FIGURE 3.45 Cheng-Mertz model for bending coupling with shear of low-rise shear walls: (a) bending; (b) shear; and (c) axial. (Reprinted from Ref. [1, pp. 913, 932, and 952] by courtesy of Marcel Dekker, Inc.)

FIGURE 3.45 Cheng-Mertz model for bending coupling with shear of low-rise shear walls: (a) bending; (b) shear; and (c) axial. (Reprinted from Ref. [1, pp. 913, 932, and 952] by courtesy of Marcel Dekker, Inc.)

FIGURE 3.46 Cheng-Lou axial hysteresis model for RC members: (a) concrete under cyclic loading and (b) reinforcement under cyclic loading. (Reprinted from Ref. [1, p. 967] by courtesy of Marcel Dekker, Inc.)

low- and high-rise shear walls [12]. Figure 3.46 is the Cheng-Lou model for axial deformation of columns and walls. These models can be used on system analysis. For example, a braced steel frame needs hysteresis models of beams, columns, and bracings. If the steel structure is also composed of trusses, the hysteresis model of trusses must be employed. The latest version of INRESB-3D-II includes the aforementioned models for both PC and supercomputer [13,14]. The program has been used for case studies to verify different instances of building damage and collapse induced by strong ground motion [15].

3.8 Concluding Remarks

This chapter has composed most of the essential analytical techniques in structural dynamics. However, some special topics are currently in vogue but not included herein. The reader may find them from the references in the areas of structural optimization subjected to dynamic forces or earthquake excitation and the soil-structure interaction on seismic resistant structures with and without control. In the current engineering design community, major efforts are based on the development of sophisticated computer programs for the analysis of complex structures. When these programs are applied to design, the relative stiffnesses of a structure's constituent members must be assumed. If the preliminary stiffnesses are misjudged, then repeated analyses, regardless of a program's sophistication, will usually not yield an improved design. Programs presently in use are actually based on conventional designs, and their application in reality is an art rather than a science. The optimum design concept has been recognized as being more rational and reliable than those that require the conventional trial and error process. This is because, for a given set of constraints, such as allowable stresses, displacements, drifts, frequencies, upper and lower bounds of member sizes, and given loading conditions, the stiffnesses of members are automatically selected through the mathematical logic (structural synthesis) written in the computer program. Consequently, the strength of constituent members is uniformly distributed, and the rigidity of every component can uniquely satisfy the demands of external loads and code requirements. The objective function may be composed of initial cost of a structure and damage cost over the life of a structure including repair cost of the structure, loss of contents, economic impact of structural damage, and cost of injuries caused by the structural damage [16-22].

Dynamic structures analyzed with fixed support are based on the assumption that the structures are built on rock or are not subjected to ground motion such as wind forces or mechanical vibration. It is well known that ground motion induces soil-structure interaction (SSI) which can significantly influence response of a superstructure built on soft soil. Thus consideration of the SSI effect on a structure with and without control is very essential. Structural control implies that performance and serviceability of a structure are controlled to maintain prescribed limits during the application of environmental loads. Structural control is achieved in several ways: with passive or active control devices or with semiactive or hybrid systems. The number of actuators and sensors is usually limited by cost and system complexity. Effectiveness of control devices depends on their optimal placement in a building. Some sample studies of SSI and optimal placement of controllers are available [23-25].

The materials presented in this chapter are mainly selected from various parts of the author's book listed as Ref. [1]. For the reader's convenience, the citations of these parts are given with page number(s). Because of page limit, the author has not cited many original publications, which, however, were listed in the bibliographies of the book's individual chapters. The author acknowledges the original contribution with gratitude.


Continuous system — Structures having constituent members with distributed mass for dynamic stiffness or finite element formulation. Deterministic — A force's time function can be specified in regular or irregular variation. Damping factor — Ratio of damping coefficient to the critical damping coefficient. Ductility factor — Ratio of maximum displacement (or rotation) to the yielding displacement (or rotation).

Eigensolution — Eigenvalues and eigenvectors of a singular matrix that are usually referred to angular frequencies and normal modes in structural dynamics. Finite element — A continuum is divided into a number of regions; each region behaves as a structural member with nodes compatible to the nodes of neighboring regions. Hysteresis model — An inelastic force-deformation relationship of a structural member subjected to cyclic loading.

Influence factor — An vector matrix to specify the force induced by ground accelerations at each floor of a building.

Multicomponent seismic components — Earthquake accelerations expressed in more than one direction such as three translations and three rotations.

Participation factor — A factor for measuring how much a given vibrating mode participates in synthesizing the structural total load.

Periodic motion — A motion repeats itself in a certain period. Harmonic motion is periodic but periodic motion is not harmonic. A combination of two harmonic motions is always a periodic motion.

Pseudo-response — The response (displacement, velocity, or acceleration) dose not reflect real time-history response but a maximum value.

Stability criterion — The numerical integration results can be conditionally stable and unconditionally stable. For unconditionally stability, the solution is not divergent even if time increment used in the integration method is large. Conditional stability, however, depends on the time increment.

Transmissibility — A ratio between the amplitude of the force transmitted to the foundation to the amplitude of the driving harmonic force.


[1] Cheng, F.Y., Matrix Analysis of Structural Dynamics — Applications and Earthquake Engineering, Marcel Dekker, Inc., New York, 2001.

[2] Wilson, E.L., Kiureghian, A.D., and Bayo, E.P., A replacement for the SRSS method in seismic analysis, J. Earthq. Eng. Struct. Dyn, 9, 187-194, 1981.

[3] Tuma, J.J. and Cheng, F.Y., Dynamics Structural Analysis, McGraw-Hill, Inc., New York, 1987.

[4] Cheng, F.Y. and Tseng, W.H., Dynamic Instability and Ultimate Capacity of Inelastic Systems Parametrically Excited by Earthquakes, Part I, NSF Report, National Technical Information Service, US Department of Commerce, Virginia, NTIS no. PB261096/AS, 1973 (143 pp.).

[5] Kanai, K., Semi-empirical formula for the seismic characteristics of the ground, Bull. Earthq. Res. Inst., Univ. Tokyo, 35, 309-325, 1957.

[6] Tajimi, H., A statistical method of determining the maximum response of a building structure during an earthquake, in Proceeding of 2nd World Conference on Earthquake Engineering, Japan, 1960, pp. 781-797.

[7] Mathews, J.H. and Howell, R.W., Complex Analysis for Mathematics and Engineering, William C. Brown Publishing, Dubuque, IA, 1996.

[8] Clough, R. and Penzien, J., Dynamics of Structures, 2nd ed., McGraw-Hill, Inc., New York, 1993, pp. 713-726.

[9] Humar, J.L., Dynamics of Structures, Prentice Hall, Englewood Cliffs, NJ, 1990, p. 728.

[10] Jain, A.K., Goel, S.C., and Hanson, R.D., Hysteretic cycles of axially loaded steel members, ASCE J. Struct. Div, 106, 1775-1795, 1980.

[11] Takeda, T., Sozen, M.A., and Nielsen, N.N., Reinforced concrete response to simulated earthquakes, ASCE J. Struc. Div., 96, 2557-2573, 1970.

[12] Cheng, F.Y. et al., Computed versus observed inelastic seismic response of low-rise RC shear walls, ASCE J. Struct. Eng., 119(11), 3255-3275, 1993.

[13] Cheng, F.Y., Ger, J.F., Li, D., and Yang, J.S., INRESB-3D-SUPII Program Listing for Supercomputer: General Purpose Program for Inelastic Analysis of RC and Steel Building Systems for 3D static and Dynamic Loads and Seismic Excitations, NSF Report, National Technical Information Service, US Department of Commerce, Virginia, NTIS no. PB97-123616, 1996 (114 pp.).

[14] Cheng, F.Y., Ger, J.F., Li, D., and Yang, J.S., INRESB-3D-SUPII Program Listing for PC: Inelastic Analysis of RC and Steel Building Systems for 3D Static and Dynamic Loads and Seismic Excitations, NSF Report, National Technical Information Service, US Department of Commerce, Virginia, NTIS no. PB97-123632, 1996 (109 pp.).

[15] Ger, J.F., Cheng, F.Y., and Lu, L.W., Collapse behavior of Pino-Suarez building during 1985 Mexico Earthquake, ASCE J. Struct. Eng., 119(3), 852-870, 1993.

[16] Cheng, F.Y., Multiobjective Optimum Design of Seismic-Resistant Structures in Recent Advances in Optimal Structural Design, S.A. Burns, ed., ASCE, Virginia, 2002, Chap. 9.

[17] Cheng, F.Y., and Li, D., Multiobjective optimization of structures with and without control, AIAA J. Quid. Control Dyn., 19(2), 392-397, 1996.

[18] Cheng, F.Y. and Li, D., Multiobjective optimization design with pareto genetic algorithm, ASCE J. Struct. Eng., 123(9), 1252-1261, 1997.

[19] Cheng, F.Y. and Ang, A.H.-S., Cost-effectiveness optimization for aseismic design criteria of RC buildings, in Proceedings of Case Studies in Optimal Design and Maintenance Planning of Civil Infrastructure Systems, D.M. Frangopol, ed., ASCE, 1998, pp. 13-25.

[20] Chang, C.C., Ger, J.R., and Cheng, F.Y., Reliability-based optimum design for UBC and nondeterministic seismic spectra, ASCE J. Struct. Eng., 120(1), 139-160, 1994.

[21] Cheng, F.Y. and Juang, D.S., Assessment of various code provisions based on optimum design of steel structures, Int. J. Earthquake Eng. Struct. Dyn., 16, 45-61, 1988.

[22] Truman, K.Z. and Cheng, F.Y., Optimum assessment of irregular three-dimensional buildings, ASCE J. Struct. Eng., 116(12), 3324-3337, 1989.

[23] Cheng, F.Y. and Suthiwong, S., Active Control for Seismic-Resistant Structures on Embedded Foundation in Layered Half-Space, NSF Report, National Technical Information Service, US Department of Commerce, Virginia, NTIS no. PB97-121345, 1996 (261 pp.).

[24] Cheng, F.Y. and Jiang, H.P., Hybrid control of seismic structures with optimal placement of control devices, ASCE J. Aerospace Eng., 11(2), 52-58, 1998.

[25] Cheng, F.Y. and Zhang, X.Z., Building structures with intelligent hybrid control and soil-structure interaction, in Proceedings of US-Korea Workshop on Smart Infrastructural Systems, C.K. Choi, F.Y. Cheng, and C.B. Yun, eds., KAIST, Korea, 2002, pp. 133-142.

Further Reading

Cheng, F.Y. and Ger, J.F., Maximum response of buildings to multi-seismic input, ASCE Dyn. Struct. 397-410, 1987.

Cheng, F.Y. and Ger, J.F., Response analysis of 3-D pipeline structures with consideration of six-component seismic input, in Proceedings of Symposium on Resent Developments in Lifeline Earthquake Engineering, American Society Mechanical Engineers (ASME) and Japan Society of Mechanical Engineers (JSME), Vol. I, pp. 257-271, 1989.

Cheng, F.Y. and Ger, J.F., The effect of multicomponent seismic excitation and direction on response behavior of 3-D structures, in Proceedings of 4th US National Conference on Earthquake Engineering, Earthquake Engineering Research Institute (EERI), Vol. 2, pp. 5-14, 1990.

Chopra, A.K., Dynamics of Structures, Prentice Hall, Englewood Cliffs, NJ, 1995.

Craig, R.R., Structural Dynamics, John Wiley, New York, 1981.

Newmark, N.M., Torsion in symmetrical buildings, in Proceedings of 4th World Conference on Earthquake Engineering, Vol. A3, Santiago, pp. 19-32, 1969.

Newmark, N.M. and Hall, W.J., Earthquake Spectra and Design, Earthquake Engineering Research Institute (EERI), Oakland, CA, 1982.

Newmark, N.M. and Rosenblueth, R., Fundamentals of Earthquake Engineering, Prentice Hall, Englewood Cliffs, NJ, 1971.

Paz, M., Structural Dynamics, 2nd ed., Van Nostrand, New York, 1985.

Penzien, J. and Watabe, M., Characteristic of 3-dimensional earthquake ground motion, J. Earthq. Eng. Struct. Dyn., 3, 365-373, 1975.

Soong, T.T. and Grigoriu, M., Random Vibration of Mechanical and Structural Systems, Prentice Hall, Englewood Cliffs, NJ, 1993.

Timoshenko, S.P., Young, D.H., and Weaver, W. Jr., Vibration Problems in Engineering, 4th ed., John Wiley, New York, 1974.

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