1106Englekirk Final Report 042107

101
1 of 52 Rational Seismic Design Procedures for Shear Wall Braced Buildings Robert E. Englekirk, Ph.D., S.E. ALL RIGHTS RESERVED WORLDWIDE. No part of this publication may be reproduced, adapted, translated, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the author.

Transcript of 1106Englekirk Final Report 042107

Page 1: 1106Englekirk Final Report 042107

1 of 52

Rational Seismic Design Procedures for Shear Wall Braced Buildings

Robert E. Englekirk, Ph.D., S.E.

ALL RIGHTS RESERVED WORLDWIDE. No part of this publication may be reproduced, adapted, translated, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the author.

Page 2: 1106Englekirk Final Report 042107

2 of 52

Table of Contents Preface Acknowledgment 1 Shake Table Test Program Overview 2 Shear Wall Design Procedures

2.1 Seismic Intensity 2.2 System Capacities and Desired Characteristics 2.3 Objective Stiffness 2.4 Objective Strength 2.5 Design Process Summary 2.6 “T” Wall Sections 3 Design Verification

3.1 Strain Based Evaluation Procedure 3.2 Inelastic Time History Analysis Procedure 3.3 Strain Based Design Verification Example 3.4 Inelastic Time History Analysis Example 3.5 Inelastic Time-History Analysis Example: UCSD “T” Wall

4 Example Design - 15 Story Building

4.1 Building Description 4.2 Conceptual Design-Rectangular Wall 4.3 Strain Based Design Verification Example-Rectangular Wall Solution 4.4 Conceptual Design Example-“T” Wall 4.5 Design Verification-“T” Wall

5 Force Based Design Procedure

5.1 Application of Code Procedures to UCSD Test Specimen 5.2 A Force Based Design Procedure Developed from Generalized Scientific Principals

Appendix A Quantification of Seismic Intensity Appendix B Identification of System Drift Limit States Appendix C Idealization of Shear Wall Stiffness Appendix D Wall Strength and Overstrength Appendix E Strain States Notation References

Page 3: 1106Englekirk Final Report 042107

3 of 52

FIGURES Figure 1 View of test specimen Figure 2 Typical residential building Figure 3 Test structure Figure 4 Analytical model – UCSD test wall Figure 5(a) Inelastic response projection – UCSD test wall Figure 5(b) Recorded top relative displacement response Figure 5(c) Inelastic projection “T” wall Figure 5(d) Recorded top relative “T” wall Figure 6 Proposed building – example design Figure 7 “T” wall alternative – plan and detail Figure 8 Relative displacement time history longitudinal direction Figure 9 Relative displacement time history transverse direction Figure 10 Relationship between base shear and displacement Figure A.1 Input ground motions Figure A.2 Acceleration response spectra of input ground motions – damping 5% Figure B.1(a) Geometry and reinforcing details-RW2 Figure B.1(b) Plan view of section indicated Figure B.2(a) RW2: Lateral load versus top displacement. Figure B.2(b) RW2: Analytical versus measured force displacement relationships Figure B.2(c) RW2: Analytical versus measured concrete strain profiles (positive displacement) Figure B.2(d) Analytical and experimental versus idealized moment curvature response Figure B.3 Toe spalling-UCSD shear wall after EQ4 – web wall – bottom of west end Figure B.4 Crack pattern after Earthquake #4 – web wall – level 2 – north side Figure C.1 Lateral load versus top horizontal displacement (1.5 m from base)[C2] Figure C.2 Lateral load versus top displacement – wall RW2[C3]

Figure C.3(a) Hysteretic behavior of test beam[C 4] Figure C.3(b) Adopted idealization describing behavior of the beam[C 5]

Figure C.4 Force-deformation relationship for a concrete wall Figure C.5 Top relative displacement response of rectangular wall – Earthquake #3 Figure C.6 Geometry and reinforcing details[C8] – “T” wall specimen

Figure C.7 Force displacement relationship for specimen described in Figure C.6 Figure C.8 Top relative displacement response of UCSD “T” wall specimen to Earthquake #2 Figure C.9 Fourier amplitude spectra of roof accelerometer Figure C.10 Top relative displacement response of UCSD “T” wall specimen to Earthquake #4 Figure D.1 Moment vs. curvature diagram Figure D.2 Hysteretic response – top displacement vs. wall base moment – rectangular shear wall Figure D.3 Idealized deformed shapes Figure D.4 Acceleration profile at maximum overturning moment Figure D.5 System shear force envelopes – rectangular UCSD wall Figure D.5 EQ4 – Acceleration profile at maximum overturning moment Figure D.6 Hysteretic response – top displacement vs. wall base shear – rectangular shear wall Figure D.7 “Plastic truss” analogy Figure E.1(a) Analytical versus measured concrete strain profiles – test specimen RW2 (Figure B.1) Figure E.1(b) Analytical and experimental versus idealized moment curvature response [E1, E2]

Figure E.2 Recorded concrete compressive strain states – UCSD rectangular wall – Earthquake #4 Figure E.3 EQ4 – Variation of neutral axis depth (dj) – base of wall

Page 4: 1106Englekirk Final Report 042107

4 of 52

ABSTRACT Earthquake induced behavior records from a 63 ft. high, 225-ton shear wall braced segment of a seven-story

building excited by ground motions on the shake table located at the Englekirk Structural Research Center at the

University of California San Diego Camp Elliott Field Station are used to refine and confirm both displacement

and force based design procedures. The emulated ground motions were those recorded during actual

earthquakes. They have spectrum velocities of 25 and 55 inches per second. Accordingly, depending on the site,

they represent earthquakes whose probabilistic recurrence (average return period) lies between 25 and 2475

years. Measured responses are compared with static, cyclically loaded test specimens. Key design parameters

are then developed as are material and system limit states. Design experience is combined with a simplified

scientific basis to produce design procedures that are not only simple but transparent. The user is invited to use

his or her knowledge of a proposed building to develop a design. Parameters are developed in a manner that

promotes this interaction between designer and process. Design verification procedures are also developed in a

form that allows the design team to parametrically evaluate the behavior of a system, thereby assuring the best

possible design and system performance.

The author describes how he arrived at a specimen design that had half of the strength recommended by

current codes and why he had confidence in the ability of the system to withstand the ground motion. The

proposed design criterion is used to develop the design of a 15-story example building.

Page 5: 1106Englekirk Final Report 042107

5 of 52

PREFACE

“I prefer the errors of enthusiasm to the indifference of wisdom.”

- Anatole France Those of us actively involved in the design and construction of buildings have for some time believed that the

produced building is excessively strong and that this excessive strength will abrogate any potential benefit. The

predominant goal of the research effort upon which these design guidelines are based is to validate our

premonition, for the strength of the tested shear wall braced system was only half of that required by modern

codes.

One of the principal objectives of these design guidelines is to develop and demonstrate the simplicity and

efficacy of design procedures that are not force based. The proposed alternative is generically categorized as

displacement based design, and its focal objective is the production a bracing system that meets the performance

objectives adopted by the designer. The mere mention of a displacement based approach to design causes

anxiety, an anxiety that is reinforced by early attempts to advance or codify such a procedure. The displacement

based design procedure developed herein is intended to dispel this anguish, for it is simple and easily applied.

This simplicity is attained by the appropriate introduction of component models which have been derived from

tests, in a manner that is appropriate to the design process, and does not obscure the design objective. Further,

the procedure is transparent for it can be easily modified to suit user predilections.

The development of component and system behavior is relegated to appendices which identify relevant

references. Unfortunately, most of the referenced texts, as an expediency and in an effort to be scientifically

defendable, become unintelligible to a design professional somewhat removed from the referenced material. To

bridge this gap, a philosophical basis for the reductions that provide the objective simplicity is also contained in

these design procedures, for it is incumbent upon the responsible design professional to understand the validity

of a design procedure, a proficiency increasingly made more difficult with each generation of new seismic

design codes and further obscured by the introduction of “facilitating” software.

Our current force based design procedures persist in wandering from their scientific base. Those elements

essential to a reintroduction of a rational basis into the force based design process are also presented.

These guidelines are appropriately applied to shear wall braced buildings whose dynamic characteristics

place them in the “velocity constant” spectrum response range, generally identified with fundamental periods

(T1) in excess of 0.5 seconds. The proposed procedures are easily modified to extend their region of

applicability into the “displacement constant” spectrum response range (T1 > 4 seconds).

Page 6: 1106Englekirk Final Report 042107

6 of 52

ACNOWLEDGEMENTS

This report, the data reduction process, as well as the research report which presents and analyzes collected data,

were made possible by a generous grant from the Charles Pankow Foundation.

The earthquake test program upon which this design procedure is based was funded entirely by the

Englekirk Center Industry Advisory Board, a group of 43 structural engineering and construction related firms

and associations based in Southern California.

Patron members include the Carpenters/Contractors Cooperation Committee, Englekirk Systems

Development, Inc., Highrise Concrete Systems, Inc. and the Charles Pankow Foundation.

Englekirk Advisory Board members who contributed time, money and moral support to the program

include: American Segmental Bridge Institute; Anderson Drilling; Baumann Engineering; Brandow & Johnston

Associates; Burkett and Wong Engineers; Charles Pankow Builders, Ltd.; Clark Pacific; Douglas E. Barnhart,

Inc.; Dywidag Systems International, USA, Inc. (DSI); Englekirk and Sabol Consulting Structural Engineers,

Inc.; EsGil Corporation; GEOCON; Gordon Forward; HILTI; Hope Engineering, Inc.; John A. Martin and

Associates; Josephson Werdowatz & Associates Incorporated; JVI, Inc.; KPFF Consulting Engineers; Matt

Construction Corporation; Morley Builders; Nabih Youssef and Associates; Oak Creek Energy Systems;

Occidental Petroleum Corporation; Pacific Southwest Structures; PCL Construction Services, Inc.; Portland

Cement Association; Precast/Prestressed Concrete Manufacturers Association of California (PCMAC);

Saiful/Bouquet Consulting Structural Engineers, Inc.; Schuff Steel-Pacific, Inc.; Structural Engineering

Association of Southern California (SEAOSC); Simon Wong Engineering, Simpson Manufacturing Co., Inc.;

Smith-Emery Company; Stedman & Dyson Structural Engineers; The Eli & Edythe L. Broad Foundation;

Twining Laboratories; UC San Diego Design and Construction; Verco Manufacturing Co.; Weidlinger

Associates, Inc.; and the Structural Engineering Association of San Diego (SEAOSD).

Construction of the seven-story structure was led by Highrise Concrete Systems, Inc. of Dallas, TX,

America's largest sub-contractor using cast-in-place tunnel form technology to build multi-story reinforced

concrete buildings throughout the U.S. Additional in-kind financial support, donated equipment, and labor was

provided by: Baumann Engineering; Dywidag Systems International, USA, Inc (DSI); HILTI; Associated Ready

Mix; California Field Ironworkers; Cemex; Concrete Reinforcing Steel Institute; Douglas E. Barnhart, Inc.;

Englekirk & Sabol, Inc.; Fontana Steel; Grace; Hanson Aggregates; Morley Builders; Pacific Southwest

Structures; Schuff Steel-Pacific Inc.; and the Southern California Ready Mix Concrete Association.

A special thanks is extended to all the Technical and Administrative Staff of the Englekirk Center, who

spent many hours making this project a success.

Page 7: 1106Englekirk Final Report 042107

7 of 52

A special thanks to Jia Wang, Kathy Lee-Choi, Ety Benichou, and Dan Shubin for their assistance in the

development of this document.

The contribution of the National Science Foundation (NSF) and Network for Earthquake Engineering

Simulation Consortium, Inc. (NEES) to earthquake engineering made possible the construction and operation of

the largest shake table in the nation without which this program would not have been possible.

Page 8: 1106Englekirk Final Report 042107

8 of 52

Figure 1 View of test specimen

SECTION 1 - Shake Table Program Overview

The test structure (Figure 1) was developed so as to emulate typical mid-rise (7- to 10-story) residential

construction. A reasonably typical residential building plan is depicted in Figure 2. The usual structural design

problem is to identify the number of shear walls that are required to brace the building. The structural engineer

will typically use the longest wall possible given functional constraints. The problem then is to determine how

many shear walls are required. The wall proposed for the test program supports seven floors at 9’-0” each. The

12 ft. long thin (6 in.) wall was selected because it has been economically constructed and satisfies the

functional requirements associated with both the subterranean parking and the residential units above. Floor

slabs were constructed at each level.

Page 9: 1106Englekirk Final Report 042107

9 of 52

(a) Residential floor plan

(b) Parking plan below

Figure 2 Typical residential building

The test structure is a slice of the 7-story residential building (Figure 2). The structural load bearing walls

also provide the lateral force-resisting system. The structure was built at full-scale as shown in Figure 3. The

initial test specimen consisted of a main 12 ft. long rectangular shear wall and two transverse walls that provide

lateral and torsional stability during the test. The east transverse wall was joined to the rectangular wall so as to

create a “T” section in a second phase of the test program. The original program included five phases; however,

only the first two phases, the rectangular wall and “T” wall, were realized. Testing was performed at the

Englekirk Structural Research Center, where the UCSD-NEES Large High-Performance Outdoor Shake Table

operates.

The construction program for the test structure is described in Figure 3. Tunnel steel forms were used for the

construction of the walls and slabs. The construction sequence included casting a level of the web and flange

walls as well as the slab at the same time. A segmental bracing pier was provided on the west end. It was precast

in three pieces and assembled afterwards using post-tensioning. The width of the web wall was 8 in. at the first

and seventh levels to enable the boundary elements to be confined. A cap beam was to have joined the wall and

the precast bracing column in the fifth, unrealized phase of the test program. The width of the web wall was 6

in. elsewhere. Both the web and the flange wall were cast into footings that were prestressed to the shake table

Page 10: 1106Englekirk Final Report 042107

10 of 52

platen. Pin-pin connections capable of transferring in-plane diaphragm forces were placed between the web

wall and the two transverse walls (Figure 3c). The gravity columns were reinforced high strength steel pin-pin

rods grouted in four inch pipes. Construction details are provided in the test report [1].

a) System elevation b) Foundation plan

c) Slotted connection d) Floor plan

N

Figure 3 Test structure

3d

3b

3c

Post-tensioned concrete column

Page 11: 1106Englekirk Final Report 042107

11 of 52

SECTION 2 - Shear Wall Design Procedures The data extracted from the UCSD shake table test of the shear wall braced system confirm that properly

constructed analytical models provide sufficient insight into the seismic response of structural systems to ground

motion to allow the effective design of this type of system. These tests also suggest that complex analytical

models similar to those proposed currently for force or displacement based design procedures[2], in addition to

being obscure and far too complex, do not necessarily promote designs which will perform well or be cost

effective. The design procedures developed herein are intended to be of the simplest form, and as transparent as

glass.

The design process can be segregated into three parts:

• A communicable description of seismic intensity,

• An understanding of system capacities which are matched to performance goals, and

• A procedure that develops a bracing system that attains system stiffness and strength objectives.

2.1 Seismic Intensity

A structural engineer uses a response spectrum to describe seismic intensity. These response spectra are

typically developed from ground motions. The engineer who elects to follow force based design procedures will

be concerned with the base shear which is developed from probable levels of accelerations. The designer who

chooses to follow a displacement based approach will be concerned with identifying system displacement. For

most buildings these desired design parameters are developed from spectrum velocities (See Appendix A). The

design procedure developed herein is displacement based, though force based alternatives are also discussed.

2.2 System Capabilities and Desired Characteristics

System performance objectives are most readily described in terms of objective levels of system displacement

(See Appendix B). System displacement objectives must be converted to an objective system spectrum

displacement Sd. Since peak displacements are generated by a response dominated by the first mode, the

conversion is easily accomplished.

Page 12: 1106Englekirk Final Report 042107

12 of 52

(1)Δ=

Γu

dS

A normalized linear mode shape ( 1.0nφ = ) suggests a participation factor (Γ ) of 1.5[3] though Chopra[4]

analytically identifies the participation factor as 1.6 for a uniform (mass and stiffness) cantilevered system.

Since our conceptual design objective is to quantify the number and composition of shear walls required to

brace a structure, select the linear (conservative) participation factor of 1.5.

It is now possible to determine the dynamic characteristics of the system required to attain performance

objectives.

0.02 (Displacement objective; See Appendix B)0.02 ( .1)1.5

0.01355 . / sec.

Δ =

=

==

=

u n

d n

n

v

v

d

h

S h Eq

hS in

SS

ω (2)

550.013

=nh

ω

If the shear wall of the test program is considered, the minimum angular frequency should be

55

0.013(756) 5.6 / sec.radians

=

=

ω

The maximum fundamental period that will appropriately limit system displacement response is

max2 (3)

6.28 5.6

1.12 sec. ( 0.89 )

=

=

= =

T

f Hz

πω

Comment: It is interesting to note that a Fourier Amplitude Spectra Analysis of 3% white noise excitations[1]

prior to Earthquake #4 identified the frequency of the primary mode as being 0.86 Hz and that the relative

displacement of the structure was 2% when subjected to Earthquake #4.

Page 13: 1106Englekirk Final Report 042107

13 of 52

2.3 Stiffness Objectives

The angular frequency, ω, or building period, T, defines the relationship between system mass, a known

quantity and system stiffness, our objective.

Most shear wall braced buildings fall into a response range in which seismic intensity is reasonably

quantified by the spectrum velocity of the criterion event(s). See Appendix A.

The relationship between spectrum velocity and spectrum displacement is

(2 )v

d

S aS

ω =

where

Sd is the spectrum displacement (in.)

ω is the angular frequency (radian/sec.)

Alternatively, since 2Tπω =

2 (2b)d

v

STSπ

=

A simple relationship that quantifies the period of a flexural response is:

0.50.89( ) (4)= Δf fT

where fΔ is the flexural component of displacement of the shear wall, in this case, subjected to a load which

represents the mass tributary to it.

2

max,max (5)

0.8fT

Δ =

In the case of the UCSD test shear wall

2

,max(1.12)

0.8 1.57

Δ =

=

f

feet

Page 14: 1106Englekirk Final Report 042107

14 of 52

Next assume that the weight which quantifies the tributary mass is uniformly distributed over the height of

the wall

64.8 9.0

7.2 / .

=

=

=

f

i

Ww

h

kips ft

where Wf represents the mass of the floor which is tributary to the wall being designed.

4

4

min,max

4

4

(6)8

(7)8

7.2(63) 8(3600)(144)(1.57)

17.4 .

0

Δ =

=

=

=

wf

w

f

e

whEIwhI

E

ft

I3

3

.35 ( 0.35 )12

24(17.4) ( 6 )0.35

10.4 .

= −

′′= =

=

we g

w

w

bl I I Appendix C

l b

l ft

Conclusion: The 12 ft. long test wall should attain our displacement objective.

1.510.4 (1.12)12

1.0 sec

T ⎛ ⎞= ⎜ ⎟⎝ ⎠

=

2.4 Objective Strength

The objective idealized flexural strength, Myi, should be that which attains the idealized yield displacement, δyi,

(Figure C.3.b).

(8)

0.000023(3600)(523,000) 43,300 .- . (3600 .- .)

yi y eM EI

in kips ft kips

φ=

==

Page 15: 1106Englekirk Final Report 042107

15 of 52

where

3

4

0.0033 (9)

0.0033 144

0.000023 . / .0.35(6)(144)

12 523,000 .

=

=

=

=

=

yw

e

l

rad in

I

in

φ

Since we have chosen to identify idealized flexural yield strength with the probable overstrength of the wall

( o nMλ )

2

( ) (10)2 2

214(68) 1.25 (60)(136)

2.8 .

wo n o s y

s

s

aM P A f d d

A

A in

λ λ⎛ ⎞ ′= + + −⎜ ⎟⎝ ⎠

= +

=

where P is the probable axial load supported by the wall (kips) and As is the steel in the boundary element.

The provided boundary reinforcing (Figure 3) was 8-#5 (2.48 2.in ); however, a moment curvature analysis

of the wall (Figure D.1) identifies yiM ( o nMλ ) as 4167 ft.-kips, and the dynamic response of the system

(Figure D.2) suggests 5000 ft.-kips as the idealized strength of the system.

Accordingly, the conceptual design procedure developed by Equation 10 should be refined so as to predict

the probable strength of the wall. Since the detailing of the wall will follow capacity based concepts, the

consequences of over reinforcing the wall should be clear.

2.5 Design Process Summary

Step 1: Identify objective seismic intensity. Quantification should be in the form of its spectrum velocity Sv.

61.5v vS C= (See Appendix A, Equation A.2).

Step 2: Identify system capability. Convert adopted drift limit state to an objective spectrum displacement.

1.5u

dS Δ= (Eq. 1)

Page 16: 1106Englekirk Final Report 042107

16 of 52

Step 3: Determine maximum period.

max2

= d

v

STSπ (Eq. 2b)

Step 4: Determine the minimum moment of inertia.

44

min 2max

( . )1440

= wwhI ftET

(11)

where w is a measure of system tributary mass per foot of wall height (kips/ft.).

hw is the height of the wall expressed in feet.

E is the modulus of elasticity (ksi).

Tmax is the maximum period (seconds).

Step 5: Size the wall. Presuming that 0.35e gI I≅ for a rectangular wall (See Appendix C).

,min 0.3334( )= e

w

Il

b (12)

where ,maxgI and b are expressed in terms of ft. (ft.4, ft.).

Step 6: Determine optimal idealized strength of the wall ( yiM ).

0.0033( )eyi

w

EIMl

= (in.-kips) (13)

where Ie is the effective moment of inertia of the wall (in.4)

E is the modulus of Elasticity (ksi) and lw is the length of the wall (in.)

Step 7: Quantify reinforcement required ( yi o nM Mλ= ).

( - )2 2

⎛ ⎞ ′= − +⎜ ⎟⎝ ⎠

wo n o s y

aM P A f d dλ λ (Eq. 10)

Page 17: 1106Englekirk Final Report 042107

17 of 52

Step 8: Consider strength of the wall based on the probable strength of all provided reinforcement.

2.6 T-Wall Sections

The design procedure for a T-wall is identical to that of a rectangular wall with the exception of the idealization

of component stiffness and strength.

Stiffness idealizations seem to be reasonably represented as a constant percentage of the gross moment of

inertia of the “T” section. See for example, Figure C.7. Supporting this hypothesis is the response of the “T”

wall to excitation on the shake table, where displacement maxima are essentially the same in either direction

(See Figure C.10).

Page 18: 1106Englekirk Final Report 042107

18 of 52

SECTION 3 - Design Verification

A two-phase approach is reasonably undertaken. A strain based assessment might be sufficient if suggested

strain states are reasonable. When strain states are large, an inelastic time history analysis is suggested.

3.1 Strain Based Evaluation Procedure

Step 1: Estimate yiΔ .

20.001Δ = w

yiw

hl

(Eq. C.4)

Step 2: Estimate postyield displacement.

(14)Δ = Δ − Δp u yi

Step 3: Estimate plastic hinge rotation, pθ .

(15)

2

( ) (16)2

4

pp

pw

p wp

ww

lh

lllh

θΔ

=−

Δ= =

Step 4: Estimate postyield curvature, pφ (in the Plastic Hinge Region).

2 ( ) (17)

pp

p

p

w

l

See Appendix El

=

=

θφ

θ

Step 5: Estimate depth of the neutral axis, c.

1

(C )(0.85)

s y

c

Pc Tf b

= ≅′β

Step 6: Estimate strain states in the plastic hinge region.

(18)

( ) (19)cu p cy

su p sy

c

d c

ε φ ε

ε φ ε

= +

= − +

Page 19: 1106Englekirk Final Report 042107

19 of 52

3.2. Inelastic Time History Analysis Procedure

Step 1: Create a fiber model of the shear wall system.

• Define bilinear material behavior models.

• Define a shear behavior model.

• Build a fiber model of the wall.

• Assign material properties to the fiber elements modeled.

Step 2: Subject model to earthquake ground motions.

3.3 Strain Based Design Verification Example: UCSD Rectangular Test Wall

Step 1: Estimate yiΔ .

2

2

0.001 ( . .4)

0.001(756) 144

4.0 .

Δ =

=

=

wyi

w

h Eq Cl

in

Step 2: Estimate postyield displacement.

( . 14)

15 4.0 11.0 .

Δ = Δ − Δ

= −=

p u yi Eq

in

Step 3: Estimate postyield rotation.

( .16)

411.0 144756

4 0.016

Δ=

=−

=

pp

ww

Eqlh

radian

θ

Step 4: Estimate postyield curvature in the plastic hinge region.

Page 20: 1106Englekirk Final Report 042107

20 of 52

2 ( .17)

2(0.016) 144

0.00022 . / .

=

=

=

pp

w

Eql

rad in

θφ

Step 5: Estimate the depth of the neutral axis, c.

1(0.85)214

0.85(0.85)(4)(8) 9.3 .

c

Pcf b

in

β=

=

=

Step 6: Estimate strain states in the plastic hinge region.

( .18)

0.00022(9.3) 0.001 0.003 . / .

( ) ( .19)

0.00022(136

cu p cy

su p sy

c Eq

in ind c Eq

ε φ ε

ε φ ε

= +

= +== − +

= 9.3) 0.002 0.03 . / .in in

− +=

Comment: Estimated strain states seem to be consistent with those reported (See Appendix E). The concrete

strain states tend to concentrate at the base of the wall (See Figure E.2). Wallace reports the attainment of

concrete strains as high as 0.01 in./in. (See Figure E.1(a)). Steel tensile strains of 0.027 in./in. were recorded in

the UCSD shear wall shake test (See Figure E.2(b)).

3.4 Inelastic Time-History Analysis Example: UCSD Rectangular Test Wall

The panel elements identified in Figure 4 represent the panels which created the finite element model used in the

inelastic time history analysis. The lowermost panels were identified as being inelastic. The remaining panels

were modeled to simulate elastic behavior. The effective moment of inertia was assumed to be 35% of the gross

moment of inertia. Reinforcing steel was concentrated in the outermost fiber.

Page 21: 1106Englekirk Final Report 042107

21 of 52

The projected relative displacement response of this model is presented in Figure 5a. The displacement

predicted is consistent with that recorded during the test (Figure 5b) both in terms of magnitude and response

characteristics. The maximum moment was 5800 ft.-kips and this, too, is consistent with the reported repeated

flexural strength of the system (See Figure D.2).

Comment: The effort used to develop Figure 5a was minimal though consistent with design procedures

advocated herein and by others. Response was sensitive to the depth of the plastic hinge p( ) here quantified as

52 inches w(0.36 ) . Observe (Figure E.2a) that this is consistent with the reported strain distribution. The

distribution of reinforcing significantly impacted the projected level of displacement, and this is reasonable

since the onset of postyield curvature, given a uniform distribution of reinforcement, occurs sooner.

Conclusion: The conceptual design seems to have been reasonably confirmed. Obviously, the design team

should test the design proposed by varying key parameters and earthquake content and only then decide whether

the proposed design is appropriate.

3.5 Inelastic Time-History Analysis Example: UCSD “T” Wall

A model similar to that described in Figure 4 was used to predict the behavior of the “T” wall section. The

projected and recorded displacement responses are shown in Figures 5c and 5d. The projected and recorded

displacements are quite similar in their initial response (t<5 seconds); observe that the maximum displacement

recorded was 8 inches (Figure 5d) as was the peak predicted displacement (Figure 5c). At this time the

previously weakened splice at the west end of level 2 failed in bond and the subsequent recorded displacements

reflect the consequences of the localized failure.

Page 22: 1106Englekirk Final Report 042107

22 of 52

Figure 4. Analytical model – UCSD test wall

Page 23: 1106Englekirk Final Report 042107

23 of 52

(a) Inelastic response projection – rectangular wall

0 5 10 15 20 25 30

-15

-10

-5

0

5

10

15

t (sec)

d (in

)

PHASE I EQ4 Top Relative Displacement Response

(b) Recorded top relative displacement response - rectangular wall

Time (sec.)

Rel

ativ

e D

ispl

acem

ent (

in.)

15

10

5

0

-5

-10

5 10 15 20 Time (sec.)

Rel

ativ

e D

ispl

acem

ent (

in.)

Page 24: 1106Englekirk Final Report 042107

24 of 52

(c) Inelastic response projection – UCSD “T” wall

Rel

ativ

e D

ispl

acem

ent (

in.)

Time (sec.)

Page 25: 1106Englekirk Final Report 042107

25 of 52

0 5 10 15 20 25 30-10

-8

-6

-4

-2

0

2

4

6

8

10

t (sec)

Dis

plac

emen

t (in

)

(d) Recorded top relative displacement response – UCSD “T” wall

Figure 5 Relative displacement – UCSD test walls when subjected

to Earthquake #4 (Figure A.1)

Time (sec.)

Rel

ativ

e D

ispl

acem

ent (

in.)

Page 26: 1106Englekirk Final Report 042107

26 of 52

SECTION 4 - Example Design – 15-Story Building The design procedures developed in the preceding section are now applied to a reasonably representative 15-

story shear wall braced residential building.

4.1 Building Description

The proposed building plan and section are described in Figure 6.

Figure 6 Proposed building plan and elevation -

example design

Page 27: 1106Englekirk Final Report 042107

27 of 52

Functional objectives and assumptions include

• Slab thickness is 8 in.

• Maximum wall length is 36 ft.

• For design purposes assume that seismic dead load is 0.16 kips/ft.2

• Assume that design spectrum velocity is 50 in./sec.

Design considerations

• Use displacement based design procedures.

• Limit concrete strain in unconfined concrete to 0.005 in./in.

• Limit strain in reinforcing steel to 10 syε .

• Limit building total drift to 2%.

• Consider rectangular walls.

• Provide an alternative design for T-wall sections.

4.2 Conceptual Design − Rectangular Wall

Follow the design process summary developed in Section 2.5.

Step 1: Identify objective seismic intensity.

50 . / sec. ( )vS in given=

Step 2: Identify system capability.

0.02u wh (given)Δ =

( . )

0.02(150)(12)1.5

24 .

udS Eq 1

in

Δ=

Γ

=

=

Step 3: Determine maximum period.

Page 28: 1106Englekirk Final Report 042107

28 of 52

max2 ( .2 )

6.28(24) 50

3 seconds

=

=

=

d

v

ST Eq bSπ

Step 4: Determine the minimum moment of inertia.

Comment: Initially in this design the objective minimum moment of inertia will be for the system

4

min 2max

( .11)1440

= wwhI EqET

where

4

min 2

4

0.16(16,000) 10

256 / .150 .4000

256(150)1440(4000)(3)

2500 .

=

=

===

=

=

f

x

w

Ww

h

kips fth ftE ksi

I

ft

Step 5: Size the shear walls.

Since the maximum length has been identified, the total width of wall is required.

3

3

120.3512(2500) ( 0.35 )0.35(36)

1.84 . ( )

=

= = −

=

gw

w

e g

It

l

I I Appendix C

ft Minimum total thickness

Conclusion: Two 12-in. by 36-ft. long walls should meet the drift/displacement objective of 2%.

Page 29: 1106Englekirk Final Report 042107

29 of 52

Step 6: Determine optimal strength of the shear wall.

Comment: Since the wall is somewhat stiffer than the minimum required, start by identifying the stiffness of the

wall to be provided.

3

3

4

0.3512

0.35(12)(36) (1728) ( 12 .)12

28,200,000 .

0.0033 =

0.0033 (4000)(28,200,000) ( ' 5 )432

862,000 .- (72,000 .

w we

w

yi y e

e

w

c

t lI

t in

inM EI

EIl

f ksi

in kips ft

=

= =

==

= =

=

φ

- )kips

Step 7: Quantify reinforcement required.

Dead load tributary to the wall is (see Figure 6)

1

2

15(40)25(0.1) 1500 ( )36(150)0.15 810 ( )

2310

= == =

W kips slabW kips wall

kips

The depth of the compression block is

0.852310

0.85(5)(12) 36.2 .

c w

Paf t

in

=′

=

=

The moment contribution provided by the shifting of the axial load is (see Equation 10):

Page 30: 1106Englekirk Final Report 042107

30 of 52

yi o n

2

( ) 2310(18 1.51)2 2

38,000 .-

( )2 2 (M M )

( )

34,000 32(1.25)(60)

14.1 . (0.27%)

w

wyi

so y

l aP

ft kipsl aM P

Ad d f

in

λλ

− = −

=

− −= =

′−

=

=

Check the cracking strength of the wall

2310 0.45 ( 0.07 )36(12)12

′= = ≅ c gP ksi f AA

'

2

6.5 0.92 ( 5 )

0.92(12)(36) (144)6

343,000 .- (28,500 . )

c c

c x

cr

P f ksi f ksiA

f S

M in kips ft kips

′+ ≅ =

=

= −

Design Objective: Provide reinforcing steel in sufficient quantities so as to attain the idealized flexural yield

strength of the wall.

Conclusion: Reinforce the ends of the shear wall with 8-#11 bars (12.5 in.2). Confirm that Myi is actually

provided using a sequential yield analysis (see Figure D.1) once the design has been verified. Use capacity

based procedures to ensure the attainment of shear strength objectives.

Page 31: 1106Englekirk Final Report 042107

31 of 52

4.3 Strain Based Design Verification Example: Rectangular Wall

Step 1: Estimate yiΔ .

2

2

0.001 ( . .4)

0.001(1800) 432

7.5 .

Δ =

=

=

wyi

w

h Eq Cl

in

Step 2: Estimate postyield displacement.

Comment: The ultimate displacement objective was 36 in. (see problem statement). Observe that providing a

wider wall than required, 12 in., as opposed to 11 in., would theoretically reduce the ultimate drift to 34.6 in., a

refinement not considered.

( . 14)

36 7.5 28.5 .

Δ = Δ − Δ

= −=

p u yi Eq

in

Step 3: Estimate plastic hinge rotation.

( .16)

428.5

1800 108 0.017

pp

ww

Eqlh

radian

θΔ

=−

=−

=

Step 4: Estimate postyield curvature.

2 ( .17)

0.017 216

0.000079 / .

pp

w

Eql

radian in

θφ =

=

=

Step 5: Estimate depth of the neutral axis

Page 32: 1106Englekirk Final Report 042107

32 of 52

1(0.85)2310

0.80(0.85)(5)(12) 56.6 .

c

Pcf b

in

β=

=

=

Step 6: Estimate strain states in the plastic hinge region.

(Eq.18)

0.000079(56.6) 0.001 0.0055 . / .

( ) (Eq.19)

cu p cy

su p sy

c

in ind c

ε φ ε

ε φ ε

= +

= +== − +

0.000079(364) 0.002 0.030 . / . 10 syin in ε

= += >

Conclusion: Strain states are higher than our adopted objectives (see Design Considerations - Section 4.1). To

reduce these strain states, the designer might increase the number of provided walls, increase their thickness, or

consider providing “T” wall sections along with a thicker flange wall.

Comment: The design of the rectangular wall solution can be quickly improved upon. A logical improvement

would be to thicken the corridor walls to 18 in. and join them to 12 in. transverse walls. This will create two

“T” sections (Figure 6). First check the efficacy of the now proposed 18 in. longitudinal walls. Δu is now

assumed to be 34.6 in. (see comment under Step 2, Section 4.3).

,12 (34.6) ( 1)18

28.2 .28.2 7.5 ( 2)

20.7 .20.7 ( 3)

1800 1080.012

u prob

p

Step

inStep

in

Step

radian

Δ =

=Δ = −

=

Θ =−

=

Page 33: 1106Englekirk Final Report 042107

33 of 52

0.012 ( 4)216

0.000056 . / .56.6(12) ( 5)

1837.7 .

p Step

rad in

c Step

in

φ =

=

=

=

0.000056(37.7) 0.001 (Eq.18) 0.003 . / .

0.000056(383) 0.002 (Eq.19) 0.023 . / .

cu

su

in in

in in

ε

ε

= +

== +=

Conclusion: This solution seems reasonable.

Page 34: 1106Englekirk Final Report 042107

34 of 52

Figure 7 “T” wall alternative - plan and detail

4.4 Conceptual Design Example: “T” Wall

The solution in the transverse direction is two “T” walls (see Figure 6).

Step 1: Determine the effective moment of inertia, Ie.

Dimensions of the proposed “T” wall are described in Figure 7.

Page 35: 1106Englekirk Final Report 042107

35 of 52

4

4

10,250

2560 ( 0.25 ; )

=

= =g

e e g

I ft

I ft I I See Appendix C

Step 2: Determine probable period of the structure.

4

4

0.5

( 256(0.5) 128 / .) ( .6)8

128(150) 8(4000)(144)(2560)

5.5 .0.89(5.5) ( .4)

2.1 sec onds

wf

e

wh w kips ft EqEI

ftT Eq

Δ = = =

=

=

==

Step 3: Determine the probable ultimate drift uΔ .

1.5 ( .1)1.5 ( )

2

4.22.1(50)

4.2 25 .

Δ =

=

=

=

=

u d

v

v

S EqT S

TS

in

π

Step 4: Determine pΔ .

25 7.5 17.5 .

Δ = Δ − Δ

= −=

p u y

in ( 7.5 .yi inΔ = Step 1, Section 4.3).

Step 5: Determine optimal strength of the wall.

( .8)

0.0033 (4000)(2560)(1728)432

135,000 .-

yi y eM EI Eq

ft kips

φ=

=

=

Page 36: 1106Englekirk Final Report 042107

36 of 52

Develop the cracking moment in the flange wall.

36(1.5)150(0.15) ( )

1215 2310 ( ; 4.2, 7)

F

S

P flangekips

P kips stem Section Step

===

Determine the amount of reinforcing required at the end of the stem wall to develop Myi (135,000 ft.-kips).

Assume that the depth of the compressive stress block (a) in the flange is 1 ft.

2

18.5(2310) (34) ( )

135,000 43,000 1.25 60(34)135,000 43,000

255036 .

o n o s y

s

s

M A f Eq.10

A

A

= in

λ λ= +

= +−

=

Stem flexural reinforcement to resist cracking moment is

4

/

6.5 5000 0.34 0.8

0.8(10,250)(12)0.8( )342

500,000 .

cr r

cr

cr

f f P A

f ksi

IMc

in kips

= +

≅ + ≅

= =

≅ −

Assume that (d − a/2) is 34 ft. (408 in.)

2500,000 20.3 .408(60)sA in= =

Comment: Vertical wall reinforcing will contribute a significant amount of resistance to cracking and the

strength of the wall.

Conclusion: Try 12-#11 bars (As=18.7 in.2).

Step 6: Estimate the shear strength required in the web following capacity based concepts.

Page 37: 1106Englekirk Final Report 042107

37 of 52

Comment: Vertical reinforcing will be required within the flange and web walls (0.25% min.). Two #6 bars @

18 in. on center meets this requirement (assume that 22 pairs of #6 bars will be provided in the flange and

possibly 12 more in the web).

2

20(1.56) 34(0.88)

61.1 .1.25(60)(61.1)

4584

s

o y

A

inT

kips

= +

==

=

λ

The axial load is

1

2

3

15(40)0.1(25) 1500 ( )34.5(150)(0.15) 776 ( )36(150)(0.225) 1215 ( )

3491

W kips SlabW kips StemW kips Flange

kips

= == == =

4584 3491 158 .0.85(5)(12)

a in+= =

Comment: This neutral axis depth is very large. Before proceeding to develop required shear resistance, check

probable strain states.

4.5 Design Verification – “T” Wall

Step 1: Strain States.

( . 6)

417.5

1800 108 0.01

pp

ww

Eqlh

radian

Δ=

=−

=

θ

Page 38: 1106Englekirk Final Report 042107

38 of 52

1

2 ( .17)

2(0.01) 432

=0.000046 / .

158 (0.000046)0.8

0.0090 . / .

pp

w

cp p

p

cu cp cy

Eql

radian inc

a

in in

θφ

ε φ

φβ

ε ε ε

=

=

=

=

=

== + (Eq.18)

0.01 . / .in in=

Comment: The neutral axis depth has caused the probable concrete strain state to exceed our adopted limit state

of 0.005 in./in. This suggests that the shell may spall. Confinement in the boundary elements over the height of

the plastic hinge region is clearly called for and should be considered in the design. The designer, given this

insight, should endeavor to improve the design. Options include

(1) Use an 18 in. stem wall.

(2) Increase the strength of the concrete in the plastic hinge region (lower two floors).

Explore Option 2: Provide high strength concrete (10,000 psi) in the lower two floors and confine the toe of the

“T” stem.

Develop strength of toe region (Confining lateral pressure, [7]0.09 ′= cf f )

Page 39: 1106Englekirk Final Report 042107

39 of 52

1 2 3

4.1 (Reference 7) 10 4.1(0.9) 13.7

'

8075 12(1.56)(60) 6952

0.85( )6952

0.85(13.7)(10) 59

cc c l

c o y s y

c

cc e

f f f

ksiC T W W W A f

kipsCa

f b

λ

′ ′= += +== + + + −

= −=

=′

=

= .7 .in

Comment: This is a conservative conclusion given the loading rate and duration of load. The developable

concrete stress should be on the order of 1.25f’cc[8]. Given this estimate of probable strength (17 ksi), the depth of

the compressive stress block becomes 41 in., and the neutral axis depth on the order of 70 in.. Observe that this

reduces the postyield concrete strain (εcp) to 0.003 in./in. and the total concrete strain within the confined core to

0.004 in./in. or well within our limit state and certainly within the ultimate strain capacities of confined concrete.

Observe also that the concrete outside of the core will probably not spall.

( .18)

0.000046(70) 0.003 . / .

0.004 . / . ( .18)

cp p

cu

c See Eq

in inin in Eq

ε φ

ε

=

===

Page 40: 1106Englekirk Final Report 042107

40 of 52

Step 2: Design verification − inelastic time history analysis.

Longitudinal Direction

Confirm that the computer output and conceptual design produce similar results.

Building Period – Conceptual Design

3

4

4

4

128 / . of each shear wall0.35

0.35(1.5)(36) 12

2041 .

( .6)8

128(150) 8(4000)(144)(2041)

6.9 .0.89(

e g

fe

w kips ftI I

ftwh EqEI

ftT

==

=

=

Δ =

=

=

= 0.56.9) 2.33 seconds=

Computer Confirmation: T = 2.5 seconds

Expected Displacement

( .2 )2

2.33(1.5)(55) 6.28

30.6 . (1.5%)

vu

T S Eq b

in

ΓΔ =

=

=

π

Computer Confirmation Δu= 31.5 in. (See Figure 8)

Page 41: 1106Englekirk Final Report 042107

41 of 52

Inelastic Time History Analysis Projections

Figure 8 Inelastic Time History Analysis Projection – Relative displacement time history longitudinal direction

Time (seconds)

Page 42: 1106Englekirk Final Report 042107

42 of 52

Transverse Direction

Building Period – Conceptual Design

4

128 / . of each shear wall0.25

2560 . ( .4.4 1)2.1 seconds ( .4.4 2)

= 2.0 seconds27.5 .

e g

u

w kips ftI I

ft Sec StepT Sec StepT

in

==

= −= −

Δ = (1.4%) ( .4.4 3)Sec Step−

Computer Confirmation = 32.5 in. (See Figure 9)

Comment: The shake table displacements were used to perform this time-history analyses (Figure A.1).

The design spectrum velocity of the table input was on the order of 55 in./sec. The spectrum velocity in the

period range of interest (2.5 seconds) is considerably higher than 55 in./sec. (See Figure A.2) Furthermore,

the impulsive nature of the driving ground motion (Figure A.1) produces an anomalous structural response

since the first inelastic excursion is the largest (see appendix A). Accordingly, one might reasonably expect

predicted (Elastic) displacements to be exceeded by those generated by this (impulsive) type of ground

motion.

Computer Confirmation

Page 43: 1106Englekirk Final Report 042107

43 of 52

Figure 9 Inelastic Time History Analysis Projections – Relative displacement time history transverse direction

Time (seconds)

Page 44: 1106Englekirk Final Report 042107

44 of 52

SECTION 5 –Force Based Design Procedures

5.1 Application of Code Procedures to UCSD Test Specimen

Current code based design procedures[9] for the UCSD shear wall building (Figure 3) would develop the design

required system strength (base shear), Vmax, as follows

Building Period

3/ 4nT 0.02(h )

0.45 second=

=

where hn is expressed in feet

v

v

55C (EarthquakeIntensity -55in./sec.)61.5

=0.89CV= W (Ref .9 Eq.105 4)RT2 WR

=

− −

=

The maximum base shear that need be considered is

amax

2.5CV W (Ref .9 Eq.105 6)R

1.3 WR

= − −

=

Design Base Shear

S1.3V WR

=

Comment: The design spectrum acceleration is 1.3g. This is somewhat lower than that suggested by the design

basis spectrum (Figure A.2).

The UCSD test building would be classified[9] as a bearing wall system. Hence, R = 4.5 and the base shear,

VS, would be

Page 45: 1106Englekirk Final Report 042107

45 of 52

S

wn s y

nD D S

w

1.3V W (code basis)4.50.29W130 kips

aM P A f (d d ') (design basis)2 2

215(72 4) 2.48(60)(10.67)12

2800 ft-kipsMV 55.6 kips (V / V 0.43)

0.67h

=

=

=

⎛ ⎞= − + −⎜ ⎟⎝ ⎠

−= +

=

φ= = =

The developable flexural strength of the wall (Figure D.1; 0.0003rad / in.φ = ) is 4850 ft-kips. This suggests that

the overstrength factor RO (Figure 10) should be

O4850R28001.73

=

=

The ductility factor Rd (Figure 10) can be developed from the probable strength given the probable period of one

second (see Figure C.9) or that proposed by the code (0.45 second).

M

E v

d

E

d

4850V 115kips0.67(63)

V C W (T=1.0 seconds)0.89W0.89(450)400 kips400R 3.5115

V 1.3(450) (T=0.45 seconds)=585kips

585R 5.1115

= =

=

=

=

=

= =

=

= =

Page 46: 1106Englekirk Final Report 042107

46 of 52

Since R is the product of RORd

R 1.73(3.5) (T 1.0seconds)6.0

R 1.73(5.1) (T 0.45seconds)8.3

= =

=

= =

=

Alternatively, Rd and RO can be developed from the UCSD test wall.

ud

yi

d

yiO

u

d O

R

15 (See Eq.C.4)4

R 3.75M

RM4167 (See Fig.D.1)28001.5

R R R3.75(1.5)5.6

Δμ = =

Δ

=

=

=

=

=

=

=

=

Conclusion: Design procedures that endeavor to prescriptively identify system period, ductility, and

overstrength significantly depart from the stated scientific basis described in Figure 10.

Page 47: 1106Englekirk Final Report 042107

47 of 52

Figure 10 Relationship between base shear and displacement[2]

Page 48: 1106Englekirk Final Report 042107

48 of 52

5.2 A Force Based Design Procedure Developed from Generalized Scientific Principals

A rational force based design procedure which adopts scientific principals simplified by an engineer’s intuitive

understanding of structural behavior is possible. The user of such a procedure must be comfortable with the fact

that strength is not directly related to building response nor performance (see Appendix D.4).

The procedure evolves from Figure 10 which has for decades been the backbone of Seismic Design. The

essential variances involve:

1. Recognizing that the Elastic Single Degree of Freedom Response Spectrum is not identical to the

base shear spectrum.

2. The period used to define the Elastic Response Parameter (Figure 10) must be developed using a

reasonable estimate of the fundamental period of the structure.

3. The development of system modifiers Rd and RO must have a consistent rational project specific

basis.

The Application of the Variances to the UCSD test specimen design would proceed as follows:

Step 1: Determine probable period of the structure.

4

fe

4

e g

f

wh (Eq.6)8EI

7.2(63) (I 0.35I )8(3600)(144)(25.2)1.08ft.

T 0.89 (Eq.4)

0.92 seconds

Δ =

= =

=

= Δ

=

Step 2: Determine Elastic design base shear (VE − Figure 10).

Comment: The objective seismic intensity was a spectrum velocity of 50 in./sec.

Page 49: 1106Englekirk Final Report 042107

49 of 52

a v

2

[4]aE

a

S S ( A.1)50(6.8)

341in. / sec.0.88g

SV W ( 1.6 )2

0.8S W0.7(450)315kips

Eq.= ω

=

=

=

Γ≅ Γ =

Step 3: Determine objective mechanism base shear (VM – Figure 10).

ud

yi

R Δ=Δ

where Δu is the selected displacement limit state (Appendix B) and Δyi is the idealized yield displacement for the

wall.

yi

u w

d

EM

d

4in. ( .C.4)

0.02h (Appendix B)0.02(756)15in.15R4

3.75VVR3153.7584kips (0.187W)

EqΔ ≅

Δ =

=

=

=

=

=

=

=

Step 4: Determine the objective moment capacity of the wall (Figure D.1).

yi M 1 wM V k h=

where Myi is the idealized yield moment (λoMn) and k1 is the effective height of the mass (Figure D.3), here

assumed to be 0.77[2.

Page 50: 1106Englekirk Final Report 042107

50 of 52

84 0 77 634075

( . )( )ft. kips

=

= −

VS of Figure 10 represents the factored design strength of the wall as determined following the usual approach

(Equation D.1). RO then is

yiO

n

MR

M4075

0.9(3000) 1.5

=

=

φ

And the design base shear

MS

O

VV (Figure 10)R84 1.5

56 kips (0.124W)

=

=

=

Comment: VS depends to a certain extent on designer predilection; it may be significantly higher than required.

As a consequence the provided strength must be used in the development of the mechanism base shear VM and

the shear used to develop the shear capacity of the system (ΩoVM). VM must be based on the provided

mechanism strength (Figure D.1).

Step 5: Determine the strength required of brittle components along the lateral load path (ΩoVM).

o MV 1.4(84) 120 kipsΩ =

=

Comment: System overstrength will depend to a certain extent on the proximity of vertical load carrying

elements[1]. An overstrength factor of 2 may seem more appropriate based on the UCSD tests, but the use of

beam shear theory should in this case be replaced by less conservative estimates of system shear strength (see

Appendix D).

Page 51: 1106Englekirk Final Report 042107

51 of 52

M

o M o

o M

c

V 84 kips step3V 120 kips ( 1.4)V 120

bd 6(138)

0.14ksi 2 f '

=Ω = Ω =Ω

=

=

Conclusion: One of the consequences of controlling the flexural strength of a shear wall is a low shear demand.

Observe that even the worst case projection of shear demand produced shear stresses that are well within the

capabilities of the wall. Observe that the reduction factor R should in this case be 5.6 (RORd).

Page 52: 1106Englekirk Final Report 042107

52 of 52

REFERENCES

[1] Panagiotou M., Restrepo J., Partial Report on the 7-Story Shake Table Test at UCSD, December 2006.

[2] Seismology Committee, Structural Engineers Association of California, Recommended Lateral Force

Requirements and Commentary, Seventh Edition. Sacramento, California, 1999.

[3] Englekirk, R. E., Seismic Design of Reinforced and Precast Concrete Buildings, Equation 3.1.18 (a),

John Wiley & Sons, Hoboken, New Jersey, 2003.

[4] Chopra, A.K., Dynamics of Structures: Theory and Applications to Earthquake Engineering, 2nd

Edition. Prentice Hall, Upper Saddle River, New Jersey, 2001

[5] Englekirk, R. E., Seismic Design of Reinforced and Precast Concrete Buildings, Equation 2.4.11, John

Wiley & Sons, Hoboken, New Jersey, 2003.

[6] Englekirk, R. E., Seismic Design of Reinforced and Precast Concrete Buildings, Section 1.3.1, John

Wiley & Sons, Hoboken, New Jersey, 2003.

[7] Englekirk, R. E., Seismic Design of Reinforced and Precast Concrete Buildings, Section 1.2.1, John

Wiley & Sons, Hoboken, New Jersey, 2003.

[8] MacGregor., J. G., Wight, J. K., Reinforced Concrete Mechanics and Design, Fourth Edition, Figure

3.2.C, Pearson Education, Inc., 2005.

[9] International Conference of Building Officials, Uniform Building Code, 1997 Edition, Whittier,

California.

Page 53: 1106Englekirk Final Report 042107

A-1

Appendix A – Quantification of Seismic Intensity

A.1 Shake Table Excitations

Most concrete shear wall braced buildings fall into a period range of between 0.5 to 4 seconds. Accordingly,

seismic intensity is most appropriately quantified by the spectrum velocity (Sv) of the event of interest.[A1] This is

consistent with current code strength based procedures since the seismic coefficient vC is directly related to the

spectrum velocity Sv and accordingly also constant within this period range. The conversion factor is

2 2a a v

vS TS TCSω π π

= = = (A.1)

where

Sv, spectrum velocity (in./sec.)

Sa, spectrum acceleration (in./sec.2)

Cv, spectrum velocity (expressed as a percent of g) for structures that fall within the constant velocity

period range (see Reference A2).

The equivalence factor ( v

v

SC

) can be developed from Equation A.1. Values of Cv are quantified for a building

whose fundamental period is one second; hence the spectrum velocity for a seismic event whose intensity is

described by Cv is:

(1) (386.4)2

vv

CS =π

61.5 (A.2)v vS C=

Comment: Strength based design procedures use the spectrum velocity coefficient, Cv, to describe the design

base shear ( vC WVRT

= ) and this is appropriate for single degree of freedom systems. It becomes increasingly

conservative for multi-degree of freedom systems. For a discussion see Reference A4.

Page 54: 1106Englekirk Final Report 042107

A-2

-1

-0.5

0

0.5

1A

ccel

erat

ion

(g

)

EQ1 EQ2 EQ3 EQ4

-50

-25

0

25

50

Vel

ocity

(in

/sec)

0 10 20-10

-5

0

5

10

t (sec)

Disp

lace

men

t

(

in)

0 10 20t (sec)

0 10 20t (sec)

0 10 20t (sec)

p p p

Figure A.1 Input UCSD shake table motions

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

2

2.5

3

T(sec)

Sa (g

)

3%g RMS White NoiseSf-vnuy-lngSf-vnuy-trNor-whox-lngNor-Sylmar-360Design

Figure A.2 Acceleration response spectra of input table motions – damping 5%

Page 55: 1106Englekirk Final Report 042107

A-3

The shake table motions used to excite the UCSD shear wall system are described in Figure A.1. The design

spectrum velocity (Figure A.2) was 50 in./sec. and that delivered by Earthquake #4 are essentially the same at a

period of one second.

The intensity of earthquake ground motion is identified for structural design purposes in terms of its

probability of exceedance which is often restated as an Average Return Period (ARP). Since earthquake

intensity is a function of the classification of the supporting soil and proximity of the site to existing faults, the

ground motions described in Figure A.1 cannot be universally identified using this probabilistic type of

classification (ARP).

Comment: The Uniform Building Code[A3] (UBC) identified levels of seismic intensity correspond to events

which have a 10% chance of exceedence in 50 years (475-year ARP). The 2006 International Building Code[A5]

(IBC) will refer to this event as a Design Basis Event (DBE). The collapse threshold event or Maximum

Considered Event (MCE) is associated with a 2% chance of exceedence in 50 years (2475-year ARP). The

relationship between these seismic events is approximated by

MCE = 1.5(DBE) (A.3)

Earthquakes #2 and #3 have spectrum velocities of about 25 in./sec. For much of Southern California this

intensity is identified with an event which has a 50% chance of exceedance in 50 years (25-year ARP).

Structural damage is not considered to be likely given the occurrence of an event of this magnitude. Structural

response should be within the idealized yield range with only minor excursions into the postyield range.

Earthquake #4 (NOR-Sylmar-360) has a peak ground acceleration of 0.82g. The recorded peak table

acceleration was 0.91g. The intensity of this ground motion would place it in the Maximum Considered Event

(MCE) range for competent soils and sites reasonably separated from major earthquake faults. The San Diego

area is typically identified with this magnitude (MCE) of ground shaking. For sites located closer to active faults

Page 56: 1106Englekirk Final Report 042107

A-4

or on less competent soils, Earthquake #4 would describe ground motion associated with the occurrence of a

Design Basis Event (DBE).

Example 1:

[A3]

[A3]

0.56 (UBC Table 16-R- soil profile)

1.6 (UBC Table 16-T 2 km-B fault)0.961.5(0.9)

55 . / .

=

= ≤===

v v C

v

v

v

C N S

NCS

in sec

The MCE spectrum velocity associated with this DBE is

MCE = 1.5(55) (Eq. A.3)

= 82.5 in./sec.

Example 2:

The DBE for a seismic zone of 0.4, CS site soil profile, located 10 km from a B fault [A3] would be

0.56(1.0)0.56(61.5) 34.4 . / sec. (DBE)1.5(34.4) 52 . / sec. (MCE)

v

v

v

CS inS in

== == =

Example 3:

The 2006 IBC [A5] follows the same process in terms of developing ground motion intensities. The Cv (UBC)

comparable is now identified as SM1 for the MCE and SD1 for the DBE. Charts or computer generated zones are

Page 57: 1106Englekirk Final Report 042107

A-5

identified in terms of their ground acceleration expressed as a percentage of “g”. S1 is the spectral acceleration at

a period of one second.

For much of the San Diego region

1 0.5S g=

For a site class SC, Fv=1.3

1 1 ( .4) 1.3(0.5) 0.65

===

M vS F S A

g

The spectrum velocity for the MCE event is

, 1 61.5(0.65) ( .2) 40 . / sec.v MS Eq A

in=

=

The DBE spectrum velocity is

,D1 , 12 ( .3)3

27 . / sec.

v v MS S Eq A

in

=

=

Given the peak table acceleration of 0.91g, the spectrum velocity which quantifies this input is on the order of

55 in./sec. Accordingly, Earthquake #4 ( 55 . / sec.vS in= ) would be considerably more powerful than

Maximum Capable Event (MCE) in the San Diego area.

Example 4:

Peak record ground motions on the order of 0.82g (Northridge Sylmar-360o- EQ #4) have not been uncommon

in the Los Angeles area. Accordingly, a criterion developed for a site class SC seems appropriate.

Page 58: 1106Englekirk Final Report 042107

A-6

1

1

1.1772 . / (MCE)(Eq A.2)0.78 (Eq A.3)48 . /

====

M

v

D

v

S gS in sec.

S gS in sec. (DBE)(Eq A.2)

Comment: Observe that given the design procedures developed, the designer may easily predict the impact this

MCE event ( 72 . /=vS in sec.) might have on a structure.

Consider the UCSD rectangular test wall. Since Δu is directly proportional to Sv, the Design Verification (section

3) process as applied to the UCSD Test Wall is easily adjusted.

7 (15)55

. .

. 4.6

1 .

( ).

u

p

p

2

19 6 in19 6 (Eq.14)

5 in15 (Eq.16)

756 36 0 021 radian

θ

Δ =

=Δ = −

=

=−

=1 (0.000186) ( )

10.40.0003 . / .0.0003(9.3) 0.0010.0038 . / .0.04 . / .

p

cu

su

5 See Appendix E

rad in (Eq.18)

in in in in

φ

ε

ε

=

== +== (Eq.19)

Observe that the predicted level of steel strain is higher than attained comparables (see Appendix E) and the

plastic hinge rotation (0.021 radian) is higher than the limit states proposed by FEMA 356[A6] (0.015 radian). The

wall should, given this intensity of event (Sv=72 in./sec.), be at its collapse threshold. A more detailed inelastic

time history should be undertaken if the wall and seismic intensity level are presumed.

Page 59: 1106Englekirk Final Report 042107

A-7

A.2 Alternative Ground Motions

Earthquake #4 represents an impulsive excitation (Figure A.1), and this type of ground motion must be

considered by a building design team. An event of longer (intense) duration is also typically considered in the

Design Verification process (Section 3.4).

The acceleration record described in Figure A.3 has been used by building design teams to represent this

longer duration type of event. Observe that multiple ground accelerations of essentially the same magnitude

occurred. Compare this record with the recorded table accelerations described in Figure A.1. The response

spectrum developed from the ground motion described in Figure A.3 is presented in Figure A.4. The spectrum

velocity for this event is 77 in./sec. (1.25 x 61.5) at a period of one second, clearly a Maximum Considered

Event (MCE) for most of California.

Time (seconds)

Figure A.3 Adjusted ground motion record Loma Prieta Corralitos – Fault Parallel Component

Acce

lerat

ion

(g)

Page 60: 1106Englekirk Final Report 042107

A-8

The obvious question is why an event such as this was not selected by the UCSD project team. The event

used in the UCSD tests was selected because structures that rely on ductility to survive large earthquakes are

impacted more by impulse type events (Figure A.1) than repeated high accelerations (Figure A.3). To appreciate

this, consider how the limited restoring force of a ductile structure will allow higher system displacements when

the first large excursion creates the displacement maxima (See Figure 5b). For a development of this topic see

references A1 and A4. It was also the opinion of the project team that subjecting the test specimen to multiple

earthquake excitations would, in effect, adequately represent the impact of the type of the (multiple) events

described in Figure A.3. Given this impact type excitation, one might reasonably view the initial excursion

(Δu=15 in.-Figure 5b) as an anomalous event, at least from a design perspective.

Figure A.4 Response spectra for the matched time history described in Figure A.3 MCE-Loma Prieta Corralitos Fault Parallel Component

Page 61: 1106Englekirk Final Report 042107

A-9

REFERENCES

[A1] Englekirk, R.E., Seismic Design of Reinforced and Precast Concrete Buildings, Figure 2.4.4, John Wiley & Sons, Hoboken, New Jersey, 2003.

[A2] International Conference of Building Officials, Uniform Building Code, Figure 16-3, 1997 Edition, Whittier, California.

[A3] International Conference of Building Officials, Uniform Building Code, 1997 Edition, Whittier, California.

[A4] Englekirk, R.E., Steel Structures Controlling Behavior through Design, Section 4.7.3, John Wiley & Sons, Hoboken, New Jersey, 1994.

[A5] International Code Council, International Building Code, 2006 Edition, Falls Church, Virginia.

[A6] Federal Emergency Management Agency, Handbook for the Seismic Evaluation of Existing Buildings - Prestandard and Commentary for the Seismic Rehabilitation of Buildings, (FEMA 356), November 2000.

Page 62: 1106Englekirk Final Report 042107

B-1

Appendix B – Identification of System Drift Limit States

Figure B.1 describes a shear wall tested by Wallace[B1]. Figure B.2 describes the behavior of this wall.[B1] The

indicated lateral drift approached 2.5%. In a later article, Orakal and Wallace[B2] decompose this overall

displacement record so as to reflect flexure only. They conclude that the “flexure only” limit state is 2.0%. The

deformation limit state is described[B2] as “buckling of the longitudinal reinforcement within the boundary

element”. The height to thickness ratio ( /w wh t ) for this wall was 36, much higher than that commonly used in

design. Further, effective confinement was not provided in the toe of the wall (Figure B.1b). The UCSD wall

reached a drift ratio of 2.0% and damage was limited to minor spalling outside of the confined core (Figure B.3).

Shear stress ratios for both walls were relatively low ( 3 cf ′± ) and as a consequence shear deformation was

negligible. Measured sliding was recorded at 0.1 in. along the construction joint (Figure B.4) in the lowermost

level of the UCSD test wall. Based on the experimental evidence available to date, the appropriate drift limit for

a shear wall seems to be at least 2.0%.

Page 63: 1106Englekirk Final Report 042107

B-2

(a) Geometry and reinforcing details

(b) Plan view at section indicated

Figure B.1 Test wall RW2[B1]

Figure B.1b

Hoop Ties Per Figure B.1a

Page 64: 1106Englekirk Final Report 042107

B-3

(a) Lateral load versus top displacement

(b) Analytical stiffness projections versus measured

force displacement relationships

Page 65: 1106Englekirk Final Report 042107

B-4

(c) Analytical and measured concrete strain profiles (positive displacement)

(d) Analytical and recorded moment curvature response

Figure B.2 Behavior of wall RW2[B1]

Page 66: 1106Englekirk Final Report 042107

B-5

Figure B.3 Toe spalling-UCSD shear wall after Earthquake #4

Web wall – bottom of West End

Figure B.4 Crack pattern after Earthquake #4

Web wall – level 2 North Side

Page 67: 1106Englekirk Final Report 042107

B-6

REFERENCES

[B1] Wallace, J. W., “Reinforced Concrete Walls: Recent Research & ACI 318-2001,” Proceedings of the 6th U.S. National Conference on Earthquake Engineering.

[B2] Orakal, K. and Wallace, J., “Flexural Modeling of Reinforced Concrete Walls,” ACI Structural Journal, Vol. 103, No. 2, March-April 2006.

Page 68: 1106Englekirk Final Report 042107

C-1

Appendix C – Idealization of Shear Wall Stiffness

The designer of a shear wall braced building must understand the stiffness characteristics of the shear wall. This

data must be developed in two forms. The conceptual design process requires that the range of effective moment

of inertia is understood. The design confirmation process requires the development of a backbone behavior

curve which reasonably bounds postyield behavior and enables the designer to develop an analytical model

appropriate to the inelastic time history confirmation of the design.

The most significant variables that impact the development of the stiffness characteristics of a shear wall are

the level of applied axial load and the predesign event condition of the shear wall. These variables are best

discussed in conjunction with a review of relevant experimental efforts.

Rectangular Wall Sections

First consider the effective moment of inertia (Ie) suggested by static cyclic tests. Axial load levels imposed on

the wall elements have a significant impact on the stiffness of the wall. Experimental data considered have an

axial load range between 0.24 c gf A′ (Figure C.1) and zero (Figure C.3). The introduction of f’c as a factor

implies that the categories are imprecisely defined. Figure C.2, for example, has an axial load of 0.11 c gf A′ if

the design concrete strength of 4 ksi is used, or 0.07 c gf A′ if the provided strength of 6 ksi is used. The

development of these design idealizations is contained in Reference C1.

Page 69: 1106Englekirk Final Report 042107

C-2

Figure C.1 Lateral load versus top horizontal displacement (1.5 m from base)

(See Reference C1 – Figure 2.4.2)

Figure C.2 Lateral load versus top displacement – Wall RW2[C3]

Page 70: 1106Englekirk Final Report 042107

C-3

(a) Hysteretic behavior of test beam[C4]

(b) Adopted idealization describing behavior of the beam[C5]

Figure C.3

Page 71: 1106Englekirk Final Report 042107

C-4

Even a cursory review of Figures C.1 through C.3 suggests that a designer knowledge factor must be

introduced into the design process. Figures C.1 and C.2 suggest that the impact of prior loading will create

considerably more softening in the moderately loaded wall (Figure C.2) than a wall subjected to higher axial

loads (Figure C.1). Prior to design event loading, one might reasonably consider the location of the building, for

a building located in a region of high seismicity will surely experience more moderate events than, say, one

located on the east coast. From these factors it seems clear that stiffness characteristics must be broadly grouped

and include a measure of designer knowledge.

It seems reasonable based on the static tests described in Figure C.1 and C.2 to adopt the following

generalization for design purposes.

Lightly loaded shear wall ( max 0.15 c gP f A′≅ ) 0.35e gI I=

Heavily loaded shear wall ( max 0.25 c gP f A′≅ ) 0.60e gI I=

A backbone curve (Figure C.4) similar to that developed in FEMA 356[C6] can be rationally developed as

proposed in Figure C.3 (b). Point B (Figure C.4) describes the idealized yield displacement (Δyi) and strength of

the shear wall (λoMn).

Figure C.4 Force-deformation relationship for a concrete wall[C6]

Page 72: 1106Englekirk Final Report 042107

C-5

The idealized yield displacement (Point B (Figure C.4)) is reasonably developed from the curvature at first

yield and an idealization of curvature distribution[C1]. Start with the curvature associated with first yield of the

reinforcing steel:

0.0022 . / .

ysy

fE

in in

ε =

and an assumed neutral axis depth (c) of wl /3.

0.0022 (Eq. 9)0.670.0033

=

=

yw

w

l

l

φ

A linear curvature/moment distribution would suggest that

2

2

(C.1)3

0.0011 (C.2)

Δ =

=

y wyi

w

w

h

hl

φ

Equation C.2 is developed from a linear moment diagram; however, neither the moment, given a first mode

distribution of force, nor the curvature, given a linear variation in moment (Table 2.4.2[C 1]), are likely to follow

the curvature distribution adopted by Equation C.1.

Wallace[C7] has proposed that

2 [C1]

2

11 (Figure 2.4.8 )40

0.0009 (C.3)

Δ =

=

yi y w

w

w

h

hl

φ

Page 73: 1106Englekirk Final Report 042107

C-6

Accordingly, a range and set of influencing parameters are established. Now a design approximation, which

matches experimental data, must be developed.

Equations C.2 and C.3 suggest that the idealized yield deflection of the behavior described in Figure C.2 is

2 2(144)48

432 . 0.48 . (Eq C.2) 0.39 . (Eq C.3)

w

w

yi

yi

hl

ininin

=

=Δ =

Δ =

Consider the UCSD test wall responding to Earthquake #2 (Figure C.5(a))

2 2(756)144

3969 . 4.4 . (Eq C.2) 3.6 . (Eq C.3)

w

w

yi

yi

hl

ininin

=

=Δ =

Δ =

Figure C.5(a), which describes the response of the UCSD test wall to Earthquake #2, suggests a reasonable

compromise and yet allows the designer to introduce prior events into the design process. Steel strain states

measured during Earthquakes #2 and #3 are identified in E.2. The recorded displacements (Figure C.5) are in

excess of 4.0 in., and peak steel strains have exceeded yield (Figure E.2). Accordingly, the quantification of

yield displacement at 4.0 in. seems reasonable. Concrete compressive strains are on the order of 0.001 in./in.

Page 74: 1106Englekirk Final Report 042107

C-7

0 5 10 15 20 25 30

-15

-10

-5

0

5

10

15

t (sec)

Dis

plac

emen

t (in

)

(a) Earthquake #2

0 5 10 15 20 25 30

-15

-10

-5

0

5

10

15

t (sec)

Dis

plac

emen

t (in

)

PHASE I - EQ3 - Roof Relative Displacement Response

(b) Earthquake #3

Figure 5. Top relative displacement response of UCSD rectangular wall

Top Relative Displacement Response

Page 75: 1106Englekirk Final Report 042107

C-8

From a design perspective the best fit appears to be a simple compromise (Equation C.4), but the designer

might reasonably consider prior history and adopt Equation C.3, especially if the stiffness of the wall is based on

a similar prior history assumption.

2

0.001 wyi

w

hl

Δ = (C.4)

Non-Rectangular Shear Wall Configurations

“T” walls are not uncommon configurations, particularly in bearing wall buildings. Wallace[C8] cyclically tested

the wall section whose plan is described in Figure C.6. The height of the wall was 144 in. The hysteretic

behavior of the wall is described in Figure C.7. The use of an idealized Moment of Inertia of 0.5 gI is overly

stiff while the experimental idealization, which corresponds to an effective Moment of Inertia of 0.2 gI , seems

soft.

Figure C.6 “T” Wall geometry and reinforcing details[C8]

Page 76: 1106Englekirk Final Report 042107

C-9

Figure C.7 Force displacement relationship for specimen described in Figure C.6[C1]

Consider the dynamic response of the UCSD “T” wall system to shaking table displacements (Figures C.8 and

C.10).

Analytically, following proposed design procedures

4

4 4

4

6,500,000

0.35

2,200,000 . (111 . )

8

. ( )

g

e g

fe

I in

I I

in ftwh (Eq.6)EI

7 2 63

=

=

Δ =

=4

0.5

8(3600)(111)(144)0. .0.89( )

0.2.

f

25 ftT (Eq.4)

44 sec.f 26 Hz

=

= Δ

==

Page 77: 1106Englekirk Final Report 042107

C-10

The response of the “T” wall to Earthquake #2 (Figure C.8) identifies this frequency (2.26 Hz) as being too

high, but the stem wall had been significantly weakened by the ground motions sustained by the stem

(rectangular) wall acting alone. A frequency of 1.5 Hz is consistent with 0.2e gI I= and, though consistent with

the response of the specimen to Earthquake #2 (Figure C.9), seems conservative.

Observe that the period of the “T” wall specimen when responding to Earthquake #4 (Figure C.10) is in

excess of one second. Once again, engineering judgment and assumptions relative to prior system excitation

must be incorporated into the design process.

An effective stiffness of 0.25Ig seems to be confirmed by the shake table test.

4

4

0.25 (Appendix C)

1,570,000 . 76 .

=

=

=

e gI I

inft

4

4

8

6.7(63)8(3600)144(76)0.33 .

fe

wh (Eq. 6)EI

ft

Δ =

=

=

0.5

0.5

0.89( )

0.89(0.45)0.52 .

fT (Eq. 4)

sec

= Δ

==

The period of the structure prior to Earthquake #3 was 0.66 second (1.5 Hz) and it softened to about 0.77

second (Figure C.9) after Earthquake #4. One might expect a wall to have already experienced a significant

earthquake but not the series of ground motions that were imposed on the test specimen. The predicted peak

displacement predicted using the proposed procedure is

Page 78: 1106Englekirk Final Report 042107

C-11

( 1)1.5

2 0.24 0.24(0.66)55 ( 0.66 sec.) 8.7 .

Δ = Γ

=

== ==

u d

v

v

S EqTS

TST

in

π

and this is the range of displacements recorded during the excitation caused by Earthquake #4 (Figure C.10).

Accordingly an adoption of an effective stiffness to 0.25 gI seems warranted.

Conclusion: The effective moment of inertia, Ie, for the conceptual design of a “T” wall should be 0.25 Ig.

0 5 10 15 20 25 30-10

-8

-6

-4

-2

0

2

4

6

8

10

t (sec)

Dis

plac

emen

t (in

)

Phase II - EQ2 - Roof Relative Displacement Response

Figure C.8 Top relative displacement response of UCSD “T”

wall specimen to Earthquake #2

Page 79: 1106Englekirk Final Report 042107

C-12

0.5 1 1.5 2 2.5 30

10

20

30

40

50

60

70

80

90

100

Frequency f (Hz)

Am

plitu

de

before EQ1before EQ2before EQ3after EQ4

f = 1.29 Hz(T=0.77 sec)

f = 1.52 Hz(T=0.66 sec)

f = 1.82 Hz(T=0.55 sec)

f = 2.11 Hz(T=0.47 sec)

2 1 0.66 0.5 0.4 0.33Period T (sec)

Figure C.9 Fourier amplitude spectra of roof accelerometer

UCSD “T” wall

Figure C.10 Top relative displacement response of UCSD “T” wall

specimen to Earthquake #4

Page 80: 1106Englekirk Final Report 042107

C-13

REFERENCES

[C1] Englekirk, R. E., Seismic Design of Reinforced and Precast Concrete Buildings, Section 2.4, John Wiley & Sons, 2003.

[C2] Seismology Committee, Structural Engineers Association of California, Recommended Lateral Force Requirements and Commentary, Sacramento, California, 17th Edition, 1999.

[C3] Taylor, C. P., Cote, P. A., and Wallace, J. W., “Design of Slender Reinforced Concrete Walls with Openings,” ACI Structural Journal, Vol. 95, No. 4, July-August 1998.

[C4] Popov, E.P., Bertero, V. V., and Krawinkler, H., “Cyclic Behavior of Three R.C. Flexural Members with High Shear,” Earthquake Engineering Research Center, University of California, Berkeley, Report No. EERC 72-5, October 1972.

[C5] Englekirk, R.E., Seismic Design of Reinforced and Precast Concrete Buildings, Section 2.1.1, John Wiley & Sons, 2003.

[C6] Federal Emergency Management Agency, Handbook for the Seismic Evaluation of Existing Buildings - Prestandard and Commentary for the Seismic Rehabilitation of Buildings, (FEMA 356), November 2000.

[C7] Taylor, C.P., Cote, P.A., and Wallace, J.W., “Design of Slender Reinforced Concrete Walls with Openings,” ACI Structural Journal, Vol.95, No. 4, July-August 1998.

[C8] Wallace, J. W., “Reinforced Concrete Walls: Recent Research & ACI 318-2001,” Proceedings of the 6th U.S. National Conference on Earthquake Engineering.

Page 81: 1106Englekirk Final Report 042107

D-1

Appendix D – Wall Strength and Overstrength

D.1 Analytical Idealizations

The identification of the strength that should be provided in a shear wall is difficult to establish during the

conceptual design process. Fortunately, performance and strength are not directly related and strength can, in the

design verification phase, be adjusted to optimize performance (see for example Reference D1).

The idealized flexural strength ( yiM ) of a shear wall should be that which attains the idealized yield

displacement, yiδ , see (Figure C.3 (b)) and one might reasonably argue be capable of sustaining the cracking

moment ( crM ) of the wall section without rupturing the steel.

(Eq.8)=yi y eM EIφ

For the UCSD rectangular test wall

3

4

0.0033

0.0033 144

0.000023 . / .0.35(6)(144)

12 523,000 .

=

=

=

=

=

yw

e

l

rad in

I

in

φ

0.000023(3600)(523,000) 43,300 .- (3600 .- )

yi y eM EI

in kips ft kips

=

==

φ

Since we have chosen to identify the idealized yield moment (Myi) with the probable overstrength of the wall

( o nMλ ), the strength developed in the flexural reinforcing should be associated with a stress of 75 ksi (1.25fy).

Page 82: 1106Englekirk Final Report 042107

D-2

( ) ( ) (Eq.10)2 2

214(68) 1.25(2.48)(60)(136)40,000 .

wo n o s y

aM P A f d d

in kips

′= − + −

= += −

λ λ

The simplification adopted in Equation 10 neglects the interior reinforcement and strength hardening of the

boundary reinforcement and as a consequence underestimates the developable strength of the wall.

The provided boundary reinforcing in the UCSD test wall was 8-#5 (2.48 in.2) and a moment curvature

analysis of the UCSD rectangular wall suggests that yiM is 50,000 in.-kips (Figure D. 1).

Figure D.1 Moment vs. curvature diagram – UCSD rectangular test wall ( 66 , 130 , 214 y uf ksi f ksi P kips= = = )

Observe that the behavior described in Figures C.1 and C.2 could reasonably be associated with the

development of a strength hardened steel stress and a bilinear elastic/perfectly plastic behavior wall model.

Accordingly, the conceptual design procedure developed in Equation 10 to quantify the amount of flexural

reinforcement should be refined so as to better predict the idealized flexural strength (Myi –λoMn) of the wall, a

refinement described in Figure D.1.

Curvature (Radian/inch)

M

(in.-kips)

yi yiM φ

Page 83: 1106Englekirk Final Report 042107

D-3

The consequences of over reinforcing a shear wall must be clear to the engineer who understands the

objectives of capacity based design. Capacity based design must be effectively introduced into the design

process if premature failure modes are to be avoided. The objective of capacity based design is to determine the

load or force the more brittle components along the lateral load path must sustain. Excessive shear, for example,

can lead to a brittle failure. To effectively implement a capacity design requires the designer to understand the

difference between the component design basis strength (ie Mn) and the force which is likely to be imposed on

the brittle element being considered. This relationship is most conveniently subdivided into two parts which are

usually identified as component ( oλ ) and system overstrength ( oΩ ).

Component overstrength involves the introduction of probable material strength as was done in Equation 10

and subsequently refined as described in Figure D.1. The so defined component overstrength factor for the

UCSD shear wall test is

O

( ) ( ') (D.1)2 2

214(68) 2.48(60)(136) 35,800 .-

(R Figure 10)

50,

wn s y

yio

n

aM P A f d d

in kipsMM

= − + −

= +=

= −

=

λ

00035,800

1.4=

System overstrength is more speculative. The reported strengths of the UCSD shear wall braced specimen

provide significant insight into system overstrength ( oΩ ). Figure D.2 relates the base moment as derived from

the recorded accelerations and displacements (P-Δ) to measured building relative displacement. Observe that the

response to Earthquake #2 (Figure D.2) seems to correspond to the idealized elastic limit state (Myi) as we have

chosen to define it. The maximum moment predicted in Figure D.2 is about 4600 ft-kips and this is reasonably

consistent with that attained as identified in Figure D.2.

Page 84: 1106Englekirk Final Report 042107

D-4

Consider next the relationship between developed base moments when the wall was subjected to Earthquake

#4 and those which represent the idealized flexural strength of the wall ( 4167 .-o nM ft kipsλ = ). The repeated

strength demand on the system seems to be on the order of 5800 ft.-kips. Indicated system overstrength is

max (D.2)

5800 4167

1.4

Ω =

=

oo n

MMλ

-15 -10 -5 0 5 10 15

-8000

-6000

-4000

-2000

0

2000

4000

6000

8000

Bas

e M

omen

t (ki

ps-ft

)

Roof Displacement (in)

EQ1EQ2EQ3EQ4Wall+Slot+ColumnsWall+SlotWall+ColumnsWall Only

Figure D.2 Hysteretic response – top displacement vs. wall base moment

Rectangular shear wall

The anomalous (see Appendix A) excursion (7500 ft.-kips) does not lend itself to a convincing

quantification. It is attributed to a variety of largely unpredictable sources, such as strain rate effects, impact

associated with the closing of large cracks, tension developed in gravity columns, and the proximity of the

orthogonal shear wall. Accordingly, significant designer input is required to appropriately select a reasonable

value of system related overstrength Ωo . The range of system flexural overstrength suggested by the UCSD

Page 85: 1106Englekirk Final Report 042107

D-5

shear wall is between 1.4 and 1.875004167

⎛ ⎞⎜ ⎟⎝ ⎠

. See reference D.2 for a detailed evaluation of system associated

flexural overstrength.

D.2 Overstrength in Shear

Shear yielding is usually avoided when possible. Shear demand is developed from the probable flexural capacity

of a shear wall and this requires that the effective height of the shear force (mass) be developed. Extant

performance based design guidelines identify the effective height of the mass (Figure D.3) for various structural

bracing programs (Figure D.3). The coefficient k1 relates the height of the structure to the effective height of a

comparable single degree of freedom system.

Figure D.3 Alternative fundamental mode shapes

Page 86: 1106Englekirk Final Report 042107

D-6

k1 is often defined as a function of the height to length ratio w w( h / ) of the shear wall which, in the case of

the UCSD test wall, is 5.25. The proposed value [D3] is 0.77. Hence, the shear associated with oo nMλΩ might be

determined for the test wall as

4167 .- (Figure D.1)63 .48.5 .86 ( Figure10)7500 .-150

o n

w

e

o n M

o o n

o o n

M ft kipsh fth ftV kips V

M ft kipsV kips

==== −

Ω ≅Ω ≅

λ

λλλ

Acceleration profiles at peak displacement response are shown in Figure D.4. These acceleration profiles are

not consistent with the primary mode shape described in Figure D.3. The story shears envelopes developed from

the various ground motions are shown in Figure D.5. The so derived level of base shear for Earthquake #4 is 260

kips. Figure 6 shows that this shear should be treated as an anomaly because it is generated by the impulsive

nature of the ground motion and not repeated as discussed in Appendix A. The system overstrength in shear (λo

Ωo) is then a function of the selected system flexural overstrength (λoΩo) and the effective height of the mass

(k1). The effective height factor (k1) associated with the repeatable base shear[D2] suggested by the UCSD shear

wall test is

1max

5800 =180

=32 .

o o nw

Mk hV

ft

Ω=

λ

This corresponds to

132=63

= 0.51

k

and in fact represents the lower bound migration of k1[D2].

Page 87: 1106Englekirk Final Report 042107

D-7

Conclusion: The use of a k1 in the vicinity of 0.77 seems non-conservative and its precise identification in the

design process is not warranted.

-1.25 -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 1.250

1

2

3

4

5

6

7

a (g)

Floo

r

t=43.76 sect=44.37 sect=45.2 sec

Figure D.4 Acceleration profile at maximum overturning moment – Earthquake #4

Page 88: 1106Englekirk Final Report 042107

D-8

0 50 100 150 200 250 3000

1

2

3

4

5

6

7

System Shear Force (kips)

Lev

el

EQ1EQ2EQ3EQ4Design

Figure D.5 System shear force envelopes – UCSD rectangular wall

-15 -10 -5 0 5 10 15-300

-200

-100

0

100

200

300

Roof Relative Lateral Displacement (in)

Syst

em B

ase

Shea

r Fo

rce

(kip

s)

Figure D.6 Hysteretic response – top displacement vs. wall base shear

UCSD rectangular wall

Page 89: 1106Englekirk Final Report 042107

D-9

D.3 Conclusions Based on UCSD Test Wall Response

Overstrength in Flexure

From a design perspective it seems most reasonable to base the quantification of the overstrength moment on a

steel stress of 130 ksi. This suggests a component moment capacity (λo Mn) of 4167 ft.-kips (Figure D.1) and

seems reasonable based on the data recorded for Earthquakes #2 and #3. System overstrength factors (Ωo) in

flexure seem reasonably quantified by 1.458004200

⎛ ⎞⎜ ⎟⎝ ⎠

.

Overstrength in Shear

The use of an effective height of 0.5hw and a flexural system overstrength factor (Ωo) of 1.4 seems reasonably

conservative and appropriate for design purposes. Neither these (180 kips) nor the extreme excursions described

in Figures D.2 and D.6 should be compared with conservative estimates of component shear strength.

Accordingly, one must consider the probable shear strength of the component as developed by plastic limit

states and not that conventionally suggested by beam shear theory. Detailing must also ensure that ductility can

be developed in shear.

UCSD Shear Wall Capacity

Outside the plastic hinge region or region of shear discontinuity the shear strength of the wall is conservatively

developed using a beam shear model with 45oΘ = [D4].

2 6 ., 136 . = 0.14(6)136 5000 = 114

c c

c

V f bd b in d inf psi

kips

′= = =

′ ≅

Page 90: 1106Englekirk Final Report 042107

D-10

0.2(60)136 #4@8 . . .8

= 204

sV in o c

kips

=

318 260 (Figure D.5)n c sV V V kips kips= + = >

Inside the region of shear discontinuity (plastic hinge region), shear friction better describes system capacity.

The nominal shear strength allowed by shear friction[D5] is defined by

0.8n v yV A f Pμ= +

Given the anomalous displacement (see Appendix A – Section A.2) at 44≅t seconds (Figure D.4) the

flexurally induced compression force ( o yTλ ) and axial load suggest that the wall will not slide as is required to

activate the shear friction mechanism. Since the compression load imposed on the shear plane was at least on the

order of 400 kips during the anomalous excursion at (t = 44 seconds) the nominal capacity in friction, acting

alone, is 320 kips and this exceeds the estimated anomalous shear demand of 260 kips. Accordingly, sliding

along construction joints should not have been expected, and none was observed.

Figure D.7 “Plastic truss” analogy

Page 91: 1106Englekirk Final Report 042107

D-11

At the shear strength limit state a compression fan will define the strength limit state in shear (Figure D.7).

The number of stirrups or, in the case of a wall, the horizontal bars engaged at the shear limit state, corresponds

to a developed angle of 25o (θ -Figure D.7). The number of engaged horizontal bars, n, is

tan 65

136(2.14) 8

36.5

=

=

=

odns

where ‘n’ is the effective number of horizontal bars crossing within a 65o shear fan emanating from the toe of

the shear wall, and ‘s’ is the spacing between these bars.

Caution: Too many designers tend to overreinforce shear walls in shear. The consequence of this action is to

promote brittle compression failures in the wall. To avoid these brittle failures the designer must limit the

amount of shear reinforcement thereby accepting or ensuring the development of shear ductility. Limiting values

of shear reinforcement can be developed from strut and tie limit states. The stress along the compression

diagonal, fcd, given a 45% angle is twice the stress developed by the horizontal reinforcement.

2cd sf v=

Strut stress limit states are generally established based on conditions at the node [D4][D6]. In the case of a shear

wall the critical node is a Tension-Tension-Compression node (T-T-C). The existence of cracks and reverse

cycle loading is believed to impact the capacity of the strut. Given the consequences of a shear induced

compression failure, it is advisable to be conservative (factor of safety of 1.5). The adopted ultimate stress in the

strut is

36(0.2)60 432 260 (Figure D.5)

sVkips kips

== >

Page 92: 1106Englekirk Final Report 042107

D-12

0.35 'cu cf f=

Since vs is based on the specified strength of the shear reinforcement, the reinforcement limit state should be

,max

0.35 ' 21.5

0.12 '

=

=

cs

s c

f v

v f

Conclusion: The shear reinforcement placed in a shear wall should not exceed 0.12f’c

D.4 Relationship between Strength and Performance

Appendix A described seismic events which should be considered by a building design team, generally

classifying them as impulse related and repeated strong motion. An inelastic time history analysis reasonably

reproduced the measured response of the UCSD test specimen as did quite a few others who participated in the

blind prediction contest sponsored by the Portland Cement Association. Figure 5 compares the projected relative

displacements (Figure 5a) with those recorded (Figure 5b). The question usually raised is the impact an increase

in system strength would have on performance. Performance is a function of displacement, hence assessed here

by comparing relative displacement levels.

The relationship between strength and performance is impacted by the type of seismic event. As can be

expected (Section A.2 of Appendix A) the stronger system, given an impulsive type of excitation, will displace

less than the less strong system. Figure D.8 describes projected response of the code basis building design–

twice as strong as the UCSD wall. The displacement is reduced to 13 inches (87%) while the shear demand is

doubled. An additional increment of strength (30% over code) results in no further reduction of projected

displacement. Given the repeated demand (Figure A.3) on system ductility the peak relative displacement

imposed on the UCSD rectangular wall is 13.3 inches (Figure D.9) while the response of the code strength level

wall is 18 inches (Figure D.10)– 35% higher, with more than twice the shear demand.

Conclusion: No benefit is derived from an increase in system strength, and this is not an atypical conclusion.

Page 93: 1106Englekirk Final Report 042107

D-13

Figure D.8 Predicted response of code level strength rectangular

wall to impulse type ground motion (Figure A.1)

Figure D.9 Predicted response of UCSD test wall to the strong repeated

ground motion described in Figure A.3

Page 94: 1106Englekirk Final Report 042107

D-14

Figure D.10 Predicted response of code level strength rectangular wall to the repeated strong ground motion described in Figure A.3

Page 95: 1106Englekirk Final Report 042107

D-15

REFERENCES

[D1] Englekirk, R.E., Seismic Design of Reinforced and Precast Concrete Buildings, Sections 4.1.2 & 4.2.1, John Wiley & Sons, Hoboken, New Jersey, 2003.

[D2] Panagiotou, M. and Restrepo, J., Partial Report on the 7-Story Shake Table Test at UCSD, December 2006.

[D3] Seismology Committee, Structural Engineers Association of California, Recommended Lateral Force Requirements and Commentary, 7th Edition, Sacramento, California, 1999.

[D4] MacGregor, J.G. and Wight, J.K., Reinforced Concrete Mechanics and Design, Fourth Edition. Pearson Education, Inc., 2005.

[D5] International Conference of Building Officials, Uniform Building Code, 1997 Edition.

[D6] American Concrete Institute, Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary (ACI 318R-05), ACI 318, Appendix A, Farmington Hills, Michigan, December 2004.

Page 96: 1106Englekirk Final Report 042107

E-1

Appendix E – Strain States

Limiting material strain states is considered a reasonable design objective. The work of Wallace[E1], summarized

in Englekirk[E2] describes the response of a thin rectangular wall subjected to a static set of cycling lateral loads.

Figure E.1 compares analytical strain and curvature predictions with measured behavior. It seems reasonable to

conclude that the following strains were attained.

Concrete – 0.01 in./in. Steel – 0.024 in./in.

(a) Analytical versus measured strain profiles

Page 97: 1106Englekirk Final Report 042107

E-2

(b) Analytical experimental and idealized moment curvature response [E1, E2]

Figure E.1

Wallace[E3] identifies the limiting behavior mechanism in this wall as “…buckling of the longitudinal

reinforcement within the boundary element…” and this seems to be a predictable limit state given the paucity of

restraining reinforcement in the boundary element (3/16 inch hoops @ 3 inches on center) [E4]. Observe that the

interior bars are not restrained and the confining pressure is less than 200 psi. Accordingly, the indicated drift

limit state (2%) and strains must be viewed as being conservative.

Strain states recorded in the UCSD shear wall test during Earthquake #4 are presented in Figure E.2. Steel

strains for both excursions (towards the east and west) are essentially the same while the peak recorded

compressive strain on the west end was 0.005 in./in. Observed damage (Figure B.2) was limited to spalling of

the cover concrete, a concrete whose quality was undoubtedly questionable given the provided 3/4 inch cover.

[E2]

Page 98: 1106Englekirk Final Report 042107

E-3

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

20

40

60

80

100

120

140

160

180

200

strain (%)

elev

atio

n (in

)

y p p

West endEast End

slab- level 1

(a) Concrete strain states

(b) Steel tensile strain envelopes

Figure E.2 Recorded strain states – UCSD rectangular wall, Earthquake #4

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040

20

40

60

80

100

120

140

160

180

200

Steel Tensile Strain (in/in)

Elev

atio

n (in

)

EQ1EQ2EQ3EQ4

First Floor Slab

Construction Joint

Layout of Web Wall Reinforcement

Steel Yield Strain

Page 99: 1106Englekirk Final Report 042107

E-4

The strain based design confirmation process (Section 3.1) requires a quantification of the depth or extent of

the plastic hinge region (lp), if strain states are to be estimated. Figure E.2 (b) suggests that the strain in the

tension reinforcing steel is fairly uniform to a height of at least 50 inches and returns to 2 or 3 times yε at a

height of 64 inches. Traditionally the plastic hinge length ( pl ) is presumed to be 0.5 wl[E5, E6]. This corresponds

to 72 inches. Figure E.2 should provide some insight into which plastic hinge length is most appropriate (0.4 wl

or 0.5 wl ) for design confirmation purposes.

15 . (Figure 5b )4.1 . (Appendix C)

(Eq.15)

210.9 720

0.015 ( 72 .)

Δ =Δ =

Δ − ΔΘ =

=

= =

Θ

u

yi

u yip

pw

p

p

inin

lh

radian l in

10.9727

0.015 ( 58 .)

0.00021 . / . ( 72 .)

0.00026 . / . ( 58 .)

=

= =

Θ=

= =

= =

p

pp

p

p p

p p

radian l in

l

rad in l in

rad in l in

φ

φ

φ

The neutral axis depth seems to be about 8 inches (Figure E.3). Accordingly,

0.00021(8)

0.0017 . / . ( 72 .)0.0021 . / . ( 58 .)

0.00021(134)

0.028 . / . ( 72 .)

0.035 . / . ( 58 .)

=

= =

= =

=

= =

= =

cp

p

cp p

sp

p

sp p

in in l inin in l in

in in l in

in in l in

ε

ε

ε

ε

Page 100: 1106Englekirk Final Report 042107

E-5

Recorded steel strains, because they are less sensitive to neutral axis depth, tend to confirm that the plastic hinge

length should be on the order of 0.5 wl (see Figure E.2(b)).

40 42 44 46 48 50 52 54 56 58 600

10

20

30

40

50

60

70

80

90

100

t (sec)

dj (i

n)EQ4 - Variation of Neutral Axis Depth - 10in from Base of Web Wall

Figure E.3 Variation of neutral axis depth as recorded during

Earthquake #4 − Base of Wall

Of particular interest is the apparent plastic or postyield strain recovery exhibited in the steel of inelastic

strains[E7] (see Figure E 2(b)); this in spite of the low level of axial load imposed on the wall. Accordingly, the

steel strain model used to develop fiber models should allow full recovery.

Peak recorded strains for the “T” wall (Phase II) are quite low as could be expected. Yield tensile strains

were recorded at flange extremities in the UCSD tests, and this suggests that the acceptable definition of

effective flange width of 8tw is conservative.

c

Page 101: 1106Englekirk Final Report 042107

E-6

REFERENCES

[E1] Taylor, C.P., Cote, P.A., and Wallace, J.W., “Design of Slender Reinforced Concrete Walls with Openings,” ACI Structural Journal, Vol. 95, No. 4, July-August 1998.

[E2] Englekirk, R.E., Seismic Design of Reinforced and Precast Concrete Buildings, Figure 2.4.4, John Wiley & Sons, Hoboken, New Jersey, 2003.

[E3] Orakal, K and Wallace, J. “Flexural Modeling of Reinforced Concrete Walls”, ACI Structural Journal, Vol. 103, No. 2, March-April 2006.

[E4] Englekirk, R.E., Seismic Design of Reinforced and Precast Concrete Buildings, Figure 2.4.3, John Wiley & Sons, Hoboken, New Jersey, 2003.

[E5] Englekirk, R.E., Seismic Design of Reinforced and Precast Concrete Buildings, Section 2.4.1.2, John Wiley & Sons, Hoboken, New Jersey, 2003.

[E6] Wallace, J. “New Methodology for Seismic Design of RC Shear Walls,” Journal of Structural Engineering, ASCE, Vol. 120, NO. 3, March 1994.

[E7] MacGregor, J.G. and Wight, J.K., Reinforced Concrete Mechanics and Design, Fourth Edition. Pearson Education, Inc., 2005.