Buildings 2012, 2(2), 107-125; doi:10.3390/buildings2020107

Article
An Introduction to the Methodology of Earthquake Resistant Structures of Uniform Response
Mark Grigorian 1,* and Carl E. Grigorian 2
1
MGA Structural Engineering Inc., 111 N. Jackson Street Glendale, CA 91206, USA
2
URS Corporation, 915 Wilshire Blvd., Los Angeles, CA 90017, USA; Email: Carl.Grigorian@urs.com
*
Author to whom correspondence should be addressed; Email: markarjan@aol.com.
Received: 9 December 2011; in revised form: 22 March 2012 / Accepted: 24 April 2012 /
Published: 2 May 2012

Abstract

: Structures of Uniform Response are special earthquake resistant frames in which members of similar groups such as beams, columns and braces of similar nature share the same demand-capacity ratios regardless of their location within the group. The fundamental idea behind this presentation is that seismic structural response is largely a function of design and construction, rather than analysis. Both strength and stiffness are induced rather than investigated. Failure mechanisms and stability conditions are enforced rather than tested. Structures of Uniform Response are expected to sustain relatively large inelastic displacements during major earthquakes. A simple technique has been proposed to control and address the gradual softening of such structures due to local/partial instabilities and formation of plastic hinges. In structures of uniform response, the magnitude and shape of distribution of lateral forces affects the distribution of story stiffness in proportion with story moments, therefore affecting the dynamic behavior of the system as a whole. Simple closed form formulae describing the nonlinear behavior of moment frames of uniform response have been proposed. While the scope of this contribution is limited to moment frames, the proposed method can successfully be extended to all types of recognized earthquake resisting systems.
Keywords:
earthquake resistant frames; structural instability; plastic analysis; stiffness degradation; sequential hinging; energy equivalency; drift control; period analysis

Notation

fmagnification factorHtotal building heightTperiod of vibration
i, jinteger coordinatesIbeam moment of inertiaUinternal energy
hstory heightJcolumn moment of inertiaVshear force
Buildings 02 00107 i001height from baseKsub frame stiffnessWsub frame weight
mnumber of storiesLspan lengthQtotal weight
nnumber of baysMbeam moment Buildings 02 00107 i002local displacement
sorder of occurrenceNcolumn moment Buildings 02 00107 i003total displacement
Cnumerical constantPjoint load Buildings 02 00107 i004drift ratio
Emodulus of elasticity Buildings 02 00107 i005beam plastic moment Buildings 02 00107 i006joint rotation
Fexternal force Buildings 02 00107 i007column plastic moment

Indexes, superscripts and the remaining symbols are defined, as they first appear in the text.

1. Introduction

The purpose of this paper is to introduce the performance of Structures of Uniform Response (SUR) under lateral loading. SUR are special frameworks in which members of similar groups such as beams, columns and braces of similar physical characteristics e.g., length, end conditions etc., share the same demand-capacity ratios regardless of their location and numbers within the group. In other words, selected groups of members develop identical levels of stress and strain under similar loading conditions.

Results of inelastic static, push-over and dynamic time-history analyses [1,2], have shown that Performance Based Plastic Design methods can successfully be applied to almost all types of code recognized earthquake resisting systems. The performance of SUR as earthquake resisting systems is directly supported by these findings. In introducing SUR the paper also presents a new analytic Performance-Based Elastic-Plastic Design method for earthquake resisting moment frames, with the ability to control their response during all phases of seismic loading, starting from zero to first yield, followed by progressive plasticity up to and including incipient collapse [3,4].

As far as it can be ascertained, the proposed drift increment and moment redistribution Equations are the only ones of their kind that can analytically estimate lateral displacements and element moments of such frames throughout both elastic as well as plastic ranges of loading. The step-by-step procedure presented in this work is particularly suitable for manual as well as spreadsheet computations. Most importantly, the proposed formulations help engineers gain insight into structural behavior of earthquake resistant SUR. All SUR formulations yield mathematically admissible initial designs within which member sizes can be modified for any reason, especially for meeting target objectives, optimizing material and construction costs without violating the prescribed performance conditions. The performance of SUR in the elastic and elastic–partially plastic stages satisfy all conditions of lower bound solutions and tends towards uniqueness as plasticity propagates through selected groups of members of the framework. SUR are generally more economical in the elastic range; and become lighter in total weight as the number of plastic hinges increase within the framework. Uniqueness implies that any upgrading of any member property can only enhance the performance of the structure beyond its targeted projections.

The methodology leading to SUR also provides a wealth of technical information that may not be readily available through traditional methods of approach. SUR in general and Moment Frames of Uniform Response (MFUR) in particular can be formulated to address the following target objectives:

  • • A prescribed drift ratio at any given loading or performance stage.

  • • A prescribed carrying capacity corresponding to any drift ratio or performance stage, including maximum allowable lateral displacement at incipient collapse.

  • • Predetermined sequences of formations of plastic hinges before collapse.

  • • Damage control in terms of the number of plastic hinges at any loading or response stage compared with number of plastic hinges at zero loading, at first yield or at incipient collapse.

  • • Reduction of the total weight of the structure to a theoretical minimum.

  • • The possibility to further enhance or the performance of the structure using moment control technologies such as brackets, haunches, end flange plates and/or proprietary devices.

1.1. Basic Design Objectives

The mathematical formulation of MFUR is based on the implementation of the following design objectives that:

  • 1. The ideal inter-story drift ratio remains constant along the height of the structure, and that lateral displacements remain a linear function of the height during all phases of loading.

  • 2. The plastic hinges are prevented from forming within columns, except at base line. Whenever possible, base line plastic hinges should form within the grade beams. Global mechanism is reached if the concept of strong-column weak-beam is considered.

  • 3. For minimum weight MFUR, the demand-capacity ratios of all members are as close to unity as possible.

1.2. Basic Design Assumptions

The methodology expounded in this presentation is based on the following design assumptions:

  • 1. Axial, shear and panel zone deformations are not coupled with flexural displacements and can be temporarily ignored for the purposes of this study.

  • 2. Groups of similar members simultaneously resist similar types of loading or combinations of loading, e.g., flexural, axial, torsional, etc.

  • 3. The shape of code specified distribution of earthquake forces remains constant during all loading phases. The shape could be triangular or determined by any rational analysis.

  • 4. Initial design is based on the fundamental period of vibration of the un-degraded structure.

  • 5. The effects of plastic hinge offsets from column center lines can be ignored.

  • 6. The possible benefits of strain hardening and yield over-strength can be ignored.

  • 7. Code level gravity loads have little or no effect on the ultimate carrying capacity of moment frames designed for moderate to severe earthquakes. However the columns should be designed in such a way as to resist gravity forces together with effects induced by plastic hinging of the beams.

  • 8. The design earthquake loads act monotonically throughout the history of loading of the structure.

  • 9. The frames are two dimensional and are constructed out of ductile materials and connection failure is prevented under all loading conditions.

  • 10. The columns remain effectively elastic during all phases of loading.

Traditional design methods for earthquake resistant moment frames begin with approximate member Properties for initial sizing, and entail several cycles of analysis before a satisfactory solution is established. The proposed procedure begins with an optimized Performance-Based Elastic-Plastic design approach that already has the code prescribed criteria built into its basic algorithm. Optimization in this context implies providing as much capacity as demand imposed on or attracted by each member of the frame. Several simple examples have been provided to illustrate the applications of the proposed procedures. While the scope of the present work is limited to moment frames, the proposed method can successfully be extended to all types of recognized lateral load resisting systems.

2. Methodology

In MFUR selected groups of beams and columns share the same drift and demand-capacity ratios.

The most fundamental step in generating a MFUR is to select the properties of its constituent elements in such a way as to achieve geometrically similar inter story drift profiles prescribed for the entire structure during all phases of loading. In other words, the most ideal lateral deformation profile for any frame is that in which the code prescribed story level displacements fall along the same straight line i.e.,

Buildings 02 00107 i008

where, Buildings 02 00107 i009. Buildings 02 00107 i010is the maximum roof or Buildings 02 00107 i011level lateral displacement at Buildings 02 00107 i012response stage. Symbol Buildings 02 00107 i013signifies increment at Buildings 02 00107 i012consecutive iteration.

Equation (1a) indicates that points of inflexion should occur at column mid-heights. By the same token, the most ideal design drift function is that where the code prescribed inter story rotations remain the same, i.e.,

Buildings 02 00107 i014

Theoretically there can be as many loading or response stages as there are beams, i.e., Buildings 02 00107 i015. However, Buildings 02 00107 i016or Buildings 02 00107 i017offer more practical options for design purposes. Rotation Buildings 02 00107 i018may be construed as the initial target drift corresponding to initial target displacement Buildings 02 00107 i019at first yield. The line diagram of a regular MFUR together with its idealized design displacement profiles, subjected to a generalized distribution of lateral forces, is presented in Figure 1. The design conditions (1a) and (1b) imply equal incremental joint rotations for all members of the frame i.e.,

Buildings 02 00107 i020
Buildings 02 00107 g001 200
Figure 1. Laterally Loaded Moment Frames of Uniform Response (MFUR) with Linearly Varying Drift Profiles.

Click here to enlarge figure

Figure 1. Laterally Loaded Moment Frames of Uniform Response (MFUR) with Linearly Varying Drift Profiles.
Buildings 02 00107 g001 1024

The global equilibrium of the structure in terms of beam stiffness and global frame rotation Buildings 02 00107 i022at any stage “s” can be expressed as:

Buildings 02 00107 i023

Where, Buildings 02 00107 i024, Buildings 02 00107 i025, Buildings 02 00107 i026and Buildings 02 00107 i027are the story level shear force, relative stiffness, end moments and moment of inertia of beam “i,j” respectively. Equation (2a) directly yields the global rotation of the structure as;

Buildings 02 00107 i028

Similarly, the racking equilibrium Equation of any representative floor in terms of its beam stifnesses may be expressed as:

Buildings 02 00107 i029

The quantities Buildings 02 00107 i030and Buildings 02 00107 i031are defined as the average and total racking moment acting on Buildings 02 00107 i032level beams at Buildings 02 00107 i012response stage respectively. Buildings 02 00107 i033, and Buildings 02 00107 i034correspond to average racking moments of grade and roof level beams respectively. Buildings 02 00107 i035is defined as the Buildings 02 00107 i032story raking moment at Buildings 02 00107 i012response stage. Equation (2c) in turn directly yields the floor level rotations as:

Buildings 02 00107 i036

However, since Buildings 02 00107 i037, then equating the global rotation Equation (2b), and floor level rotation, Equation (2d), gives:

Buildings 02 00107 i038 Buildings 02 00107 i039 (3a)

Substituting for Buildings 02 00107 i040and expanding the right hand side of Equation (3a), it gives:

Buildings 02 00107 i041

Equation (3b) can be satisfied only if the following mathematical conditions are met:

Buildings 02 00107 i042 Buildings 02 00107 i043 Buildings 02 00107 i044 Buildings 02 00107 i045 (3c)

In physical terms, the condition of uniform drift requires that the sum of the stiffnesses of beams of each floor be selected in proportion with the average racking moments of that floor, i.e.,

Buildings 02 00107 i046…. Buildings 02 00107 i047 (3d)

Equation (3d) also describes a state of uniform demand-capacity for the beams of the structure. In practical terms, uniform demand-capacity implies providing as much capacity as demand imposed on or attracted by each member of the frame. Since equal joint rotations also imply zero moments at column mid-points, then the racking equilibrium of the frame in terms of column moments of any representative floor “m” or “i” may be expressed as;

Buildings 02 00107 i048

or

Buildings 02 00107 i049

Where, Buildings 02 00107 i050, Buildings 02 00107 i051, and Buildings 02 00107 i052are the relative stiffness, end moments and moment of inertia of column “i,j” respectively. Equation (4b) yields the floor level rotations as:

Buildings 02 00107 i053

Buildings 02 00107 i054is the drift component of the Buildings 02 00107 i032level floor, due to deformations of the columns of the same level. Comparing the two sides of Equations (4a) and (4c) yields the conditions of uniform response or demand-capacity for the columns of the subject frame as:

Buildings 02 00107 i055…… Buildings 02 00107 i056 (4d)

Equations (3d) and (4d) together fulfill the condition of compatible drift angles along the height of the frame and verify the statement of the methodology of MFUR presented at the beginning of this section. The applications of the proposed approach are elaborated through parametric examples in the forthcoming sections.

3. Story Level Elastic-Plastic Displacement Response

The total drift of any story in terms of its racking moments at any response stage may be computed as;

Buildings 02 00107 i057

or

Buildings 02 00107 i058 Buildings 02 00107 i059 (5b)

In the absence of gravity loads, the force-deformation relationship or the drift increment Equation (5b) may be expanded to include the effects of member plasticity and story level axial forces, [5,6] i.e.,

Buildings 02 00107 i060

However, since drift increment is a function of stiffness degradation as well as sequence of formation of plastic hinges, it becomes necessary to relate beam stiffness factors Buildings 02 00107 i061to their sequence of formation of plastic hinges or response stage “s”, by means of subscript ”r”, rather than their location “j”. This is achieved by replacing Buildings 02 00107 i061with Buildings 02 00107 i062and Buildings 02 00107 i063with Buildings 02 00107 i064and incorporating the symbol Buildings 02 00107 i065and Buildings 02 00107 i066in Equation (5c) in order to include the effects of formation or prevention of formation of plastic hinges at the ends of beams “ Buildings 02 00107 i067”.i.e.,

Buildings 02 00107 i068

Equation (5d) now represents both the elastic as well as plastic deformations of the subject moment frame within a single, seamless expression, where by definition, the smaller the subscript “s” the stiffer the beam it represents. Buildings 02 00107 i069 for Buildings 02 00107 i070and implies structural damage and/or loss of stiffness with respect to beam Buildings 02 00107 i071 Buildings 02 00107 i072for Buildings 02 00107 i073. In mathematical terms, Buildings 02 00107 i072for Buildings 02 00107 i074and Buildings 02 00107 i069for Buildings 02 00107 i075Similarly the symbol Buildings 02 00107 i066has Buildings 02 00107 i076been introduced to relate column stifnesses Buildings 02 00107 i077to effects of formation of plastic hinges in the adjoining beams “i,j” and “i,j-1. Buildings 02 00107 i078for Buildings 02 00107 i079and Buildings 02 00107 i080, otherwise Buildings 02 00107 i081 Buildings 02 00107 i082is the force magnification function. Buildings 02 00107 i083and Buildings 02 00107 i084are the total axial load and the critical axial load of level “i” at Buildings 02 00107 i012response stage respectively. In reality, since the drift ratio is constant, it would be sufficient to compute Buildings 02 00107 i085for the simplest representative level, i.e., the roof, where, Buildings 02 00107 i086. Equation (5d) reduces to:

Buildings 02 00107 i087

Buildings 02 00107 i088is the stiffness of the Buildings 02 00107 i089level framework at Buildings 02 00107 i012response stage.

3.1. Demonstrative Example I

Consider the lateral displacements of the four bay (n = 4), three story (m = 3) moment frame of Figure 2a, subjected to a uniform distribution of lateral forces Buildings 02 00107 i090and axial joint forces Buildings 02 00107 i091for all “i” and Buildings 02 00107 i092for all other“i,j”.

Buildings 02 00107 g002 200
Figure 2. Example I, Moment Frame Loading and Racking Moment.

Click here to enlarge figure

Figure 2. Example I, Moment Frame Loading and Racking Moment.
Buildings 02 00107 g002 1024

The cross section and moment of resistance of beams and columns of each level are required to be constant, e.g., Buildings 02 00107 i093and Buildings 02 00107 i094for all “j”. Buildings 02 00107 i095and Buildings 02 00107 i096for all other “j”. The primary purpose of this exercise is to generate a MFUR by computing the quantities Buildings 02 00107 i097 Buildings 02 00107 i098and Buildings 02 00107 i099in terms of their corresponding values Buildings 02 00107 i100and Buildings 02 00107 i101(at roof level) respectively. The distribution of story level racking moments Buildings 02 00107 i102is shown in Figure 2b. The total racking moments Buildings 02 00107 i031can now be computed as 4.5 Buildings 02 00107 i103, 7.0 Buildings 02 00107 i103, 3.5 Buildings 02 00107 i103 and 1.0 Buildings 02 00107 i103 for the grade, 1st, 2nd and roof level beams respectively. Since Buildings 02 00107 i104is uniform for each level and the quantity Buildings 02 00107 i105is constant for the entire frame, then for Buildings 02 00107 i106 Buildings 02 00107 i107and Buildings 02 00107 i108. And, as a result, Equation (3d) reduces to Buildings 02 00107 i109 i.e., Buildings 02 00107 i110 Buildings 02 00107 i111 Buildings 02 00107 i112and Buildings 02 00107 i113By the same token, since Buildings 02 00107 i114simplifies Equation (4d) to Buildings 02 00107 i115then: Buildings 02 00107 i116 Buildings 02 00107 i117 Buildings 02 00107 i118The third level (roof) drift of the newly generated MFUR at s = 1 can now be expressed as:

Buildings 02 00107 i119

Assuming that the design decisions; Buildings 02 00107 i120 Buildings 02 00107 i121for all other “j” an Buildings 02 00107 i122d satisfy the Strong column-weak beam requirements, then for Buildings 02 00107 i123 L = h, Buildings 02 00107 i124and Buildings 02 00107 i125, Equation (6a) becomes; Buildings 02 00107 i076

Buildings 02 00107 i126

Therefore; Buildings 02 00107 i127 Buildings 02 00107 i128and

Buildings 02 00107 i129Finally, if the target drift Buildings 02 00107 i130is not to exceed Buildings 02 00107 i131where the subscript Y signifies first yield, i.e., Buildings 02 00107 i132then, the design representative moment of inertia becomes Buildings 02 00107 i133

4. Story Level Elastic-Plastic Moment Response

The elastic-plastic displacement response of the moment frame, Equation (5c), is directly influenced by the redistribution of forces in the members of the structure. For instance, if the magnitude of the end moments of beam “i,j” at Buildings 02 00107 i012response stage is given by; Buildings 02 00107 i134, then by substituting for Buildings 02 00107 i135from Equation (2d) gives:

Buildings 02 00107 i136

and

Buildings 02 00107 i137

as the Moment Redistribution or Plasticity Progression equations of the beams and columns of the subject frame at any given response stage respectively. At ultimate loading or incipient collapse the quantities Buildings 02 00107 i138and Buildings 02 00107 i139become Buildings 02 00107 i140and Buildings 02 00107 i141respectively.

The ultimate carrying capacity of regular moment frames is usually computed using the virtual work method of plastic analysis, [7] which eventually results in static equilibrium Equations that involve the global overturning moments Buildings 02 00107 i142of the system at incipient collapse, e.g., considering the plastic collapse of the moment frame of Figure 3d, through formation of plastic hinges at beam ends only, and conforming to a uniform virtual side sway of inclination θ = 1, it gives:

Buildings 02 00107 i143

For the particular conditions of Example I, the long hand solution of Equation (8a) gives; [3.75 + 2.75 + 1.5] Buildings 02 00107 i144= 2[1.0 + 3.5 + 7.0 + 4.5] Buildings 02 00107 i145or Buildings 02 00107 i146as the ultimate load carrying capacity of the subject moment frame. However in case of MFUR, the racking equilibrium Equation of any story, Equation (2c), can also be used to achieve the same results, i.e.,

Buildings 02 00107 i147
Buildings 02 00107 i148

Where, Buildings 02 00107 i099is the plastic moment of resistance of the stiffest beam of the Buildings 02 00107 i032level framing. Since the pre-assigned uniformity ratios Buildings 02 00107 i149are constant for all “i”, then dividing Equations (8a) and (8b) by each other, reaffirms the condition of uniform strength at incipient collapse, i.e.,

Buildings 02 00107 i150
Buildings 02 00107 g003 200
Figure 3. Progression of Plasticity in Moment Frame of Uniform Response.

Click here to enlarge figure

Figure 3. Progression of Plasticity in Moment Frame of Uniform Response.
Buildings 02 00107 g003 1024

Equation (8d) can be used to compute the plastic moments of resistance of beams of any level “i” in terms of plastic moments of resistance of the uppermost level beams. For the MFUR of the preceding example this gives; Buildings 02 00107 i151 Buildings 02 00107 i152 Buildings 02 00107 i153and Buildings 02 00107 i154Substituting Buildings 02 00107 i155and Buildings 02 00107 i156in Equation (8c) yields; Buildings 02 00107 i157or Buildings 02 00107 i158a result already established using Equation (8a) above. This result implies that:

The ultimate load carrying capacity of an ( Buildings 02 00107 i159 MFUR with moment resisting grade beams under lateral loading of apex value Buildings 02 00107 i160 is Buildings 02 00107 i161

In physical terms, the plastic failure load of MFUR with moment resisting grade beams is independent of the number of stories and the distribution profile of the lateral forces. Equation (7a) can now be used to establish the first, r = 1, increment of loading that causes formation of the first set of plastic hinges in the beams of the stiffest bay of the structure. Since the plastic hinges of the beams of any bay “j” form simultaneously, it would suffice to first study the distribution of moments of any level “i” and then extend the results to beams of other bays by simple proportioning as indicated by Equation (8d), i.e.,

Buildings 02 00107 i162

Substituting for Buildings 02 00107 i163, Buildings 02 00107 i164and Buildings 02 00107 i165in Equation (9a) after some rearrangement, it gives the amount of force needed to produce the first set of plastic hinges in the stiffest beam of the Buildings 02 00107 i011level:

Buildings 02 00107 i166

Now bearing in mind that by virtue of Equation (7a) moments generated in the Buildings 02 00107 i167beam (x > s) of any level can be expressed in terms of the maximum moments of the stiffest beam of that level i.e., Buildings 02 00107 i168and that the sequence of formation of the plastic hinges of any level is the same as the sequence of decreasing values of stiffnesses of the beams of the same floor, then the plastic moment of resistance of the stiffest element s = 1 and moment of resistance of the next stiffest element s = 2 can be computed as Buildings 02 00107 i169and Buildings 02 00107 i170respectively. Therefore, the balance of bending moment needed to elevate the moment of resistance of beam s = 2 to Buildings 02 00107 i171can be computed as Buildings 02 00107 i172whence the amount of additional force required to generate plastic hinges at the ends of the next stiffest beam may be generalized as:

Buildings 02 00107 i173

Since the sum of the incremental forces Buildings 02 00107 i174should add up to the ultimate load Buildings 02 00107 i175,then summing both sides of Equation (9c) over all “n” iterations gives:

Buildings 02 00107 i176

Substituting for Buildings 02 00107 i177 Buildings 02 00107 i178and Buildings 02 00107 i179in Equation (9d) leads to the previously established solution; Buildings 02 00107 i180.

Equations (5e) and (9c) indicate that each stage of propagation of plastic hinges characterized by s = 1, 2…, n may be construed as a target design point or a state of stable damage with respect to fully elastic or fully plastic conditions of the structure. The final stage also represents a minimum weight, unique [8] state of plastic design since it satisfies the prescribed yield criteria, and static equilibrium as well as the selected boundary support conditions at incipient collapse. This implies that the proposed scheme also provides an envelope of several initial designs within which member sizes could be rearranged for any purpose while observing the prescribed performance conditions.

4.1. Demonstrative Example II

Use Equations (9c) and (5e) to study the nonlinear behavior of the MFUR of the previous example and compute the total internal energy of the system in terms of the drift function Buildings 02 00107 i181Given; Buildings 02 00107 i182 Buildings 02 00107 i183 Buildings 02 00107 i184 Buildings 02 00107 i185J = 1.2I, Buildings 02 00107 i186, Buildings 02 00107 i187, h = L and Buildings 02 00107 i188Hence from Equation (9c):

Buildings 02 00107 i189

In other words:

Buildings 02 00107 i190
Buildings 02 00107 i191
Buildings 02 00107 i192
Buildings 02 00107 i193

Therefore, Buildings 02 00107 i194 Buildings 02 00107 i195 Buildings 02 00107 i196, and as expected Buildings 02 00107 i197, reconfirms the validity of the failure load formula Buildings 02 00107 i198

It is instructive to note that because of the particular sequence of formation of plastic hinges, all columns remain intact up to and including completion of stage two, i.e., Buildings 02 00107 i199for s = 1 and s = 2. After culmination of stage two, first columns j = 0 and j = 1 together at the beginning of s = 3, next columns j = 2 at the end of s = 3, then columns j = 3 and j = 4 after culmination of s = 4 lose their stiffness, due to formation of plastic hinges at their adjoining beam ends. Equation (5e) for drift increment becomes:

Buildings 02 00107 i200

Or in numerical terms:

Buildings 02 00107 i201= Buildings 02 00107 i202

Buildings 02 00107 i203= Buildings 02 00107 i204

Buildings 02 00107 i205= Buildings 02 00107 i206 (11b)

Buildings 02 00107 i207= Buildings 02 00107 i208

Therefore; Buildings 02 00107 i209 Buildings 02 00107 i210 Buildings 02 00107 i211and Buildings 02 00107 i212The combined numerical results of groups of Equations (10b) and (11b), are presented in Figure 4 as the nonlinear load-displacement relationship of the subject MFUR. Equations (5c) and (9e) together provide useful design information that neither elastic nor plastic methods of analysis can offer on their own, for instance the maximum lateral displacement of the example frame at first yield and incipient collapse can be estimated as:

Buildings 02 00107 i213

and

Buildings 02 00107 i214

respectively. Furthermore, it was demonstrated that the sequences of formations of plastic hinges could be controlled by selecting the relative stiffness of groups of similar beams in accordance with certain target decisions.

Buildings 02 00107 g004 200
Figure 4. Load-Displacement relationship, Demonstrative Example II.

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Figure 4. Load-Displacement relationship, Demonstrative Example II.
Buildings 02 00107 g004 1024

5. Energy Computations For MFUR

The total accumulative internal energy of any stable structural system due to elastic-plastic deformations of its constituent elements at any response stage can be computed as:

Buildings 02 00107 i215

However, since in MFUR the drift ratio Buildings 02 00107 i216is constant, it would suffice to compute the internal energy of any representative level, such as that of the Buildings 02 00107 i011level, and then compute the rest by simple proportioning. The energy Equation corresponding to level “m” may be computed as:

Buildings 02 00107 i217

Furthermore, since the ratio of internal energies of any two stories is the same as the ratio of average racking moments of the same two stories, i.e., Buildings 02 00107 i218, then the total energy of the system may be expressed as:

Buildings 02 00107 i219

Recalling that the sum of the average racking moments is equal to the sum of the story racking moments as well as the global overturning moment, i.e., Buildings 02 00107 i220then Equation (12c) reduces to its most practical form:

Buildings 02 00107 i221

Equation (12d) implies that:

The ratio of total internal energy of MFUR to that of anyone of its levels, such as the roof, is equal to the ratio of the global overturning moment to the overturning moment of that (roof) Level.

A proof of the validity of this statement is presented in Appendix I. The energy quantity Buildings 02 00107 i222for the preceding example can be worked out via Equation (12b) or as the total area under the force-displacement (push-over) curve of Figure 4, i.e., Buildings 02 00107 i2236.9512 Buildings 02 00107 i224. Equation (12d) can then be used to compute the total internal energy of the entire system as; Buildings 02 00107 i225 Buildings 02 00107 i22655.6096 Buildings 02 00107 i224, or in terms of ultimate values; Buildings 02 00107 i227and Buildings 02 00107 i228at incipient collapse as; Buildings 02 00107 i229 Buildings 02 00107 i230can also be looked upon as an indication of the capacity of the structure to absorb external energy.

5.1.1. Stiffness Degradation

An understanding of the rate and sequence of degradation of story level stiffnesses Buildings 02 00107 i231, is a priori to estimating the momentary periods, Buildings 02 00107 i232, of vibrations of the system at any response stage “s”. Progressive plasticity tends to degrade the global stiffness and modify the dynamic characteristics of statically indeterminate structures under monotonically increasing lateral forces. The effects of stiffness degradation are more pronounced in MFUR since many members of similar characteristics either, fail, become inactive or develop plastic hinges simultaneously. The natural period of vibration of each stage of global loss of stiffness increases with advancing stages of loading until the structure ceases to resist external forces. As the rate of degradation of global stiffness is a function of increasing number of plastic hinges, Equations (5d) and (5e) may be rearranged to assess the gradual loss of global stiffness in terms of sequential formation of plastic hinges.

To demonstrate the use of Equations (5d) and (5e), consider the long hand stiffness analysis of the 3rd story of the MFUR of example I for all four stages of response. The results of this rather cumbersome exercise for Buildings 02 00107 i179may be summarized as; Buildings 02 00107 i233 Buildings 02 00107 i234, Buildings 02 00107 i235 Buildings 02 00107 i236, Buildings 02 00107 i237 Buildings 02 00107 i238and Buildings 02 00107 i239, where Buildings 02 00107 i240However, since in MFUR the distribution of story level stiffness is also a function of the story level shears, it would be reasonable to seek a simpler method of computing for the story level stiffnesses in terms of shear force ratios at different stages of loading, i.e.,

Buildings 02 00107 i241

A derivation of Equation (13a) is presented in the Appendix II. This Equation implies that:

The ratio of stiffness of any two floors of an MFUR is proportional to the ratio of shear forces of the two levels multiplied by the inverse ratio of their heights.

The use of Equation (13) can be demonstrated by the following simple computations:

Buildings 02 00107 i242, Buildings 02 00107 i243, etc.

To further illustrate the applications of the proposed solutions consider the deterioration of the story level stiffnesses of the example frame at all four stages of loading. Since changes in the global drift angle are influenced directly by degradation of story level stiffnesses, then the story level stiffness of any response stage can be associated with the drift angle of the same stage. Therefore, the results of group of Equations (11b) can be used to compute all Buildings 02 00107 i244by simple, numerical proportioning e.g.,

Buildings 02 00107 i245 Buildings 02 00107 i246 Buildings 02 00107 i247 Buildings 02 00107 i248 Buildings 02 00107 i249 Buildings 02 00107 i250 Buildings 02 00107 i251

Buildings 02 00107 i252 Buildings 02 00107 i253 Buildings 02 00107 i254 Buildings 02 00107 i255 Buildings 02 00107 i256 Buildings 02 00107 i257 Buildings 02 00107 i258 (13b)

Buildings 02 00107 i259 Buildings 02 00107 i260 Buildings 02 00107 i261 Buildings 02 00107 i262 Buildings 02 00107 i263 Buildings 02 00107 i264 Buildings 02 00107 i265

This set of numbers complete the stiffness degradation matrix of the subject example frame.

5.1.2. Period Analysis

The dynamically induced seismic forces of MFUR are highly sensitive to variations in the fundamental period of vibrations as well as the shape and magnitude of the pre-assigned drift profile. Both of these issues are briefly discussed in this section. The period analysis presented herein is based on the following basic assumptions:

  • 1. The normalized displacement function Buildings 02 00107 i266remains unchanged throughout the loading history of the structure. In other words, loss of stiffness changes only the magnitude of lateral displacements, but not the deformed shape Buildings 02 00107 i267of the system.

  • 2. The lateral displacement profile of the frame is a function of the single variable Buildings 02 00107 i216for all stages of loading, i.e., all displacement profiles Buildings 02 00107 i268follow the same linear shape function as their normalized displacement function Buildings 02 00107 i269

  • 3. Dominant mode shapes remain unchanged during formation of plastic hinges and that the coupling of modes in the inelastic range can be neglected.

  • 4. The first mode of the first un-degraded stage is the most dominant mode of all response stages and that the first mode of each stage is the dominant mode of that stage.

The fundamental period of vibrations of any stable MFUR corresponding to any particular response stage can be expressed as: Buildings 02 00107 i270, where, the generalized stiffness and mass of the s Buildings 02 00107 i271response stage are defined as:

Buildings 02 00107 i272and Buildings 02 00107 i273respectively (14a)

Assuming Buildings 02 00107 i274and substituting Buildings 02 00107 i275and Buildings 02 00107 i276in Equation (14a) gives;

Buildings 02 00107 i277

Observing that the normalized displacement function is independent of sequence of formation of plastic hinges, and that the term Buildings 02 00107 i278and the ratio Buildings 02 00107 i279are constant for all “s”, then Buildings 02 00107 i280can be expressed in terms of the single variable, Buildings 02 00107 i088, i.e.,

Buildings 02 00107 i281

Where, C is a numerical constant. Equation (14c) indicates that MFUR can be treated as single degree of freedom (SDOF) systems for all “s”, and all practical purposes. Since both C and Buildings 02 00107 i282are independent of “s”, once the fundamental period of vibrations of the un-degraded structure, Buildings 02 00107 i283, is determined the corresponding values for each degraded stage can be worked out through simple proportioning,

Buildings 02 00107 i284

To demonstrate the applications of Equations (14b) and (14d), consider the variations of fundamental period of vibrations of the example frame with respect to changes in the global stiffness of the structure. Equation (14b) gives:

Buildings 02 00107 i285 Buildings 02 00107 i286

Equation (14d) can now be used to determine Buildings 02 00107 i287 Buildings 02 00107 i288and Buildings 02 00107 i289as:

Buildings 02 00107 i290 Buildings 02 00107 i291

and

Buildings 02 00107 i292

The elongation of the natural period of vibrations is associated with the loss of global stiffness. Degradation of the stiffness reduces the rate of change of demand as well as the reserve capacity of the remaining (intact) structure. As the reserve capacity and the rate of change of demand diminish, the structure becomes softer until it fails through a collapse mechanism.

5.1.3. Energy Equivalency

Since by virtue of ( Buildings 02 00107 i293), the global stiffness of the structure is in direct proportion with each story level stiffness, and that by definition, Equation (1a), the lateral displacements are a function of the single variable Buildings 02 00107 i216for all “s” then MFUR may be looked upon as SDOF systems for all practical purposes.

In short; MFUR may be treated as statically determinate, SDOF structures.

This implies that Equation (12d) can be used in conjunction with Housner’s [9] equal energy concept for SDOF structures in order to formulate the demand-capacity relationship of the subject frame in terms of its seismic shears and the corresponding internal energy generated within the structure at any response stage “s”, i.e.,

Buildings 02 00107 i294 Buildings 02 00107 i295 (15)

Buildings 02 00107 i296 and Buildings 02 00107 i297are the spectral acceleration and the energy equivalency factors respectively [10]. Buildings 02 00107 i298and are defined as the period dependant ductility and ductility reduction factors respectively. If the quantity Buildings 02 00107 i299is interpreted as the seismic capacity of the structure, then the right hand side of Equation (15), may be looked upon as the seismic demand or equivalent total dynamic input energy of the system.

5.2. Demonstrative Example III

Compute the base shear of the moment frame of example I subjected to seismically induced lateral forces of uniform intensity F at first yield, s = 1, and at incipient collapse, s = 4, in terms of the following design data;

Prescribed design drift ratios Buildings 02 00107 i300and Buildings 02 00107 i301radians, total structural self weigh Q = 3W and un-medium degraded range fundamental period of vibrations, Buildings 02 00107 i302, where, Buildings 02 00107 i303is the site specific design spectral response acceleration parameter. Substituting for the corresponding quantities and the total internal energies Buildings 02 00107 i304at first yield and Buildings 02 00107 i305at incipient collapse in Equation (15), and bearing in mind that Buildings 02 00107 i306, it gives:

Buildings 02 00107 i307

and

Buildings 02 00107 i308

respectively.

6. Conclusions

It was shown that SUR in general and MFUR in particular are ideally suited for Performance Based Elastic-Plastic Design. A number of new, closed form formulae for understanding the response of MFUR were presented. The proposed methodology lends itself well to controlling the sequential response of MFUR due to monotonically increasing lateral forces. MFUR approach results in Unique Minimum Weight solutions for lateral force resisting moment frames designed to perform as intended at any prescribed response stage. Because of their predetermined characteristics, MFUR can be treated as statically determinate, SDOF structures ideally suited for energy equivalency analysis. It was demonstrated through simple parametric examples that the proposed procedures provide useful design information that neither elastic nor plastic methods of analysis can offer on their own. Furthermore, it was shown that the sequences of formations of plastic hinges could be controlled by selecting the relative stiffness of groups of similar beams in accordance with predetermined performance objectives. The more significant theoretical aspects of MFUR were summarized in the following statements:

  • • In MFUR selected groups of beams and columns share the same drift and Demand-capacity ratios.

  • • The ultimate load carrying capacity of an ( Buildings 02 00107 i159MFUR with moment resisting grade beams under lateral loading of apex value Buildings 02 00107 i160is Buildings 02 00107 i161

  • • The ratio of total internal energy of MFUR to that of anyone of its levels, such as the roof, is equal to the ratio of the global overturning moment to the overturning moment of that (roof) level.

  • • The ratio of stiffness of any two floors of an MFUR is proportional to the ratio of shear forces of the two levels multiplied by the inverse ratio of their heights.

  • • MFUR may be treated as statically determinate, SDOF structures.

Most importantly, the proposed formulations may help engineers gain insight into structural behavior of lateral load resisting SUR and lend themselves well to manual as well as spreadsheet computations.

In closing it should be emphasized that the presented approach may be used for the preliminary economical performance based design of lateral force resisting moment frames. The final design may need to be checked and verified using an appropriate inelastic dynamic or static analysis.

References

  1. Goel, S.C.; Liao, W.; Bayat, M.R.; Chao, S. Performance-based plastic design method for earthquake resistant structures. Struct. Des. Tall Spec. Build. 2010, 19, 115–137.
  2. Mazzolani, F.; Pilosu, V. Theory and Design of Seismic Resisting Moment Frames; Taylor & Francis: Oxford, UK, 1996.
  3. Grigorian, M.; Grigorian, C. Performance control for seismic design of moment frames. J. Construct. Steel Res. 2011, 67, 1106–1114, doi:10.1016/j.jcsr.2011.02.001.
  4. Grigorian, M.; Grigorian, C. Recent developments in plastic design analysis of steel moment frames. J. Construct. Steel Res. 2012. in press.
  5. Grigorian, M. On the lateral response of regular high-rise frames. Struct. Des. Tall Spec. Build. 1993, 2, 233–252, doi:10.1002/tal.4320020305.
  6. Grigorian, M.; Grigorian, C. Preliminary minimum weight design of moment frames under lateral loading. AISC Eng. J. 1988, 25, 129–136.
  7. Beedle, L.S. Plastic Design of Steel Frames; John Wiley & Sons: Hoboken, NJ, USA, 1985.
  8. Neal, B.G. The Plastic Methods of Structural Analysis; Chapman & Hall Ltd.: London, UK, 1963.
  9. Housner, G.W. Limit Design of Structures to Resist Earthquakes. In Proceedings of the 1st World Conference on Earthquake Engineering, Berkeley, CA, USA, 1956; Earthquake Engineering Research Institute (EERI): Berkeley, CA, USA, 1956.
  10. Chopra, A.K.; Goel, R.K. Capacity-demand diagram methods for estimating seismic deformations of inelastic structures. Earthq. Spectra 1999, 15, 637–655, doi:10.1193/1.1586065.

Appendix I

In order to verify the validity of the total internal energy expression, Buildings 02 00107 i309, consider the ratio of internal energy of any level “i” to that of the roof level “m”, i.e.,

Buildings 02 00107 i310

However, since the drift ratio is constant, i.e., Buildings 02 00107 i311, then Buildings 02 00107 i312Substituting for Buildings 02 00107 i313in Equation (16a), it gives Buildings 02 00107 i314The total internal energy of the system may also be computed as the sum of internal energies of the “m” individual levels, i.e.,

Buildings 02 00107 i315

Appendix II

The local Force- displacement relationship of any story level “i” can be expressed in terms of its local drift angle as Buildings 02 00107 i316, and for the roof level as Buildings 02 00107 i317. Observing that by definition, Buildings 02 00107 i318, then the division ( Buildings 02 00107 i319gives; Buildings 02 00107 i320

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