Maximum values of static lateral deflections 
High-rise buildings are constructed everywhere in the world. The height and size of high-rise buildings get larger and larger. The structural design of high-rise buildings depends on dynamic analysis for winds and earthquakes. Since today performance of computer progresses remarkably, almost structural designers use the software of computer for the structural design of high-rise buildings. Hence, after that the structural plane and outline of high-rise buildings are determined, the structural design of high-rise buildings which checks structural safety for the individual structural members is not necessary outstanding structural ability by the use of structural software on the market. However, it is not exaggeration to say that the performance of high-rise buildings is almost determined in the preliminary design stages which work on multifaceted examinations of the structural form and outline. The structural designer is necessary to gap exactly the whole picture in this stage. The static and dynamic structural behaviors of high-rise buildings are governed by the distributions of transverse shear stiffness and bending stiffness per each storey. Therefore, in the preliminary design stages of high-rise buildings a simple but accurate analytical method which reflects easily the structural stiffness on the whole situation is more suitable than an analytical method which each structural member is indispensable to calculate such as FEM.
There are many simplified analytical methods which are applicable for high-rise buildings. Since high-rise buildings are composed of many structural members, the main treatment for the simplification is to be replaced with a continuous simple structural member equivalent to the original structures. This equivalently replaced continuous member is the most suitable to use the one-dimensional rod theory.
Since the dynamic behavior of high-rise buildings is already stated to govern by the shear stiffness and bending stiffness determined from the structural property. The deformations of high-rise buildings are composed of the axial deformation, bending deformation, transverse shear deformation, shear-lag deformation, and torsional deformation. The problem is to be how to take account of these deformations under keeping the simplification.
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 Bernonlli-Euler beam theory is unsuitable for the modeling of high-rise buildings.
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 is assumed to be linear distributed in the transverse cross section.
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 closed section.
Reissner  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 appears in high-rise buildings the same as wing of aircrafts. Especially the shear-lag is remarkable 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 deformation. In generally, high-rise buildings have doubly symmetric structural forms from viewpoint 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-dimensional rod theory which can consider simply the above deformations. This theory is called the one-dimensional extended rod theory.
The previous works for continuous method are surveyed as follows: Beck  analyzed coupled shear walls by means of beam model. Heidenbrech et al.  indicated an approximate analysis of wall-frame structures and the equivalent stiffness for the equivalent beam. Dynamic analysis of coupled shear walls was studied by Tso et al. , Rutenberg [10, 11], Danay et al. , and Bause . Cheung and Swaddiwudhipong  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 aforementioned continuous approaches have not been presented as a closed-form solution for tube structures with variable stiffness due to the variation of frame members and bracings. In this chapter high-rise buildings are expressed as tube structures in which three dimensional frame structures are included naturally.
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. Assuming 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 torsional 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.
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 in-plane stiffness of the slabs. The displacement composes, , and in the x-, y-, and z-directions on the middle surface of the tube structures as
in which and longitudinal and transverse displacement components in the x-and y-directions on the axial point, respectively; rotational angle on the axial point along the z-axis; shear-lag coefficient in the flanges; 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  and for the web sections sine distribution [5, 6] is assumed:
in which the positive of takes for the flange being the positive value of the y-axis and vice versa and are hafe width of equivalent flange and web sections, as shown in Figure 1.
The governing equation of tube structures is proposed by means of the following Hamilton’s principle.
in which the kinetic energy; the strain energy; the potential energy produced by the external loads; and the variational operator taken during the indicated time interval.
Using linear relationship between strain and displacement, the following expressions are obtained.
in which dashes indicate the differentiation with respect x and the differentiations with respect y and z are expressed as
The relationships between stress and strain are used well-known engineering expression for one-dimentional structural member of the frame structure.
in which E is Yound modulus and G shear modulus.
Assuming the above linear stress-strain relation, the strain energy is given by
in which the total height of the tube structure; the shear coefficient; and and the sectional stiffnesses. These sectional stiffnesses vary discontinuously with respect to for a variable tube structure and are defined as
in which the total cross-sectional area of columns per story.
The kinetic energy, , for the time interval from to is
in which the dot indicates differentiation with respect to time and mass density of the tube structure. Now assuming that the variation of the displacements and rotation at and is negligible, the variation may be written as
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
in which and components of external loads in the x- and y-directions per unit are, respectively;, and damping coefficients for longtitudinal and transverse motions, respectively. The substitution of Eqs. (1) and (2) into Eq. (24) yields
in which, , , and are defined as
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, vanishes. Hence, Eq. (25) reduces to
together with the associated boundary conditions at and.
2.2. Governing equations for torsional moment
The displacement components for current tube structures subjected to torsional moments, , around the x-axis are expressed by
in which the displacement component in the z-direction on the tube structures; torsional angle; and warping function. Using the same manner as the aforementioned development, the differential equation of motion for current tube structures can be obtained
together with the association boundary conditions
2.3. Sectional constants
in which 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, and are given as and.
2.4. Equivalent transverse shear stiffness κGA
When the tube structure are composed of frame and bracing, the equivalent transverse shear stiffness κGA for each story is given by
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 and for each side of the web surfaces, respectively, are given by
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 and, respectively; the second and third terms indicate the shear deformation of only the columns and beams in the current web-frame, respectively. and the web's cross-sectional area of a column and of a beam, respectively. and 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 neglected, these terms must vanish. Furthermore, the span length; the cross-sectional area of a brace; the Young's modulus; and the incline of the brace. The coefficient indicates the effective number of brace and takes for a brace resisting only tension and for two brace resisting tension and compression, as shown in Figure 3.
2.5. Equivalent bending stiffness EI
The equivalent bending stiffness 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.
in which, and Young modulus, moment of inertia and the cross section of the ith column; and 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 equations with variable coefficients due to the variation of structural members and forms in the longitudinal direction. Furthermore, although the equations of motion and boundary conditions for the vertical displacement u are uncoupled from the other displacement components, the governing equations take coupled from concerning variables, , and.
Takabatake [2, 3] presented the uncoupled equations as shown in section 7 by introducing positively appropriate approximations into the coupled equations and proposed a closed-form solution. For usual tube structures this method produces reasonable results. However the analytical approach deteriorates on the accuracy of numerical results for high-rise buildings 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, rotational angle, and shear-lag displacement.
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:
in which the finite difference mesh;, , ,... represent displacements at the (i-1)th, ith, and (i+1)th mesh points, respectively, as shown in Figure 4; and and the lateral load and moment, respectively, at the ith mesh point. In the above equations, the rigidities, ,... at the pivotal mesh point i are taken as the mean value of the rigidities of current prototype tube structures located in the mesh region, in which the mesh region is defined as each half height between the mesh point i and the adjoin mesh points, i-1 and i+1, namely from to, as shown in Figure 5. Hence, the stiffness at a mesh point i is evaluated
in which, ,.., and, ,..., the effective story heights and story rigidities, located in the mesh region, respectively; and 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 considered for two cases: free and constrained. Hence, from Eqs. (36) to (38)
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.
Since the number of each boundary condition of the base and top for, and is one, respectively, the imaginary number of the boundary mesh in finite differences can be taken one for each displacement component at each boundary.
in which, , and 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
in which the mesh point locates on the boundary point at the free end of; the mesh point is imaginary point adjoining the mesh point; and the mesh point is inner point adjoining the mesh point. Solving the above eqations for the variables, , at the imaginary point, we have
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 but backward difference for, because is unsolvable in the use of the central difference.
Solving the above eqations for the variables, , , we have
Static solutions are obtained by solving a system of linear, homogeneous, simultaneous algebraic 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.
in which the matrix A is the total stiffness matrix summed the individual stiffness matrix 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
Let us consider the axial forces of columns. The axial stress of the tube structure is given from Eq. (12) by
Hence, the axial force in the ith column with the column’s sectional area is
for columns in flange surfaces,
for columns in web surfaces, and
for corner columns, in which, and, lower and upper coordinate values of the half between the ith column and both adjacent columns, respectively, and the cross-sectional area of the ith column divided by the sum of half spans between the ith column and the both adjacent columns.
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, , , and are expressed as
The finite difference expressions of the above equations reduce to eigenvalue problem for, , and.
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 ith mesh point is given in Figure 11. The ith natural frequencies can be obtained from the ith eigenvalue.
5. Forced transverse vibrations by finite difference method
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 total 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 at the base may be written in the matrix form as
in which mass matrix; the damping coefficient matrix; and, , and are the relative acceleration vector, the relative velocity vector and relative displacement vector, respectively, measured from the base. unit vector. It is assumed that the dynamic deflection vector and the rotational angle vector may be written as
in which the j-th participation coefficient; and the j-th eigenfunctions for and, respectively; the j-th dynamic response depending on time; and 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, we have
Now, Eq. (86) may be rewritten as
Multiplying the above equation by and substituting the reduced equation into Eq. (93), we have
Here, the damping coefficient matrix and the participation coefficients are assumed to satisfy the following expressions:
in which is the ith damping constant. Thus, Eq. (94) may be reduced to
The general solution of Eq. (98) is
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 frame-tubes. 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:
the dynamic loads are taken from El Centro 1940 NS, Taft 1952 EW, and Hachinohe 1968 NS, in which each maximum acceleration is 200 m/s2;
the damping ratio for the first mode of the frame-tubes is0.02, and the higher damping ratio for the n-th mode is;
the weight of each floor is 9.807 kN/m-2 and the mass of the frame-tube is considered to be only floor's weight;
in the modal analysis, the number of modes for the participation coefficients is taken five into consideration as five.
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 DEMOS 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 NASTRAN 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 discrepancy in shear-lag is negligible in practice.
|Maximum static lateral deflection (m)|
|Maximum shear-lag (m)|
Figure 16 shows the distribution of axial forces of model T2. A discrepancy between the results 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 because the axial forces in tube structures are designed from the axial forces on the flange surfaces, being always larger than those on the web surfaces.
|Natural frequency (rad/s)|
6.3. Free vibration results
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 discontinuously, 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 accelerations and of the maximum overturning moments for model T7. Thus, the proposed approximate 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 behavior caused by the part attached to the bracing.
|Maximum dynamic lateral deflection (m)|
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 convergence is obtained by the number of mesh points, being equal to the number of stories of the tube structures.
|Maximum story shear (kN)|
|Maximum overturning moment (MN m)|
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  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 equations concerning, , and, 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, , in Eq. (32), the differential of rotational angle with respect to may be written as
in which is defined as
Eq. (102) is a sixth-order partial differential equation with variable coefficients with respect to. 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
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 . Since Eq. (103) is very simple equation, the free transverse vibration analysis is developed by means of Eq. (103). To simplify the future expression, the following notation is introduced
Hence, Eq. (104) may be rewritten
The aforementioned equation suggests that in the simplified equation the transverse shear stiffness 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
Using the separation method of variables, is expressed as
Substituting the above equation into Eq. (106), the equation for free vibrations becomes
in which the coefficients, and, are defined as
in which and are defined as
in which and are defined as
Thus, since for a uniform tube structure Eq. (108) becomes a fourth-order differential equation with constant coefficients, the general solution is
in which to are integral constants and and are defined as
in which and are
Meanwhile, for the free transverse vibration is from Eq. (99)
The integration of the aforementioned equation becomes
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
Substituting Eq. (115) into the aforementioned boundary conditions, the equation determining a nondimensional constant (knL)2 corresponding to the nth natural frequency is obtained as
in which and are defined as
The value of (knL)2 is determined from Eq. (135) as follows:
STEP 2. From Eq. (112), determine.
STEP 3. Assume the value of (knL)2.
STEP 4. From Eq. (120), determine.
Hence, the value of (knL)2 depends on the slenderness ratios, and, of the uniform tube structure. So, for practical uses, the value of (knL)2 for the given values and can be presented previously as shown in Figure 21. Numerical results show that the values of (knL)2 depend mainly on and are negligible for the variation of. When increases, the value of (knL)2 approaches the value of the well-known Bernoulli-Euler beam. The practical tube structures take a value in the region from 0.1 to 5.
Thus, substituting the value of (knL)2 into Eq. (111), the nth natural frequency, , of the tube structure is
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 nth natural function,
7.3. Natural frequency of variable tube structures
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 replacing the variable tube structure with a pseudo uniform tube structure having an appropriate reference stiffness.
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
in which unknown coefficients; and
in which the coefficients, and, are defined as
Hence, the natural frequency of the variable tube structure is obtained from solving eigenvalue 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, , is 90 m; the base is rigid; Young’s modulus 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 replacing 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 natural frequencies in the preliminary stages of the design.
|NATURAL FREQUENCIES (rad/s)|
|UNIFORM FRAME TUBE||VARIABLE FRAME TUBE|
8. Expansion of one-dimensional extended rod theory
In order to carry out approximate analysis for a large scale complicated structure such as a high-rise building in the preliminary design stages, the use of equivalent rod theory is very effective. Rutenberg , Smith and Coull , Tarjan and Kollar  presented approximate calculations based on the continuum method, in which the building structure stiffened by an arbitrary combination of lateral load-resisting subsystems, such as shear walls, frames, coupled shear walls, and cores, are replaced by a continuum beam. Georgoussis  proposed to asses frequencies of common structural bents including the effect of axial deformation in the column members for symmetrical buildings by means of a simple shear-flexure model based on the continuum approach. Tarian and Kollar  presented the stiffnesses of the replacement sandwich beam of the stiffening system of building structures.
Takabatake et al. [2-6, 26-28] developed a simple but accurate one-dimensional extended rod theory which takes account of longitudinal, bending, and transverse shear deformation, as well as shear-lag. In the preceding sections the effectiveness of this theory has been demonstrated by comparison with the numerical results obtained from a frame analysis on the basis of FEM code NASTRAN for various high-rise buildings, tube structures and mega structures.
The equivalent one-dimensional extended rod theory replaces the original structure by a model of one-dimensional rod with an equivalent stiffness distribution, appropriate with regard to the global behavior. Difficulty arises in this modeling due to the restricted number of freedom of the equivalent rod; local properties of each structural member cannot always be properly represented, which leads to significant discrepancy in some cases. The one-dimensional idealization is able to deal only with the distribution of stiffness and mass in the longitudinal direction, possibly with an account of the averaged effects of transverse stiffness variation. In common practice, however, structures are composed of a variety of members or structural parts, often including distinct constituents such as a frame-wall or coupled wall with opening. Overall behavior of such a structure is significantly affected by the local distribution of stiffness. In addition, the individual behavior of each structural member 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 extended rod theory to take into account of the effect of transverse variations in individual member stiffness.
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 two-dimensional approximation, structural components with different stiffness and mass distribution 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 components 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 better approximation of the structural behavior. The efficiency of this theory has been demonstrated from numerical results for exemplified building structures of distinct components. This theory may be applicable to soil-structure interaction problems involving the effect of multi-layered or non-uniform grounds.
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 explained 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 boundary 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 other is the continuous condition corresponding to longitudinally adjoining constituents. Thus, Takabatake et al.  proved the efficiency of the two-dimensional extended rod theory to the general structures with setbacks.
Two-dimensional extended rod theory has been presented for simply analyzing a large or complicated structure with setback in which the stiffness and mass due to the existence of setback vary rapidly in the longitudinal and transverse directions. The effectiveness of this theory has been demonstrated from numerical results for exemplified numerical models. The transverse-wise distribution of longitudinal stress for structures with setbacks has been clarified to behave remarkable nonlinear behavior. Since the structural form of high-rise buildings with setbacks is frequently adapted in the world, the incensement of stress distribution occurred locally due to setback is very important for structural designers. The present theory may estimate such nonlinear stress behaviors in the preliminary design stages. The further development of the present theory will be necessary to extend to the three-dimensional extended rod theory which is applicable to a complicated building with three dimensional behaviors due to the eccentric station of many earthquake-resistant structural members, such as shear walls with opening.
9. Current problem of existing high-rise buildings
High-rise buildings have relative long natural period from the structural form. This characteristic is considered to be the most effective to avoid structural damages due to earthquake actions. However, when high-rise buildings subject to the action of the earthquake wave included the excellent long period components, a serious problem which the lateral deflection is remarkably large is produced in Japan. This phenomenon is based on resonance between the long period of high-rise buildings and the excellent long period of earthquake wave.
The 2011 Tohoku Earthquake (M 9.0) occurred many earthquake waves, which long period components are distinguished, on everywhere in Japan. These earthquake waves occur many physical and mental damages to structures and people living in high-rise buildings. The damage occurs high-rise buildings existing on all parts of Japan which appears long distance from the source. People entertain remarkable doubt about the ability to withstand earthquakes of high-rise buildings. This distrust is an urgent problem to people living and working in high-rise buildings. Existing high-rise buildings are necessary to improve urgently earthquake resistance. This section presents about an urgency problem which many existing high-rise buildings face a technical difficulty.
Let us consider dynamic behavior for one plane-frame of a high-rise building, as shown in Figure 24.
This plane frame is composed of uniform structural members. The sizes of columns and beams are □-800 x 800 x 25 (BCP) and H-400 x 300 x 11 x 18 (SN400B), respectively. The inertia moment of beams takes twice due to take into account of slab stiffness. This plane frame is a part of a three-dimensional frame structure with the span 6 m between adjacent plane-frames. The width and height are 36 m and 120 m, respectively. The main data used in numerical calculations are given in Table 8. Four kinds of earthquake waves are given in Table 9. El-Centro 1940 NS is converted the velocity to 0.5 m/s; JMA Kobe 1995 NS is the original wave with the maximum velocity 0.965 m/s; Shinjuku 2011 NS is the original wave with the maximum velocity 0.253 m/s; and Urayasu 2011 NS is the original wave with the maximum velocity 0.317 m/s. Figures 25(a) to 25(d) indicate time histories of accelerations for the four earthquake waves. In these earthquake waves, Shinjuku 2011 NS and Urayasu 2011 NS are obtained from K-net system measured at the 2011 Tohoku Earthquake. These earthquake waves are considered as earthquake waves included the excellent long periods. The excellent periods obtained from the Fourier spectrum of Shinjuku 2011 NS and Urayasu 2011 NS earthquake waves are 1.706 s and 1.342 s, respectively. The maximum acceleration and maximum velocity of these earthquake waves are shown in Table 9.
|Structure shape||Width:||@6 m x 6 = 36 m|
|Height:||@4 m x 30 floors = 120 m|
|Weight per floor (kN/m2)||12|
|Young modulus E (N/m2)||2.06 x 1011|
|Shear modulus G (N/m2)||7.92 x 1010|
Figure 26(a) shows the dynamic maximum lateral displacement subjected to the four kinds of earthquake waves. The maximum dynamic lateral displacement subject to Urayasu 2011 NS is remarkable larger than in the other earthquake waves. Figures 26(b) and (c) indicate the maximum shear force and overturning moment of the plane high-rise building subject to these earthquake actions, respectively. Earthquake wave Urayasu 2011 NS which includes long period components influence remarkable dynamic responses on the current high-rise building.
|Earthquake Wave||Maximum Accelerationm/s2||Maximum Velocitym/s|
|EL-CENTRO 1940 NS|
JMA KOBE 1995 NS
SHINJUKU 2011 NS
URAYASU 2011 NS
It is very difficult to sort out this problem. If the existing structure stiffens the transverse shear rigidity of overall or selected stories, the dynamic responses produced by the earthquake wave included excellently long period decrease within initial design criteria for dynamic calculations. However, inversely the dynamic responses produced by both EL-Centro 1940 NS with the maximum velocity 0.5 m/s and JMA Kobe 1995 NS exceed largely over the initial design criteria. The original design is based on flexibility which is the most characteristic of high-rise buildings. This flexibility brings an effect which lowers dynamic responses produced by earthquake actions excluding long period components. Now, changing the structural stiffness from relatively soft to hard, this effect is lost and the safety of the high-rise building becomes dangerous for earthquake waves excluding the long period components.
Author has not in this stage a clear answer to this problem. This problem includes two situations. The first point is to find out the appropriate distribution of the transverse stiffness. The variation of the transverse stiffness is considered to stiffen or soften. In general, existing high-rise buildings are easily stiffening then softening. However, there is a strong probability that the stiffening of the transverse shear stiffness exceeds the allowable limit for the lateral deflection, story shear force, and overturning moment in the dynamic response subjected to earthquake waves used in original structural design. Therefore, the softening of the transverse shear stiffness used column isolation for all columns located on one or more selected story is considered to be effective. It is clarified from author’s numerical computations that the isolated location is the most effective at the midheight. The second point is to find out an effective seismic retrofitting to existing high-rise buildings without the movement of people living and working in the high-rise building. These are necessary to propose urgently these measures for seismic retrofitting of existing high-rise buildings subject to earthquake waves included excellently long wave period. This subject will be progress to ensure comfortable life in high-rise buildings by many researchers.
A simple but accurate analytical theory for doubly symmetric frame-tube structures has been presented by applying ordinary finite difference method to the governing equations proposed by the one-dimensional extended rod theory. From the numerical results, the present theory has been clarified to be usable in the preliminary design stages of the static and dynamic analyses for a doubly symmetric single or double frame-tube with braces, in practical use. Furthermore, it will be applicable to hyper high-rise buildings, e.g. over 600m in the total height, because the calculation is very simple and very fast. Next the approximate method for natural frequencies of high-rise buildings is presented in the closed-form solutions. This method is very simple and effective in the preliminary design stages. Furthermore, the two-dimensional extended rod theory is introduced as for the expansion of the one-dimensional extended rod theory. Last it is stated to be urgently necessary seismic retrofitting for existing high-rise buildings subject to earthquake wave included relatively long period.