# An Introduction to the Methodology of Earthquake Resistant Structures of Uniform Response

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## Abstract

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## Notation

f | magnification factor | H | total building height | T | period of vibration |

i, j | integer coordinates | I | beam moment of inertia | U | internal energy |

h | story height | J | column moment of inertia | V | shear force |

height from base | K | sub frame stiffness | W | sub frame weight | |

m | number of stories | L | span length | Q | total weight |

n | number of bays | M | beam moment | local displacement | |

s | order of occurrence | N | column moment | total displacement | |

C | numerical constant | P | joint load | drift ratio | |

E | modulus of elasticity | beam plastic moment | joint rotation | ||

F | external force | column plastic moment |

## 1. Introduction

- • 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

- 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

- 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.

## 2. Methodology

**Figure 1.**Laterally Loaded Moment Frames of Uniform Response (MFUR) with Linearly Varying Drift Profiles.

## 3. Story Level Elastic-Plastic Displacement Response

#### 3.1. Demonstrative Example I

^{st}, 2

^{nd}and roof level beams respectively. Since is uniform for each level and the quantity is constant for the entire frame, then for and . And, as a result, Equation (3d) reduces to i.e., and By the same token, since simplifies Equation (4d) to then: The third level (roof) drift of the newly generated MFUR at s = 1 can now be expressed as:

## 4. Story Level Elastic-Plastic Moment Response

#### 4.1. Demonstrative Example II

## 5. Energy Computations For MFUR

#### 5.1.1. Stiffness Degradation

#### 5.1.2. Period Analysis

- 1. The normalized displacement function remains 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 of the system.
- 2. The lateral displacement profile of the frame is a function of the single variable for all stages of loading, i.e., all displacement profiles follow the same linear shape function as their normalized displacement function
- 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.

#### 5.1.3. Energy Equivalency

#### 5.2. Demonstrative Example III

## 6. Conclusions

- • In MFUR selected groups of beams and columns share the same drift and Demand-capacity ratios.
- • The ultimate load carrying capacity of an ( MFUR with moment resisting grade beams under lateral loading of apex value is
- • 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.

## References

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**2010**, 19, 115–137. [Google Scholar] - Mazzolani, F.; Pilosu, V. Theory and Design of Seismic Resisting Moment Frames; Taylor & Francis: Oxford, UK, 1996. [Google Scholar]
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## Appendix I

## Appendix II

© 2012 by the authors; licensee MDPI, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Grigorian, M.; Grigorian, C.E.
An Introduction to the Methodology of Earthquake Resistant Structures of Uniform Response. *Buildings* **2012**, *2*, 107-125.
https://doi.org/10.3390/buildings2020107

**AMA Style**

Grigorian M, Grigorian CE.
An Introduction to the Methodology of Earthquake Resistant Structures of Uniform Response. *Buildings*. 2012; 2(2):107-125.
https://doi.org/10.3390/buildings2020107

**Chicago/Turabian Style**

Grigorian, Mark, and Carl E. Grigorian.
2012. "An Introduction to the Methodology of Earthquake Resistant Structures of Uniform Response" *Buildings* 2, no. 2: 107-125.
https://doi.org/10.3390/buildings2020107