**2. Formulation of the one-dimensional extended rod theory for high-rise buildings**

Frame tubes with braces and/or shear walls are replaced with an equivalent beam. Assum‐ ing that in-plane floor's stiffness is rigid, the individual deformations of outer and inner tubes in tube-in-tube are restricted. Hence, the difference between double tube and single tube depends on only the values of bending stiffness, transverse shear stiffness, and torsion‐ al stiffness. Therefore, for the sake of simplicity, consider a doubly symmetric single tube structure, as shown in Figure 1. Cartesian coordinate system, *x, y, z* is employed, in which the axis *x* takes the centroidal axis, and the transverse axes *y* and *z* take the principal axes of the tube structures. Since the lateral deformation and torsional deformation for a doubly symmetric tube structure are uncouple, the governing equations for these deformations can be formulated separately for simplicity.

**Figure 1.** Doubly symmetric tube structure

shear deformation, shear-lag deformation, and torsional deformation. The problem is to be

There are many rod theories. The most simple rod theory is Bernonlli-Euler beam theory which may treat the bending deformation excluding the transverse shear deformation. The

The transverse resistance of the frame depends on the bending of each structural member consisted of the frame. Therefore, the transverse deformation always occurs corresponding to the transverse stiffness κGA. Since the transverse shear deformation is independent of the bending deformation of the one-dimensional rod, this shear deformation cannot neglect as for equivalent rod theory. This deformation behavior can be expressed by Timoshenko beam theory. Timoshenko beam theory may consider both the bending and the transverse shear deformation of high-rise buildings. The transverse deformation in Timoshenko beam theory

Usual high-rise buildings have the form of the three-dimensional structural frame. Therefore the structures produce the three dimensional behaviors. The representative dissimilarity which is differ from behavior of plane frames is to cause the shear-lag deformation. The shear-lag deformation is noticed in bending problem of box form composed of thin-walled

Reissner [1] presented a simplified beam theory including the effect of the shear-lag in the Bernonlli-Euler beam for bending problem of box form composed of thin-walled member. In this theory the shear-lag is considered only the flange of box form. This phenomenon ap‐ pears in high-rise buildings the same as wing of aircrafts. Especially the shear-lag is remark‐ able in tube structures of high-rise buildings and occurs on the flange sides and web ones of the tube structures. The shear-lag occurs on all three-dimensional frame structures to a greater or lesser degree. Thus the one-dimensional rod theory which is applicable to analyze simply high-rise buildings is necessary to consider the longitudinal deformation, bending deformation, transverse shear deformation, shear-lag deformation, and torsional deforma‐ tion. In generally, high-rise buildings have doubly symmetric structural forms from view‐ point the balance of facade and structural simplicity. Therefore the torsional deformation is considered to separate from the other deformations. Takabatake [2-6] presented a one-di‐ mensional rod theory which can consider simply the above deformations. This theory is

The previous works for continuous method are surveyed as follows: Beck [7] analyzed cou‐ pled shear walls by means of beam model. Heidenbrech et al. [8] indicated an approximate analysis of wall-frame structures and the equivalent stiffness for the equivalent beam. Dy‐ namic analysis of coupled shear walls was studied by Tso et al. [9], Rutenberg [10, 11], Da‐ nay et al. [12], and Bause [13]. Cheung and Swaddiwudhipong [14] presented free vibration of frame shear wall structures. Coull et al. [15, 16] indicated simplified analyses of tube structures subjected to torsion and bending. Smith et al. [17, 18] proposed an approximate method for deflections and natural frequencies of tall buildings. However, the aforemen‐ tioned continuous approaches have not been presented as a closed-form solution for tube

how to take account of these deformations under keeping the simplification.

236 Advances in Vibration Engineering and Structural Dynamics

is assumed to be linear distributed in the transverse cross section.

called the one-dimensional extended rod theory.

closed section.

Bernonlli-Euler beam theory is unsuitable for the modeling of high-rise buildings.

#### **2.1. Governing equations for lateral forces**

Consider a motion of the tube structure subjected to lateral external forces such as winds and earthquakes acting in the *y*-direction, as shown in Figure 1. The deformation of the tube structures is composed of axial deformation, bending, transverse shear deformation, and shear-lag, in which the in-plane distortion of the cross section is neglected due to the inplane stiffness of the slabs. The displacement composes *<sup>U</sup>*¯(*<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>*, *<sup>t</sup>*) , *V*¯(*<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>*, *<sup>t</sup>*) , and *<sup>W</sup>*¯(*<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>*, *<sup>t</sup>*) in the *x-, y-,* and *z-*directions on the middle surface of the tube structures as

$$\overline{\mathbf{U}}\begin{pmatrix} \mathbf{x}, \ y, \ z, t \end{pmatrix} = \mu(\mathbf{x}, t) + y\phi(\mathbf{x}, t) + \boldsymbol{\varphi}^\*(\mathbf{y}, \ z)\boldsymbol{\mu}^\*(\mathbf{x}, t) \tag{1}$$

$$
\overline{V}(\mathbf{x}, \ y, z, t) = v(\mathbf{x}, t) \tag{2}
$$

*<sup>δ</sup><sup>I</sup>* <sup>=</sup>*δ∫<sup>t</sup>*<sup>1</sup> *t*2

*<sup>ε</sup><sup>x</sup>* <sup>=</sup> <sup>∂</sup>*<sup>U</sup>*¯

*<sup>γ</sup>xz* <sup>=</sup> <sup>∂</sup>*<sup>U</sup>*¯ ∂ *z* + ∂*W*¯

*<sup>γ</sup>xy* <sup>=</sup> <sup>∂</sup>*<sup>U</sup>*¯ ∂ *y* + ∂*V*¯

one-dimentional structural member of the frame structure.

in which *E* is Yound modulus and *G* shear modulus.

<sup>2</sup> + *EI* <sup>∗</sup>(*u* ∗′)

<sup>2</sup> + *EI*(*ϕ*′ )

obtained.

*<sup>U</sup>* <sup>=</sup> <sup>1</sup> 2 *∫* 0 *L*

*EA*(*u* ′ )

spect *y* and *z* are expressed as

in which *T* = the kinetic energy; *U* = the strain energy; *V* = the potential energy produced by the external loads; and *δ* = the variational operator taken during the indicated time interval.

Using linear relationship between strain and displacement, the following expressions are

<sup>∂</sup> *<sup>x</sup>* <sup>=</sup>*<sup>ϕ</sup>* <sup>+</sup> *<sup>φ</sup>* <sup>∗</sup>

<sup>∂</sup> *<sup>x</sup>* <sup>=</sup>*<sup>φ</sup>* <sup>∗</sup>

in which dashes indicate the differentiation with respect *x* and the differentiations with re‐

The relationships between stress and strain are used well-known engineering expression for

*φ*,*<sup>y</sup>* \* <sup>=</sup> <sup>∂</sup>*<sup>φ</sup>* \*

*φ*,*z* \* <sup>=</sup> <sup>∂</sup>*<sup>φ</sup>* \* ∂ *z*

Assuming the above linear stress-strain relation, the strain energy *U* is given by

<sup>2</sup> + *κGF* <sup>∗</sup>(*u* <sup>∗</sup>)

,*y*

,*z*

(*T* −*U* −*V* )*dt* =0 (6)

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239

<sup>∂</sup> *<sup>x</sup>* <sup>=</sup>*<sup>u</sup>* ′ <sup>+</sup> *<sup>y</sup>ϕ*′ <sup>+</sup> *<sup>φ</sup>* <sup>∗</sup>*<sup>u</sup>* ∗′ (7)

*u* <sup>∗</sup> + *v* ′ (8)

*u* <sup>∗</sup> (9)

(11)

<sup>∂</sup> *<sup>y</sup>* (10)

*σ<sup>x</sup>* = *Eε<sup>x</sup>* (12)

*τxy* =*Gγxy* (13)

*τxz* =*Gτxz* (14)

*u* ∗′ + *κGA*(*v* ′ + *ϕ*)<sup>2</sup> *dx* (15)

<sup>2</sup> + 2*ES* <sup>∗</sup>*ϕ*′

$$
\overline{W}(\mathbf{x}, \ y, z, t) = 0 \tag{3}
$$

in which *u* and *v* = longitudinal and transverse displacement components in the *x*-and *y*-di‐ rections on the axial point, respectively; *ϕ* = rotational angle on the axial point along the *z*axis; *u* \* = shear-lag coefficient in the flanges; *φ* \* (*x*, *y*)= shear-lag function indicating the distribution of shear-lag. These displacements and shear-lag coefficient are defined positive as the positive direction of the coordinate axes. However, the rotation is defined positive as counterclockwise along the *z* axis, as shown in Figure 2. The shear-lag function for the flange sections is used following function given by Reissner [1] and for the web sections sine distribution [5, 6] is assumed:

$$\varphi^\*(y, z) = \pm \left[1 - \left(\frac{z}{b\_1}\right)^2\right] \text{ for flange} \tag{4}$$

$$
\langle \varphi^\*(y, z) \rangle = \sin \left( \frac{\pi}{b\_2} \right) \text{ for web} \tag{5}
$$

in which the positive of ± takes for the flange being the positive value of the *y*-axis and vice versa *b* 1 and *b* 2 are hafe width of equivalent flange and web sections, as shown in Figure 1.

**Figure 2.** Positive direction of rotation

The governing equation of tube structures is proposed by means of the following Hamil‐ ton's principle.

A Simplified Analytical Method for High-Rise Buildings http://dx.doi.org/10.5772/51158 239

$$
\delta I = \delta \int\_{t\_1}^{t\_2} (T - \mathcal{U} - V)dt = 0 \tag{6}
$$

in which *T* = the kinetic energy; *U* = the strain energy; *V* = the potential energy produced by the external loads; and *δ* = the variational operator taken during the indicated time interval.

shear-lag, in which the in-plane distortion of the cross section is neglected due to the inplane stiffness of the slabs. The displacement composes *<sup>U</sup>*¯(*<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>*, *<sup>t</sup>*) , *V*¯(*<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>*, *<sup>t</sup>*) , and *<sup>W</sup>*¯(*<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>*, *<sup>t</sup>*) in the *x-, y-,* and *z-*directions on the middle surface of the tube structures as

in which *u* and *v* = longitudinal and transverse displacement components in the *x*-and *y*-di‐ rections on the axial point, respectively; *ϕ* = rotational angle on the axial point along the *z*-

distribution of shear-lag. These displacements and shear-lag coefficient are defined positive as the positive direction of the coordinate axes. However, the rotation is defined positive as counterclockwise along the *z* axis, as shown in Figure 2. The shear-lag function for the flange sections is used following function given by Reissner [1] and for the web sections sine

> *z b*1 ) 2

> > *b*2

in which the positive of ± takes for the flange being the positive value of the *y*-axis and vice

The governing equation of tube structures is proposed by means of the following Hamil‐

are hafe width of equivalent flange and web sections, as shown in Figure 1.

= shear-lag coefficient in the flanges; *φ* \*

238 Advances in Vibration Engineering and Structural Dynamics

*φ* <sup>∗</sup>(*y*, *z*)= ± 1−(

*<sup>φ</sup>* <sup>∗</sup>(*y*, *<sup>z</sup>*)=sin( *<sup>π</sup> <sup>y</sup>*

axis; *u* \*

versa *b* 1 and *b* 2

distribution [5, 6] is assumed:

**Figure 2.** Positive direction of rotation

ton's principle.

*<sup>U</sup>*¯(*<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>*, *<sup>t</sup>*)=*u*(*x*, *<sup>t</sup>*) <sup>+</sup> *<sup>y</sup>ϕ*(*x*, *<sup>t</sup>*) <sup>+</sup> *<sup>φ</sup>* <sup>∗</sup>(*y*, *<sup>z</sup>*)*<sup>u</sup>* <sup>∗</sup>(*x*, *<sup>t</sup>*) (1)

*<sup>V</sup>*¯(*<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>*, *<sup>t</sup>*)=*v*(*x*, *<sup>t</sup>*) (2)

*<sup>W</sup>*¯(*<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>*, *<sup>t</sup>*)=0 (3)

(*x*, *y*)= shear-lag function indicating the

for flange (4)

) for web (5)

Using linear relationship between strain and displacement, the following expressions are obtained.

$$
\varepsilon\_{\pm} = \frac{\partial \overline{\boldsymbol{L}}}{\partial \boldsymbol{\omega}} = \boldsymbol{\mu}^{\prime} + \boldsymbol{y} \boldsymbol{\phi}^{\prime} + \boldsymbol{q}^{\ast} \boldsymbol{u}^{\ast \prime} \tag{7}
$$

$$
\Delta\mathcal{V}\_{xy} = \frac{\partial \overline{\mathcal{U}}}{\partial \underline{\mathcal{Y}}} + \frac{\partial \overline{\mathcal{V}}}{\partial \underline{\mathcal{X}}} = \phi + \boldsymbol{\varphi} \overset{\*}{\mathop{\rm on}}\_{,y} \boldsymbol{\mu}^{\*} + \boldsymbol{\upsilon}^{\prime} \tag{8}
$$

$$
\gamma\_{zz} = \frac{\partial \overline{\mathcal{U}}}{\partial z} + \frac{\partial \overline{\mathcal{W}}}{\partial x} = \boldsymbol{\rho}^\* \boldsymbol{\Gamma}\_{,z} \boldsymbol{\mu}^\* \tag{9}
$$

in which dashes indicate the differentiation with respect *x* and the differentiations with re‐ spect *y* and *z* are expressed as

$$
\rho \overset{\*}{\underset{,y}{\*}} = \frac{\partial \rho \overset{\*}{\underset{,y}{\rightleftharpoons}}}{\partial y} \tag{10}
$$

$$
\rho \stackrel{\*}{\varphi}\_{z}^{\*} = \frac{\partial \rho \stackrel{\*}{\varphi}^{\*}}{\partial z} \tag{11}
$$

The relationships between stress and strain are used well-known engineering expression for one-dimentional structural member of the frame structure.

$$
\sigma\_{\mathbf{x}} = \mathbf{E} \,\varepsilon\_{\mathbf{x}} \tag{12}
$$

$$
\pi\_{xy} = G\gamma\_{xy} \tag{13}
$$

$$
\pi\_{xz} = G \pi\_{zx} \tag{14}
$$

in which *E* is Yound modulus and *G* shear modulus.

Assuming the above linear stress-strain relation, the strain energy *U* is given by

$$\text{AL} = \frac{1}{2} \int\_0^L \left[ EA(\boldsymbol{\mu}^\prime)^2 + EI(\boldsymbol{\phi}^\prime)^2 + EI^\*(\boldsymbol{\mu}^{\*\prime})^2 + \kappa GF^\*(\boldsymbol{\mu}^\*)^2 + 2ES^\*\boldsymbol{\phi}^\*\boldsymbol{\mu}^{\*\prime} + \kappa GA(\boldsymbol{\nu}^\prime + \boldsymbol{\phi})^2 \right] d\mathbf{x} \tag{15}$$

in which *L* = the total height of the tube structure; *k* = the shear coefficient; and *A*, *I*, *I* \* , *S* \* , and *F* \* = the sectional stiffnesses. These sectional stiffnesses vary discontinuously with re‐ spect to *x* for a variable tube structure and are defined as

$$A = \iint dydz = \sum A\_c$$

in which *px* and *py* = components of external loads in the *x-* and *y-*directions per unit are, respectively; *cu* , and *cv* = damping coefficients for longtitudinal and transverse motions, re‐

*<sup>δ</sup><sup>u</sup>* \* <sup>+</sup> *Pyδ<sup>v</sup>* <sup>−</sup>*cuu*˙ *<sup>δ</sup><sup>u</sup>* <sup>−</sup>*cvv*˙*δv*)*dx* (25)

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241

*Px* =*∫∫ pxdydz* (26)

*Py* =*∫∫ pydydz* (27)

*m*=*∫∫ px ydydz* (28)

(*Pxδu* + *mδϕ* + *Pyδv* −*cuu*˙ *δu* −*cvv*˙*δv*)*dx* (30)

¨ <sup>+</sup> *cvv*˙ <sup>−</sup> *<sup>κ</sup>GA*(*<sup>v</sup>* ′ <sup>+</sup> *<sup>ϕ</sup>*) ′−*Py* =0 (32)

*δϕ* :*ρIϕ*¨ + *ρS* \* *u*¨ \* −(*EIϕ*′ + *ES* \* *u* \*′)′ + *κGA*(*v* ′ + *ϕ*) −*m*=0 (33)

*dydz* =0 (29)

)′−*Px* =0 (31)

)′ + *κGF* \* *u* \* =0 (34)

van‐

spectively. The substitution of Eqs. (1) and (2) into Eq. (24) yields

(*Pxδ<sup>u</sup>* <sup>+</sup> *<sup>m</sup>δϕ* <sup>+</sup> *<sup>m</sup>*\*

are defined as

*m*\*

<sup>=</sup>*∫∫ px<sup>φ</sup>* \*

Since for a doubly symmetric tube structure the distribution of the shear-lag function on the flange and web surfaces confronting each other with respect to *z* axis is asymmetric, *m*\*

Substituting Eqs. (15), (28), and (30) into Eq. (6), the differential equations of motion can be

*<sup>δ</sup><sup>u</sup>* :*ρAu*¨ <sup>+</sup> *cuu*˙ <sup>−</sup>(*EAu* ′

*δu* \* :*ρI* \* *u*¨ \* + *ρS* \**ϕ*¨ −(*EI* \* *u* \*′ + *ES* \**ϕ*′

together with the associated boundary conditions at *x* =0 and *x* = *L* .

*δV* = −*∫* 0 *L*

in which *Px* , *Py* , *m* , and *m*\*

ishes. Hence, Eq. (25) reduces to

obtained

*δV* = −*∫* 0 *L*

*δv* :*ρAv*

$$I = \iint y^2 dy dz\tag{17}$$

$$A^\* = \iint \boldsymbol{\varphi}^\* d\boldsymbol{y} d\boldsymbol{z} = \iint \boldsymbol{\uprho}\_f^\* + \boldsymbol{\uprho}\_w^\* \mathbf{J} d\boldsymbol{y} d\boldsymbol{z} \tag{18}$$

$$I \stackrel{\*}{=} \text{|f} \begin{Bmatrix} \wp \ \text{\*} \end{Bmatrix}^2 dydz = 2t\_2 \int\_{-b\_1}^{b\_1} \begin{Bmatrix} \wp\_f \ \text{\*} \end{Bmatrix}^2 dz + 2t\_1 \int\_{-b\_2}^{b\_2} \begin{Bmatrix} \wp\_w \ \text{\*} \end{Bmatrix}^2 dy \tag{19}$$

$$\text{L'} = \text{ff} \ y \,\varphi^\* \, dydz = 2t\_2 \Big|\_{{-b\_1}}^{{b\_1}} b\_2 \varphi\_f^\* dz + 2t\_2 \Big|\_{{-b\_2}}^{{b\_2}} y \,\varphi\_w^\* dy \tag{20}$$

$$F^\* = \iint \begin{pmatrix} \wp\_{,z}^\* \end{pmatrix}^2 dydz + \iint \begin{pmatrix} \wp\_{,y}^\* \end{pmatrix}^2 dydz = \iint \begin{pmatrix} \wp\_{f,z}^\* + \wp\_{w,z}^\* \end{pmatrix}^2 dydz + \iint \begin{pmatrix} \wp\_{f,y}^\* + \wp\_{w,y}^\* \end{pmatrix}^2 dydz\tag{21}$$

in which ∑ *Ac* = the total cross-sectional area of columns per story.

The kinetic energy, *T* , for the time interval from *t*1 to *t*2 is

$$T = \int\_{t\_1}^{t\_2} \left| \frac{1}{2} \int\_0^L \left[ \rho A(\dot{u})^2 + \rho I(\dot{\phi})^2 + \rho I^\*(\dot{u}^\*)^2 + 2\rho S^\* \dot{\dot{\phi}} \dot{u}^\* + \rho A(\dot{v})^2 \mathbf{J} dx \right] dt \tag{22}$$

in which the dot indicates differentiation with respect to time and *ρ* = mass density of the tube structure. Now assuming that the variation of the displacements and rotation at *t* =*t*<sup>1</sup> and *t* =*t*2 is negligible, the variation *δT* may be written as

$$\delta T = -\int\_{t\_1}^{t\_2} \left| \int\_0^L \left[ \rho A \ddot{u} \delta u + \rho I \ddot{\phi} \delta \phi + \rho I \dot{\ddot{u}}^\* \ddot{u} \delta u^\* + \rho S \left( \ddot{\phi} \delta u^\* + \ddot{u}^\* \delta \phi \right) + \rho A \ddot{\ddot{v}} \delta v \right] dx \right| dt \tag{23}$$

When the external force at the boundary point (top for current problem) prescribed by the mechanical boundary condition is absent, the variation of the potential energy of the tube structures becomes

$$
\delta \delta V = -\int\_0^L \left[ \iiint (p\_x \delta \overline{U} + p\_y \delta \overline{V}) dy dz \right] d\mathbf{x} + \int\_0^L (c\_u \dot{u} \delta u + c\_v \dot{v} \delta v) d\mathbf{x} \tag{24}
$$

in which *px* and *py* = components of external loads in the *x-* and *y-*directions per unit are, respectively; *cu* , and *cv* = damping coefficients for longtitudinal and transverse motions, re‐ spectively. The substitution of Eqs. (1) and (2) into Eq. (24) yields

$$
\delta V = -\int\_0^L \left( P\_x \delta u + m \delta \phi + m \stackrel{\*}{\sigma} \delta u \stackrel{\*}{\ } + P\_y \delta v - c\_u \dot{u} \delta u - c\_v \dot{v} \delta v \right) d\mathbf{x} \tag{25}
$$

in which *Px* , *Py* , *m* , and *m*\* are defined as

in which *L* = the total height of the tube structure; *k* = the shear coefficient; and *A*, *I*, *I* \*

*I* =*∫∫y* <sup>2</sup>

*dydz* = *∬ φ<sup>f</sup>*

*b*1 (*φf* \* ) 2

> *b*1 *b*2*φ<sup>f</sup>* \*

> > \* <sup>+</sup> *<sup>φ</sup>w*,*<sup>z</sup>* \* )<sup>2</sup>

in which the dot indicates differentiation with respect to time and *ρ* = mass density of the tube structure. Now assuming that the variation of the displacements and rotation at *t* =*t*<sup>1</sup>

*δu* \* + *ρS* \*

When the external force at the boundary point (top for current problem) prescribed by the mechanical boundary condition is absent, the variation of the potential energy of the tube

> 0 *L*

(*ϕ*¨ *δu* \* + *u*¨ \*

*u*¨ \*

*∫∫*(*pxδU*¯ <sup>+</sup> *pyδV*¯)*dydz dx* <sup>+</sup> *<sup>∫</sup>*

*dydz* =2*t*2*∫*−*b*<sup>1</sup>

*dydz* =2*t*2*∫*−*b*<sup>1</sup>

*dydz* = *∬*(*φ<sup>f</sup>* ,*<sup>z</sup>*

spect to *x* for a variable tube structure and are defined as

*A*\* = *∬φ* \*

= *∬*(*φ* \*)<sup>2</sup>

= *∬ yφ* \*

\* )<sup>2</sup>

The kinetic energy, *T* , for the time interval from *t*1 to *t*2 is

*ρA*(*u*˙)

and *t* =*t*2 is negligible, the variation *δT* may be written as

*ρAu*¨ *δu* + *ρIϕ*¨ *δϕ* + *ρI* \*

in which ∑ *Ac* = the total cross-sectional area of columns per story.

<sup>2</sup> + *ρI*(*ϕ* ⋅ ) 2 + *ρI* \* (*u*˙ \* ) <sup>2</sup> + 2*ρS* \* *ϕ* ⋅

*I* \*

240 Advances in Vibration Engineering and Structural Dynamics

*S* \*

*dydz* + *∬*(*φ*,*<sup>y</sup>*

= the sectional stiffnesses. These sectional stiffnesses vary discontinuously with re‐

\* <sup>+</sup> *<sup>φ</sup><sup>w</sup>*

*<sup>d</sup> <sup>z</sup>* <sup>+</sup> <sup>2</sup>*t*1*∫*−*b*<sup>2</sup> *b*2 (*φ<sup>w</sup>* \*)2

*dz* +2*t*2*∫*−*b*<sup>2</sup> *b*2 *yφ<sup>w</sup>* \*

*dydz* + *∬*(*φ<sup>f</sup>* ,*<sup>y</sup>*

*u*˙ \* + *ρA*(*v*˙)

*δϕ*) + *ρAv*

(*cuu*˙ *δu* + *cvv*˙*δv*)*dx* (24)

*<sup>A</sup>*=*∫∫dydz* <sup>=</sup>∑ *Ac* (16)

*dydz* (17)

\* <sup>+</sup> *<sup>φ</sup>w*,*<sup>y</sup>* \* )<sup>2</sup>

\* *dydz* (18)

*d y* (19)

*dy* (20)

*dydz* (21)

<sup>2</sup> *dx*}*dt* (22)

¨*δv dx*}*dt* (23)

and *F* \*

*F* \*

= *∬*(*φ*,*<sup>z</sup>* \*)2

> *<sup>T</sup>* <sup>=</sup>*∫ <sup>t</sup>*<sup>1</sup> *t*2 { 1 2 *∫* 0 *L*

*<sup>δ</sup><sup>T</sup>* <sup>=</sup> <sup>−</sup>*∫<sup>t</sup>*<sup>1</sup> *t*2 {*∫* 0 *L*

structures becomes

*δV* = −*∫* 0 *L* , *S* \* ,

$$P\_x = \iint p\_x dydz \tag{26}$$

$$P\_y = \iint p\_y dydz \tag{27}$$

$$m = \iint p\_x y dy dz \tag{28}$$

$$
\hat{\rho}m^\* = \iint p\_x \rho \rho^\* dydz = 0\tag{29}
$$

Since for a doubly symmetric tube structure the distribution of the shear-lag function on the flange and web surfaces confronting each other with respect to *z* axis is asymmetric, *m*\* van‐ ishes. Hence, Eq. (25) reduces to

$$
\delta V = -\int\_0^L \left\{ P\_x \delta u + m \delta \phi + P\_y \delta v - c\_u \dot{\mu} \delta u - c\_v \dot{\nu} \delta v \right\} dx \tag{30}
$$

Substituting Eqs. (15), (28), and (30) into Eq. (6), the differential equations of motion can be obtained

$$
\delta\delta u : \rho A \ddot{u} + c\_u \dot{u} - \text{(E}A u \text{ }')' - P\_\chi = 0 \tag{31}
$$

$$\delta\boldsymbol{\sigma} : \rho A \ddot{\boldsymbol{\nu}} + c\_v \dot{\boldsymbol{\nu}} - \mathsf{f} \kappa \mathsf{G} A (\boldsymbol{\nu}^\prime + \phi \boldsymbol{\lambda}) \mathbf{I}^\prime - P\_y = 0 \tag{32}$$

$$\delta\phi : \rho I \ddot{\phi} + \rho S \ast \ddot{u} \, ^\*- \{ EI \phi \, ^' + ES \ast ^\*u \, ^\*\prime \} + \kappa GA \{ v \, ^' + \phi \} - m = 0 \tag{33}$$

$$\delta u^\* : \rho I \ast^\* \ddot{u}^\* + \rho S \ast^\* \ddot{\phi} - \{EI \ast u^\* \,' + ES \ast^\* \phi \}' + \kappa GF \ast^\* u^\* = 0 \tag{34}$$

together with the associated boundary conditions at *x* =0 and *x* = *L* .

$$
u\_{--} = 0 \quad \text{or} \quad EA \\
u' = 0 \quad \tag{35}$$

*<sup>S</sup>* \* <sup>=</sup> <sup>2</sup>

*<sup>F</sup>* \* <sup>=</sup> <sup>4</sup>

as *Af* =4*t*1*b*1 and *Aw* =4*t*2*b*2 .

**2.4. Equivalent transverse shear stiffness κGA**

stiffness κGA for each story is given by

1 (*κGA*) *frame*

respectively. ∑ *<sup>c</sup>*

=

and ∑ *<sup>b</sup>*

*<sup>h</sup>* ( <sup>1</sup> ∑ *Kc* + 1 ∑ *Kb* )

(*κGA*)

<sup>12</sup>*<sup>E</sup>* <sup>+</sup>

*brace* <sup>=</sup> *<sup>h</sup>*

<sup>3</sup> *<sup>b</sup>*2*Af* <sup>+</sup>

3(*b*1)2 *Af* <sup>+</sup>

*b*2

*π* 2 2*b*<sup>2</sup>

in which *Af* = the total cross-sectional area of columns in the flanges and webs, respectively, per story. For the cross-section of tube structures, as shown in Figure 1, *Af* and *Aw* are given

When the tube structure are composed of frame and bracing, the equivalent transverse shear

in which ∑ is taken the summation of equivalent transverse shear stiffnesses of web frame and of braces per story. The shear stiffnesses of web frame and web double-brace ∑ (*κGA*) *frame* and ∑ (*κGA*)*brace* for each side of the web surfaces, respectively, are given by

1

(*κGAcw*) <sup>+</sup>

∑ *c*

<sup>ℓ</sup> *kbraceABEB*cos<sup>2</sup>

in which the first term on the right side of Eq. (49) indicates the deformation of the frame with the stiffnesses of columns and beams *Kc* and *Kb* , respectively; the second and third terms indicate the shear deformation of only the columns and beams in the current webframe, respectively. *Acw* and *Abw* = the web's cross-sectional area of a column and of a beam,

the current story of the frame tube. If the shear deformations of columns and beams are ne‐ glected, these terms must vanish. Furthermore, ℓ= the span length; *AB* = the cross-sectional area of a brace; *EB* = the Young's modulus; and *θ<sup>B</sup>* = the incline of the brace. The coefficient *kbrace* indicates the effective number of brace and takes *kbrace* =1 for a brace resisting only ten‐ sion and *kbrace* =2 for two brace resisting tension and compression, as shown in Figure 3.

*κGA*=∑ (*κGA*) *frame* + ∑ (*κGA*)*brace* (48)

*h*

(ℓ*κGAbw*)

*θ<sup>B</sup>* (50)

(49)

∑ *b*

= the sums of columns and beams, respectively, in a web-frame at

*<sup>π</sup> Aw* (46)

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243

<sup>2</sup> *Aw* (47)

$$
v\_{\
u} = 0 \text{ or } \times GA(v^{'} + \phi) = 0\tag{36}$$

$$
\phi\_- = 0 \text{ or } \operatorname{EI}\phi\_-^{'} + \operatorname{ES} \ast \operatorname{\ast} \iota^{\ast \prime} = 0 \tag{37}
$$

$$
\mu \ast = 0 \text{ or } EI \ast \mu \ast \prime + ES \ast \phi \prime = 0 \tag{38}
$$

#### **2.2. Governing equations for torsional moment**

The displacement components for current tube structures subjected to torsional moments, *mx* , around the *x*-axis are expressed by

$$
\overline{\overline{U}} = \overline{w}(y, z) \boldsymbol{\Theta}^{\cdot}(\boldsymbol{x}, t) \tag{39}
$$

$$
\overline{V} = -z\Theta
\tag{40}
$$

$$
\overline{\mathcal{W}} = \mathcal{y}\mathcal{O} \tag{41}
$$

in which *<sup>W</sup>*¯= the displacement component in the *z-*direction on the tube structures; *<sup>θ</sup>* <sup>=</sup> tor‐ sional angle; and *<sup>w</sup>*¯(*<sup>y</sup>*, *<sup>z</sup>*)= warping function. Using the same manner as the aforementioned development, the differential equation of motion for current tube structures can be obtained

$$\delta\Theta : \rho I\_p \ddot{\Theta} - \{G\} \Theta \,\, ^\dagger \mathfrak{r} - m\_\text{x} = 0 \tag{42}$$

together with the association boundary conditions

$$
\Theta = 0 \quad \text{or} \quad GJ\Theta' = \boldsymbol{m}\_{\boldsymbol{x}\boldsymbol{l}} \tag{43}
$$

at *x* =0 and *L* , in which *GJ* = the torsional stiffness.

#### **2.3. Sectional constants**

The sectional constants are defined by Eqs. (16) to (21). For doubly symmetric single-tube structures as shown in Figure 1, these sectional constants are simplified as follow.

$$A^\* = 0\tag{44}$$

$$I \stackrel{\*}{=} \frac{8}{15}A\_{\uparrow} + \frac{1}{2}A\_{w} \tag{45}$$

A Simplified Analytical Method for High-Rise Buildings http://dx.doi.org/10.5772/51158 243

$$\mathcal{S}^\* = \frac{2}{3} b\_2 A\_{\circ} + \frac{b\_2}{\pi} A\_w \tag{46}$$

$$F \triangleq \frac{4}{3(b\_1)^2} A\_f + \frac{\pi^2}{2b\_2^2} A\_w \tag{47}$$

in which *Af* = the total cross-sectional area of columns in the flanges and webs, respectively, per story. For the cross-section of tube structures, as shown in Figure 1, *Af* and *Aw* are given as *Af* =4*t*1*b*1 and *Aw* =4*t*2*b*2 .

#### **2.4. Equivalent transverse shear stiffness κGA**

*u EAu* 0 or 0 = =¢ (35)

*v* =0 or *κ GA*(*v* ′ + *ϕ*) =0 (36)

*ϕ* =0 *or EIϕ*′ + *ES* \* *u* \*′ =0 (37)

*u* \* =0 or *EI* \* *u* \* ′ + *ES* \**ϕ* ′=0 (38)

(*x*, *t*) (39)

)'−*mx* =0 (42)

=0 (44)

<sup>2</sup> *Aw* (45)

0 or *GJ m xL* q= q = (43)

*<sup>V</sup>*¯ <sup>=</sup> <sup>−</sup> *<sup>z</sup><sup>θ</sup>* (40)

*<sup>W</sup>*¯= *<sup>y</sup><sup>θ</sup>* (41)

The displacement components for current tube structures subjected to torsional moments,

in which *<sup>W</sup>*¯= the displacement component in the *z-*direction on the tube structures; *<sup>θ</sup>* <sup>=</sup> tor‐ sional angle; and *<sup>w</sup>*¯(*<sup>y</sup>*, *<sup>z</sup>*)= warping function. Using the same manner as the aforementioned development, the differential equation of motion for current tube structures can be obtained

'

The sectional constants are defined by Eqs. (16) to (21). For doubly symmetric single-tube

1

structures as shown in Figure 1, these sectional constants are simplified as follow.

*A*\*

*I* \* = 8 <sup>15</sup> *Af* <sup>+</sup>

*<sup>U</sup>*¯ <sup>=</sup>*<sup>w</sup>*¯(*<sup>y</sup>*, *<sup>z</sup>*)*<sup>θ</sup>* '

*δθ* :*ρIpθ*¨ <sup>−</sup>(*GJ <sup>θ</sup>* '

**2.2. Governing equations for torsional moment**

together with the association boundary conditions

at *x* =0 and *L* , in which *GJ* = the torsional stiffness.

**2.3. Sectional constants**

*mx* , around the *x*-axis are expressed by

242 Advances in Vibration Engineering and Structural Dynamics

When the tube structure are composed of frame and bracing, the equivalent transverse shear stiffness κGA for each story is given by

$$
\kappa GA = \sum \left( \kappa GA \right)\_{frame} + \sum \left( \kappa GA \right)\_{brane} \tag{48}
$$

in which ∑ is taken the summation of equivalent transverse shear stiffnesses of web frame and of braces per story. The shear stiffnesses of web frame and web double-brace ∑ (*κGA*) *frame* and ∑ (*κGA*)*brace* for each side of the web surfaces, respectively, are given by

$$\frac{1}{\left(\kappa GA\right)\_{frame}} = \frac{h\left(\frac{1}{\sum K\_c} + \frac{1}{\sum K\_b}\right)}{12E} + \frac{1}{\sum\_c \left(\kappa GA\_{cw}\right)} + \frac{h}{\sum\_b \left(\theta \kappa GA\_{bw}\right)}\tag{49}$$

$$\left(\kappa GA\right)\_{brane} = \frac{h}{\mathcal{U}} k\_{brane} A\_B E\_B \cos^2 \theta\_B \tag{50}$$

in which the first term on the right side of Eq. (49) indicates the deformation of the frame with the stiffnesses of columns and beams *Kc* and *Kb* , respectively; the second and third terms indicate the shear deformation of only the columns and beams in the current webframe, respectively. *Acw* and *Abw* = the web's cross-sectional area of a column and of a beam, respectively. ∑ *<sup>c</sup>* and ∑ *<sup>b</sup>* = the sums of columns and beams, respectively, in a web-frame at the current story of the frame tube. If the shear deformations of columns and beams are ne‐ glected, these terms must vanish. Furthermore, ℓ= the span length; *AB* = the cross-sectional area of a brace; *EB* = the Young's modulus; and *θ<sup>B</sup>* = the incline of the brace. The coefficient *kbrace* indicates the effective number of brace and takes *kbrace* =1 for a brace resisting only ten‐ sion and *kbrace* =2 for two brace resisting tension and compression, as shown in Figure 3.

**Figure 3.** Brace resisting tension and compression

#### **2.5. Equivalent bending stiffness** *EI*

The equivalent bending stiffness *EI* for each storey is determined from the total sum of the moment of inertia about the *z*-axis of each column located on the storey.

$$EE = \sum\_{i=1}^{n} E\_i \left[ I\_{0i} + A\_i e\_i^2 \right] \tag{51}$$

given by the governing equation coupled about the lateral displacements *v* , rotational angle

Using ordinary central finite differences, the finite difference expressions of the current equilibrium equations, obtained from the equations of motion Eqs. (32)-(34), may be written,

> *κGA* <sup>2</sup>*<sup>Δ</sup> <sup>ϕ</sup>i*−<sup>1</sup> <sup>+</sup>

<sup>2</sup>*<sup>Δ</sup> vi*+1 <sup>−</sup> *<sup>κ</sup>GA*

(*ES* \*)' <sup>2</sup>*<sup>Δ</sup> ui*−<sup>1</sup>

*<sup>Δ</sup>* <sup>2</sup> <sup>+</sup>

<sup>2</sup>*<sup>Δ</sup> <sup>ϕ</sup>i*+1 <sup>+</sup> <sup>−</sup> *ES* \*

\* =0

(*κGA*)'

<sup>2</sup>*<sup>Δ</sup> <sup>ϕ</sup>i*−<sup>1</sup> <sup>+</sup> <sup>−</sup> *ES* \*

(*EI*)'

(*EI* \*)' <sup>2</sup>*<sup>Δ</sup> ui*−<sup>1</sup> \* +

(*EI* \*)' <sup>2</sup>*<sup>Δ</sup> ui*+1

load and moment, respectively, at the *i*th mesh point. In the above equations, the rigidities *κGA* , *EI* ,... at the pivotal mesh point *i* are taken as the mean value of the rigidities of cur‐ rent prototype tube structures located in the mesh region, in which the mesh region is de‐ fined as each half height between the mesh point *i* and the adjoin mesh points, *i-*1 and *i+*1, namely from (*xi* + *xi*−1) / 2 to (*xi* + *xi*+1) / 2 , as shown in Figure 5. Hence, the stiffness *k*(*i*)

> *ai*1*ki*<sup>1</sup> + *ai*2*ki*<sup>2</sup> + ⋯ + *ainkin h*(*i*)

in which *ai*1 , *ai*2 ,.., *ain* and *ki*1 , *ki*2 ,..., *kin* = the effective story heights and story rigidities, lo‐ cated in the mesh region, respectively; and *h*(*i*)= the current mesh region for the pivotal mesh point *i*. The first mesh region in the vicinity of the base is defined as region from the

*<sup>Δ</sup>* <sup>2</sup> <sup>+</sup>

*<sup>Δ</sup>* <sup>2</sup> <sup>+</sup>

*<sup>Δ</sup>* <sup>2</sup> <sup>−</sup>

*i*th, and (*i*+1)th mesh points, respectively, as shown in Figure 4; and *Pyi*

2*κGA <sup>Δ</sup>* <sup>2</sup> *vi*

*<sup>Δ</sup>* <sup>2</sup> <sup>−</sup>

*<sup>Δ</sup>* <sup>2</sup> *<sup>ϕ</sup><sup>i</sup>* <sup>+</sup> ( <sup>2</sup>*EI* \*

2*ES* \*

<sup>2</sup>*<sup>Δ</sup> <sup>ϕ</sup>i*+1 <sup>=</sup>*Pyi*

\* <sup>+</sup> ( <sup>2</sup>*EI*

(*ES* \*)' <sup>2</sup>*<sup>Δ</sup> ui*+1

*<sup>Δ</sup>* <sup>2</sup> <sup>+</sup> *<sup>κ</sup>GA*)*ϕ<sup>i</sup>*

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\* <sup>=</sup>*mzi*

)*ui* \*

and *mzi* = the lateral

*<sup>Δ</sup>* <sup>2</sup> <sup>+</sup> *<sup>κ</sup>GF* \*

, *vi*+1 ,... represent displacements at the (*i*-1)th,

(52)

245

(53)

(54)

at a

(55)

.

(*κGA*)' <sup>2</sup>*<sup>Δ</sup> vi*−<sup>1</sup> <sup>+</sup>

*<sup>ϕ</sup><sup>i</sup>* <sup>−</sup> *<sup>κ</sup>GA <sup>Δ</sup>* <sup>2</sup> <sup>+</sup>

(*EI*)'

<sup>2</sup>*<sup>Δ</sup> vi*+1 <sup>+</sup> <sup>−</sup> *EI*

<sup>2</sup>*<sup>Δ</sup> <sup>ϕ</sup>i*−<sup>1</sup> <sup>+</sup> <sup>−</sup> *EI* \*

<sup>2</sup>*<sup>Δ</sup> <sup>ϕ</sup>i*+1 <sup>+</sup> <sup>−</sup> *EI* \*

*k*(*i*)=

base to the mid-height between the mesh points 1 and 2.

*ϕ* , and shear-lag displacement *u* \*

<sup>−</sup> *<sup>κ</sup>GA <sup>Δ</sup>* <sup>2</sup> <sup>+</sup>

−(*κGA*)'

*<sup>Δ</sup>* <sup>2</sup> <sup>+</sup>

respectively, as follows:

<sup>−</sup> *<sup>κ</sup>GA*

+ 2*ES* \* *<sup>Δ</sup>* <sup>2</sup> *ui* \* + *κGA*

<sup>−</sup> *ES* \* *<sup>Δ</sup>* <sup>2</sup> <sup>+</sup>

<sup>+</sup> <sup>−</sup> *ES* \* *<sup>Δ</sup>* <sup>2</sup> <sup>−</sup>

mesh point *i* is evaluated

<sup>2</sup>*<sup>Δ</sup> vi*−<sup>1</sup> <sup>+</sup> <sup>−</sup> *EI*

(*ES* \*)'

(*ES* \*)'

in which *Δ* = the finite difference mesh; *vi*−1 , *vi*

in which *Ei* , *I*0*<sup>i</sup>* and *Ai* = Young modulus, moment of inertia and the cross section of the *i*th column; and *ei* = the distance measured from the *z-*axis.

## **3. Static analysis by the finite defference method**

### **3.1. Expression of static analysis**

The governing equations for the one-dimensional extended rod theory are differential equa‐ tions with variable coefficients due to the variation of structural members and forms in the longitudinal direction. Furthermore, although the equations of motion and boundary condi‐ tions for the vertical displacement *u* are uncoupled from the other displacement compo‐ nents, the governing equations take coupled from concerning variables *v* , *ϕ* , and *u* \* .

Takabatake [2, 3] presented the uncoupled equations as shown in section 7 by introducing positively appropriate approximations into the coupled equations and proposed a closedform solution. For usual tube structures this method produces reasonable results. However the analytical approach deteriorates on the accuracy of numerical results for high-rise build‐ ings with the rapid local variations of transverse shear stiffness and/or braces. Especially the difference appears on the distributions of not dynamic deflection but story acceleration and storey shear force. It is limit to express these rapid variations by a functional expression. So, the above governing equations are solved by means of the finite difference method.

The equations of motion and boundary conditions for the longitudinal displacement *u* are uncoupled from the other displacement components. So we consider only the lateral motion given by the governing equation coupled about the lateral displacements *v* , rotational angle *ϕ* , and shear-lag displacement *u* \* .

Using ordinary central finite differences, the finite difference expressions of the current equilibrium equations, obtained from the equations of motion Eqs. (32)-(34), may be written, respectively, as follows:

**Figure 3.** Brace resisting tension and compression

244 Advances in Vibration Engineering and Structural Dynamics

**2.5. Equivalent bending stiffness** *EI*

in which *Ei*

, *I*0*<sup>i</sup>*

**3.1. Expression of static analysis**

The equivalent bending stiffness *EI* for each storey is determined from the total sum of the

*Ei I*0*<sup>i</sup>* + *Ai*

The governing equations for the one-dimensional extended rod theory are differential equa‐ tions with variable coefficients due to the variation of structural members and forms in the longitudinal direction. Furthermore, although the equations of motion and boundary condi‐ tions for the vertical displacement *u* are uncoupled from the other displacement compo‐

Takabatake [2, 3] presented the uncoupled equations as shown in section 7 by introducing positively appropriate approximations into the coupled equations and proposed a closedform solution. For usual tube structures this method produces reasonable results. However the analytical approach deteriorates on the accuracy of numerical results for high-rise build‐ ings with the rapid local variations of transverse shear stiffness and/or braces. Especially the difference appears on the distributions of not dynamic deflection but story acceleration and storey shear force. It is limit to express these rapid variations by a functional expression. So,

The equations of motion and boundary conditions for the longitudinal displacement *u* are uncoupled from the other displacement components. So we consider only the lateral motion

nents, the governing equations take coupled from concerning variables *v* , *ϕ* , and *u* \*

the above governing equations are solved by means of the finite difference method.

*ei*

and *Ai* = Young modulus, moment of inertia and the cross section of the *i*th

<sup>2</sup> (51)

.

moment of inertia about the *z*-axis of each column located on the storey.

*EI* =∑ *i*=1 *n*

column; and *ei* = the distance measured from the *z-*axis.

**3. Static analysis by the finite defference method**

<sup>−</sup> *<sup>κ</sup>GA <sup>Δ</sup>* <sup>2</sup> <sup>+</sup> (*κGA*)' <sup>2</sup>*<sup>Δ</sup> vi*−<sup>1</sup> <sup>+</sup> *κGA* <sup>2</sup>*<sup>Δ</sup> <sup>ϕ</sup>i*−<sup>1</sup> <sup>+</sup> 2*κGA <sup>Δ</sup>* <sup>2</sup> *vi* −(*κGA*)' *<sup>ϕ</sup><sup>i</sup>* <sup>−</sup> *<sup>κ</sup>GA <sup>Δ</sup>* <sup>2</sup> <sup>+</sup> (*κGA*)' <sup>2</sup>*<sup>Δ</sup> vi*+1 <sup>−</sup> *<sup>κ</sup>GA* <sup>2</sup>*<sup>Δ</sup> <sup>ϕ</sup>i*+1 <sup>=</sup>*Pyi* (52) <sup>−</sup> *<sup>κ</sup>GA* <sup>2</sup>*<sup>Δ</sup> vi*−<sup>1</sup> <sup>+</sup> <sup>−</sup> *EI <sup>Δ</sup>* <sup>2</sup> <sup>+</sup> (*EI*)' <sup>2</sup>*<sup>Δ</sup> <sup>ϕ</sup>i*−<sup>1</sup> <sup>+</sup> <sup>−</sup> *ES* \* *<sup>Δ</sup>* <sup>2</sup> <sup>+</sup> (*ES* \*)' <sup>2</sup>*<sup>Δ</sup> ui*−<sup>1</sup> \* <sup>+</sup> ( <sup>2</sup>*EI <sup>Δ</sup>* <sup>2</sup> <sup>+</sup> *<sup>κ</sup>GA*)*ϕ<sup>i</sup>* + 2*ES* \* *<sup>Δ</sup>* <sup>2</sup> *ui* \* + *κGA* <sup>2</sup>*<sup>Δ</sup> vi*+1 <sup>+</sup> <sup>−</sup> *EI <sup>Δ</sup>* <sup>2</sup> <sup>+</sup> (*EI*)' <sup>2</sup>*<sup>Δ</sup> <sup>ϕ</sup>i*+1 <sup>+</sup> <sup>−</sup> *ES* \* *<sup>Δ</sup>* <sup>2</sup> <sup>−</sup> (*ES* \*)' <sup>2</sup>*<sup>Δ</sup> ui*+1 \* <sup>=</sup>*mzi* (53) <sup>−</sup> *ES* \* *<sup>Δ</sup>* <sup>2</sup> <sup>+</sup> (*ES* \*)' <sup>2</sup>*<sup>Δ</sup> <sup>ϕ</sup>i*−<sup>1</sup> <sup>+</sup> <sup>−</sup> *EI* \* *<sup>Δ</sup>* <sup>2</sup> <sup>+</sup> (*EI* \*)' <sup>2</sup>*<sup>Δ</sup> ui*−<sup>1</sup> \* + 2*ES* \* *<sup>Δ</sup>* <sup>2</sup> *<sup>ϕ</sup><sup>i</sup>* <sup>+</sup> ( <sup>2</sup>*EI* \* *<sup>Δ</sup>* <sup>2</sup> <sup>+</sup> *<sup>κ</sup>GF* \* )*ui* \* <sup>+</sup> <sup>−</sup> *ES* \* *<sup>Δ</sup>* <sup>2</sup> <sup>−</sup> (*ES* \*)' <sup>2</sup>*<sup>Δ</sup> <sup>ϕ</sup>i*+1 <sup>+</sup> <sup>−</sup> *EI* \* *<sup>Δ</sup>* <sup>2</sup> <sup>−</sup> (*EI* \*)' <sup>2</sup>*<sup>Δ</sup> ui*+1 \* =0 (54)

in which *Δ* = the finite difference mesh; *vi*−1 , *vi* , *vi*+1 ,... represent displacements at the (*i*-1)th, *i*th, and (*i*+1)th mesh points, respectively, as shown in Figure 4; and *Pyi* and *mzi* = the lateral load and moment, respectively, at the *i*th mesh point. In the above equations, the rigidities *κGA* , *EI* ,... at the pivotal mesh point *i* are taken as the mean value of the rigidities of cur‐ rent prototype tube structures located in the mesh region, in which the mesh region is de‐ fined as each half height between the mesh point *i* and the adjoin mesh points, *i-*1 and *i+*1, namely from (*xi* + *xi*−1) / 2 to (*xi* + *xi*+1) / 2 , as shown in Figure 5. Hence, the stiffness *k*(*i*) at a mesh point *i* is evaluated

$$k\_{\langle i\rangle} = \frac{a\_{i1}k\_{i1} + a\_{i2}k\_{i2} + \dots + a\_{in}k\_{in}}{h\_{\langle i\rangle}} \tag{55}$$

in which *ai*1 , *ai*2 ,.., *ain* and *ki*1 , *ki*2 ,..., *kin* = the effective story heights and story rigidities, lo‐ cated in the mesh region, respectively; and *h*(*i*)= the current mesh region for the pivotal mesh point *i*. The first mesh region in the vicinity of the base is defined as region from the base to the mid-height between the mesh points 1 and 2.

Now, the boundary conditions for a doubly-symmetric tube structure are assumed to be fixed at the base and free, except for the shear-lag, at the top. The shear-lag at the top is con‐ sidered for two cases: free and constrained. Hence, from Eqs. (36) to (38)

$$\mathbf{v} = \mathbf{0} \text{ at } \mathbf{x} = \mathbf{0} \tag{56}$$

Let us consider the finite difference expression for the boundary conditions (56) - (61). Since tube structures are replaced with an equivalent cantilever in the one-dimensional extended rod theory, the inner points for finite difference method take total numbers *m* as shown in

respectively, the imaginary number of the boundary mesh in finite differences can be taken

The finite differences expressions for the boundary conditions (56)-(58) at the base (*x=0*) are

\* represent quantities at the base.

*vbase* =0 (62)

*ϕbase* =0 (63)

*u*\**base* =0 (64)

is one,

247

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Figure 6, in which the mesh point *m* locates on the boundary point at *x* = *L* .

one for each displacement component at each boundary.

**Figure 5.** Equivalent rigidity in finite difference method [6]

**Figure 6.** Inner points and imaginary point

in which *vbase* , *ϕbase* , and *ubase*

Since the number of each boundary condition of the base and top for *v* , *ϕ* and *u* \*

$$
\phi = 0 \text{ at } \ge = 0 \tag{57}
$$

$$
\mu^\ast = 0 \text{ at } \ge = \text{ 0} \tag{58}
$$

$$
v^{\\\dagger} + \phi = 0 \text{ at } \propto = L \tag{59}$$

$$ES\stackrel{"\ast}{\
u}^{"\ast} + EI\stackrel{\div}{\phi}^{\cdot} = 0 \text{ at } \text{x} = L \tag{60}$$

$$E I \, ^\ast u^\ast \, ^\ast + E S \phi \stackrel{\circ}{=} 0 \text{ at } \text{x} = L \text{ (Shear} - \text{lag is free.)}\tag{61a}$$

$$
\mu^\* = 0 \text{ at } x \; = \, L \left( \text{ Shear} - \text{lag is constraint. } \right) \tag{61b}
$$

**Figure 4.** Mesh point in finite difference method [6]

Let us consider the finite difference expression for the boundary conditions (56) - (61). Since tube structures are replaced with an equivalent cantilever in the one-dimensional extended rod theory, the inner points for finite difference method take total numbers *m* as shown in Figure 6, in which the mesh point *m* locates on the boundary point at *x* = *L* .

Since the number of each boundary condition of the base and top for *v* , *ϕ* and *u* \* is one, respectively, the imaginary number of the boundary mesh in finite differences can be taken one for each displacement component at each boundary.

**Figure 5.** Equivalent rigidity in finite difference method [6]

**Figure 6.** Inner points and imaginary point

Now, the boundary conditions for a doubly-symmetric tube structure are assumed to be

fixed at the base and free, except for the shear-lag, at the top. The shear-lag at the top is con‐

*v x* = = 0 at 0 (56)

*ϕ* =0 at *x* = 0 (57)

*u x* \* 0 at 0 = = (58)

+ *ϕ* =0 at *x* = *L* (59)

=0 at *x* = *L* ( Shear−lag is free.) (61a)

*u xL* \* 0 at Shear lag is constraint. = = - ( ) (61b)

=0 at *x* = *L* (60)

sidered for two cases: free and constrained. Hence, from Eqs. (36) to (38)

*v* '

*u* \*' + *EIϕ*'

*ES* \*

*EI* \* *u*\* '+ *ESϕ*'

246 Advances in Vibration Engineering and Structural Dynamics

**Figure 4.** Mesh point in finite difference method [6]

The finite differences expressions for the boundary conditions (56)-(58) at the base (*x=0*) are

$$
\sigma\_{\text{base}} = 0 \tag{62}
$$

$$
\phi\_{base} = 0\tag{63}
$$

$$
\mu^\*\_{base} = 0\tag{64}
$$

in which *vbase* , *ϕbase* , and *ubase* \* represent quantities at the base. On the other hand, using central difference method, the finite difference expressions for the boundary conditions (59), (60), and (61a), in case where the shear-lag is free at the top(*x=L*), are expressed as

$$\left[ -\upsilon\_{m-1} + \upsilon\_{m+1} \right] \frac{1}{2\Delta} + \phi\_m = 0 \tag{65}$$

*ϕm*+1 =*ϕm*−<sup>1</sup> +

point from *1* to *m*.

vector, respectively.

shear-lag is constrained at the top.

**3.2. Axial forces of columns**

Hence, the axial force *Ni*

for columns in flange surfaces,

for columns in web surfaces, and

from Eq. (12) by

*um*

2*ES* \* *EI um*−<sup>1</sup>

Static solutions are obtained by solving a system of linear, homogeneous, simultaneous alge‐ braic equation (77) with respect to unknown displacement components at the internal mesh points. In finite difference method the equilibrium equations are formulated on each inner

in which the matrix **A** is the total stiffness matrix summed the individual stiffness ma‐ trix at each mesh point. **v** and **P** are the total displacement vector and total external load

Figure 7 shows stencil of equilibrium equations at a general inner point *i*. Figure 8 shows stencil of equilibrium equations at inner point *1* adjoining the base. Figure 9 shows stencil of equilibrium equations at inner point *i=m* for the case that the shear-lag is free at the top. Figure 10 shows stencil of equilibrium equations at inner point *i=m* for the case that the

Let us consider the axial forces of columns. The axial stress *σx* of the tube structure is given

(*z*)*u* \*'

in the *i*th column with the column's sectional area *Ai*

*z*2 *ti u* \*'

(*x*, *t*) + *φ* \*

*Ai* <sup>±</sup> *<sup>z</sup>* <sup>−</sup> *<sup>z</sup>* <sup>3</sup>

<sup>2</sup> (*y*<sup>2</sup> <sup>+</sup> *<sup>y</sup>*1)*Ai*

3*bz* 2 *z*1

*σ<sup>x</sup>* = *E yϕ*'

*Ni* = *E*{*yϕ*'

*Ni* <sup>=</sup> *<sup>E</sup>* <sup>1</sup>

\* (75)

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\* =0 (76)

*A v* =*P* (77)

(*x*, *t*) (78)

} (79)

*ϕ*' (80)

is

$$EI\left[ -\phi\_{m-1} + \phi\_{m+1} \right] \frac{1}{\Delta\Delta} + ES\left[ -\mu\_{m-1}^\* + \mu\_{m+1}^\* \right] \frac{1}{\Delta\Delta} = 0\tag{66}$$

$$\mathbb{E}\,ES\,\Big"\mathbb{I}-\phi\_{m-1}+\phi\_{m+1}\Big|\frac{1}{2\Delta}+\mathbb{E}\,I\,\Big"\mathbb{I}-\boldsymbol{u}\_{m-1}^{\*}+\boldsymbol{u}\_{m+1}^{\*}\Big|\frac{1}{2\Delta}=0\tag{67}$$

in which the mesh point *m* locates on the boundary point at the free end of *x* = *L* ; the mesh point *m* + 1 is imaginary point adjoining the mesh point *m* ; and the mesh point *m*−1 is inner point adjoining the mesh point *m* . Solving the above eqations for the variables *vm*+1 , *ϕm*+1 , *um*+1 \* at the imaginary point *m* + 1 , we have

$$
\Delta v\_{m+1} = v\_{m-1} - \Delta \Lambda \cdot \phi\_m \tag{68}
$$

$$
\phi\_{m+1} = \phi\_{m-1} \tag{69}
$$

$$
\mu\_{m+1}^\* = \mu\_{m-1}^\* \tag{70}
$$

On the other hand, the finite difference expressions for boundary conditions (59), (60), and (61b), in case where the shear-lag is constraint at the top, use the central diference for *v* and *ϕ* but backward difference for *u* \* , because *um*+1 \* is unsolvable in the use of the central difference.

$$\left[ -\upsilon\_{m-1} + \upsilon\_{m+1} \right] \frac{1}{2\Delta} + \phi\_m = 0 \tag{71}$$

$$EI\left[ -\phi\_{m-1} + \phi\_{m+1} \right] \frac{1}{2\Delta} + ES\left[ -\mu\_{m-1}^\* + \mu\_m^\* \right] \frac{1}{\Delta} = 0\tag{72}$$

$$
\mu\_m^\* = 0\tag{73}
$$

Solving the above eqations for the variables *vm*+1 , *ϕm*+1 , *um* \* , we have

$$
\upsilon\_{m+1} = \upsilon\_{m-1} - \mathcal{Q}\Lambda \cdot \phi\_m \tag{74}
$$

$$
\phi\_{m+1} = \phi\_{m-1} + \frac{2ES}{EI} \mathbf{u}\_{m-1}^\* \tag{75}
$$

$$
\mu\_m^\* = 0\tag{76}
$$

Static solutions are obtained by solving a system of linear, homogeneous, simultaneous alge‐ braic equation (77) with respect to unknown displacement components at the internal mesh points. In finite difference method the equilibrium equations are formulated on each inner point from *1* to *m*.

$$\begin{array}{c} A \ v = \mathcal{P} \end{array} \tag{77}$$

in which the matrix **A** is the total stiffness matrix summed the individual stiffness ma‐ trix at each mesh point. **v** and **P** are the total displacement vector and total external load vector, respectively.

Figure 7 shows stencil of equilibrium equations at a general inner point *i*. Figure 8 shows stencil of equilibrium equations at inner point *1* adjoining the base. Figure 9 shows stencil of equilibrium equations at inner point *i=m* for the case that the shear-lag is free at the top. Figure 10 shows stencil of equilibrium equations at inner point *i=m* for the case that the shear-lag is constrained at the top.

#### **3.2. Axial forces of columns**

On the other hand, using central difference method, the finite difference expressions for the boundary conditions (59), (60), and (61a), in case where the shear-lag is free at the top(*x=L*),

1

<sup>2</sup>*<sup>Δ</sup>* <sup>+</sup> *ES* \* <sup>−</sup>*um*−<sup>1</sup> \* <sup>+</sup> *um*+1 \* <sup>1</sup>

<sup>2</sup>*<sup>Δ</sup>* <sup>+</sup> *EI* \* <sup>−</sup>*um*−<sup>1</sup> \* <sup>+</sup> *um*+1 \* <sup>1</sup>

in which the mesh point *m* locates on the boundary point at the free end of *x* = *L* ; the mesh point *m* + 1 is imaginary point adjoining the mesh point *m* ; and the mesh point *m*−1 is inner point adjoining the mesh point *m* . Solving the above eqations for the variables *vm*+1 , *ϕm*+1 ,

On the other hand, the finite difference expressions for boundary conditions (59), (60), and (61b), in case where the shear-lag is constraint at the top, use the central diference for *v* and *ϕ*

1

<sup>2</sup>*<sup>Δ</sup>* <sup>+</sup> *ES* \* <sup>−</sup>*um*−<sup>1</sup>

\* <sup>+</sup> *um* \* 1

\*

, we have

*vm*+1 =*vm*−<sup>1</sup> −2*Δ* ⋅*ϕ<sup>m</sup>* (74)

−*vm*−<sup>1</sup> + *vm*+1

1

*um*

*EI* −*ϕm*−<sup>1</sup> + *ϕm*+1

Solving the above eqations for the variables *vm*+1 , *ϕm*+1 , *um*

<sup>2</sup>*<sup>Δ</sup>* <sup>+</sup> *<sup>ϕ</sup><sup>m</sup>* =0 (65)

*vm*+1 =*vm*−<sup>1</sup> −2*Δ* ⋅*ϕ<sup>m</sup>* (68)

*ϕm*+1 =*ϕm*−<sup>1</sup> (69)

*um*+1 \* <sup>=</sup>*um*−<sup>1</sup> \* (70)

<sup>2</sup>*<sup>Δ</sup>* <sup>+</sup> *<sup>ϕ</sup><sup>m</sup>* =0 (71)

\* =0 (73)

*<sup>Δ</sup>* =0 (72)

, because *um*+1 \* is unsolvable in the use of the central difference.

<sup>2</sup>*<sup>Δ</sup>* =0 (66)

<sup>2</sup>*<sup>Δ</sup>* =0 (67)

−*vm*−<sup>1</sup> + *vm*+1

1

1

*EI* −*ϕm*−<sup>1</sup> + *ϕm*+1

248 Advances in Vibration Engineering and Structural Dynamics

*ES* \* <sup>−</sup>*ϕm*−<sup>1</sup> <sup>+</sup> *<sup>ϕ</sup>m*+1

\* at the imaginary point *m* + 1 , we have

but backward difference for *u* \*

are expressed as

*um*+1

Let us consider the axial forces of columns. The axial stress *σx* of the tube structure is given from Eq. (12) by

$$
\sigma\_{\mathbf{x}} = \mathbb{E} \left[ y \phi^{\uparrow}(\mathbf{x}, t) + \phi^{\star}(\mathbf{z}) u^{\star}(\mathbf{x}, t) \right] \tag{78}
$$

Hence, the axial force *Ni* in the *i*th column with the column's sectional area *Ai* is

$$N\_i = E\left\{ y\phi^\cdot A\_i \pm \left[ z - \frac{z^3}{3b\_z^2} \right]\_{z1}^{z2} t\_i u^\cdot \right\} \tag{79}$$

for columns in flange surfaces,

$$N\_i = E\left[\frac{1}{2}(y\_2 + y\_1)A\_i\phi\right] \tag{80}$$

for columns in web surfaces, and

$$N\_i = E\left\{ y\phi^\cdot A\_i \pm \left[ z - \frac{z^3}{3b\_z^2} \right]\_{z1}^{z2} t\_i u^\cdot \right\} + E\left[ \frac{1}{2} (y\_2^2 - y\_1^2) \phi^\cdot \right] t\_i \tag{81}$$

for corner columns, in which *y*1 , *y*1 and *y*1 , *z*<sup>2</sup> = lower and upper coordinate values of the half between the *i*th column and both adjacent columns, respectively, and *ti* = the cross-sec‐ tional area *A*<sup>i</sup> of the *i*th column divided by the sum of half spans between the *i*th column and the both adjacent columns.


**Figure 9.** Stencil of equilibrium equations at inner point *i=m* for the case that the shear-lag is free at the top

**Figure 10.** Stencil of equilibrium equations at inner point *i=m* for the case that the shear-lag is constrained at the top

Consider free transverse vibrations of the current doubly-symmetric tube structures by

(*x*, *t*) are expressed as

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**4. Free transverse vibrations by finite difference method**

means of the finite difference method. Now, *v*(*x*, *t*) , *ϕ*(*x*, *t*) , and *u* \*

**Figure 7.** Stencil of equilibrium equations at inner point *i*


**Figure 8.** Stencil of equilibrium equations at inner point *1*


*Ni* = *E*{*yϕ*'

250 Advances in Vibration Engineering and Structural Dynamics

**Figure 7.** Stencil of equilibrium equations at inner point *i*

**Figure 8.** Stencil of equilibrium equations at inner point *1*

tional area *A*<sup>i</sup>

the both adjacent columns.

*Ai* <sup>±</sup> *<sup>z</sup>* <sup>−</sup> *<sup>z</sup>* <sup>3</sup>

3*bz* 2 *z*1

*z*2 *ti u* \*'

for corner columns, in which *y*1 , *y*1 and *y*1 , *z*<sup>2</sup> = lower and upper coordinate values of the half between the *i*th column and both adjacent columns, respectively, and *ti* = the cross-sec‐

} <sup>+</sup> *<sup>E</sup>* <sup>1</sup> <sup>2</sup> (*y*<sup>2</sup> <sup>2</sup> <sup>−</sup> *<sup>y</sup>*<sup>1</sup> 2

of the *i*th column divided by the sum of half spans between the *i*th column and

)*ϕ*' *ti* (81)

**Figure 9.** Stencil of equilibrium equations at inner point *i=m* for the case that the shear-lag is free at the top


**Figure 10.** Stencil of equilibrium equations at inner point *i=m* for the case that the shear-lag is constrained at the top

### **4. Free transverse vibrations by finite difference method**

Consider free transverse vibrations of the current doubly-symmetric tube structures by means of the finite difference method. Now, *v*(*x*, *t*) , *ϕ*(*x*, *t*) , and *u* \* (*x*, *t*) are expressed as

$$
\bar{v}(\mathbf{x},t) = \bar{v}(\mathbf{x},t) \exp\{i\omega t\} \tag{82}
$$

**5. Forced transverse vibrations by finite difference method**

¨} <sup>+</sup> *cv* {*v*˙}<sup>−</sup> (*κGA*(*<sup>v</sup>* '

in which *M* = mass matrix; *cv* = the damping coefficient matrix; and {*v*

flection vector {*v*} and the rotational angle vector {*ϕ*} may be written as

{*v*}=∑ *j*=1 *n βj* {*v*} *<sup>j</sup> qj*

{*ϕ*}=∑ *j*=1 *n βj* {*ϕ*} *<sup>j</sup> qj*

0 at the base may be written in the matrix form as

in which *β<sup>j</sup>* = the *j-*th participation coefficient; {*v*} *<sup>j</sup>*

into Eq. (89) and multiplying the reduced equation by {*v*}

*<sup>i</sup> β<sup>i</sup> q* ¨ *i* (*t*) + {*v*} *i <sup>T</sup> cv* {*v*}

> *i* " *qi* <sup>+</sup> {*v*} *i <sup>T</sup>* (*κGA*)'

*ω* <sup>2</sup> *M* {*v*}

*<sup>T</sup>* (*κGA*) {*v*}

and *ϕ* , respectively; *qj*

{*v*} *i <sup>T</sup> M* {*v*}

{*v*} *i*

Now, Eq. (86) may be rewritten as

Multiplying the above equation by {*v*}

we have

{*v*} *i <sup>T</sup> M* {*v*} *i βi q* ¨ *i* (*t*) + {*v*} *i <sup>T</sup> c* {*v*} *i βi q*˙*i* (*t*) + *β<sup>i</sup>*

*M* {*v*

*v* ¨

Forced lateral vibration of current tube structures may be obtained easily by means of modal analysis for elastic behavior subject to earthquake motion. Applying the finite difference method into Eq. (32), the equation of motion of current tube structures with distributed properties may be changed to discreet structure with degrees of freedom three times the to‐ tal number of mesh points because each mesh point has three freedoms for the displacement components. Hence, Eq. (32) for current tube structure, subjected to earthquake acceleration

the relative acceleration vector, the relative velocity vector and relative displacement vector, respectively, measured from the base. {1}= unit vector. It is assumed that the dynamic de‐

number of degrees of freedom taken into consideration here. Substituting Eqs. (90) and (91)

{*v*} *i* '

> *i* ' + {*ϕ*} *i*

*ω* 2{*v*} *i <sup>T</sup> M* {*v*} *i qi* = −{*v*} *i <sup>T</sup> M* {1}*v* ¨

*<sup>i</sup>* − *κGA*({*v*}

*i*

*<sup>i</sup> β<sup>i</sup> q*˙*<sup>i</sup>* (*t*)−

*qi β<sup>i</sup>* = −{*v*}

+ *ϕ*))' = − *M* {1}{*v*

(*t*)= the *j*-th dynamic response depending on time *t* ; and *n* = the total

*i*

*i*

*<sup>T</sup>* , we have

*<sup>T</sup> M* {1} *v* ¨ 0

*<sup>T</sup>* and substituting the reduced equation into Eq. (93),

¨

(*t*) (90)

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(*t*) (91)

and {*ϕ*} *<sup>j</sup>* = the *j*-th eigenfunctions for *v*

) ' =0 (93)

0} (89)

¨} , {*v*˙} , and {*v*} are

(92)

<sup>0</sup> (94)

$$
\phi(\mathbf{x}, t) = \overline{\phi}(\mathbf{x}, t) \exp\{i\omega t\} \tag{83}
$$

$$
\hat{u}^\*(\mathbf{x}, t) = \bar{\hat{u}}^\*(\mathbf{x}, t) \exp\{i\omega t\} \tag{84}
$$

Substituting the above equations into the equations of motion for the free transverse vibra‐ tion obtained from Eqs. (32)-(39), the equations for free vibrations become

$$\left[\delta\nu: \omega^2 m\bar{\upsilon} + \mathsf{f}\_{\kappa} \mathsf{G} A \{\bar{\upsilon}^\cdot + \overline{\phi}\} \right] = 0 \tag{85}$$

$$\delta\phi \colon \omega^2 \rho I \overline{\phi} + \omega^2 \rho S \stackrel{\*}{\overline{u}}^\* + \{EI\overline{\phi}\} \text{---} \kappa GA (\overline{v}^\cdot + \overline{\phi}) + \{ES \stackrel{\*}{\overline{u}}^\*\overline{u}^\*\} \text{=} 0 \tag{86}$$

$$\left(\delta\boldsymbol{u}^{\ast}.\boldsymbol{\omega}^{2}\rho\boldsymbol{S}^{\ast}\overline{\boldsymbol{\phi}}+\boldsymbol{\omega}^{2}\rho\boldsymbol{I}^{\ast}\overline{\boldsymbol{u}}^{\ast}+\left(\boldsymbol{E}\boldsymbol{I}^{\ast}\overline{\boldsymbol{u}}^{\ast}\right)\right)\cdot\left(\boldsymbol{E}\boldsymbol{S}^{\ast}\overline{\boldsymbol{\phi}}\right)-\kappa\boldsymbol{G}\boldsymbol{F}^{\ast}\overline{\boldsymbol{u}}^{\ast}=\boldsymbol{0}\tag{87}$$

The finite difference expressions of the above equations reduce to eigenvalue problem for *v*¯ , *<sup>ϕ</sup>*¯ , and *u*¯\* .

$$
\begin{bmatrix} \mathbf{A} - \omega^2 \mathbf{B} \end{bmatrix} \mathbf{v} = \mathbf{0} \tag{88}
$$

Here the matrix **A** is the total stiffness matrix as given in Eq. (78). On the other hand, the matrix **B** is total mass matrix which is the sum of individual mass matrix. The individual mass matrix at the *i*th mesh point is given in Figure 11. The *it*h natural frequencies *ω<sup>i</sup>* can be obtained from the *i*th eigenvalue.


**Figure 11.** Individual mass matrix at mesh point *i*

## **5. Forced transverse vibrations by finite difference method**

*v*(*x*, *t*)=*v*¯(*x*, *t*)exp{*iωt*} (82)

*<sup>ϕ</sup>*(*x*, *<sup>t</sup>*)=*ϕ*¯(*<sup>x</sup>*, *<sup>t</sup>*)exp{*iωt*} (83)

<sup>+</sup> *<sup>ϕ</sup>*¯) <sup>+</sup> (*ES* \*

)−*κGF* \*

*u*¯\*

*B v* =0 (88)

(*x*, *t*)exp{*iωt*} (84)

<sup>+</sup> *<sup>ϕ</sup>*¯) '=0 (85)

*u*¯\*')'=0 (86)

=0 (87)

can be

*u* \*

*δv* :*ω* <sup>2</sup>

*ρS* \*

*ρI* \*

*<sup>ϕ</sup>*¯ <sup>+</sup> *<sup>ω</sup>* <sup>2</sup>

*<sup>ρ</sup>Iϕ*¯ <sup>+</sup> *<sup>ω</sup>* <sup>2</sup>

*δϕ* :*ω* <sup>2</sup>

252 Advances in Vibration Engineering and Structural Dynamics

*δu* \* :*ω* <sup>2</sup> *ρS* \*

obtained from the *i*th eigenvalue.

**Figure 11.** Individual mass matrix at mesh point *i*

*<sup>ϕ</sup>*¯ , and *u*¯\*

.

(*x*, *t*)=*u*¯\*

tion obtained from Eqs. (32)-(39), the equations for free vibrations become

*mv*¯ + *κGA*(*v*¯'

*u*¯\* + (*EI* \*

*A*−*ω* <sup>2</sup>

*<sup>u</sup>*¯\* <sup>+</sup> (*EIϕ*¯)'−*κGA*(*v*¯'

*u*¯\*') '

The finite difference expressions of the above equations reduce to eigenvalue problem for *v*¯ ,

Here the matrix **A** is the total stiffness matrix as given in Eq. (78). On the other hand, the matrix **B** is total mass matrix which is the sum of individual mass matrix. The individual

mass matrix at the *i*th mesh point is given in Figure 11. The *it*h natural frequencies *ω<sup>i</sup>*

+ (*ES* \* *ϕ*¯'

Substituting the above equations into the equations of motion for the free transverse vibra‐

Forced lateral vibration of current tube structures may be obtained easily by means of modal analysis for elastic behavior subject to earthquake motion. Applying the finite difference method into Eq. (32), the equation of motion of current tube structures with distributed properties may be changed to discreet structure with degrees of freedom three times the to‐ tal number of mesh points because each mesh point has three freedoms for the displacement components. Hence, Eq. (32) for current tube structure, subjected to earthquake acceleration *v* ¨ 0 at the base may be written in the matrix form as

$$\mathbf{f}\mathbf{M}\mathbf{J}\mathbf{\ddot{\boldsymbol{\upsilon}}\boldsymbol{\upsilon}} + \mathbf{f}\mathbf{c}\_{\upsilon}\mathbf{J}\mathbf{\dot{\boldsymbol{\upsilon}}\boldsymbol{\upsilon}} - \mathbf{f}\mathbf{\dot{\boldsymbol{\chi}}}\mathbf{c}\mathbf{A}\mathbf{A}\mathbf{\dot{\boldsymbol{\upsilon}}\boldsymbol{\upsilon}} + \boldsymbol{\phi}\mathbf{\dot{\boldsymbol{\beta}}}\mathbf{\dot{\boldsymbol{\upsilon}}} = -\mathbf{f}\mathbf{M}\mathbf{J}\mathbf{\dot{\mathbf{1}}}\mathbf{1}\mathbf{\dot{\boldsymbol{\upsilon}}\boldsymbol{\upsilon}}\_{\Gamma}\tag{89}$$

in which *M* = mass matrix; *cv* = the damping coefficient matrix; and {*v* ¨} , {*v*˙} , and {*v*} are the relative acceleration vector, the relative velocity vector and relative displacement vector, respectively, measured from the base. {1}= unit vector. It is assumed that the dynamic de‐ flection vector {*v*} and the rotational angle vector {*ϕ*} may be written as

$$\{\upsilon\} = \sum\_{j=1}^{n} \beta\_j \{\upsilon\} \,\, \_j\eta\_j(t) \tag{90}$$

$$\{\{\phi\}\} = \sum\_{j=1}^{n} \beta\_j \{\phi\} \, \, \_j \eta\_j(t) \tag{91}$$

in which *β<sup>j</sup>* = the *j-*th participation coefficient; {*v*} *<sup>j</sup>* and {*ϕ*} *<sup>j</sup>* = the *j*-th eigenfunctions for *v* and *ϕ* , respectively; *qj* (*t*)= the *j*-th dynamic response depending on time *t* ; and *n* = the total number of degrees of freedom taken into consideration here. Substituting Eqs. (90) and (91) into Eq. (89) and multiplying the reduced equation by {*v*} *i <sup>T</sup>* , we have

$$\begin{aligned} \{\boldsymbol{\upsilon}\}\_i^T \{\boldsymbol{\mathsf{M}}\} \{\boldsymbol{\upsilon}\}\_i \beta\_i \overset{\scriptstyle \boldsymbol{\alpha}}{\boldsymbol{q}}\_i \{\boldsymbol{t}\} + \{\boldsymbol{\upsilon}\}\_i^T \{\boldsymbol{\mathsf{c}}\_\upsilon\} \{\boldsymbol{\upsilon}\}\_i \beta\_i \overset{\scriptstyle \boldsymbol{\alpha}}{\boldsymbol{q}}\_i \{\boldsymbol{t}\} - \\ \{\boldsymbol{\upsilon}\}\_i^T \{\boldsymbol{\kappa}\} \{\boldsymbol{\upsilon}\}\_i^{\scriptscriptstyle \boldsymbol{q}} \boldsymbol{q}\_i + \{\boldsymbol{\upsilon}\}\_i^T \{\boldsymbol{\kappa}\} \boldsymbol{\epsilon} \boldsymbol{M}^{\scriptscriptstyle \boldsymbol{q}} \{\boldsymbol{\upsilon}\}\_i^{\scriptscriptstyle \boldsymbol{q}} \boldsymbol{q}\_i \} \beta\_i = -\{\boldsymbol{\upsilon}\}\_i^T \{\boldsymbol{\mathsf{M}}\} \{\boldsymbol{1}\}\_i \overset{\scriptstyle \boldsymbol{\alpha}}{\boldsymbol{\upsilon}}\_0 \end{aligned} \tag{92}$$

Now, Eq. (86) may be rewritten as

$$\left[\omega^2 \mathbf{\tilde{f}} M \mathbf{J} \{\boldsymbol{\upsilon}\}\_i - \left[\kappa GA \{\{\boldsymbol{\upsilon}\}\_i^\cdot + \{\boldsymbol{\phi}\}\_i\right] \right] = 0 \tag{93}$$

Multiplying the above equation by {*v*} *i <sup>T</sup>* and substituting the reduced equation into Eq. (93), we have

$$\{\boldsymbol{\upsilon}\}\_{i}^{T}\mathbf{\mathsf{f}}\mathbf{M}\mathbf{J}\{\boldsymbol{\upsilon}\}\_{i}\boldsymbol{\beta}\_{i}\boldsymbol{\dot{q}}\_{i}(t) + \{\boldsymbol{\upsilon}\}\_{i}^{T}\mathbf{\mathsf{f}}\mathbf{J}\{\boldsymbol{\upsilon}\}\_{i}\boldsymbol{\beta}\_{i}\boldsymbol{\dot{q}}\_{i}(t) + \boldsymbol{\beta}\_{i}\boldsymbol{\omega}^{T}\{\boldsymbol{\upsilon}\}\_{i}^{T}\mathbf{\mathsf{f}}\mathbf{M}\mathbf{J}\{\boldsymbol{\upsilon}\}\_{i}\boldsymbol{q}\_{i} = -\{\boldsymbol{\upsilon}\}\_{i}^{T}\mathbf{\mathsf{f}}\mathbf{M}\mathbf{J}\{\boldsymbol{\upsilon}\}\_{0}^{\top} \tag{94}$$

Here, the damping coefficient matrix *cv* and the participation coefficients *β<sup>i</sup>* are assumed to satisfy the following expressions:

$$\frac{\{\upsilon\}\_i^T \{c\_v\} \{v\_i\}}{\{\upsilon\}\_i^T \{M\} \{v\_i\}} = \mathcal{Z}h\_i \omega\_i \tag{95}$$

**2.** the dynamic loads are taken from El Centro 1940 NS, Taft 1952 EW, and Hachinohe

**3.** the damping ratio for the first mode of the frame-tubes is *h*<sup>1</sup> = 0.02, and the higher

**4.** the weight of each floor is 9.807 kN/m-2 and the mass of the frame-tube is considered to

**5.** in the modal analysis, the number of modes for the participation coefficients is taken

;

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1968 NS, in which each maximum acceleration is 200 m/s2

damping ratio for the *n*-th mode is *h*<sup>1</sup> =*h*1*ω<sup>n</sup>* / *ω*1 ;

be only floor's weight;

**Figure 12.** Numerical models [6]

five into consideration as five.

$$\beta\_i = \frac{\{\upsilon\}\_i^T \mathbf{[MJ]} \mathbf{1}\{\mathbf{1}\}}{\{\upsilon\}\_i^T \mathbf{[MJ]} \{\upsilon\_i\}}\tag{96}$$

in which *hi* is the *i*th damping constant. Thus, Eq. (94) may be reduced to

$$
\ddot{\ddot{q}}\_i(t) + 2h\_i \omega\_i \dot{q}\_i(t) + \omega\_i^2 q\_i(t) = -\ddot{\ddot{v}}\_0 \tag{97}
$$

The general solution of Eq. (98) is

$$q\_i(t) = \exp\{-h\_i\omega\_i t\} \left(\mathbf{C}\_1 \sin\omega\_{Di} t + \mathbf{C}\_2 \cos\omega\_{Di} t\right) - \frac{1}{\omega\_{Di}} \Big|\_{0}^{t} \exp\{-h\_i\omega\_i (t-\tau)\} \sin\omega\_{Di} (t-\tau)\ddot{\bar{v}}\_0 d\tau \tag{98}$$

in which *ωDi* =*ω<sup>i</sup>* 1−*hi* 2 and *C*1 and *C*<sup>2</sup> are constants determined from the initial conditions. The Duhamel integral in Eq. (98) may be calculated approximately by means of Paz [19] or Takabatake [2].

## **6. Numerical results by finite difference method**

#### **6.1. Numerical models**

Numerical models for examining the simplified analysis proposed here have are shown in Figure 12. These numerical models are determined to find out the following effects: (1) the effect of the aspect ratio of the outer and inner tubes; (2) the effect of omitting the corners; and (3) the effect of bracing. **Model T1** is a doubly symmetric single frame-tube prepared for comparison with the numerical results of the doubly symmetric frame-double-tube. **T7** and **T8** are made up steel reinforced concrete frame-tubes, and the other models are steel frametubes. The total number of stories is 30. The difference between models **T2** to **T5** concerns the number of story and span attached bracing. The members of the single and double tubes are shown in Figures 13 and 14.

In the numerical computation, the following assumptions are made:

**1.** the static lateral force is a triangularly distributed load, as shown in Figures 13 and 14;



**Figure 12.** Numerical models [6]

Here, the damping coefficient matrix *cv* and the participation coefficients *β<sup>i</sup>* are assumed to

} =2*hi*

*ω<sup>i</sup>* (95)

} (96)

<sup>0</sup> (97)

(*t* −*τ*)*v* ¨

<sup>0</sup>*dτ* (98)

{*v*} *i <sup>T</sup> cv* {*vi* }

{*v*} *i <sup>T</sup> <sup>M</sup>* {*vi*

> *β<sup>i</sup>* = {*v*} *i <sup>T</sup> M* {1}

> > *ωi q*˙*i* (*t*) + *ω<sup>i</sup>* 2 *qi* (*t*)= −*v* ¨

*t* + *C*2cos*ωDi*

**6. Numerical results by finite difference method**

In the numerical computation, the following assumptions are made:

*q* ¨ *i* (*t*) + 2*hi* {*v*} *i <sup>T</sup> <sup>M</sup>* {*vi*

is the *i*th damping constant. Thus, Eq. (94) may be reduced to

*<sup>t</sup>*) <sup>−</sup> <sup>1</sup> *ωDi ∫* 0 *t*

The Duhamel integral in Eq. (98) may be calculated approximately by means of Paz [19] or

Numerical models for examining the simplified analysis proposed here have are shown in Figure 12. These numerical models are determined to find out the following effects: (1) the effect of the aspect ratio of the outer and inner tubes; (2) the effect of omitting the corners; and (3) the effect of bracing. **Model T1** is a doubly symmetric single frame-tube prepared for comparison with the numerical results of the doubly symmetric frame-double-tube. **T7** and **T8** are made up steel reinforced concrete frame-tubes, and the other models are steel frametubes. The total number of stories is 30. The difference between models **T2** to **T5** concerns the number of story and span attached bracing. The members of the single and double tubes

**1.** the static lateral force is a triangularly distributed load, as shown in Figures 13 and 14;

exp −*hi*

*ωi*

and *C*1 and *C*<sup>2</sup> are constants determined from the initial conditions.

(*t* −*τ*) sin*ωDi*

satisfy the following expressions:

254 Advances in Vibration Engineering and Structural Dynamics

The general solution of Eq. (98) is

*ωi*

*t*)(*C*1sin*ωDi*

2

in which *hi*

*qi*

(*t*)=exp( −*hi*

in which *ωDi* =*ω<sup>i</sup>* 1−*hi*

**6.1. Numerical models**

are shown in Figures 13 and 14.

Takabatake [2].

MOS is negligible, in practice. The ratios are those of the values obtained from the present theory to the corresponding values from the three-dimensional frame analysis using NAS‐ TRAN and DEMOS. The distributions of the static lateral displacements are shown in Figure 15. These numerical results show the simplified analysis is in good agreement, in practice, with the results of three-dimensional frame analysis using NASTRAN and DEMOS. Since the shear-lag is far smaller than the transverse deflections, as shown in Table 2 the discrep‐

**Maximum static lateral deflection (m)**

T1 0.441 0.430 1.026 T2 0.327 0.343 0.953 T3 0.307 0.318 0.965 T4 0.299 0.319 0.937 T5 0.312 0.330 0.945 T6 0.329 0.311 1.058 T7 0.151 0.158 0.956 T8 0.157 0.166 0.946

**Maximum shear-lag (m)**

T1 0.0145 0.0079 1.835 T2 0.0149 0.0085 1.753 T7 0.0090 0.0052 1.731 T8 0.0103 0.0028 3.679

Figure 16 shows the distribution of axial forces of model **T2**. A discrepancy between the re‐ sults obtained from the proposed theory and those from three-dimensional frame analysis is found. However, this discrepancy is within 10 % and is also allowable for practical use be‐ cause the axial forces in tube structures are designed from the axial forces on the flange sur‐

Frame analysis (3)

Frame analysis (3)

Ratio(2)/(3) (4)

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Ratio(2)/(3) (4)

ancy in shear-lag is negligible in practice.

**Table 1.** Maximum values of static lateral deflections [6]

**Table 2.** Maximum values of static shear-lags [6]

Present theory (2)

Present theory (2)

faces, being always larger than those on the web surfaces.

Model (1)

Model (1)

**Figure 13.** Member of numerical models T1 and T5 [6]

**Figure 14.** Member of numerical models T7 and T8 [6]

#### **6.2. Static numerical results**

First, the static numerical results are stated. Tables 1 and 2 show the maximum values of the static lateral displacements and shear-lags, calculated from the present theory, NASTRAN and DEMOS, in which a discrepancy between results obtained from NASTRAN and DE‐ MOS is negligible, in practice. The ratios are those of the values obtained from the present theory to the corresponding values from the three-dimensional frame analysis using NAS‐ TRAN and DEMOS. The distributions of the static lateral displacements are shown in Figure 15. These numerical results show the simplified analysis is in good agreement, in practice, with the results of three-dimensional frame analysis using NASTRAN and DEMOS. Since the shear-lag is far smaller than the transverse deflections, as shown in Table 2 the discrep‐ ancy in shear-lag is negligible in practice.


**Table 1.** Maximum values of static lateral deflections [6]

**Figure 13.** Member of numerical models T1 and T5 [6]

256 Advances in Vibration Engineering and Structural Dynamics

**Figure 14.** Member of numerical models T7 and T8 [6]

First, the static numerical results are stated. Tables 1 and 2 show the maximum values of the static lateral displacements and shear-lags, calculated from the present theory, NASTRAN and DEMOS, in which a discrepancy between results obtained from NASTRAN and DE‐

**6.2. Static numerical results**


**Table 2.** Maximum values of static shear-lags [6]

Figure 16 shows the distribution of axial forces of model **T2**. A discrepancy between the re‐ sults obtained from the proposed theory and those from three-dimensional frame analysis is found. However, this discrepancy is within 10 % and is also allowable for practical use be‐ cause the axial forces in tube structures are designed from the axial forces on the flange sur‐ faces, being always larger than those on the web surfaces.

**Natural frequency (rad/s)**

Second (4)

Frame analysis 2.058 6.223 11.076 16.020 20.826 Ratio 1.011 1.005 1.007 0.997 0.990

Present theory 2.137 6.197 11.694 15.924 21.740 Frame analysis 2.138 6.290 11.687 16.198 22.062 Ratio 1.000 0.985 1.001 0.983 0.985

Present theory 2.192 6.186 11.112 16.805 21.334 Frame analysis 2.152 6.296 11.224 16.813 21.756 Ratio 1.019 0.983 0.990 1.000 0.981

Present theory 2.126 6.266 11.559 16.073 21.390 Frame analysis 2.100 6.260 11.464 16.136 21.636 Ratio 1.012 1.001 1.008 0.996 0.989

Present theory 2.055 6.045 11.631 16.053 22.019 Frame analysis 2.147 6.369 11.848 16.662 22.610 Ratio 0.957 0.949 0.982 0.963 0.974

Present theory 3.458 9.920 17.924 25.863 32.626 Frame analysis 3.462 10.037 17.983 26.246 34.675 Ratio 0.999 0.988 0.997 0.985 0.941

Present theory 3.401 9.811 17.825 25.786 32.696 Frame analysis 3.382 9.856 17.729 26.028 34.561 Ratio 1.006 0.995 1.005 0.991 0.946

Secondly, consider the natural frequencies. Table 3 shows the natural frequencies of the above-mentioned numerical models. It follows that, in practical use, the simplified analysis gives in excellent agreement with the results obtained from the three-dimensional frame analysis using NASTRAN and DEMOS. Since the transverse stiffness of the bracing is far larger than for frames, the transverse stiffness of current frame-tube with braces varies dis‐ continuously, particularly at the part attached to the bracing. However, such discontinuous

Thirdly, let us present dynamic results. The maximum values of dynamic deflections, story shears, and overturning moments are shown in Tables 4-6, respectively. Figure 17 shows the distribution of the maximum dynamic deflections and of the maximum story shear forces of model **T7** for El Centro 1949 NS. Figure 18 indicates the distribution of the absolute accelera‐

Third (5)

Fourth (6)

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Fifth (7)

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259

First (3)

Model (1)

T3

T4

T5

T6

T7

T8

**6.3. Free vibration results**

**6.4. Dynamic results**

Analytical methods (2)

**Table 3.** Natural frequencies [6]. Note. Ratio = present theory/frame analysis

and local variation due to bracing can be expressed by the present theory.

**Figure 15.** Distribution of static lateral deflection [6]

**Figure 16.** Axial force [6]



**Table 3.** Natural frequencies [6]. Note. Ratio = present theory/frame analysis

### **6.3. Free vibration results**

**Figure 15.** Distribution of static lateral deflection [6]

258 Advances in Vibration Engineering and Structural Dynamics

**Figure 16.** Axial force [6]

Model (1)

T1

Analytical methods (2)

**Natural frequency (rad/s)**

Second (4)

Present theory 1.998 6.077 10.942 15.759 20.397 Frame analysis 2.062 6.211 11.048 15.907 20.648 Ratio 0.969 0.978 0.990 0.991 0.988 T2 Present theory 2.080 6.255 11.150 15.977 20.614

Third (5)

Fourth (6)

Fifth (7)

First (3)

Secondly, consider the natural frequencies. Table 3 shows the natural frequencies of the above-mentioned numerical models. It follows that, in practical use, the simplified analysis gives in excellent agreement with the results obtained from the three-dimensional frame analysis using NASTRAN and DEMOS. Since the transverse stiffness of the bracing is far larger than for frames, the transverse stiffness of current frame-tube with braces varies dis‐ continuously, particularly at the part attached to the bracing. However, such discontinuous and local variation due to bracing can be expressed by the present theory.

#### **6.4. Dynamic results**

Thirdly, let us present dynamic results. The maximum values of dynamic deflections, story shears, and overturning moments are shown in Tables 4-6, respectively. Figure 17 shows the distribution of the maximum dynamic deflections and of the maximum story shear forces of model **T7** for El Centro 1949 NS. Figure 18 indicates the distribution of the absolute accelera‐ tions and of the maximum overturning moments for model **T7**. Thus, the proposed approxi‐ mate theory is in good agreement with the results of the three-dimensional frame analysis using NASTRAN and DEMOS in practice. These excellent agreements may be estimated from participation functions as shown in Figure 19. The present one-dimensional extended rod theory used the finite difference method can always express discontinuous and local be‐ havior caused by the part attached to the bracing.

gence is obtained by the number of mesh points, being equal to the number of stories of the

**Maximum story shear (kN)**

El Centro NS 5482 6659 0.823 Hachinohe NS 12494 10003 1.249 Taft EW 4972 4835 1.028

El Centro NS 11464 11082 1.035 Hachinohe NS 17260 17309 0.997 Taft EW 8414 8071 1.043

El Centro NS 12239 11768 1.040 Hachinohe NS 18937 19378 0.977 Taft EW 9248 9012 1.026

El Centro NS 14749 12258 1.203 Hachinohe NS 27498 21084 1.304 Taft EW 9316 9307 1.001

El Centro NS 11484 11180 1.027 Hachinohe NS 18172 17515 1.038 Taft EW 8865 8659 1.024

El Centro NS 11562 12160 0.951 Hachinohe NS 17632 20270 0.870 Taft EW 9807 9150 1.072

El Centro NS 38746 37167 1.042 Hachinohe NS 56153 53642 1.047 Taft EW 56731 54819 1.035

El Centro NS 41306 40109 1.030 Hachinohe NS 59595 57957 1.028 Taft EW 49004 44718 1.096

**Maximum overturning moment (MN m)**

Present theory (3)

El Centro NS 3.233 3.991 0.810 Hachinohe NS 6.099 5.599 1.089 Taft EW 2.731 2.863 0.954

Frame analysis (4)

Ratio(3)/(4) (5)

Frame analysis (4)

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Ratio(3)/(4) (5)

Present theory (3)

tube structures.

Model (1)

T1

T2

T3

T4

T5

T6

T7

T8

**Table 5.** Maximum story shears [6]

Model (1)

T1

Earthquake type (2)

Earthquake type (2)


**Table 4.** Maximum dynamic lateral deflections [6]

The above-mentioned numerical computations are obtained from that the total number of mesh points, including the top, is 60. Figure 20 shows the convergence characteristics of the static and dynamic responses for model **T7**, due to the number of mesh points. The conver‐ gence is obtained by the number of mesh points, being equal to the number of stories of the tube structures.


**Table 5.** Maximum story shears [6]

tions and of the maximum overturning moments for model **T7**. Thus, the proposed approxi‐ mate theory is in good agreement with the results of the three-dimensional frame analysis using NASTRAN and DEMOS in practice. These excellent agreements may be estimated from participation functions as shown in Figure 19. The present one-dimensional extended rod theory used the finite difference method can always express discontinuous and local be‐

**Maximum dynamic lateral deflection (m)**

Present theory (3)

El Centro NS 0.263 0.293 0.898 Hachinohe NS 0.453 0.411 1.102 Taft EW 0.213 0.208 1.024

El Centro NS 0.311 0.291 1.069 Hachinohe NS 0.420 0.419 1.002 Taft EW 0.212 0.213 0.995

El Centro NS 0.327 0.329 0.994 Hachinohe NS 0.460 0.465 0.989 Taft EW 0.205 0.206 0.995

El Centro NS 0.318 0.324 0.981 Hachinohe NS 0.515 0.469 1.098 Taft EW 0.196 0.201 0.975

El Centro NS 0.327 0.315 1.038 Hachinohe NS 0.454 0.429 1.058 Taft EW 0.207 0.212 0.976

El Centro NS 0.297 0.321 0.925 Hachinohe NS 0.424 0.437 0.970 Taft EW 0.211 0.207 1.019

El Centro NS 0.155 0.150 1.033 Hachinohe NS 0.202 0.195 1.036 Taft EW 0.221 0.215 1.028

El Centro NS 0.158 0.154 1.026 Hachinohe NS 0.232 0.235 0.987 Taft EW 0.219 0.210 1.043

The above-mentioned numerical computations are obtained from that the total number of mesh points, including the top, is 60. Figure 20 shows the convergence characteristics of the static and dynamic responses for model **T7**, due to the number of mesh points. The conver‐

Frame analysis (4)

Ratio(3)/(4) (5)

havior caused by the part attached to the bracing.

260 Advances in Vibration Engineering and Structural Dynamics

Earthquake type (2)

Model (1)

T1

T2

T3

T4

T5

T6

T7

T8

**Table 4.** Maximum dynamic lateral deflections [6]



**Figure 18.** Distribution of absolute acceleration and overturning moment [6]

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**Figure 19.** Participation functions [6]

**Table 6.** Maximum overturning moments [6]

**Figure 17.** Distribution of dynamic lateral deflection and story shear force [6]

**Figure 18.** Distribution of absolute acceleration and overturning moment [6]

**Figure 19.** Participation functions [6]

**Maximum overturning moment (MN m)**

Present theory (3)

El Centro NS 7.118 6.531 1.090 Hachinohe NS 9.829 9.413 1.044 Taft EW 4.928 4.849 1.016

El Centro NS 8.090 7.972 1.015 Hachinohe NS 11.277 11.122 1.014 Taft EW 5.129 5.070 1.012

El Centro NS 8.140 7.727 1.053 Hachinohe NS 13.827 10.983 1.259 Taft EW 4.984 5.021 0.993

El Centro NS 7.879 7.315 1.077 Hachinohe NS 10.778 10.250 1.052 Taft EW 5.053 5.007 1.009

El Centro NS 6.566 8.070 0.814 Hachinohe NS 9.593 11.431 0.839 Taft EW 4.968 5.091 0.976

El Centro NS 22.148 21.574 1.027 Hachinohe NS 29.914 28.929 1.034 Taft EW 32.186 31.675 1.016

El Centro NS 21.662 21.659 1.000 Hachinohe NS 32.693 33.462 0.977 Taft EW 28.779 28.246 1.019

Frame analysis (4)

Ratio(3)/(4) (5)

Model (1)

T2

T3

T4

T5

T6

T7

T8

**Table 6.** Maximum overturning moments [6]

**Figure 17.** Distribution of dynamic lateral deflection and story shear force [6]

Earthquake type (2)

262 Advances in Vibration Engineering and Structural Dynamics

in which *I*

ten as

*EI v* ‴′ + *ρAv*

*<sup>κ</sup>GA* <sup>+</sup> *<sup>ρ</sup><sup>I</sup>* \*

*ρI* ^ *<sup>κ</sup>GA* ( <sup>−</sup>*Py*

*ρI* ^ *<sup>κ</sup>GA* ( <sup>−</sup>*Py*

> ^ *v* ‴‴ + *EI* \* *κGF* \*

+( *<sup>ρ</sup><sup>I</sup>*

<sup>+</sup> *<sup>ρ</sup><sup>I</sup>* \* *κGF* \*

<sup>−</sup> *EI* \* *κGF* \*

<sup>−</sup> *EI* \* *<sup>κ</sup>GF* \* *EI*

^

is defined as

¨ <sup>+</sup> *cvv*˙ <sup>−</sup>*Py* <sup>−</sup>*ρ<sup>I</sup> <sup>v</sup>*

*<sup>κ</sup>GF* \* )( <sup>−</sup> *Py*

governing equation is given

+ *ρAv*

Hence, Eq. (104) may be rewritten

*EI v* ″″

¨ ″ <sup>−</sup> *EI*

+ *ρAv*

¨ <sup>+</sup> *cvv*˙)

*EI* ^ *<sup>κ</sup>GA* ( <sup>−</sup>*Py*

¨ <sup>+</sup> *cvv*˙ <sup>−</sup> *Py* <sup>−</sup>*ρ<sup>I</sup> <sup>v</sup>*

+ *ρAv*

+ *ρAv*

*<sup>κ</sup>GA* ( <sup>−</sup> *<sup>P</sup>* ″

¨ <sup>+</sup> *cvv*˙)⋅⋅ <sup>+</sup> *<sup>ρ</sup><sup>I</sup>* \*

¨ <sup>+</sup> *cvv*˙)⋅⋅⋅− *<sup>ρ</sup><sup>I</sup>* \*

*I* ^ <sup>=</sup> *<sup>I</sup>* <sup>1</sup><sup>−</sup> (*<sup>S</sup>* \*)2

*<sup>y</sup>* + *ρAv*

^ *v* ¨ ″″

*<sup>κ</sup>GF* \* *EI*

¨ <sup>+</sup> *cvv*˙)''''

*<sup>κ</sup>GF* \* *<sup>ρ</sup><sup>I</sup>* ^ *v* ″ ⋅⋅⋅⋅ +

+ *ρAv*

¨ ″ <sup>−</sup> *EI*

plify the future expression, the following notation is introduced

*<sup>κ</sup>GA* (<sup>1</sup> <sup>+</sup>

( ) <sup>1</sup>

k =k

*GA GA GA I*

Differentiating Eq. (33) with respect to *x* and substituting Eqs. (32), (99), and (100) into the result, the equation of motion expressed in terms with the transverse deflection may be writ‐

¨ ″ <sup>+</sup> *cvv*˙ ″) <sup>−</sup> *EI* \*

<sup>−</sup> *<sup>ρ</sup><sup>I</sup>* \* *κGF* \*

> ^ *v* ¨ ″″

*EI* \* *<sup>κ</sup>GF* \* *<sup>ρ</sup><sup>I</sup>*

=0

*κGAI* \* *IκGF* \* \_

)( <sup>−</sup> *<sup>P</sup>* ″

*<sup>y</sup>* + *ρAv*

Eq. (102) is a sixth-order partial differential equation with variable coefficients with respect to *x* . In order to simplify the future development, considering only bending, transverse shear deformation, shear lag, inertia, and rotatory inertia terms in Eq. (102), a simplified

The equation neglecting the underlined term in Eq. (103) reduces to the equation of motion of Timoshenko beam theory, for example, Eq. (9.49) Craig [20]. Since Eq. (103) is very simple equation, the free transverse vibration analysis is developed by means of Eq. (103). To sim‐

\* <sup>1</sup> \*

k + k

*GF I*

*<sup>κ</sup>GF* \* ( <sup>−</sup> *<sup>P</sup>* ″

*EI* ^ *<sup>κ</sup>GA* ( <sup>−</sup>*<sup>P</sup>* ″

*<sup>I</sup> <sup>I</sup>* \* (101)

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*y* + *ρAv*

*<sup>y</sup>* + *ρAv*

¨ ″ <sup>+</sup> *cvv*˙ ″ )

¨ ″ <sup>+</sup> *cvv*˙ ″)⋅⋅″

¨ ″ <sup>+</sup> *cvv*˙ ″) =0 (103)

(102)

265

(104)

**Figure 20.** Convergence characteristics [6]

## **7. Natural frequencies by approximate method**

#### **7.1. Simplification of governing equation**

In the structural design of high-rise buildings, structure designers want to grasp simply the natural frequencies in the preliminary design stages. Takabatake [3] presented a general and simple analytical method for natural frequencies to meet the above demands. This section explains about this simple but accurate analytical method.

The one-dimensional extended rod theory for the transverse motion takes the coupled equa‐ tions concerning *v* , *ϕ* , and *u* \* , as given in Eqs. (32) to (34). Now consider the equation of motion expressed in terms with the lateral deflection. Neglecting the differential term of the transverse shear stiffness, *κGA* , in Eq. (32), the differential of rotational angle with respect to *x* may be written as

$$\phi = -\stackrel{\circ}{\upsilon}^{\circ} + \frac{1}{\kappa GA} \{-P\_y + \rho A \stackrel{\circ}{\upsilon} + c\_{\upsilon} \dot{\upsilon}\} \tag{99}$$

From (33) and (34), *u* \* becomes

$$
\mu \ast = \frac{1}{\kappa G F^\*} \frac{I^\*}{S^\*} \mathbf{\hat{I}} \stackrel{\frown}{\ast} \mathbf{\hat{I}} \dot{\phi} \dot{I} \dot{\phi} - E \stackrel{\frown}{I} \dot{\phi} \mathbf{\hat{I}} \stackrel{\frown}{\ast} + \kappa G A \begin{Bmatrix} \mathbf{\hat{v}} \ \mathbf{\hat{z}} + \phi \end{Bmatrix} \tag{100}
$$

in which *I* ^ is defined as

**Figure 20.** Convergence characteristics [6]

264 Advances in Vibration Engineering and Structural Dynamics

to *x* may be written as

From (33) and (34), *u* \* becomes

**7.1. Simplification of governing equation**

**7. Natural frequencies by approximate method**

explains about this simple but accurate analytical method.

*ϕ* = −*v* '' +

*I* \* *<sup>S</sup>* \* *<sup>ρ</sup><sup>I</sup>* ^ *ϕ*¨ −*EI* ^

*<sup>u</sup>* \* <sup>=</sup> <sup>1</sup> *κGF* \*

1 *<sup>κ</sup>GA* ( <sup>−</sup>*Py*

In the structural design of high-rise buildings, structure designers want to grasp simply the natural frequencies in the preliminary design stages. Takabatake [3] presented a general and simple analytical method for natural frequencies to meet the above demands. This section

The one-dimensional extended rod theory for the transverse motion takes the coupled equa‐ tions concerning *v* , *ϕ* , and *u* \* , as given in Eqs. (32) to (34). Now consider the equation of motion expressed in terms with the lateral deflection. Neglecting the differential term of the transverse shear stiffness, *κGA* , in Eq. (32), the differential of rotational angle with respect

<sup>+</sup> *<sup>ρ</sup>Av*¨ <sup>+</sup> *cvv*˙) (99)

*ϕ*″ + *κGA*(*v* ′ + *ϕ*) (100)

$$\stackrel{\wedge}{I} = I \left[ 1 - \frac{(\mathcal{S} \ \* \prime)^2}{II \ \*} \right] \tag{101}$$

Differentiating Eq. (33) with respect to *x* and substituting Eqs. (32), (99), and (100) into the result, the equation of motion expressed in terms with the transverse deflection may be writ‐ ten as

$$\begin{split} &EI\boldsymbol{\nu}^{\prime\prime} + \rho A\bar{\boldsymbol{\nu}} + c\_{\boldsymbol{\nu}}\boldsymbol{\bar{\nu}} - P\_{y} - \rho I\boldsymbol{\bar{\nu}}^{\prime\prime} - \frac{EI}{\kappa GA} (-P\_{y} + \rho A\bar{\boldsymbol{\nu}}^{\prime\prime} + c\_{\boldsymbol{\nu}}\boldsymbol{\bar{\nu}}^{\prime}) - \frac{EI}{\kappa GA} \boldsymbol{\bar{\nu}}^{\prime\prime} (-P\_{y} + \rho A\bar{\boldsymbol{\nu}}^{\prime\prime} + c\_{\boldsymbol{\nu}}\boldsymbol{\bar{\nu}}^{\prime}) \\ &+ \left(\frac{\rho I}{\kappa GA} + \frac{\rho I}{\kappa GF} \boldsymbol{\bar{\nu}}^{\prime} \right) (-P\_{y} + \rho A\bar{\boldsymbol{\nu}} + c\_{\boldsymbol{\nu}}\boldsymbol{\bar{\nu}})^{\prime\prime} + \frac{\rho I}{\kappa GF} \boldsymbol{\bar{\nu}}^{\prime} \boldsymbol{\bar{\nu}}^{\prime} - \frac{\rho I}{\kappa GF} \boldsymbol{\bar{\nu}}^{\prime} \frac{EI}{\kappa GA} (-P\_{y} + \rho A\bar{\boldsymbol{\nu}}^{\prime} + c\_{\boldsymbol{\nu}}\boldsymbol{\bar{\nu}}^{\prime})^{\prime\prime} \\ &+ \frac{\rho I}{\kappa GF}^{\prime\ast} \cdot \frac{\rho I}{\kappa GA} (-P\_{y} + \rho A\bar{\boldsymbol{\nu}} + c\_{\boldsymbol{\nu}}\boldsymbol{\bar{\nu}})^{\prime\prime} - \frac{\rho I}{\kappa GF}^{\prime\ast} \cdot \rho \overset{\scriptstyle\$$

Eq. (102) is a sixth-order partial differential equation with variable coefficients with respect to *x* . In order to simplify the future development, considering only bending, transverse shear deformation, shear lag, inertia, and rotatory inertia terms in Eq. (102), a simplified governing equation is given

$$EI\boldsymbol{\upsilon}^{\prime\prime} + \rho A \boldsymbol{\bar{\upsilon}} + c\_{\upsilon} \boldsymbol{\dot{\upsilon}} - P\_y - \rho I \boldsymbol{\bar{\upsilon}}^{\prime\prime} - \frac{EI}{\kappa GA} \left( 1 + \underbrace{\frac{\kappa GA \boldsymbol{I}^{\ast} \ast}{I \kappa GF^{\ast}}}\_{} \right) \left( -P\_{\phantom{\boldsymbol{\upsilon}}} + \rho A \boldsymbol{\bar{\upsilon}}^{\prime\prime} + c\_{\upsilon} \boldsymbol{\dot{\upsilon}}^{\prime} \right) = 0 \tag{103}$$

The equation neglecting the underlined term in Eq. (103) reduces to the equation of motion of Timoshenko beam theory, for example, Eq. (9.49) Craig [20]. Since Eq. (103) is very simple equation, the free transverse vibration analysis is developed by means of Eq. (103). To sim‐ plify the future expression, the following notation is introduced

$$\kappa \left( \underline{\kappa GA} \right) = \kappa GA \frac{1}{1 + \frac{\kappa GA \ I^\*}{\kappa G \ F^\* I}} \tag{104}$$

Hence, Eq. (104) may be rewritten

$$EI\,\mathbf{v}'''' + \rho \, A\ddot{\mathbf{v}} + c\_\mathbf{v}\dot{\mathbf{v}} - P\_y - \rho \, I\,\ddot{\mathbf{v}}'' - \frac{EI}{\left(\underline{\mathbf{\underline{\mathbf{\tilde{\mathbf{E}}}}}} \mathbf{\underline{\mathbf{\tilde{\mathbf{E}}}}} \mathbf{\underline{\mathbf{\tilde{\mathbf{E}}}}} \mathbf{\tilde{\mathbf{v}}''} + \rho A\dot{\mathbf{v}}'' + c\_\mathbf{v}\dot{\mathbf{v}}''\right) = 0\tag{105}$$

2 2 2 \* 0 0 0 111

(112)

267

(115)

*<sup>ρ</sup><sup>I</sup>* (113)

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l = (114)

l= a (116)

l= a (117)

<sup>l</sup> (118)

æ ö æöæö ç ÷ = + ç÷ç÷ ç ÷ <sup>l</sup> èøèø l l è ø

<sup>=</sup> *<sup>L</sup> <sup>ρ</sup><sup>A</sup>*

( )

k

*E I*

 and *λ*0 are pseudo slenderness ratios of the tube structures, depending on the bending stiffness and the transverse shear stiffness, respectively. Since for a variable tube structure the coefficients, *b* and *c* , are variable with respect to *x* , it is difficult to solve analytically Eq. (108). So, first we consider a uniform tube structure where these coefficients become con‐ stant. The solution for a variable tube structure will be presented by means of the Galerkin

Thus, since for a uniform tube structure Eq. (108) becomes a fourth-order differential equa‐

( ) 2 \*

1 1 *kL L*

> ( ) 2 \*

2 2 *kL L*

\* 1

a =

0 1 1 ˆ 2 + a

*G A*

ˆ

*λ*0 \*

0

*L*

in which *λ*<sup>0</sup>

*λ*0 \*

method.

in which *α*<sup>1</sup>

\* and *α*<sup>2</sup> \* are

\*

and *λ*0 are defined as

tion with constant coefficients, the general solution is

in which *C*1 to *C*4 are integral constants and *λ*1 and *λ*2 are defined as

The aforementioned equation suggests that in the simplified equation the transverse shear stiffness *κGA* must be replaced with the modified transverse shear stiffness .

#### **7.2. Undamped free transverse vibrations**

Let us consider undamped free transverse vibration of high-rise buildings. The equation for undamped free transverse vibrations is written from Eq. (105) as

$$\mathbf{v''} - \frac{\mathfrak{p}\,A}{E\,I} \left[ \frac{\mathfrak{p}\,I}{\mathfrak{p}\,A} + \frac{EI}{\left(\underline{\mathbf{k}\,G}\underline{\mathbf{A}}\right)} \right] \ddot{\mathbf{v''}} + \frac{\mathfrak{p}\,A}{E\,I} \ddot{\mathbf{v}} = \mathbf{0} \tag{106}$$

Using the separation method of variables, *v*(*x*, *t*) is expressed as

$$\nu(\boldsymbol{x},t) = \Phi(\boldsymbol{x})e^{\boldsymbol{\upbeta}\boldsymbol{\upgamma}t} \tag{107}$$

Substituting the above equation into Eq. (106), the equation for free vibrations becomes

$$
\partial \Phi + b \Phi'''' + c \,\!\!\!\/ \Phi = 0 \tag{108}
$$

in which the coefficients, *b* and *c* , are defined as

$$b = \frac{\left(k \, L\right)^4}{L^2} \left(\frac{1}{\hat{\lambda}\_0}\right)^2\tag{109}$$

$$c = -k \, ^{4}\tag{110}$$

in which *k* <sup>4</sup> and *λ* ^ 0 are defined as

$$k^4 = \frac{\rho A}{EI} \omega^2 \tag{111}$$

$$\left(\frac{1}{\hat{\lambda}\_0}\right)^2 = \left(\frac{1}{\lambda\_0^\*}\right)^2 + \left(\frac{1}{\lambda\_0}\right)^2\tag{112}$$

in which *λ*<sup>0</sup> \* and *λ*0 are defined as

( ) ( ) 0 *v y y v*

The aforementioned equation suggests that in the simplified equation the transverse shear

Let us consider undamped free transverse vibration of high-rise buildings. The equation for

( ) <sup>0</sup> *A I EI A <sup>v</sup> v v EI A GA E I* é ù r r <sup>r</sup> ¢¢¢¢ - + += ê ú ¢¢ ê ú <sup>r</sup> <sup>k</sup> ë û

Substituting the above equation into Eq. (106), the equation for free vibrations becomes

( ) <sup>2</sup> <sup>4</sup> 2

*k L*

*L* æ ö <sup>=</sup> ç ÷ ç ÷

*<sup>k</sup>* <sup>4</sup> <sup>=</sup> *<sup>ρ</sup><sup>A</sup>*

*b*

0 1 ˆ

*GA* ¢¢¢¢ +r + - -r - - +r + = ¢¢ ¢¢ ¢¢ ¢¢ <sup>k</sup> && & && && & (105)

&& && (106)

è ø <sup>l</sup> (109)

*c* = −*k* <sup>4</sup> (110)

*EI <sup>ω</sup>* <sup>2</sup> (111)

(107)

(108)

*EI EI v Av c v P I v P Av c v*

stiffness *κGA* must be replaced with the modified transverse shear stiffness .

undamped free transverse vibrations is written from Eq. (105) as

Using the separation method of variables, *v*(*x*, *t*) is expressed as

in which the coefficients, *b* and *c* , are defined as

0 are defined as

in which *k* <sup>4</sup>

 and *λ* ^

**7.2. Undamped free transverse vibrations**

266 Advances in Vibration Engineering and Structural Dynamics

$$
\lambda\_0^\* = L \sqrt{\frac{\rho A}{\rho I}}\tag{113}
$$

$$\lambda\_0 = L \sqrt{\frac{\left(\underline{\kappa \, G \, A}{\,}\right)}{E \, I}}\tag{114}$$

*λ*0 \* and *λ*0 are pseudo slenderness ratios of the tube structures, depending on the bending stiffness and the transverse shear stiffness, respectively. Since for a variable tube structure the coefficients, *b* and *c* , are variable with respect to *x* , it is difficult to solve analytically Eq. (108). So, first we consider a uniform tube structure where these coefficients become con‐ stant. The solution for a variable tube structure will be presented by means of the Galerkin method.

Thus, since for a uniform tube structure Eq. (108) becomes a fourth-order differential equa‐ tion with constant coefficients, the general solution is

$$\Phi = C\_1 \cos \lambda\_1 \mathbf{x} + C\_2 \sin \lambda\_1 \mathbf{x} + C\_3 \sin h \lambda\_2 \mathbf{x} + C\_4 \cosh \lambda\_2 \mathbf{x} \tag{115}$$

in which *C*1 to *C*4 are integral constants and *λ*1 and *λ*2 are defined as

$$
\lambda\_\text{\tiny\Delta} = \frac{\left(kL\right)^2}{L} \alpha\_\text{\tiny\Delta}^\* \tag{116}
$$

$$
\lambda\_2 = \frac{\left(kL\right)^2}{L} \alpha\_2^\* \tag{117}
$$

in which *α*<sup>1</sup> \* and *α*<sup>2</sup> \* are

$$
\alpha\_1^\* = \frac{1}{\hat{\lambda}\_0} \sqrt{\frac{1+\alpha}{2}} \tag{118}
$$

$$
\alpha\_2^\* = \frac{1}{\hat{\lambda}\_0} \sqrt{\frac{-1 + \alpha}{2}} \tag{119}
$$

*v* ′ + *ϕ* =0 at *x* = *L* (127)

=0 at *x* = *L* (130a)

*u xL* \* 0 at ¢ = = (130b)

F= = 0 at 0 *x* (131)

F= = ( *x dx x L* ) 0 at ò (133)

2 corresponding to the *n*th natural frequency is ob‐

¯

2cosh*λ*2*L* ) =0 (135)

F+ F = = ¢ <sup>k</sup> ò (132)

F + F= = ¢¢ <sup>k</sup> (134)

<sup>2</sup> (sinh*λ*2*L* + *k*

=0 at *x* = *L* (128)

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269

=0 at *x* = *L* (129)

*E S* \* *u*\*

*EI* \* *u*\*

*ϕ* ′

(130a). Using Eq. (107), these boundary conditions are rewritten as

Eqs. (128) and (129) reduce to

ing a nondimensional constant (*k*n*L*)

<sup>2</sup>cos*λ*1*L* −*k*

<sup>2</sup> are defined as

¯

<sup>1</sup>sin*λ*1*L* ) + (*α*<sup>2</sup>

tained as

in which *k*

(*α*1 \* )<sup>2</sup> <sup>−</sup> <sup>1</sup> *λ*0 <sup>2</sup> (*k* ¯

> ¯ <sup>1</sup> and *k* ¯

′ + *E I ϕ*′

′ + *ES* \**ϕ*′

Hence the boundary conditions for current problem become as Eqs. (124), (125), (127), and

( ) ( ) ( ) <sup>4</sup> 0 at 0 *EI x k x dx x*

( ) ( ) ( ) <sup>4</sup> 0 at *EI x k x xL GA*

Substituting Eq. (115) into the aforementioned boundary conditions, the equation determin‐

\* )<sup>2</sup> + 1 *λ*0

*GA*

in which

$$\alpha = \sqrt{1 + \frac{4}{\left(kL\right)^4 \left(\frac{1}{\hat{\lambda}\_o}\right)^4}}\tag{120}$$

Meanwhile, *ϕ*' for the free transverse vibration is from Eq. (99)

$$\boldsymbol{\phi}' = -\boldsymbol{\nu}'' + \frac{\boldsymbol{\rho}\,\boldsymbol{A}}{\left(\underline{\mathbf{\underline{\boldsymbol{K}}}}\underline{\mathbf{\underline{\boldsymbol{A}}}}\right)}\ddot{\boldsymbol{\nu}}\tag{121}$$

in which *κGA* is replaced with . The substitution of Eqs. (108) and (111) into the afore‐ mentioned equation yields

$$\phi'(x,t) = -\left[\Phi(x) + k^4 \frac{EI}{(\underline{\underline{\kappa}GA})} \Phi(x)\right] e^{\phi x^\*} \tag{122}$$

The integration of the aforementioned equation becomes

$$\Phi(\mathbf{x},t) = -\left[\Phi'(\mathbf{x}) + k^4 \frac{EI}{\left(\underline{\mathbf{x}GA}\right)} \int \Phi(\mathbf{x})d\mathbf{x}\right]e^{i\mathbf{m}t} \tag{123}$$

The boundary conditions for the current tube structures are assumed to be constrained for all deformations at the base and free for bending moment, transverse shear and shear-lag at the top. Hence the boundary conditions are rewritten from Eqs. (35) to (38) as

$$\mathbf{v} = \mathbf{0} \quad \text{at} \quad \mathbf{x} = \mathbf{0} \tag{124}$$

$$
\phi \text{=} 0 \text{ at } \ge 0 \tag{125}
$$

$$
\mu^\* = 0 \quad \text{at} \quad \varkappa = 0 \tag{126}
$$

$$\begin{array}{c} v \stackrel{\cdot}{\cdot} + \phi = 0 \quad \text{at} \quad \mathbf{x} = L \end{array} \tag{127}$$

$$E \stackrel{\circ}{S} \stackrel{\bullet}{\ast} \stackrel{\circ'}{\ast} + E \stackrel{\circ'}{I} \stackrel{\circ'}{\phi} = 0 \text{ at } \propto \text{L} \tag{128}$$

$$EI\,\,{}^{\ast}u\,{}^{\ast'} + ES\,{}^{\ast}\phi\,{}^{\prime} = 0 \text{ at } \,\,{}x = L \tag{129}$$

Eqs. (128) and (129) reduce to

\* 2

268 Advances in Vibration Engineering and Structural Dynamics

a= +

for the free transverse vibration is from Eq. (99)

in which

Meanwhile, *ϕ*'

mentioned equation yields

The integration of the aforementioned equation becomes

a =

0 1 1 ˆ 2 - +a

( )

<sup>4</sup> <sup>1</sup>

4

( ) *<sup>A</sup> v v GA*

in which *κGA* is replaced with . The substitution of Eqs. (108) and (111) into the afore‐

( ) ( ) ( ) ( ) <sup>4</sup> , *EI i t xt x k x dx e GA* <sup>w</sup> é ù

the top. Hence the boundary conditions are rewritten from Eqs. (35) to (38) as

The boundary conditions for the current tube structures are assumed to be constrained for all deformations at the base and free for bending moment, transverse shear and shear-lag at

f =- F + F ê ú ¢ <sup>k</sup> ë û <sup>ò</sup> (123)

*v x* = = 0 at 0 (124)

*ϕ* =0 at *x* =0 (125)

*u x* \* 0 at 0 = = (126)

4

0

1 <sup>ˆ</sup> *kL*

æ ö ç ÷ è ø l

<sup>l</sup> (119)

<sup>r</sup> f =- + ¢ ¢¢ <sup>k</sup> && (121)

(120)

(122)

$$\boldsymbol{\phi}^{'} = \boldsymbol{0} \quad \text{at} \quad \boldsymbol{\chi} = \boldsymbol{L} \tag{130a}$$

$$
\mu^{\ast \prime} = 0 \quad \text{at} \ \propto = L \tag{130b}
$$

Hence the boundary conditions for current problem become as Eqs. (124), (125), (127), and (130a). Using Eq. (107), these boundary conditions are rewritten as

$$
\Phi = 0 \quad \text{at} \quad \text{x} = 0 \tag{131}
$$

$$\Phi'(\mathbf{x}) + k^4 \frac{EI}{\left(\underline{\mathbf{x}GA}\right)} \int \Phi(\mathbf{x})d\mathbf{x} = 0 \quad \text{at} \quad \mathbf{x} = \mathbf{0} \tag{132}$$

$$\int \Phi(\mathbf{x})d\mathbf{x} = 0 \quad \text{at} \ \mathbf{x} = L \tag{133}$$

$$\Phi''(\mathbf{x}) + k^4 \frac{EI}{\left(\underline{\mathbf{x}GA}\right)} \Phi(\mathbf{x}) = 0 \text{ at } \mathbf{x} = L \tag{134}$$

Substituting Eq. (115) into the aforementioned boundary conditions, the equation determin‐ ing a nondimensional constant (*k*n*L*) 2 corresponding to the *n*th natural frequency is ob‐ tained as

$$\left[ \left( a\_1^\* \right)^2 - \frac{1}{\lambda\_0^2} \right] \left( \bar{k}\_2 \cos \lambda\_1 L \quad - \bar{k}\_1 \sin \lambda\_1 L \right) + \left[ \left( a\_2^\* \right)^2 + \frac{1}{\lambda\_0^2} \right] \left( \sinh \lambda\_2 L \quad + \bar{k}\_2 \cosh \lambda\_2 L \right) = 0 \tag{135}$$

in which *k* ¯ <sup>1</sup> and *k* ¯ <sup>2</sup> are defined as

$$\bar{k}\_1 = -\frac{\alpha\_2^\* + \frac{1}{\alpha\_2^\* \lambda\_0^2}}{\alpha\_1^\* + \frac{1}{\alpha\_1^\* \lambda\_0^2}}\tag{136}$$

(139)

271

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Now, neglecting the effect of the shear lag, the solutions proposed here agree with the re‐

The natural frequency of a uniform tube structure has been proposed in closed form. For a variable tube structure the proposed results give the approximate natural frequency by re‐ placing the variable tube structure with a pseudo uniform tube structure having an appro‐

On the other hand, the natural frequency for a variable tube structure is presented by means

sults for a uniform Timoshenko beam presented by Herrmann [21] and Young [22].

**7.3. Natural frequency of variable tube structures**

priate reference stiffness.

**Figure 21.** Values of (*knL* )2 [3]

of the Galerkin method. So, Eq. (108) may be rewritten as

$$\bar{k}\_2 = -\frac{-\frac{\bar{k}\_1}{\alpha\_1^\*} \cos \lambda\_1 L\_1 + \frac{1}{\alpha\_2^\*} \cosh \lambda\_2 L\_2}{-\frac{1}{\alpha\_1^\*} \sin \lambda\_1 L\_1 + \frac{1}{\alpha\_2^\*} \sinh \lambda\_2 L\_2} \tag{137}$$

The value of (*k*n*L*) 2 is determined from Eq. (135) as follows:

STEP 1. From Eqs. (113) and (114), determine *λ*<sup>0</sup> \* and *λ*0 .

STEP 2. From Eq. (112), determine *λ* ^ 0 .

STEP 3. Assume the value of (*k*n*L*) 2 .

STEP 4. From Eq. (120), determine *α* .

STEP 5. From Eqs. (116) to (119), determine *λ*1 , *λ*2 , *α*<sup>1</sup> \* and *α*<sup>2</sup> \* , respectively.

STEP 6. From Eqs. (136) and (137), calculate *k* ¯ <sup>1</sup> and *k* ¯ 2 .

STEP 7. Substitute these vales into Eq. (135) and find out the value of (*k*n*L*) <sup>2</sup> satisfying Eq. (135) with trial and error.

Hence, the value of (*k*n*L*) 2 depends on the slenderness ratios, *λ*<sup>0</sup> \* and *λ*0 , of the uniform tube structure. So, for practical uses, the value of (*k*n*L*) <sup>2</sup> for the given values *λ*<sup>0</sup> \* and *λ*<sup>0</sup> can be pre‐ sented previously as shown in Figure 21. Numerical results show that the values of (*k*n*L*) 2 depend mainly on *λ*<sup>0</sup> and are negligible for the variation of *λ*<sup>0</sup> \* . When *λ*0 increases, the value of (*k*n*L*) 2 approaches the value of the well-known Bernoulli-Euler beam. The practical tube structures take a value in the region from *λ*<sup>0</sup> = 0.1 to *λ*<sup>0</sup> = 5.

Thus, substituting the value of (*k*n*L*)2 into Eq. (111), the *n*th natural frequency, *ωn* , of the tube structure is

$$
\cos\_u = \frac{\left(k\_u L\right)^2}{L^2} \sqrt{\frac{EI}{\rho A}}\tag{138}
$$

Using Figure 21, the structural engineers may easily obtain from the first to tenth natural frequencies and also grasp the relationships among these natural frequencies.

The *n*th natural function, , corresponding to the *n*th natural frequency is

$$\Phi\_n(\mathbf{x}) = -\overline{k}\_2 \cos \lambda\_\mathbf{r} \mathbf{x} + \overline{k}\_1 \sin \lambda\_\mathbf{r} \mathbf{x} + \sin \hbar \lambda\_\mathbf{r} \mathbf{x} + \overline{k}\_2 \cos \hbar \lambda\_\mathbf{r} \tag{139}$$

Now, neglecting the effect of the shear lag, the solutions proposed here agree with the re‐ sults for a uniform Timoshenko beam presented by Herrmann [21] and Young [22].

### **7.3. Natural frequency of variable tube structures**

*k* ¯ <sup>1</sup> = −

− *k* ¯ 1 *α*1

− 1 *α*1

*k* ¯ <sup>2</sup> = −

STEP 1. From Eqs. (113) and (114), determine *λ*<sup>0</sup>

STEP 5. From Eqs. (116) to (119), determine *λ*1 , *λ*2 , *α*<sup>1</sup>

STEP 6. From Eqs. (136) and (137), calculate *k*

2

structure. So, for practical uses, the value of (*k*n*L*)

Thus, substituting the value of (*k*n*L*)2

depend mainly on *λ*<sup>0</sup> and are negligible for the variation of *λ*<sup>0</sup>

structures take a value in the region from *λ*<sup>0</sup> = 0.1 to *λ*<sup>0</sup> = 5.

The value of (*k*n*L*)

2

270 Advances in Vibration Engineering and Structural Dynamics

STEP 2. From Eq. (112), determine *λ*

STEP 4. From Eq. (120), determine *α* .

STEP 3. Assume the value of (*k*n*L*)

(135) with trial and error.

Hence, the value of (*k*n*L*)

of (*k*n*L*) 2

tube structure is

*α*2 \* +

*α*1 \* +

\* cos*λ*1*L* +

\* sin*λ*1*L* +

is determined from Eq. (135) as follows:

^ 0 .

2 .

1 *α*2 \* *λ*0 2

1 *α*1 \* *λ*0 2

> 1 *α*2

> 1 *α*2

> > \* and *λ*0 .

¯ <sup>1</sup> and *k* ¯ 2 .

depends on the slenderness ratios, *λ*<sup>0</sup>

( ) 2 2 *n*

*k L EI L A*

Using Figure 21, the structural engineers may easily obtain from the first to tenth natural

r

*n*

w =

frequencies and also grasp the relationships among these natural frequencies.

The *n*th natural function, , corresponding to the *n*th natural frequency is

sented previously as shown in Figure 21. Numerical results show that the values of (*k*n*L*)

approaches the value of the well-known Bernoulli-Euler beam. The practical tube

STEP 7. Substitute these vales into Eq. (135) and find out the value of (*k*n*L*)

\* cos*h λ*2*L*

\* sin*h λ*2*L*

\* and *α*<sup>2</sup> \*

, respectively.

\*

<sup>2</sup> for the given values *λ*<sup>0</sup>

\*

into Eq. (111), the *n*th natural frequency, *ωn* , of the

(136)

(137)

<sup>2</sup> satisfying Eq.

and *λ*<sup>0</sup> can be pre‐

2

(138)

and *λ*0 , of the uniform tube

. When *λ*0 increases, the value

\*

The natural frequency of a uniform tube structure has been proposed in closed form. For a variable tube structure the proposed results give the approximate natural frequency by re‐ placing the variable tube structure with a pseudo uniform tube structure having an appro‐ priate reference stiffness.

**Figure 21.** Values of (*knL* )2 [3]

On the other hand, the natural frequency for a variable tube structure is presented by means of the Galerkin method. So, Eq. (108) may be rewritten as

$$\left[EI\,\Phi'''' - \alpha^2\right] \left[\rhd A\Phi - \rho \, A\left(\frac{L}{\widehat{\lambda}\_0}\right)^2 \,\Phi''\right] = 0\tag{140}$$

The cross sections of columns and beams in the variable frame tube shown in Figure 22 vary in three steps along the height. On the other hand, the uniform frame tube is assumed to be

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273

Table 7 shows the natural frequencies of the uniform and variable frame tubes, in which the approximate solution for the variable frame tube indicates the value obtained from replac‐ ing the variable frame tube with a pseudo uniform frame tube having the stiffnesses at the lowest story. The results obtained from the proposed method show excellent agreement with the three-dimensional frame analysis using FEM code NASTRAN. The approximate solution for the variable frame tube is also applicable to determine approximately the natu‐

the stiffness at the midheight (*L*/2) of the variable tube structure.

ral frequencies in the preliminary stages of the design.

**Figure 22.** Numerical model of frame tube [3]

is expressed by a power series expansion as follows

$$\Phi\left(\mathbf{x}\right) = \sum\_{n=1}^{\infty} c\_n \cdot \Phi\_n \tag{141}$$

in which *cn* = unknown coefficients; and *(x)* = functions satisfying the specified boundary conditions of the variable tube structure. Approximately, *(x)* take the natural function of the pseudo uniform tube structure, as given in Eq. (139). Applying Eq. (141) into Eq. (140), the Galerkin equations of Eq. (140) become

$$\delta \mathcal{L}\_m : \sum\_{n=1}^{\alpha} c\_n \left( A\_{mn} - \omega^{-2} B\_{mn} \right) = 0 \tag{142}$$

in which the coefficients, *Amn* and *Bmn* , are defined as

$$A\_{mn} = \int\_0^L EI \Phi\_n'''' \,\Phi\_m \,d\mathbf{x} \tag{143}$$

$$B\_{\mu\nu} = \int\_0^L \otimes A \Phi\_n \Phi\_{\mu\nu} \, d\mathbf{x} - \int\_0^L \otimes A \left(\frac{L}{\hat{\lambda}\_0}\right) \Phi\_{\nu}^{\star\star\star} \Phi\_{\mu\nu} d\mathbf{x} \tag{144}$$

Hence, the natural frequency of the variable tube structure is obtained from solving eigen‐ value problem of Eq. (142).

#### **7.4. Numerical results for natural frequencies**

The natural frequencies for doubly symmetric uniform and variable tube structures have been presented by means of the analytical and Galerkin methods, respectively. In order to examine the natural frequencies proposed here, numerical computations were carried out for a doubly symmetric steel frame tube, as shown in Figure 22. This frame tube equals to the tube structure used in the static numerical example in the Section 6, except for with or without bracing at 15 and 16 stories. The data used are as follows: the total story is 30; each story height is 3 m; the total height, *L* , is 90 m; the base is rigid; Young's modulus *E* of the material used is 2.05 x 1011 N/m2 . The weight per story is 9.8 kN/m2 x 18 m x 18 m = 3214 kN.

The cross sections of columns and beams in the variable frame tube shown in Figure 22 vary in three steps along the height. On the other hand, the uniform frame tube is assumed to be the stiffness at the midheight (*L*/2) of the variable tube structure.

Table 7 shows the natural frequencies of the uniform and variable frame tubes, in which the approximate solution for the variable frame tube indicates the value obtained from replac‐ ing the variable frame tube with a pseudo uniform frame tube having the stiffnesses at the lowest story. The results obtained from the proposed method show excellent agreement with the three-dimensional frame analysis using FEM code NASTRAN. The approximate solution for the variable frame tube is also applicable to determine approximately the natu‐ ral frequencies in the preliminary stages of the design.

**Figure 22.** Numerical model of frame tube [3]

(140)

(141)

(143)

(144)

x 18 m x 18 m = 3214 kN.

*Bmn*) =0 (142)

is expressed by a power series expansion as follows

*<sup>δ</sup>cm* :∑ *n*=1 *∞*

in which the coefficients, *Amn* and *Bmn* , are defined as

the Galerkin equations of Eq. (140) become

272 Advances in Vibration Engineering and Structural Dynamics

value problem of Eq. (142).

material used is 2.05 x 1011 N/m2

**7.4. Numerical results for natural frequencies**

in which *cn* = unknown coefficients; and *(x)* = functions satisfying the specified boundary conditions of the variable tube structure. Approximately, *(x)* take the natural function of the pseudo uniform tube structure, as given in Eq. (139). Applying Eq. (141) into Eq. (140),

*cn*(*Amn* <sup>−</sup>*<sup>ω</sup>* <sup>2</sup>

Hence, the natural frequency of the variable tube structure is obtained from solving eigen‐

The natural frequencies for doubly symmetric uniform and variable tube structures have been presented by means of the analytical and Galerkin methods, respectively. In order to examine the natural frequencies proposed here, numerical computations were carried out for a doubly symmetric steel frame tube, as shown in Figure 22. This frame tube equals to the tube structure used in the static numerical example in the Section 6, except for with or without bracing at 15 and 16 stories. The data used are as follows: the total story is 30; each story height is 3 m; the total height, *L* , is 90 m; the base is rigid; Young's modulus *E* of the

. The weight per story is 9.8 kN/m2


plays an important role from the standpoint of structural design. So, Takabatake [29, 30] propose two-dimensional extended rod theory as an extension of the one-dimensional ex‐ tended rod theory to take into account of the effect of transverse variations in individual

A Simplified Analytical Method for High-Rise Buildings

http://dx.doi.org/10.5772/51158

275

Figure 23 illustrates the difference between the one- and two-dimensional extended rod theories in evaluating the local stiffness distribution of structural components. In the twodimensional approximation, structural components with different stiffness and mass distri‐ bution are continuously connected. On the basis of linear elasticity, governing equations are derived from Hamilton's principle. Use is made of a displacement function which satisfies continuity conditions across the boundary surfaces between the structural components.

Two-dimensional extended rod theory has been presented for simply analyzing a large or complicated structure such as a high-rise building or shear wall with opening. The principle of this theory is that the original structure comprising various different structural compo‐ nents is replaced by an assembly of continuous strata which has stiffness equivalent to the original structure in terms of overall behavior. The two-dimensional extended rod theory is an extended version of a previously proposed one-dimensional extended rod theory for bet‐ ter approximation of the structural behavior. The efficiency of this theory has been demon‐ strated from numerical results for exemplified building structures of distinct components. This theory may be applicable to soil-structure interaction problems involving the effect of

On the other hand, the exterior of tall buildings has frequently the shape with many setback parts. On such a building the local variation of stress is considered to be very remarkable due the existence of setback. This nonlinear phenomenon of stress distribution may be ex‐ plained by two-dimensional extended rod theory but not by one-dimensional extended rod theory. In order to treat exactly the local stress variation due to setback, the proper boun‐ dary condition in the two-dimensional extended rod theory must separate into two parts. One part is the mechanical boundary condition corresponding to the setback part and the

**Figure 23.** The difference between one- and two-dimensional rod theories [29]

multi-layered or non-uniform grounds.

member stiffness.

**Table 7.** Natural frequencies of uniform and variable frame tubes [3]
