no abrupt variation in the cross-section of the beam so that the Euler–Bernoulli theory is valid. The method is based on the integration of the exact shape functions which are derived from the solution of the axial deformation problem of a non-uniform bar and the bending problem of a non-uniform beam which are both formulated in terms of the two displacement components. The governing differential equations are uncoupled with variable coefficients and are solved within the framework of the analog equation concept. According to this, the two differential equations with variable coefficients are replaced by two linear ones pertaining to the axial and transverse deformation of a substitute beam with unit axial and bending stiffness, respectively, under ideal load distributions. The key point of the method is the evaluation of the two ideal loads which in this work is achieved by approximating them by two polynomials. More specifically, the axial ideal load is approximated by a linear polynomial while the transverse one by a cubic polynomial. The numerical implementation of the method is simple, and the results are compared favorably to those obtained by exact solutions available in literature. Keywords Stiffness matrix · Mass matrix · Non-uniform · Bernoulli–Euler beam · Winkler foundation 1 Introduction Structural members of variable mass and stiffness properties are usually receive attention to civil, mechanical and aeronautical engineering. Among them, non-uniform beams (prismatic and non-prismatic) play an important role since are often used, not only to serve architectural design, but also to improve the strength of particular regions and/or to reduce the cross-sectional dimensions for more economical solutions. Analysis of structures using the finite element method is well established. Many formulations exist for complex elements, but simple elements remain popular since they are usually well tested and easy to implement into an analysis program. The beam’s finite elements are usually derived assuming displacement functions. These functions are typically polynomials (Hermitian, Lagrangian), where the order of the polynomial is selected based upon compatibility requirements, desired accuracy and allowable computational cost. In Euler– Bernoulli beam theory, a cubic polynomial is typically used, for a uniform beam, since satisfies the differential equation of beam bending equilibrium. Similarly, a linear polynomial is used to develop the exact axial stiffness matrix for a uniform bar, since that polynomial satisfies the differential equation for axial deformation. However, G. C. Tsiatas (B) Institute of Structural Analysis & Seismic Research, School of Civil Engineering, National Technical University of Athens, 15773 Athens, Greece E-mail: [email protected] URL: http://users.ntua.gr/gtsiatas

G. C. Tsiatas

for non-uniform beams, the above polynomials are insufficient because they give only approximate solutions of the equilibrium differential equations [1]. During last decades, there was an extensive study in evaluating exact static and dynamic matrices of beam elements on elastic foundation. The two main approaches are based on either the use of approximate shape functions (see, e.g., [2–4]) or the development exact ones based on the solution of the differential equations (see, e.g., [5–9]). More specifically, Eisenberger and Yankelevsky [5] presented an exact stiffness matrix for a uniform beam resting on a Winkler foundation, while Sen et al. [6] used the Green’s functions for a nonuniform taper beam with various kinds of loading and boundary conditions in terms of the four normalized fundamental solutions, and Razaqpur and Shah [7] derived a finite element using the differential equation of a uniform beam on a two-parameter elastic foundation. A generalized Bernoulli/Timoshenko finite uniform beam element on a two-parameter elastic foundation was presented by Morfidis and Avramidis [8], which is based on the exact solution of the differential equation governing displacements, and possesses the ability for an optional consideration of shear deformations, semirigid connections and rigid offsets, while Kim et al. [9] proposed an improved numerical method to perform the spatially coupled elastic and stability analyses of non-symmetric and open/closed thin-walled beam-columns with and without two types of elastic foundation. In this work, a new efficient method to develop exact stiffness and mass matrices is presented. The method is general and is applied successfully to the evaluation of exact stiffness and mass matrices of a non-uniform Bernoulli–Euler beam resting on an elastic Winkler foundation. The non-uniformity may result from variable cross-section and/or from inhomogeneous linearly elastic material. It is assumed that there is no abrupt variation in the cross-section of the beam so that the Euler–Bernoulli theory is valid. The shape functions are derived from the solution of the governing differential equations, that is, the axial deformation problem of a non-uniform bar and the bending problem of a non-uniform beam. Both problems are formulated in terms of the two displacement components. The governing differential equations are uncoupled with variable coefficients and are solved within the framework of the analog equation concept. According to this, the two differential equations with variable coefficients are replaced by two linear ones pertaining to the axial and transverse deformation of a substitute beam with unit axial and bending stiffness, respectively, under ideal load distributions. The key point of the method is the evaluation of the two ideal loads which in this work is achieved by approximating them by two polynomials. More specifically, the axial ideal load is approximated by a linear polynomial while the transverse one by a cubic polynomial. The numerical implementation of the method is simple, and the results are compared favorably to those obtained by exact solutions available in literature. 2 Problem statement and numerical solution 2.1 Governing equations of the beam Consider an initially straight beam of length l having variable axial stiffness EA and bending stiffness EI, which may result from variable cross-section, A = A(x), and/or from inhomogeneous linearly elastic material, E = E(x); I = I (x) is the moment of inertia of the cross-section. The x-axis coincides with the neutral axis of the beam, which is bent in its plane of symmetry x z under the combined action of the distributed loads px = px (x) and pz = pz (x) in the axial and transverse direction, respectively. We assume that there is no abrupt variation in cross-section of the beam so that the Euler–Bernoulli theory is valid [10]. The beam is resting on a linear elastic foundation, and the equilibrium equations in terms of the displacements can expressed as [11] EA(x)u,x x + [EA(x)] ,x u,x = − px (x) EI(x)w,x x x x +2 [EI(x)] ,x w,x x x + [EI(x)] ,x x w,x x +k(x)w = pz (x)

(1) (2)

where u and w are the axial and transverse displacements, respectively, and k(x) is the linear Winkler foundation parameter. The most general boundary conditions associated with the problem which can include elastic support or restrain can be stated below α1 u (0) + α2 N (0) = α3 , α¯ 1 u (l) + α¯ 2 N (l) = α¯ 3 β1 w (0) + β2 V (0) = β3 , β¯1 w (l) + β¯2 V (l) = β¯3 γ1 w,x (0) + γ2 M (0) = γ3 , γ¯1 w,x (l) + γ¯2 M (l) = γ¯3

(3a,b) (4a,b) (5a,b)

A new efficient method to evaluate exact stiffness

bz (x )

pz (x )

px (x ) EA (x ), EI (x )

EA = EI = 1 u (x ), w(x )

u (x ), w(x )

k (x )

bx (x )

(a)

(b)

Fig. 1 Real problem (a) and ideal problem (b)

where ak , a¯ k , βk , β¯k , γk , γ¯k (k = 1, 2, 3) are given constants. It is apparent that all types of the conventional boundary conditions (clamped, simply supported, free or guided edge) can be derived from these equations by specifying appropriately these functions (e.g., for a clamped edge it is a1 = a¯ 1 = β1 = β¯1 = γ1 = γ¯1 = 1). The axial force, the shear force and the bending moment can be expressed as N (x) = EA(x)u,x V (x) = − [EI(x)] ,x w,x x −EI(x)w,x x x M(x) = −EI(x)w,x x

(6a) (6b) (6c)

respectively. 2.2 The analog equation concept According to the concept of the analog equation [12], the real problem (see Fig. 1a) is replaced by an ideal one (see Fig. 1b). The original governing differential Eqs. (1), and (2) with variable coefficients of the real problems are replaced by the following two linear ones u,x x = bx (x) w,x x x x = bz (x)

(7a) (7b)

pertaining to the axial and transverse deformation of a substitute beam with unit axial and bending stiffness, respectively, under the ideal load distributions bx (x) and bz (x). The two problems (real and ideal) have the same geometry, the same boundary conditions and the same solution. The key point of the method is the evaluation of the two ideal loads which in this work is achieved by approximating them by two polynomials. More specifically, the axial ideal load is approximated by a linear polynomial while the transverse one by a cubic polynomial. That is, bx (x) = 6a1 x + 2a2 bz (x) = 840a3 x 3 + 360a4 x 2 + 120a5 x + a6

(8a) (8b)

Substitution of Eqs. (8) into (7) leads to the analog equations of the ideal problem u,x x (x) = 6a1 x + 2a2 w,x x x x (x) = 840a3 x 3 + 360a4 x 2 + 120a5 x + 24a6

(9a) (9b)

The first analog Eq. (9a) is integrated two times, with respect to x, resulting in the axial displacement and its derivative u(x) = a1 x 3 + a2 x 2 + c1 x + c2 u,x (x) = 3a1 x 2 + 2a2 x + c1

(10a) (10b)

while the second one (9b) is integrated four times, also with respect to x, resulting in the transverse displacement and its derivatives w(x) = a3 x 7 + a4 x 6 + a5 x 5 + a6 x 4 + c3 x 3 + c4 x 2 + c5 x + c6 w,x (x) = 7a3 x 6 + 6a4 x 5 + 5a5 x 4 + 4a6 x 3 + 3c3 x 2 + 2c4 x + c5 w,x x (x) = 42a3 x 5 + 30a4 x 4 + 20a5 x 3 + 12a6 x 2 + 6c3 x + 2c4 w,x x x (x) = 210a3 x 4 + 120a4 x 3 + 60a5 x 2 + 24a6 x + 6c3

(11a) (11b) (11c) (11d)

G. C. Tsiatas

The 12 unknown constants (ai , ci i = 1, 2, . . . 6) appearing in Eqs. (10), and (11) are evaluated from a linear system of 12 equations. The first six equations are derived by applying the first equation of the real problem (1) at two points xi (i = 1, 2) of the beam (e.g., x1 = l/3 and x2 = 2l/3) EA(xi )u,x x (xi ) + [EA(xi )] ,x u,x (xi ) = − px (xi ), i = 1, 2

(12)

and the second Eq. (2) at four points x j ( j = 3, 4, 5, 6) of the beam (e.g., x3 = 0.2L , x4 = 0.4L , x5 = 0.6L and x6 = 0.8L)   EI(x j )w,x x x x (x j ) + 2 EI(x j ) ,x w,x x x (x j )   + EI(x j ) ,x x w,x x (x j ) + k(x j )w(x j ) = pz (x j ), j = 3, 4, 5, 6 (13) The six additional equations result from the boundary conditions (3, 4, 5) of the problem. Using Eqs. (12) and (13), the boundary conditions of the problem and the discretized Eqs. (10) and (11) at the six points xi (i = 1, 2) and x j ( j = 3, 4, 5, 6) lead to the formulation of the following set of 12 simultaneous equation ⎡ ⎤ 0 0 0 B71 (x1 ) 0 0 0 0 0 B11 (x1 ) B21 (x1 ) 0 ⎢ B11 (x2 ) B21 (x2 ) 0 ⎥ 0 0 0 B71 (x2 ) 0 0 0 0 0 ⎢ ⎥ 0 B33 (x3 ) B34 (x3 ) B35 (x3 ) B36 (x3 ) 0 0 B39 (x3 ) B310 (x3 ) B311 (x3 ) B312 (x3 ) ⎥ ⎢0 ⎢0 0 B33 (x4 ) B34 (x4 ) B35 (x4 ) B36 (x4 ) 0 0 B39 (x4 ) B310 (x4 ) B311 (x4 ) B312 (x4 ) ⎥ ⎢ ⎥ ⎢0 0 B (x ) B (x ) B (x ) B (x ) 0 0 B39 (x5 ) B310 (x5 ) B311 (x5 ) B312 (x5 ) ⎥ 33 5 34 5 36 5 35 5 ⎢ ⎥ ⎢0 0 B33 (x6 ) B34 (x6 ) B35 (x6 ) B36 (x6 ) 0 0 B39 (x6 ) B310 (x6 ) B311 (x6 ) B312 (x6 ) ⎥ ⎢ ⎥ ⎢0 ⎥ 0 0 0 0 0 C77 C78 0 0 0 0 ⎢ ⎥ ⎢ C81 ⎥ C82 0 0 0 0 C87 C88 0 0 0 0 ⎢ ⎥ ⎢0 ⎥ 0 0 0 0 0 0 0 C99 C910 0 C912 ⎢ ⎥ 0 C103 C104 C105 C106 0 0 C109 C1010 C1011 C1012 ⎥ ⎢0 ⎣0 ⎦ 0 0 0 0 0 0 0 0 C C 0 0

0 ⎤ ⎡

C123 C124 ⎤ ⎡ px (x1 ) a1 ⎢ a2 ⎥ ⎢ px (x2 ) ⎥ ⎢ ⎥ ⎢ p (x ) ⎥ ⎢ a3 ⎥ ⎢ z 3 ⎥ ⎢ a ⎥ ⎢ p (x ) ⎥ ⎢ 4⎥ ⎢ z 4 ⎥ ⎢ a ⎥ ⎢ pz (x5 ) ⎥ ⎥ ⎢ 5⎥ ⎢ ⎢ a6 ⎥ ⎢ pz (x6 ) ⎥ ⎥ ⎥ ⎢ ⎢ ×⎢ ⎥ = ⎢α ⎥ ⎥ ⎢ c1 ⎥ ⎢ 3 ⎥ ⎢ c2 ⎥ ⎢ α¯ 3 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ c3 ⎥ ⎢ β3 ⎥ ⎢ ⎥ ⎢¯ ⎥ ⎢ c4 ⎥ ⎢ β3 ⎦ ⎣c ⎦ ⎣γ 5

c6

C125

C126

0

0

C129

1110

1111

C1210

C1211

0

(14)

3

γ¯3

where B11 (x) = 6xEA(x) + 3x 2 [EA(x)] ,x B21 (x) = 2EA(x) + 2x [EA(x)] ,x , B71 (x) = [EA(x)] ,x B33 (x) = 840x 3 EI(x) + 420x 4 [EI(x)] ,x +42x 5 [EI(x)] ,x x +x 7 k(x) B34 (x) = 360x 2 EI(x) + 240x 3 [EI(x)] ,x +30x 4 [EI(x)] ,x x +x 6 k(x) B35 (x) = 120xEI(x) + 120x 2 [EI(x)] ,x +20x 3 [EI(x)] ,x x +x 5 k(x) B36 (x) = 24EI(x) + 48x [EI(x)] ,x +12x 2 [EI(x)] ,x x +x 4 k(x)

(15a) (15b,c) (16a) (16b) (16c) (16d)

B39 (x) = 12 [EI(x)] ,x +6x [EI(x)] ,x x +x 3 k(x), B310 (x) = 2 [EI(x)] ,x x +x 2 k(x), B311 (x) = xk(x), B312 (x) = k(x)

(16e,f) (16g,h)

C77 = α2 EA(0), C78 = α1 , C81 = α¯ 1l 3 + α¯ 2 3l 2 EA(l), C82 = α¯ 1l 2 + α¯ 2 2lEA(l) C88 = α¯ 1 , C99 = −6β2 EI(0), C910 = −2β2 [EI(0)] ,x , C912 = β1

(17a-c) (18a-d)

A new efficient method to evaluate exact stiffness

C103 = β¯1l 7 − 42l 5 β¯2 [EI(l)] ,x −210l 4 β¯2 EI(l) C104 = β¯1l 6 − 30l 4 β¯2 [EI(l)] ,x −120l 3 β¯2 EI(l) C105 = β¯1l 5 − 20l 3 β¯2 [EI(l)] ,x −60l 2 β¯2 EI(l) C106 = β¯1l 4 − 12l 2 β¯2 [EI(l)] ,x −24l β¯2 EI(l) C109 = β¯1l 3 − 6l β¯2 [EI(l)] ,x −6β¯2 EI(l) C1010 = β¯1l 2 − 2β¯2 [EI(l)] ,x , C1011 = β¯1l, C1012 = β¯1 C1110 = −2γ2 EI(0), C1111 = γ1

(19a) (19b) (19c) (19d) (19e) (19f-h) (20a,b)

C123 = 7l γ¯1 − 42γ¯2 l EI(l), C124 = 6l γ¯1 − 30γ¯2 l EI(l)

(21a,b)

C125 = 5l γ¯1 − 20γ¯2 l EI(l), C126 = 4l γ¯1 − 12γ¯2 l EI(l)

(21c,d)

C129 = 3l γ¯1 − 6γ¯2 lEI(l), C1210 = 2l γ¯1 − 2γ¯2 EI(l), C1211 = γ¯1

(21e-g)

6

5

4

5

3

3

4

2

2

Solving the linear system of Eq. (14) for the constants ai , ci (i = 1, 2, . . . 6), the displacements and their derivatives are computed using Eqs. (9), (10) and (11). 2.3 Shape functions According to the finite element method (FEM), the axial and transverse displacements are approximated by a set of nodal displacements u i (i = 1, 2, . . . 6) and a corresponding set of shape functions ψi (x)(i = 1, 2, . . . 6), as below u(x, t) = u 1 ψ1 (x) + u 4 ψ4 (x) w(x, t) = u 2 ψ2 (x) + u 3 ψ3 (x) + u 5 ψ5 (x) + u 6 ψ6 (x)

(22a) (22b)

respectively, where u 1 , u 2 are the axial nodal displacements, u 2 , u 5 the transverse nodal displacements and u 3 , u 6 the rotations at the beam ends. The shape function ψi (x) describe the effect of the nodal displacement u i = 1 on the shape of the beam when u j = 0 with j  = i. In this study, the shape functions are derived from the solution of the axial deformation problem of a non-uniform bar and the bending problem of a non-uniform beam which are described by the following differential equations EA(x)u,x x + [EA(x)] ,x u,x = 0 EI(x)w,x x x x +2 [EI(x)] ,x w,x x x + [EI(x)] ,x x w,x x +k(x)w = 0

(23a) (23b)

2.4 Strain energy and stiffness matrix The total elastic strain energy of the beam element can be expressed as U = Ua + Ub + Uspring

(24)

where 1 Ua = 2

L EA(x) [u,x (x, t)]2 dx,

(25a)

EI(x) [w,x x (x, t)]2 dx,

(25b)

k(x) [w(x, t)]2 dx

(25c)

0

Ub =

1 2

L 0

Uspring =

1 2

L 0

G. C. Tsiatas

is the elastic energy due to axial deformation, the elastic energy due to bending deformation and the elastic energy from the Winkler foundation, respectively. Equation (24) using Eqs. (25) and (22) takes the form 1 U = 2

L

 2 EA(x) u 1 ψ1 (x) + u 4 ψ4 (x) dx

0

1 + 2

L

 2 EI(x) u 2 ψ2 (x) + u 3 ψ3 (x) + u 5 ψ5 (x) + u 6 ψ6 (x) dx

0

+

1 2

L k(x) [u 2 ψ2 (x) + u 3 ψ3 (x) + u 5 ψ5 (x) + u 6 ψ6 (x)]2 dx

(26)

0

which after appropriate differentiation gives the following coefficients of the stiffness matrix

L ki j =

EA(x)ψi (x)ψ j (x)dx, i, j = 1, 4

(27a)

0

L  EI(x)ψi (x)ψ j (x) + k(x)ψi (x)ψi (x) dx, i, j = 2, 3, 5, 6 ki j =

(27b)

0

2.5 Kinetic energy and mass matrix The kinetic energy of the beam element can be expressed as 1 K = 2

L

 ˙ t)]2 + [w(x, m(x) [u(x, ˙ t)]2 dx

(28)

0

The above relation after the substitution of Eq. (22) results in 1 K = 2

L

  m(x) [u˙ 1 ψ1 (x) + u˙ 4 ψ4 (x)]2 dx

0

1 + 2

L m(x) [u˙ 2 ψ2 (x) + u˙ 3 ψ3 (x) + u˙ 5 ψ5 (x) + u˙ 6 ψ6 (x)]2 dx

(29)

0

which after appropriate differentiation gives the following coefficients of the mass matrix

L mi j =

m(x)ψi (x)ψ j (x)dx, i, j = 1, 4 and i, j = 2, 3, 5, 6

(30)

0

3 Numerical results On the basis of the numerical procedure presented in the previous section, a MAPLETM code has been written and numerical results for certain uniform and non-uniform beams have been obtained, which illustrate the applicability, effectiveness and accuracy of the proposed method.

A new efficient method to evaluate exact stiffness

L

i

j

hj

k

hk

Fig. 2 Beam with variable mass and stiffness properties Table 1 Stiffness matrix coefficients of the non-uniform beam of Example 3.1 Stiffness matrix

k11

k22

k32

k62

k33

k63

k55

k65

k66

Present (×103 ) Exact (×103 )

2,380.617 2,380.447

51.927 51.925

173.090 173.083

86.546 86.542

668.630 668.611

196.819 196.805

51.927 51.925

−86.546 −86.542

235.912 235.903

Table 2 Mass matrix coefficients of the non-uniform beam of Example 3.1 Mass matrix

m 11

m 22

m 32

m 62

m 33

m 63

m 44

m 55

m 65

m 66

Present

2.102

2.393

2.057

−0.502

2.371

0.751

1.068

1.141

−0.626

0.423

L

hj

i

j

k

hk

Fig. 3 Beam with variable mass and stiffness properties resting on an elastic foundation

3.1 Stiffness and mass matrices of a non-uniform beam In the first example, a non-uniform beam of constant width b and linearly varying height h is considered (see Fig. 2). For this case, the data are E = 33 × 106 kN/m2 , b = 0.5 m, h j = 1.0 m, h k = 0.5 m, L = 5.0 m and ρ = 2.5 ton/m3 . In Table 1, numerical results for the evaluated coefficients of the stiffness matrix (using only one element, N = 1) are presented which are in excellent agreement as compared to the exact ones. Moreover, the coefficients of the corresponding mass matrix are tabulated in Table 2.

3.2 Stiffness and mass matrices of uniform and non-uniform beams resting on elastic foundation In this second example, beams resting on elastic foundation are considered (see Fig. 3). First a uniform beam was analyzed in order to compare the results with exact ones available in literature [5]. For this case, the data are E = 33 × 106 kN/m2 , b = 0.5 m, h j = h k = 1.0 m, L = 5.0 m, k = 100, 000 × b kN/m and ρ = 2.5 ton/m3 . In Table 3, numerical results for the evaluated coefficients of the stiffness matrix using only one element (N = 1) are presented which are in excellent agreement with exact ones [5]. Moreover, the coefficients of the corresponding mass matrix are tabulated in Table 4. Afterwards, the same beam was analyzed changing the height at the kth end (h k = 0.75 m). As the results depend on the number of elements, a convergence test was carried out. The results for the stiffness matrix Table 3 Stiffness matrix coefficients of the uniform beam of Example 3.2 Stiffness matrix

k11

k22

k32

k62

k33

k63

k55

k65

k66

Present (×103 ) Exact (×103 )

3,300 3,300

222.870 222.870

393.391 393.391

293.269 293.269

1,157.313 1,157.312

507.490 507.490

222.870 222.870

−393.391 −393.391

1,157.313 1,157.313

G. C. Tsiatas

Table 4 Mass matrix coefficients of the uniform beam of Example 3.2 Mass matrix

m 11

m 22

m 32

m 62

m 33

m 63

m 44

m 55

m 65

m 66

Present

2.083

2.224

1.535

−0.871

1.380

−1.012

2.083

2.224

−1.535

1.380

Table 5 Stiffness matrix coefficients of the non-uniform beam of Example 3.2 Stiffness matrix (×103 )

Present N =1 Present (×103 ) N = 3

k11

k22

k32

k62

k33

k63

k55

k65

k66

2,867.750 2,867.749

184.304 184.304

324.938 324.938

156.061 156.061

96.661 96.661

317.008 317.007

170.312 170.312

−238.070 −238.070

618.061 618.061

Table 6 Mass matrix coefficients of the non-uniform beam of Example 3.2 Mass matrix

m 11

m 22

m 32

m 62

m 33

m 63

m 44

m 55

m 65

m 66

Present

2.088

2.213

1.650

−0.632

1.638

−0.826

1.568

1.611

−0.998

0.794

Table 7 First four non-dimensional natural frequencies ω¯ n for uniform cantilever beams of Example 3.3 n

Weaver et al. [14]

Huang and Li [13] (N = 10)

Present (N = 5)

Present (N = 10)

1 2 3 4

3.5160 22.0345 61.6972 120.9019

3.5160 22.0345 61.6972 120.9019

3.5160 22.0455 61.9188 122.3197

3.5160 22.0352 61.7129 121.0171

coefficients are shown in Table 5 for one (N = 1) and three elements (N = 3). It can be easily seen that only one element is sufficient in order to gain convergence solution. Moreover, the corresponding mass matrix of the non-uniform beam is tabulated in Table 6.

3.3 Free vibration of uniform and non-uniform beams The final example is devoted to the vibration of uniform and non-uniform beams subject to various boundary conditions. The first beam studied √ was a uniform cantilever beam. The evaluated first four non-dimensional natural frequencies ω¯ n = ωn L 2 ρ A/EI are shown in Table 7 as compared to those from an integral equation solution [13] and the exact ones [14]. The accuracy of the obtained results is very good, and the convergence is rapid. As N increases from 5 to 10, the numerical results of the first two natural frequencies are identical to the exact ones up to three decimals, which indicate the efficiency of the proposed method. The accuracy, however, drops for the third and fourth natural frequencies. In particular for the fourth natural frequency, the deviation from the exact value is 1.2 % for N = 5 and drops to 0.1 % for N = 10 elements. Afterwards, a non-uniform beam with a cross-section of constant width and linearly varying height, i.e., 3 A/A0 = 1 + √ aξ, I /I0 = (1 + aξ ) was analyzed. The corresponding non-dimensional natural frequencies ω¯ n = ωn L 2 ρ A0 /EI0 are also calculated and listed in Table 8 for the clamped–pinned beam and in Table 9 for the clamped–clamped beam. In the same tables, results derived previously by the FEM [15] and an integral equation solution [13] are also presented. By comparison, the results of the proposed method agree very well with the existing ones. Table 8 First three non-dimensional natural frequencies ω¯ n for non-uniform C–P beams of Example 3.3 a

n

Cortinez and Laura [15]

Huang and Li [13] (N = 10)

Present (N = 5)

Present (N = 10)

−0.1

1 2 3 1 2 3

14.92 – – 15.997 – –

14.849 47.637 99.172 15.969 52.237 109.202

14.853 47.765 100.247 15.973 52.374 110.362

14.849 47.645 99.245 15.969 52.246 109.282

+0.1

A new efficient method to evaluate exact stiffness

Table 9 First three non-dimensional natural frequencies ω¯ n for non-uniform C–C beams of Example 3.3 a

n

Cortinez and Laura [15]

Huang and Li [13] (N = 10)

Present (N = 5)

Present (N = 10)

−0.1

1 2 3 1 2 3

– – – 23.521 – –

21.241 58.550 114.780 23.480 64.721 126.878

21.253 58.784 116.368 23.492 64.980 128.633

21.242 58.565 114.894 23.480 64.738 127.003

+0.1

4 Conclusions In this work, a new efficient method was developed for the evaluation of exact stiffness and mass matrices. The method was applied successfully to the problem of non-uniform Bernoulli–Euler beams resting on an elastic Winkler foundation. The main conclusions that can be drawn from this investigation are as follows: • • • •

The method is general since the non-uniformity may result either from variable cross-section or from inhomogeneous linearly elastic material without any change in the solution procedure. The method can be easily modified to develop beam elements including shear deformation effects. The numerical method is efficient since a small number of elements is adequate to obtain accurate results for non-uniform beams. The examples presented indicate the versatility of the new method, the simplicity it offers in the modeling of uniform and non-uniform beams and finally its reliability, due to the fact that it is based on the exact solution of the differential governing equations.

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