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MCE 516 – SEISMIC ASSESSMENT AND STRENGTHENING OF REINFORCED CONCRETE BUILDINGS

Assoc.Prof.Dr. Emre AKIN

Adnan Menderes Unv.

Civil Engineering Dept.

Introduction

• Earthquake (Seismic) Loads: Loads which are produced due to the inertia of the structure which tries to remain its position when subjected to a ground motion (where magnitude changes rapidly along with the time). Therefore, the seismic loads are actually dynamic.

Total mass of the building:

𝑚 = 𝑖=1𝑁 𝑚𝑖

𝑇𝑛 =2𝜋

𝑤𝑛

Assoc.Prof.Dr. Emre Akın

m1

mi

mN mi: mass of the «i»th story of an N-story building (all the story mass is assumed to be lumped at the mass center)

Single Degree of Freedom (SDOF) representation

(at the fundamental

vibration mode)

Gro

un

d

Acc

. (t)

Time, t

ui(t

)

Time, tLateral

displacement of «it»th story (function of time)

mk: rigiditywn: natural (angular) vibration frequency at the fund. modeTn: natural vibration period at the fund. modeh: equivalent height

Gro

un

d

Acc

. (t)

Time, t

u(t

)

Time, t

Lateral displacement of SDOF system:

fs(t)=k.u(t)

Lateral load:

Vb(t)=fs(t)

Mb(t)=fs(t)×h

Base shear:

Introduction-SDOF Representation

• Earthquake (Seismic) Loads: Loads which are produced due to the inertia of the structure which tries to remain its position when subjected to a ground motion (where magnitude changes rapidly along with the time). Therefore, the seismic loads are actually dynamic.


m1

mi

mN

uN= q1× φN1𝑢 =

𝑢1

𝑢𝑖𝑢𝑁

= 𝑞1

𝜙1

𝜙𝑖𝜙𝑁 1

= 𝑞1. 𝜙 1 : displacement vector

𝑚 =𝑚1 ⋯ 0⋮ ⋱ ⋮0 ⋯ 𝑚𝑁

: mass matrix

𝑘 =𝑘1 ⋯ 0⋮ ⋱ ⋮0 ⋯ 𝑘𝑁

: rigidity matrix

𝑀1 = 𝜙 1𝑇 𝑚 𝜙 1: modal mass contributing to the fundamental mode

𝐾1 = 𝜙 1𝑇 𝐾 𝜙 1: rigidity corresponding to the fundamental mode

𝑤12 =

𝐾1

𝑀1: vibration frequency at the fundamental mode 𝑇1 =

2𝜋

𝑤1: fundamental period

ui= q1× φi1

u1= q1× φ11

Note: In a structure that is subjected to dynamic ground motion, the lateral displacements are functions of time and geometry, which are represented by generalized displacement, q(t) and shape function φ(x); u(x,t)= q(t). φ(x)Here we use the peak value of generalized displacement for any ground motion record: q(t)= q1. («1» represents that it corresponds to the fundamental mode of vibration).

Here we represent the lateral displacements by only those contributing at the fundamental mode of vibration. This may only be done for regular buildings where the lateral response is governed by the first (fundemantal) mode. Any irregularity may increase the effect of higher modes on the lateral response, where mode superposition should be applied.

q1M1

K1 T1

Simply «m» in the other slides

Simply «k» in the other slides

Mentioned as «Tn» in the other slides

Single-d

egree-of-freed

om

(SDO

F)Mu

lti-

deg

ree-

of-

free

do

m (

MD

OF)

Introduction

• As shown in the previous slide, the earthquakes induce forces and displacements in the structures. These displacements and forces are directly related by the system stiffness for elastic systems. However, the relationship between seismic forces and displacements (especially in case of cyclic loading) are much more complex for the structures responding inelastically. And this relationship depends on the displacement level, displacement history, etc.

Linear elastic response Inelastic cyclic (hysteretic) response


f

u

k1

Introduction

• The seismic design and assessment has traditionally been based on forces (load based design or assessment). The main reason for this is that the traditional design against other effects (dead+live loads, wind forces, etc.) are load based. A certain level of strength is provided (in design or strengthening) to deal with these effects.

• However, we know that strength is less important when the seismic events are considered, since the structures and their members/sections/materials experience inelastic deformations under these events and ductility becomes critical.

• Anyway, if the load based approach is used in design or assessment for practical purposes, the inelastic response or ductility should be somehow taken into account.


Introduction-Load Based Design/Assessment

Pseudo Acceleration: 𝐴 = 𝑤𝑛2𝑢𝑚𝑎𝑥

Relationship between rigidity and mass: 𝑘 = 𝑚.𝑤𝑛2

Then the lateral load becomes: 𝑓𝑠 = 𝑚.𝑤𝑛2𝑢𝑚𝑎𝑥 = 𝑚. 𝐴

• The structure is now subjected to an equivalent static seismic load (fs=m.A) which corresponds to the ultimate effect of the seismic actions on the structure (dynamic action is represented as an equivalent static action). Pseudo acceleration is taken from a design spectrum and termed as spectral acceleration (Sa) in elastic load-based design or assessment.

m

Gro

un

d

Acc

. (t)

Time, t

u(t

)

Time, t

fs=k.umax

Vb=fs=m.A

Mb(t)

umax


Introduction

• In this explained procedure, the structure is assumed as elastic. Actually the structure is pushed into nonlinear inelastic response when subjected to a significant ground motion record. If the structure is designed considering certain seismic code regulations provided for highly ductile members (capacity design and seismic detailing), the structure will have the ability to produce a significant inelastic response without a serious degradation of strength (a significant amount of energy can be dissipated through inelastic actions).

• In the beginning of design, if the engineer accepts to fulfill the requirements for a (highly) ductile design, he/she may take a reduced design base shear by knowing that the structure will respond inelastically (lower inelastic forces) instead of attaining high elastic forces.


mf=k.u

Vb=f=m.A

Mb(t)

Introduction


mf=k.u

Vb=f=m.A

Mb(t)

u

𝑓𝑦 =𝑓𝑒𝑅𝑦

𝜇 =𝑢𝑚𝑎𝑥𝑢𝑦

Reduction Factor

Displacement Ductility Factor

𝑘 =𝑓𝑒

𝑢𝑒=

𝑓𝑦

𝑢𝑦→ 𝑅𝑦 =

𝑓𝑒

𝑓𝑦=

𝑢𝑒

𝑢𝑦= 𝜇

𝑢𝑒

𝑢𝑚𝑎𝑥

TEC (2018):If the rigidity of structure is low (T>TB), then equal displacement rule (ue=umax) may be assumed: Ry=μNote that equal area rule applies otherwise: Ry=1+(μ-1)(T/TB)

f

u

fe

fy

uy ue

Elastic Demand

Inelastic Demand

umax

k1

f

u

fe

fy

uy ue=umax

Equ

al Disp

lacemen

t Ru

le

k1

Introduction

• Elastic Base Shear : 𝑉𝑏 = 𝑚.𝐴 =𝐴

𝑔.𝑊 = 𝑐𝑒 .𝑊

• Inelastic Base Shear: 𝑉𝑏 = 𝑐𝑠.𝑊 ; 𝑐𝑠= 𝛼. 𝑐𝑒 → 𝑉𝑏 = 𝛼. 𝑐𝑒 .𝑊 → 𝛼 =1

𝑅𝑦

This may also be expressed as 𝑉𝑏 =𝑆𝑎𝑒

𝑔

1

𝑅𝑦𝑊 =

𝑆𝑎𝑖

𝑔𝑊


mf=k.u

Vb=f=m.A

Mb(t)

Elastic Base Shear Coefficient

Inelastic Base Shear Coefficient

Sa (or A)

T (Period)TA TB

Elastic Design Spectrum, Sae (T)

Inelastic (Reduced) Design Spectrum, Sai (T)

Sae

Note for «Equal Displacement Rule»: The time-history analyses of structures whose fundamental periods are in the range of 0.6-2.0 seconds have shown that the maximum seismic displacements of elastic and inelastic systems (with identical stiffness, mass and resulting elastic period) are very similar.

Introduction

• After the base shear is calculated, this base shear is distributed to the story levels. This distribution is proportional to the product of height and mass at different story levels, which is compatible with the displaced shape of preferred inelastic mechanism. This inelastic mechanism may be defined as «beam-end plastic hinges + column-base plastic hinges» and «shear wall-base plastic hinges».

Beam side-sway mechanism

Introduction

• The distribution of base shear over the building height according to TEC (2019):

Then the story shear force is distributed between the different lateral load resisting members (columns and shear walls) of that story in proportion to their elastic stiffness.

Introduction-Fundamental Period

• For a SDOF representation of the building: 𝑇 = 2𝜋𝑚

𝑘(TEC, 2019)

• In some codes: 𝑇 = 𝐶1 𝐻𝑛0.75 where C1 depends on structural system and Hn is the building height

• In TEC (2019): C1=0.1 for reinforced concrete buildings

C1=0.08 for steel buildings

C1=0.07 for all other buildings

where Ct=C1

Introduction

• In design, assuming that a well-designed structure possess ductility (i.e. can deform inelastically to the required deformations imposed by the earthquake without a significant loss of strength), the structure is let to damage but not collapse (check performance objective). This is achieved by seismic load reduction in load based design.

• In doing so, an economical benefit is provided from the reduced construction cost by considering that the design-level earthquakes are rare events (with a certain probability of exceedence in the defined periods).

• In load based assessment, since the existing structure (which may not have the ductile design details) is evaluated, no seismic load reduction is applied (Ry=1).


Reference: Priestley, M.J.N.; Calvi, G.M.; Kowalsky, M.J. 2007, «Displacement-Based Seismic Design of Structures», 720 p. , USS Press, Pavia Italy.

Introduction

• Performance Levels:

▫ Immediate Occupancy: The structure is safe to occupy after the earthquake and expected to experience negligible structural and minor non-structural damage. Only minor cracking without out-of-plane offsets are acceptable for the infill walls. Repair cost should be limited (<15 percent of new construction).

▫ Life Safety: The structure can be re-occupied only after the repair and strengthening of the building. The lateral load resisting members (columns, shear walls) may suffer major damage but no partial collapse is accepted. Extensive cracking is distributed throughout the infill walls and some isolated crushing may be observed.

▫ Collapse Prevention: The re-occupancy of the building is not safe. There is an extensive structural and non-structural damage and the structural system may suffer partial collapse. There is a significant degradation in stiffness and strength of structural members.


Introduction

Imm

edia

te

Occ

up

ancy

, IO

Life

Saf

ety,

LS

Co

llap

se

Pre

ven

tio

n, C

P

CA

PAC

ITY

DEFORMATION

Turkish Earthquake Code, TEC (2019)

Boundary for minimum damage (IO?)

Boundary for visible, controlled damage (LS)

Boundary for significant damage and ollapse prevention (CP)

Besides: KK: Continuous Occupancy (serviciability of the building) (IO?)


Introduction

• Performance Objectives:

Peer’s (Pacific Earthquake Engineering Research Center) Performance Based EQ Engineering Methodology


Introduction

• The seismic code provisions mainly aim for minimizing the hazard to human life and increase the expected performance of the buildings which should be balanced considering economy.

• The main uncertainties or variables that the seismic code should somehow take into consideration:

▫ Characteristics of ground motion

▫ Soil type

▫ Structural properties

▫ Construction practice

▫ Material properties

• The seismic code covers not only design of new building as resistant against anticipated earthquakes, but also performance assessment and strengthening of existing buildings against anticipated earthquakes.


[The frequency content and characteristics of the ground motion record (accelerations) may be significantly different in different soil conditions.]

[Even a structure with certain vibration period may respond in a different manner in case of different ground motions or frequency content. Vibration period of the structure is correlated with the frequency content of the ground motion and thus soil conditions with regard to the expected damage]