1106Englekirk Final Report 042107
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Transcript of 1106Englekirk Final Report 042107
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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.
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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
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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
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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.
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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).
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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.
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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.
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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.
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(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
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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
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.
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(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.
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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
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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 =
=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=
==
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)
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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)
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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).
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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
= +
= − +
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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.
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.
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.
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Figure 4. Analytical model – UCSD test wall
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(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.)
24 of 52
(c) Inelastic response projection – UCSD “T” wall
Rel
ativ
e D
ispl
acem
ent (
in.)
Time (sec.)
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.)
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
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.
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%.
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):
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.
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
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 =−
=
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.
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.
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=
=
=
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.
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
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 )
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
=
===
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)
41 of 52
Inelastic Time History Analysis Projections
Figure 8 Inelastic Time History Analysis Projection – Relative displacement time history longitudinal direction
Time (seconds)
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
43 of 52
Figure 9 Inelastic Time History Analysis Projections – Relative displacement time history transverse direction
Time (seconds)
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
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
= =
=
=
=
=
= =
=
= =
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.
47 of 52
Figure 10 Relationship between base shear and displacement[2]
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.
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.
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).
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).
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.
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.
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%
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
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
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.
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.
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)
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
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.
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%.
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
B-3
(a) Lateral load versus top displacement
(b) Analytical stiffness projections versus measured
force displacement relationships
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]
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
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.
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.
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]
C-3
(a) Hysteretic behavior of test beam[C4]
(b) Adopted idealization describing behavior of the beam[C5]
Figure C.3
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]
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
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.
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
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]
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
==
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
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
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
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.
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).
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
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.
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
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
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].
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
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
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
′= = =
′ ≅
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
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
== >
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.
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
D-14
Figure D.10 Predicted response of code level strength rectangular wall to the repeated strong ground motion described in Figure A.3
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.
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
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]
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
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
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
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.