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.
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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
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
(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.)
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(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
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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.
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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%.
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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):
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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.
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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
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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.
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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.
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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.
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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
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
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.