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Analysis of deep beams and shear walls by finite strip method with C0 continuous displacement functions
Thin-Walled Structures 32 (1998) 289–303
Analysis of deep beams and shear walls by finite strip method with C 0 continuous displacement functions Y.K. Cheung*, F.T.K. Au, D.Y. Zheng Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, PR China
Abstract This paper presents a new finite strip method for the analysis of deep beams and shear walls. The essence of the method lies in the adoption of displacement functions possessing the right amount of continuity at the ends as well as at locations of abrupt changes of thickness. The concept of periodic extension in Fourier series is utilized to improve the accuracy of the stresses at the strip ends. The equilibrium conditions at locations of abrupt changes of thickness are taken into account by the incorporation of piecewise linear correction functions. As these displacement functions are built up from harmonic functions with appropriate corrections, they possess both the advantages of fast convergence of harmonic functions as well as appropriate order of continuity. Numerical results also show that the method is versatile, efficient and accurate. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Deep beam; Shear wall; Finite strip method; C 0-continuous function
1. Introduction In the analysis of plane elasticity problems by the finite strip method, the choice of longitudinal interpolation functions is of utmost importance. In the semi-analytical finite strip method [1], the interpolation functions have to satisfy a priori the boundary conditions in the longitudinal direction and they are normally the eigen-functions of vibration or stability of an Euler beam. In the spline finite strip method [2], C2* Corresponding author. 0263-8231/98/$ - see front matter 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 9 8 ) 0 0 0 2 4 - X
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Y.K. Cheung et al. / Thin-Walled Structures 32 (1998) 289–303
continuous B3-spline functions are chosen instead. However, these methods are not capable of dealing with problems involving abrupt changes of thickness because of the excessive continuity inherent in the longitudinal interpolation functions. More recently, Cheung and Kong [3] developed a new set of displacement functions for the analysis of two-dimensional elasticity problems. Computed shape functions are used longitudinally whereas Lagrangian shape functions are adopted in the transverse direction. In this way, C 0 continuity across all boundaries between adjacent strips is maintained. A strip with abrupt changes in rigidity is made up of component strips stitched together at the ends. This paper presents a new finite strip method for the analysis of deep beams and shear walls. The use of finite strips with abrupt changes in rigidity is made possible by displacement functions possessing the right amount of continuity at the ends as well as at locations of abrupt changes of thickness. The concept of the periodic extension of Fourier series is extended beyond the normal half-range expansions to ensure that the first derivatives and hence the stresses at the end points of the halfrange are well defined. In addition, piecewise linear augmenting functions are superimposed on the basic harmonic functions making the resulting interpolation function C 0-continuous at locations of abrupt changes of thickness, so that the equilibrium conditions there are satisfied. The application of similar approaches to beams with abrupt changes of cross-sections has been rather successful [4]. The chosen interpolation functions therefore possess both the advantages of fast convergence of harmonic functions and the appropriate order of continuity at the locations of abrupt changes of sectional properties. The method allows the extensive use of matrix notations and programming is rather straightforward. The present method can cope with strips with abrupt changes of rigidity as well as rectangular domains with re-entrant corners. A few numerical examples are given to demonstrate the versatility, efficiency and accuracy of the method. 2. Basic harmonic functions An arbitrary function f(y) continuous over the interval [0,l] may be written in terms of the basic trigonometric functions as f (y) ⫽
a0 ⫹ 2
冘冋 ⬁
ancos
n⫽1
ny ny ⫹ bnsin l l
册
(1)
in which periodic extension is implicit [5]. This form of periodic extension is superior to the normal form of half-range expansions involving either odd or even components, as the function and its derivatives are well defined at the end points of the interval. In the analysis of cantilever deep beams and shear walls, it is often more convenient to adopt an equivalent set of displacement functions 兵Y¯m(y),m ⫽ 1,2,…其 with zero values at the base, i.e.
冋
Y¯2n ⫺ 1(y) ⫽ 1 ⫺ cos
ny ny ⫹ sin l l
册
n ⫽ 1,2,…,r/2
(2)
Y.K. Cheung et al. / Thin-Walled Structures 32 (1998) 289–303
冋
Y¯2n(y) ⫽ 1 ⫺ cos
ny ny ⫺ sin l l
册
n ⫽ 1,2,…,r/2
291
(3)
This set of functions is a complete set to describe an arbitrary function f(y) in which f(y)兩y=0 ⫽ 0. It is important to note that, unlike vibration functions obtained from cantilever beams, the slope of the resulting displacement function may be nonzero at the starting point. This is essential to structures in which shear effect is significant.
3. C 0 modified harmonic function set A shear wall can be sub-divided into finite strips extending from the base to the top. Fig. 1 shows a typical strip of width b comprising s segments of different thicknesses t ⫽ t1,t2,…,ts with step change of thickness at locations y ⫽ y2,y3,…,ys. The displacements u and v in x and y directions, respectively, at an arbitrary point (x,y) within the strip can be written in series form as
冘 冘 r
u(x,y) ⫽
Ymu(y)M(x)um
(4)
Ymv(y)M(x)vm
(5)
m⫽1 r
v(x,y) ⫽
m⫽1
where Ymu(y) and Ymv(y) are the mth terms of the series part of the displacement function, M(x) is a row vector containing the shape functions, and um and vm are vectors containing the corresponding displacement parameters associated with the nodal lines. They are further elaborated below in Eqs. (6)–(10). Ymu(y) ⫽ Y¯m(y)
(6)
Ymv(y) ⫽ Y¯m(y) ⫹ Y˜m(y)
(7)
Fig. 1. A cantilever strip comprising s segments of thicknesses t1,t2,…, ts and with step changes in thickness at locations y2,y3,…,ys.
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Y.K. Cheung et al. / Thin-Walled Structures 32 (1998) 289–303
冋
M(x) ⫽ 1 ⫺
册
x x b b
(8)
um ⫽ [u1 u2]Tm
(9)
vm ⫽ [v1 v2]Tm
(10)
The functions Ymu(y) for displacements in the x direction can be simply taken as the functions Y¯m(y) defined in Eqs. (2) and (3). However, the functions Ymv(y) used to describe displacements in y direction should be capable of satisfying the equilibrium conditions at step change of thickness, and hence piecewise linear augmenting functions Y˜m(y) are incorporated to satisfy C 0-continuity conditions at locations y ⫽ y2,y3,…,ys. In a plane stress problem, the equilibrium condition at a step change of thickness can be written as Y m⬘ (t)兩y ⫽ yj ⫺ 0 ⫽
tj
Y ⬘m(y)兩y ⫽ yj ⫹ 0 (j ⫽ 2,3,…,s)
tj ⫺ 1
(11)
The properties of the augmenting function Y˜m(y) can be obtained from Eqs. (8) and (11), i.e. Y˜ m⬘ (y)兩y ⫽ yj ⫺ 0 ⫽
tj tj ⫺ 1
Y˜ ⬘m(y)兩y ⫽ yj ⫹ 0 ⫹
冉
tj tj ⫺ 1
冊
⫺ 1 Y¯ ⬘m(y)兩y ⫽ yj
(j
(12)
⫽ 2,3,…,s) If we set f j ⫽ Y˜m(y)兩y ⫽ yj and hk ⫽ yk ⫹ 1 ⫺ yk, then the augmenting function Y˜m(y) can be written as:
冘
s⫹1
Y˜m(y) ⫽
fj lj (y)
(13)
j⫽1
where y ⫺ yj ⫺ 1 yj ⫺ yj ⫺ 1
l (y) ⫽ y ⫺ y y ⫺0y
j⫹1
j
j
j⫹1
yj ⫺ 1 ⱕ y ⱕ yj (omitted if j ⫽ 1) yj ⱕ y ⱕ yj ⫹ 1 (omitted if j ⫽ s ⫹ 1)
(14)
y ⰻ [yj ⫺ 1,yj ⫹ 1]
Introducing the parameters j and j defined as follows
j ⫽
tj tj ⫺ 1
(j ⫽ 2,3,…,s)
(15)
Y.K. Cheung et al. / Thin-Walled Structures 32 (1998) 289–303
j ⫽ (j ⫺ 1)Y¯ ⬘m(y)兩y ⫽ yj
293
(j ⫽ 2,3,…,s)
(16)
Eq. (12) can be written as − hj fj − 1 + [hj + hj − 1j ]fj − hj − 1j fj + 1 = hj − 1hj j
(j = 2,3,…,s)
(17)
The augmenting function is also prescribed to be zero at the two ends, i.e. f 1 ⫽ fs ⫹ 1 ⫽ 0
(18)
Eqs. (17) and (18) represent (s ⫹ 1) conditions whereby the augmenting function Y˜m(y) can be uniquely determined.
4. Finite strip formulation The displacement functions given in Eqs. (4) and (5) can be expressed in matrix form as
冘 r
U(x,y) ⫽
Nm(x,y)␦m
(19)
m⫽1
where U(x,y) is the vector containing displacements at the point (x,y), Nm(x,y) is the mth the shape function matrix and ␦m is the displacement vector associated with it. Their explicit forms are given below: U(x,y) ⫽
再 冎 u(x,y)
(20)
v(x,y)
␦m ⫽ [u1v1u2v2]Tm
冊 冉
Nm(x,y) ⫽
1⫺
x Y (y) b mu 0
(21) 0
冉 冊
x Y (y) b mu
x 1 ⫺ Ymv(y) b
0
0
x Y (y) b mv
(22)
Following the normal formulation procedures, the strain vector ⑀(x,y) can be written in terms of the strain matrices Bm(x,y) as
冘 r
⑀(x,y) ⫽ Diff·U(x,y) ⫽ Diff·
m⫽1
冘 r
Nm(x,y)␦m ⫽
Bm(x,y)␦m
(23)
m⫽1
where the explicit forms of the differential operator Diff and the strain matrix Bm(x,y) are
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∂ ∂x
∂ ∂y
Diff ⫽ 0
0
∂ ∂y
∂ ∂x
(24)
1 Y b mu
冉 冊 冉 冊 ⫺
Bm(x,y) ⫽
0
1⫺
x ⬘ Y b mu
0
1⫺ ⫺
1 Y b mu
x ⬘ Y b mv
1 Y b mv
0
0
x ⬘ Y b mv
(25)
x ⬘ 1 Y Y b mu b mv
The equilibrium equation can be written as
冘 r
Kmn␦n ⫽ Fm m ⫽ 1,2,…,r
(26)
n⫽1
where Kmn is a stiffness sub-matrix and Fm is a load sub-vector. The stiffness submatrix Kmn is obtained from integrating over the volume V of the strip, i.e.
冕
Kmn ⫽ BTmDBndV
(27)
V
The elasticity matrix D for an orthotropic material used in plane stress problems is
冤 冥 k1 k2 0
D ⫽ k2 k3 0
(28)
0 0 k4
where the elements are expressed in terms of the moduli of elasticity Ex and Ey, the shear modulus Gxy and the Poisson’s ratio x and y k1 ⫽
xEy Ex Ey , k2 ⫽ , k3 ⫽ , k ⫽ Gxy 1 ⫺ xy 1 ⫺ xy 1 ⫺ xy 4
(29–32)
Substituting Eqs. (25) and (28) into Eq. (27), the explicit form of the stiffness sub-matrix Kmn can be worked out as
Y.K. Cheung et al. / Thin-Walled Structures 32 (1998) 289–303
3k1I1 + k4b2I2 3b
Kmn =
−
k2I5 + k4I6 2
6k1I1 + k4b2I2 − 6b −
k2I5 − k4I6 2
295
k 2 I3 − k 4 I4 2
6k1I1 + k4b2I2 6b
−
k3b2I7 + 3k4I8 3b
k2I5 − k4I6 2
k3b2I7 − 6k4I8 6b
k2I3 − k4I4 2
3k1I1 + k4b2I2 3b
k2I3 − k4I4 2
k3b2I7 − 6k4I8 6b
k2I5 + k4I6 2
−
k2I3 + k4I4 2
−
k3b2I7 + 3k4I8 3b
(33) where
冕
冕
冕
冕
(34–37)
冕
冕
冕
冕
(38–41)
l
l
l
l
I1 = tYmuYnudy, I2 = tY ⬘muY ⬘nudy, I3 = tYmuY ⬘nvdy, I4 = tY ⬘muYnvdy, 0 l
0 l
0 l
0 l
I5 = tY ⬘mvYnudy, I6 = tYmvY ⬘nudy, I7 = tY ⬘mvY ⬘nvdy, I8 = tYmvYnvdy, 0
0
0
0
5. Numerical examples A number of numerical examples are given to demonstrate the versatility of the present method. Results obtained by the present method are compared with solutions published before if available or solutions obtained by finite element method [6]. As some of the examples published before were presented using imperial units, these units are omitted altogether for simplicity. 5.1. Example 1: rectangular shear wall supported on two wide columns Fig. 2 shows a rectangular shear wall supported on two wide columns of the same thickness. The material is isotropic with Young’s modulus E of 471 000 and Poisson’s ratio of 0.375. A horizontal point load of 20 is applied to its top left corner. This structure can be considered as a shear wall with a large rectangular opening at the base. This problem has been studied by various researchers [3,7]. The shear wall was discretized into 16 strips of equal width and solved by the present method using 16 terms. The first category of strips are the four strips going through the two wide columns, and they run from the base to the top. The other category of 12 strips are those above the opening. The boundary conditions at the three edges adjacent to the opening are natural boundary conditions. As this formulation is of the Rayleigh– Ritz type, the natural boundary conditions are therefore automatically dealt with. However, in the formulation of the stiffness matrices for the 12 strips above the
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Fig. 2.
Shear wall supported on two wide columns.
opening, integration is only carried out over the solid height above the opening. The problem was also solved by the finite element method using 2528 QM6 incompatible elements [6]. The vertical stresses over the horizontal cross-section at y ⫽ 4 are shown in Fig. 3 and compared with the finite element solution [6] as well as those from finite strips with computed shape functions [3]. Sharp variation of stresses is observed as the cross-section is close to the wall-frame interfaces. Very good agreement is observed between the present results and the reference results. 5.2. Example 2: deep cantilever beam with step change in thickness Fig. 4 shows a deep cantilever beam made of isotropic material with step change in thickness. The Young’s modulus E is 1.0 and Poisson’s ratio is 0.15. The beam supports a downward unit point load at the top right corner. Two schemes of discretization were tried. The deep beam was respectively discretized into four and 10 strips of equal widths. The number of terms used was varied to study the convergence of the present method. The deflections at two points of the tip are summarized in Table 1 and compared with the reference solution obtained using 1800 QM6 incompatible elements [6]. Very good agreement is observed. Incidentally, the results obtained from the coarse mesh comprising four strips are only 1.6% less than the
Y.K. Cheung et al. / Thin-Walled Structures 32 (1998) 289–303
297
Fig. 3. Shear wall supported on two wide columns, distribution of vertical stresses (y) over the horizontal cross section at y = 4.
Fig. 4.
Cantilever deep beam with step change in thickness.
reference solution. The stresses computed using 10 strips and 10 terms are presented in Figs. 5 and 6. In particular, Fig. 5 shows the bending stresses at the vertical crosssections just adjacent to the step change in thickness. The bending stresses along the centre line of the lowest strip (i.e. at x ⫽ 9.5) are shown in Fig. 6. Both figures show very good agreement between the present results and the finite element results [6].
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Y.K. Cheung et al. / Thin-Walled Structures 32 (1998) 289–303
Table 1 Deep cantilever beam with step change in thickness, deflections at two points of the tip Scheme used in the present method 4 strips, 12 terms 4 strips, 14 terms 4 strips, 16 terms 4 strips, 18 terms 4 strips, 20 terms 10 strips, 12 terms 10 strips, 14 terms 10 strips, 16 terms 10 strips, 18 terms 10 strips, 20 terms FEM [6]
Deflection at x = 5, y = 20
212.88 212.94 212.97 212.99 213.01 215.54 215.61 215.63 215.65 215.68 216.31
Deflection at x = 10, y = 20
208.86 208.91 208.94 208.96 208.97 211.58 211.67 211.68 211.71 211.74 212.36
Fig. 5. Cantilever deep beam with step change in thickness, bending stresses (y) adjacent to step change in thickness.
5.3. Example 3: deep cantilever beam with step change in depth Fig. 7 shows a deep cantilever beam of unit thickness but with step change in depth. The material is isotropic with a Young’s modulus E of 1.0 and Poisson’s ratio of 0.15. The beam is subjected to a unit point load at the top of the cantilever tip. It is discretized into 10 strips of equal width. Twenty terms were used in the present analysis. The beam was also analysed by the finite element method [6] using 3200 QM6 incompatible elements. Figs. 8 and 9 show the bending stresses at the root (y ⫽ 0) and the section at y ⫽ 9.5. The present method is capable of predicting stress concentration around the re-entrant corner, and good agreement is observed between the present results and the reference solution.
Y.K. Cheung et al. / Thin-Walled Structures 32 (1998) 289–303
Fig. 6.
299
Deep cantilever beam with step change in thickness, bending stresses (y) along the line x = 9.5.
Fig. 7.
Cantilever deep beam with step change in depth.
5.4. Example 4: asymmetrical coupled shear wall with step change in thickness Fig. 10(a) shows an asymmetrical coupled shear wall with step change in thickness previously studied by Chan and Cheung [8] by higher order finite elements. The wall is subjected to a unit horizontal uniformly distributed load at the left side. The wall is assumed to be made of isotropic material having Young’s modulus E of 463 000 and Poisson’s ratio of 0.0. In the analysis of coupled shear walls, the spandrel beams and openings between the coupled shear walls are very often represented by an orthotropic continuum with equivalent elastic properties. Referring to Fig. 10(b), the equivalent Young’s moduli of the continuum in the x and y direc-
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Y.K. Cheung et al. / Thin-Walled Structures 32 (1998) 289–303
Fig. 8. Cantilever deep beam with step change in depth, bending stresses (y) at the base (y = 0).
Fig. 9.
Cantilever deep beam with step change in depth, bending stresses (y) at y = 9.5.
tions, as well as the equivalent shear rigidity of the continuum [8] are given, respectively, by Ex ⫽
d d E E, Ey ⫽ 0, G ⫽ . 2 h h c⬘ ⫹ 2.4 d
冉冊
where c⬘ ⫽ c ⫹ d is the effective span length of the spandrel beam. The adjustment to the span length of the spandrel beam is to allow for the fact that the rigid-end condition could not possibly occur immediately at the junction of the wall and beam.
Y.K. Cheung et al. / Thin-Walled Structures 32 (1998) 289–303
301
Fig. 10. (a) Asymmetrical coupled shear wall of step change in thickness; (b) Geometric dimensions of the openings and spandrel beams.
The equivalent properties of the continuum are therefore calculated as Ex ⫽ 115 750, Ey ⫽ 0 and G ⫽ 2251.9. In the analysis by the present method, the shear wall was first discretized into 14 strips of equal widths. In addition, a very coarse mesh comprising only three strips (one for the equivalent continuum and two for the walls) was tried. The number of terms used in the analysis ranges from 12 to 20. The problem was also solved using 1512 QM6 incompatible finite elements [6]. Table 2 shows the convergence of the deflection at the top right corner of the shear wall Table 2 Asymmetrical coupled shear wall with step change in thickness, deflection at top right corner Scheme used in the present method 3 strips, 12 terms 3 strips, 14 terms 3 strips, 16 terms 3 strips, 18 terms 3 strips, 20 terms 14 strips, 12 terms 14 strips, 14 terms 14 strips, 16 terms 14 strips, 18 terms 14 strips, 20 terms FEM [6]
Deflection at x = 7, y = 21 9.6485 9.6489 9.6492 9.6494 9.6495 9.6887 9.6909 9.6922 9.6928 9.6936 9.6974
× × × × × × × × × × ×
10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3
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Y.K. Cheung et al. / Thin-Walled Structures 32 (1998) 289–303
Fig. 11. Coupled shear wall with step change in thickness, horizontal deflections along the right edge (x = 7).
as compared with the finite element results [6]. The horizontal deflection of the right side of the wall is shown in Fig. 11. The vertical stresses at the horizontal cross section at y ⫽ 3.375 are plotted in Fig. 12. The vertical stresses along the centre line of the rightmost strip (i.e. at x ⫽ 6.75) are shown in Fig. 13. It is observed that results from the present method agree very well with those obtained by the finite element method [6] and the higher order finite elements [8]. The accuracy of the deflections obtained from the very coarse mesh is also striking.
Fig. 12. Coupled shear wall with step change in thickness, bending stress (y) at horizontal section at y = 3.375.
Y.K. Cheung et al. / Thin-Walled Structures 32 (1998) 289–303
303
Fig. 13. Coupled shear wall with step change in thickness, bending stresses (y) along vertical line at x = 6.75.
6. Conclusions A new finite strip method has been developed for the analysis of deep beams and shear walls. The use of piecewise linear augmenting functions in conjunction with the basic harmonic functions results in displacement functions with the appropriate order of continuity at the locations of abrupt changes of sectional properties. It therefore allows the use of continuous finite strips with step changes in thickness. The present method can also cope with rectangular domains with re-entrant corners. A few numerical examples have been given to demonstrate the versatility, efficiency and accuracy of the method.
References [1] Cheung YK. Finite strip method in structural analysis. Oxford: Pergamon Press, 1977. [2] Fan SC. Spline finite strip in structural analysis. PhD thesis, The University of Hong Kong, Hong Kong, 1982. [3] Cheung YK, Kong J. An accurate finite strip for analyzing deep beams and shear walls. Commun Numer Methods Engng 1995;11:643–53. [4] Au FTK, Zheng DY, Cheung YK. Vibration and stability of non-uniform beams with abrupt changes of cross-section by using C1 modified beam vibration functions. Applied Mathematical Modeling, in press. [5] Liu JY, Zheng DY, Mei ZJ. Practical integral transforms in engineering. Huazhong University of Science and Technology Press, 1995. [6] Cosmos User Manual. Structural research and analysis cooperation. Santa Monica, CA 90404, USA, 1991. [7] Cheung YK, Swaddiwudhipong S. Analysis of frame shear wall structures using finite strip elements. Proc Instn Civ Engrs 1978;65:517–35. [8] Chan HC, Cheung YK. Analysis of shear walls using higher order finite elements. Building Environ 1979;14:217–24.
Analysis of deep beams and shear walls by finite strip method with C 0 continuous displacement functions Y.K. Cheung*, F.T.K. Au, D.Y. Zheng Department of Civil Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, PR China
Abstract This paper presents a new finite strip method for the analysis of deep beams and shear walls. The essence of the method lies in the adoption of displacement functions possessing the right amount of continuity at the ends as well as at locations of abrupt changes of thickness. The concept of periodic extension in Fourier series is utilized to improve the accuracy of the stresses at the strip ends. The equilibrium conditions at locations of abrupt changes of thickness are taken into account by the incorporation of piecewise linear correction functions. As these displacement functions are built up from harmonic functions with appropriate corrections, they possess both the advantages of fast convergence of harmonic functions as well as appropriate order of continuity. Numerical results also show that the method is versatile, efficient and accurate. 1998 Elsevier Science Ltd. All rights reserved. Keywords: Deep beam; Shear wall; Finite strip method; C 0-continuous function
1. Introduction In the analysis of plane elasticity problems by the finite strip method, the choice of longitudinal interpolation functions is of utmost importance. In the semi-analytical finite strip method [1], the interpolation functions have to satisfy a priori the boundary conditions in the longitudinal direction and they are normally the eigen-functions of vibration or stability of an Euler beam. In the spline finite strip method [2], C2* Corresponding author. 0263-8231/98/$ - see front matter 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 2 6 3 - 8 2 3 1 ( 9 8 ) 0 0 0 2 4 - X
290
Y.K. Cheung et al. / Thin-Walled Structures 32 (1998) 289–303
continuous B3-spline functions are chosen instead. However, these methods are not capable of dealing with problems involving abrupt changes of thickness because of the excessive continuity inherent in the longitudinal interpolation functions. More recently, Cheung and Kong [3] developed a new set of displacement functions for the analysis of two-dimensional elasticity problems. Computed shape functions are used longitudinally whereas Lagrangian shape functions are adopted in the transverse direction. In this way, C 0 continuity across all boundaries between adjacent strips is maintained. A strip with abrupt changes in rigidity is made up of component strips stitched together at the ends. This paper presents a new finite strip method for the analysis of deep beams and shear walls. The use of finite strips with abrupt changes in rigidity is made possible by displacement functions possessing the right amount of continuity at the ends as well as at locations of abrupt changes of thickness. The concept of the periodic extension of Fourier series is extended beyond the normal half-range expansions to ensure that the first derivatives and hence the stresses at the end points of the halfrange are well defined. In addition, piecewise linear augmenting functions are superimposed on the basic harmonic functions making the resulting interpolation function C 0-continuous at locations of abrupt changes of thickness, so that the equilibrium conditions there are satisfied. The application of similar approaches to beams with abrupt changes of cross-sections has been rather successful [4]. The chosen interpolation functions therefore possess both the advantages of fast convergence of harmonic functions and the appropriate order of continuity at the locations of abrupt changes of sectional properties. The method allows the extensive use of matrix notations and programming is rather straightforward. The present method can cope with strips with abrupt changes of rigidity as well as rectangular domains with re-entrant corners. A few numerical examples are given to demonstrate the versatility, efficiency and accuracy of the method. 2. Basic harmonic functions An arbitrary function f(y) continuous over the interval [0,l] may be written in terms of the basic trigonometric functions as f (y) ⫽
a0 ⫹ 2
冘冋 ⬁
ancos
n⫽1
ny ny ⫹ bnsin l l
册
(1)
in which periodic extension is implicit [5]. This form of periodic extension is superior to the normal form of half-range expansions involving either odd or even components, as the function and its derivatives are well defined at the end points of the interval. In the analysis of cantilever deep beams and shear walls, it is often more convenient to adopt an equivalent set of displacement functions 兵Y¯m(y),m ⫽ 1,2,…其 with zero values at the base, i.e.
冋
Y¯2n ⫺ 1(y) ⫽ 1 ⫺ cos
ny ny ⫹ sin l l
册
n ⫽ 1,2,…,r/2
(2)
Y.K. Cheung et al. / Thin-Walled Structures 32 (1998) 289–303
冋
Y¯2n(y) ⫽ 1 ⫺ cos
ny ny ⫺ sin l l
册
n ⫽ 1,2,…,r/2
291
(3)
This set of functions is a complete set to describe an arbitrary function f(y) in which f(y)兩y=0 ⫽ 0. It is important to note that, unlike vibration functions obtained from cantilever beams, the slope of the resulting displacement function may be nonzero at the starting point. This is essential to structures in which shear effect is significant.
3. C 0 modified harmonic function set A shear wall can be sub-divided into finite strips extending from the base to the top. Fig. 1 shows a typical strip of width b comprising s segments of different thicknesses t ⫽ t1,t2,…,ts with step change of thickness at locations y ⫽ y2,y3,…,ys. The displacements u and v in x and y directions, respectively, at an arbitrary point (x,y) within the strip can be written in series form as
冘 冘 r
u(x,y) ⫽
Ymu(y)M(x)um
(4)
Ymv(y)M(x)vm
(5)
m⫽1 r
v(x,y) ⫽
m⫽1
where Ymu(y) and Ymv(y) are the mth terms of the series part of the displacement function, M(x) is a row vector containing the shape functions, and um and vm are vectors containing the corresponding displacement parameters associated with the nodal lines. They are further elaborated below in Eqs. (6)–(10). Ymu(y) ⫽ Y¯m(y)
(6)
Ymv(y) ⫽ Y¯m(y) ⫹ Y˜m(y)
(7)
Fig. 1. A cantilever strip comprising s segments of thicknesses t1,t2,…, ts and with step changes in thickness at locations y2,y3,…,ys.
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冋
M(x) ⫽ 1 ⫺
册
x x b b
(8)
um ⫽ [u1 u2]Tm
(9)
vm ⫽ [v1 v2]Tm
(10)
The functions Ymu(y) for displacements in the x direction can be simply taken as the functions Y¯m(y) defined in Eqs. (2) and (3). However, the functions Ymv(y) used to describe displacements in y direction should be capable of satisfying the equilibrium conditions at step change of thickness, and hence piecewise linear augmenting functions Y˜m(y) are incorporated to satisfy C 0-continuity conditions at locations y ⫽ y2,y3,…,ys. In a plane stress problem, the equilibrium condition at a step change of thickness can be written as Y m⬘ (t)兩y ⫽ yj ⫺ 0 ⫽
tj
Y ⬘m(y)兩y ⫽ yj ⫹ 0 (j ⫽ 2,3,…,s)
tj ⫺ 1
(11)
The properties of the augmenting function Y˜m(y) can be obtained from Eqs. (8) and (11), i.e. Y˜ m⬘ (y)兩y ⫽ yj ⫺ 0 ⫽
tj tj ⫺ 1
Y˜ ⬘m(y)兩y ⫽ yj ⫹ 0 ⫹
冉
tj tj ⫺ 1
冊
⫺ 1 Y¯ ⬘m(y)兩y ⫽ yj
(j
(12)
⫽ 2,3,…,s) If we set f j ⫽ Y˜m(y)兩y ⫽ yj and hk ⫽ yk ⫹ 1 ⫺ yk, then the augmenting function Y˜m(y) can be written as:
冘
s⫹1
Y˜m(y) ⫽
fj lj (y)
(13)
j⫽1
where y ⫺ yj ⫺ 1 yj ⫺ yj ⫺ 1
l (y) ⫽ y ⫺ y y ⫺0y
j⫹1
j
j
j⫹1
yj ⫺ 1 ⱕ y ⱕ yj (omitted if j ⫽ 1) yj ⱕ y ⱕ yj ⫹ 1 (omitted if j ⫽ s ⫹ 1)
(14)
y ⰻ [yj ⫺ 1,yj ⫹ 1]
Introducing the parameters j and j defined as follows
j ⫽
tj tj ⫺ 1
(j ⫽ 2,3,…,s)
(15)
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j ⫽ (j ⫺ 1)Y¯ ⬘m(y)兩y ⫽ yj
293
(j ⫽ 2,3,…,s)
(16)
Eq. (12) can be written as − hj fj − 1 + [hj + hj − 1j ]fj − hj − 1j fj + 1 = hj − 1hj j
(j = 2,3,…,s)
(17)
The augmenting function is also prescribed to be zero at the two ends, i.e. f 1 ⫽ fs ⫹ 1 ⫽ 0
(18)
Eqs. (17) and (18) represent (s ⫹ 1) conditions whereby the augmenting function Y˜m(y) can be uniquely determined.
4. Finite strip formulation The displacement functions given in Eqs. (4) and (5) can be expressed in matrix form as
冘 r
U(x,y) ⫽
Nm(x,y)␦m
(19)
m⫽1
where U(x,y) is the vector containing displacements at the point (x,y), Nm(x,y) is the mth the shape function matrix and ␦m is the displacement vector associated with it. Their explicit forms are given below: U(x,y) ⫽
再 冎 u(x,y)
(20)
v(x,y)
␦m ⫽ [u1v1u2v2]Tm
冊 冉
Nm(x,y) ⫽
1⫺
x Y (y) b mu 0
(21) 0
冉 冊
x Y (y) b mu
x 1 ⫺ Ymv(y) b
0
0
x Y (y) b mv
(22)
Following the normal formulation procedures, the strain vector ⑀(x,y) can be written in terms of the strain matrices Bm(x,y) as
冘 r
⑀(x,y) ⫽ Diff·U(x,y) ⫽ Diff·
m⫽1
冘 r
Nm(x,y)␦m ⫽
Bm(x,y)␦m
(23)
m⫽1
where the explicit forms of the differential operator Diff and the strain matrix Bm(x,y) are
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∂ ∂x
∂ ∂y
Diff ⫽ 0
0
∂ ∂y
∂ ∂x
(24)
1 Y b mu
冉 冊 冉 冊 ⫺
Bm(x,y) ⫽
0
1⫺
x ⬘ Y b mu
0
1⫺ ⫺
1 Y b mu
x ⬘ Y b mv
1 Y b mv
0
0
x ⬘ Y b mv
(25)
x ⬘ 1 Y Y b mu b mv
The equilibrium equation can be written as
冘 r
Kmn␦n ⫽ Fm m ⫽ 1,2,…,r
(26)
n⫽1
where Kmn is a stiffness sub-matrix and Fm is a load sub-vector. The stiffness submatrix Kmn is obtained from integrating over the volume V of the strip, i.e.
冕
Kmn ⫽ BTmDBndV
(27)
V
The elasticity matrix D for an orthotropic material used in plane stress problems is
冤 冥 k1 k2 0
D ⫽ k2 k3 0
(28)
0 0 k4
where the elements are expressed in terms of the moduli of elasticity Ex and Ey, the shear modulus Gxy and the Poisson’s ratio x and y k1 ⫽
xEy Ex Ey , k2 ⫽ , k3 ⫽ , k ⫽ Gxy 1 ⫺ xy 1 ⫺ xy 1 ⫺ xy 4
(29–32)
Substituting Eqs. (25) and (28) into Eq. (27), the explicit form of the stiffness sub-matrix Kmn can be worked out as
Y.K. Cheung et al. / Thin-Walled Structures 32 (1998) 289–303
3k1I1 + k4b2I2 3b
Kmn =
−
k2I5 + k4I6 2
6k1I1 + k4b2I2 − 6b −
k2I5 − k4I6 2
295
k 2 I3 − k 4 I4 2
6k1I1 + k4b2I2 6b
−
k3b2I7 + 3k4I8 3b
k2I5 − k4I6 2
k3b2I7 − 6k4I8 6b
k2I3 − k4I4 2
3k1I1 + k4b2I2 3b
k2I3 − k4I4 2
k3b2I7 − 6k4I8 6b
k2I5 + k4I6 2
−
k2I3 + k4I4 2
−
k3b2I7 + 3k4I8 3b
(33) where
冕
冕
冕
冕
(34–37)
冕
冕
冕
冕
(38–41)
l
l
l
l
I1 = tYmuYnudy, I2 = tY ⬘muY ⬘nudy, I3 = tYmuY ⬘nvdy, I4 = tY ⬘muYnvdy, 0 l
0 l
0 l
0 l
I5 = tY ⬘mvYnudy, I6 = tYmvY ⬘nudy, I7 = tY ⬘mvY ⬘nvdy, I8 = tYmvYnvdy, 0
0
0
0
5. Numerical examples A number of numerical examples are given to demonstrate the versatility of the present method. Results obtained by the present method are compared with solutions published before if available or solutions obtained by finite element method [6]. As some of the examples published before were presented using imperial units, these units are omitted altogether for simplicity. 5.1. Example 1: rectangular shear wall supported on two wide columns Fig. 2 shows a rectangular shear wall supported on two wide columns of the same thickness. The material is isotropic with Young’s modulus E of 471 000 and Poisson’s ratio of 0.375. A horizontal point load of 20 is applied to its top left corner. This structure can be considered as a shear wall with a large rectangular opening at the base. This problem has been studied by various researchers [3,7]. The shear wall was discretized into 16 strips of equal width and solved by the present method using 16 terms. The first category of strips are the four strips going through the two wide columns, and they run from the base to the top. The other category of 12 strips are those above the opening. The boundary conditions at the three edges adjacent to the opening are natural boundary conditions. As this formulation is of the Rayleigh– Ritz type, the natural boundary conditions are therefore automatically dealt with. However, in the formulation of the stiffness matrices for the 12 strips above the
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Fig. 2.
Shear wall supported on two wide columns.
opening, integration is only carried out over the solid height above the opening. The problem was also solved by the finite element method using 2528 QM6 incompatible elements [6]. The vertical stresses over the horizontal cross-section at y ⫽ 4 are shown in Fig. 3 and compared with the finite element solution [6] as well as those from finite strips with computed shape functions [3]. Sharp variation of stresses is observed as the cross-section is close to the wall-frame interfaces. Very good agreement is observed between the present results and the reference results. 5.2. Example 2: deep cantilever beam with step change in thickness Fig. 4 shows a deep cantilever beam made of isotropic material with step change in thickness. The Young’s modulus E is 1.0 and Poisson’s ratio is 0.15. The beam supports a downward unit point load at the top right corner. Two schemes of discretization were tried. The deep beam was respectively discretized into four and 10 strips of equal widths. The number of terms used was varied to study the convergence of the present method. The deflections at two points of the tip are summarized in Table 1 and compared with the reference solution obtained using 1800 QM6 incompatible elements [6]. Very good agreement is observed. Incidentally, the results obtained from the coarse mesh comprising four strips are only 1.6% less than the
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297
Fig. 3. Shear wall supported on two wide columns, distribution of vertical stresses (y) over the horizontal cross section at y = 4.
Fig. 4.
Cantilever deep beam with step change in thickness.
reference solution. The stresses computed using 10 strips and 10 terms are presented in Figs. 5 and 6. In particular, Fig. 5 shows the bending stresses at the vertical crosssections just adjacent to the step change in thickness. The bending stresses along the centre line of the lowest strip (i.e. at x ⫽ 9.5) are shown in Fig. 6. Both figures show very good agreement between the present results and the finite element results [6].
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Table 1 Deep cantilever beam with step change in thickness, deflections at two points of the tip Scheme used in the present method 4 strips, 12 terms 4 strips, 14 terms 4 strips, 16 terms 4 strips, 18 terms 4 strips, 20 terms 10 strips, 12 terms 10 strips, 14 terms 10 strips, 16 terms 10 strips, 18 terms 10 strips, 20 terms FEM [6]
Deflection at x = 5, y = 20
212.88 212.94 212.97 212.99 213.01 215.54 215.61 215.63 215.65 215.68 216.31
Deflection at x = 10, y = 20
208.86 208.91 208.94 208.96 208.97 211.58 211.67 211.68 211.71 211.74 212.36
Fig. 5. Cantilever deep beam with step change in thickness, bending stresses (y) adjacent to step change in thickness.
5.3. Example 3: deep cantilever beam with step change in depth Fig. 7 shows a deep cantilever beam of unit thickness but with step change in depth. The material is isotropic with a Young’s modulus E of 1.0 and Poisson’s ratio of 0.15. The beam is subjected to a unit point load at the top of the cantilever tip. It is discretized into 10 strips of equal width. Twenty terms were used in the present analysis. The beam was also analysed by the finite element method [6] using 3200 QM6 incompatible elements. Figs. 8 and 9 show the bending stresses at the root (y ⫽ 0) and the section at y ⫽ 9.5. The present method is capable of predicting stress concentration around the re-entrant corner, and good agreement is observed between the present results and the reference solution.
Y.K. Cheung et al. / Thin-Walled Structures 32 (1998) 289–303
Fig. 6.
299
Deep cantilever beam with step change in thickness, bending stresses (y) along the line x = 9.5.
Fig. 7.
Cantilever deep beam with step change in depth.
5.4. Example 4: asymmetrical coupled shear wall with step change in thickness Fig. 10(a) shows an asymmetrical coupled shear wall with step change in thickness previously studied by Chan and Cheung [8] by higher order finite elements. The wall is subjected to a unit horizontal uniformly distributed load at the left side. The wall is assumed to be made of isotropic material having Young’s modulus E of 463 000 and Poisson’s ratio of 0.0. In the analysis of coupled shear walls, the spandrel beams and openings between the coupled shear walls are very often represented by an orthotropic continuum with equivalent elastic properties. Referring to Fig. 10(b), the equivalent Young’s moduli of the continuum in the x and y direc-
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Fig. 8. Cantilever deep beam with step change in depth, bending stresses (y) at the base (y = 0).
Fig. 9.
Cantilever deep beam with step change in depth, bending stresses (y) at y = 9.5.
tions, as well as the equivalent shear rigidity of the continuum [8] are given, respectively, by Ex ⫽
d d E E, Ey ⫽ 0, G ⫽ . 2 h h c⬘ ⫹ 2.4 d
冉冊
where c⬘ ⫽ c ⫹ d is the effective span length of the spandrel beam. The adjustment to the span length of the spandrel beam is to allow for the fact that the rigid-end condition could not possibly occur immediately at the junction of the wall and beam.
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301
Fig. 10. (a) Asymmetrical coupled shear wall of step change in thickness; (b) Geometric dimensions of the openings and spandrel beams.
The equivalent properties of the continuum are therefore calculated as Ex ⫽ 115 750, Ey ⫽ 0 and G ⫽ 2251.9. In the analysis by the present method, the shear wall was first discretized into 14 strips of equal widths. In addition, a very coarse mesh comprising only three strips (one for the equivalent continuum and two for the walls) was tried. The number of terms used in the analysis ranges from 12 to 20. The problem was also solved using 1512 QM6 incompatible finite elements [6]. Table 2 shows the convergence of the deflection at the top right corner of the shear wall Table 2 Asymmetrical coupled shear wall with step change in thickness, deflection at top right corner Scheme used in the present method 3 strips, 12 terms 3 strips, 14 terms 3 strips, 16 terms 3 strips, 18 terms 3 strips, 20 terms 14 strips, 12 terms 14 strips, 14 terms 14 strips, 16 terms 14 strips, 18 terms 14 strips, 20 terms FEM [6]
Deflection at x = 7, y = 21 9.6485 9.6489 9.6492 9.6494 9.6495 9.6887 9.6909 9.6922 9.6928 9.6936 9.6974
× × × × × × × × × × ×
10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3 10−3
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Fig. 11. Coupled shear wall with step change in thickness, horizontal deflections along the right edge (x = 7).
as compared with the finite element results [6]. The horizontal deflection of the right side of the wall is shown in Fig. 11. The vertical stresses at the horizontal cross section at y ⫽ 3.375 are plotted in Fig. 12. The vertical stresses along the centre line of the rightmost strip (i.e. at x ⫽ 6.75) are shown in Fig. 13. It is observed that results from the present method agree very well with those obtained by the finite element method [6] and the higher order finite elements [8]. The accuracy of the deflections obtained from the very coarse mesh is also striking.
Fig. 12. Coupled shear wall with step change in thickness, bending stress (y) at horizontal section at y = 3.375.
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303
Fig. 13. Coupled shear wall with step change in thickness, bending stresses (y) along vertical line at x = 6.75.
6. Conclusions A new finite strip method has been developed for the analysis of deep beams and shear walls. The use of piecewise linear augmenting functions in conjunction with the basic harmonic functions results in displacement functions with the appropriate order of continuity at the locations of abrupt changes of sectional properties. It therefore allows the use of continuous finite strips with step changes in thickness. The present method can also cope with rectangular domains with re-entrant corners. A few numerical examples have been given to demonstrate the versatility, efficiency and accuracy of the method.
References [1] Cheung YK. Finite strip method in structural analysis. Oxford: Pergamon Press, 1977. [2] Fan SC. Spline finite strip in structural analysis. PhD thesis, The University of Hong Kong, Hong Kong, 1982. [3] Cheung YK, Kong J. An accurate finite strip for analyzing deep beams and shear walls. Commun Numer Methods Engng 1995;11:643–53. [4] Au FTK, Zheng DY, Cheung YK. Vibration and stability of non-uniform beams with abrupt changes of cross-section by using C1 modified beam vibration functions. Applied Mathematical Modeling, in press. [5] Liu JY, Zheng DY, Mei ZJ. Practical integral transforms in engineering. Huazhong University of Science and Technology Press, 1995. [6] Cosmos User Manual. Structural research and analysis cooperation. Santa Monica, CA 90404, USA, 1991. [7] Cheung YK, Swaddiwudhipong S. Analysis of frame shear wall structures using finite strip elements. Proc Instn Civ Engrs 1978;65:517–35. [8] Chan HC, Cheung YK. Analysis of shear walls using higher order finite elements. Building Environ 1979;14:217–24.