# MCE 516 SEISMIC ASSESSMENT AND STRENGTHENING OF REINFORCED ... ¢â‚¬¢The...

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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=1 𝑁 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)

G ro

u n

d

A cc

. ( t)

Time, t

u i (t

)

Time, tLateral

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

m k: rigidity wn: natural (angular) vibration frequency at the fund. mode Tn: natural vibration period at the fund. mode h: equivalent height

G ro

u n

d

A cc

. ( 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.

Assoc.Prof.Dr. Emre Akın

m1

mi

mN uN= q1× φN1

𝑢 =

𝑢1

𝑢𝑖 𝑢𝑁

= 𝑞1

𝜙1

𝜙𝑖 𝜙𝑁 1

= 𝑞1. 𝜙 1 : displacement vector

𝑚 = 𝑚1 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ 𝑚𝑁

: mass matrix

𝑘 = 𝑘1 ⋯ 0 ⋮ ⋱ ⋮ 0 ⋯ 𝑘𝑁

: rigidity matrix

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

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

𝑤1 2 =

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

Sin gle-d

egree-o f-freed

o m

(SD O

F)M u

lt i-

d eg

re e-

o f-

fr ee

d o

m (

M D

O F)

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

Assoc.Prof.Dr. Emre Akın

f

u

k 1

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.

Assoc.Prof.Dr. Emre Akın

Introduction-Load Based Design/Assessment

Pseudo Acceleration: 𝐴 = 𝑤𝑛 2𝑢𝑚𝑎𝑥

Relationship between rigidity and mass: 𝑘 = 𝑚.𝑤𝑛 2

Then the lateral load becomes: 𝑓𝑠 = 𝑚.𝑤𝑛 2𝑢𝑚𝑎𝑥 = 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

G ro

u n

d

A cc

. ( t)

Time, t

u (t

)

Time, t

fs=k.umax

Vb=fs=m.A

Mb(t)

umax

Assoc.Prof.Dr. Emre Akın

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.

Assoc.Prof.Dr. Emre Akın

mf=k.u

Vb=f=m.A

Mb(t)

Introduction

Assoc.Prof.Dr. Emre Akın

m f=k.u

Vb=f=m.A

Mb(t)

u

𝑓𝑦 = 𝑓𝑒 𝑅𝑦

𝜇 = 𝑢𝑚𝑎𝑥 𝑢𝑦

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=μ Note that equal area rule applies otherwise: Ry=1+(μ-1)(T/TB)

f

u

fe

fy

uy ue

Elastic Demand

Inelastic Demand

umax

k 1

f

u

fe

fy

uy ue=umax

Eq u

al D isp

lacem en

t R u

le

k 1

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

𝑅𝑦 𝑊 =

𝑆𝑎𝑖

𝑔 𝑊

Assoc.Prof.Dr. Emre Akın

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

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