#Qr#ixuwkhu#uhsurgxfwlrq#ru#glvwulexwlrq#lv#shuplwwhg1

Frs|uljkwhg#pdwhuldo#olfhqvhg#wr#Xqlyhuvlw|#ri#Wrurqwr#e|#Fodulydwh#Dqdo|wlfv#+XV,#OOF/#vxevfulswlrqv1whfkvwuhhw1frp/#grzqordghg#rq#534<038064#49=3;=64#.3333#e|##Xqlyhuvlw|#ri#Wrurqwr#Xvhu1
80 ACI 318-19: BUILDING CODE REQUIREMENTS FOR STRUCTURAL CONCRETE

CODE COMMENTARY
where ȕdns shall be the ratio of maximum factored sustained VLPSOL¿FDWLRQLWFDQEHDVVXPHGWKDWȕdns = 0.6. In this case,
axial load to maximum factored axial load associated with Eq. (6.6.4.4.4a) becomes (EI)Hৼ = 0.25EcIg.
the same load combination and I in Eq. (6.6.4.4.4c) is calcu- In reinforced concrete columns subject to sustained
lated according to Table 6.6.3.1.1(b) for columns and walls. loads, creep transfers some of the load from the concrete to
the longitudinal reinforcement, increasing the reinforcement
stresses. In the case of lightly reinforced columns, this load
transfer may cause the compression reinforcement to yield
SUHPDWXUHO\UHVXOWLQJLQDORVVLQWKHH൵HFWLYHEI. Accordingly,
both the concrete and longitudinal reinforcement terms in Eq.
(6.6.4.4.4b) are reduced to account for creep.

6.6.4.5 0RPHQWPDJQL¿FDWLRQPHWKRG1RQVZD\IUDPHV R6.6.4.5 0RPHQWPDJQL¿FDWLRQPHWKRG1RQVZD\IUDPHV

6.6.4.5.1 The factored moment used for design of columns

and walls, McVKDOOEHWKH¿UVWRUGHUIDFWRUHGPRPHQWM2
DPSOL¿HGIRUWKHH൵HFWVRIPHPEHUFXUYDWXUH

Mc įM2 (6.6.4.5.1)

DOFXODWHGE\
6.6.4.5.20DJQL¿FDWLRQIDFWRUįVKDOOEHFDOFXODWHGE\ R6.6.4.5.27K
R6.6.4.5.27KHIDFWRULQ(T  LVWKHVWL൵QHVV
tion factor ࢥK, which is based on the probability of
reduction
reng h of a sing
understrength singlee iis
isolated slender column. Studies
CP
δ= ≥ 1.0 (6.6.4.5.2) reported d in Mirza za et al. (1987)
( LQGLFDWH WKDW WKH VWL൵QHVV
Pu
1− redu on fa
reduction factor ࢥK and the cr cross-sectional strength reduction
0.75 Pc ࢥ factors
fact s do not have ve the sam
same values. These studies suggest
൵QHV UHGXFWLRQ
WKH VWL൵QHVV LRQ IDFWR
IDFWRU ࢥK for an isolated column
should be 0. oth tied aand spiral columns. In the case of
0.75 for both
DPX WRU\ HWKHFROX
DPXOWLVWRU\IUDPHWKHFROXPQDQGIUDPHGHÀHFWLRQVGHSHQG
on the average concrete strength, which is higher than the
strength of the concr
concrete in the critical single understrength
mn. For this reason, the value of ࢥK implicit in I values
column.
663
in 6.6.3.1.1 is 0.875.

6.6.4.5.3 Cm shall be in accordance with (a) or (b): R6.6.4.5.3 The factor Cm is a correction factor relating the
actual moment diagram to an equivalent uniform moment
(a) For columns without transverse loads applied between GLDJUDP7KHGHULYDWLRQRIWKHPRPHQWPDJQL¿HUDVVXPHV
supports that the maximum moment is at or near midheight of the
column. If the maximum moment occurs at one end of the
column, design should be based on an equivalent uniform
M1
CP = 0.6 − 0.4 (6.6.4.5.3a) moment CmM2 that leads to the same maximum moment at or
M2 QHDUPLGKHLJKWRIWKHFROXPQZKHQPDJQL¿HG MacGregor
et al. 1970).
where M1/M2 is negative if the column is bent in single The sign convention for M1/M2 has been updated to follow
curvature, and positive if bent in double curvature. M1 the right hand rule convention; hence, M1/M2 is negative
corresponds to the end moment with the lesser absolute if bent in single curvature and positive if bent in double
value. FXUYDWXUH7KLVUHÀHFWVDVLJQFRQYHQWLRQFKDQJHIURPWKH
(b) For columns with transverse loads applied between 2011 Code.
supports. In the case of columns that are subjected to transverse
loading between supports, it is possible that the maximum
CP = 1.0 (6.6.4.5.3b) moment will occur at a section away from the end of the
member. If this occurs, the value of the largest calculated
moment occurring anywhere along the member should be
used for the value of M2 in Eq. (6.6.4.5.1). Cm is to be taken
as 1.0 for this case.

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 #Qr#ixuwkhu#uhsurgxfwlrq#ru#glvwulexwlrq#lv#shuplwwhg1
Frs|uljkwhg#pdwhuldo#olfhqvhg#wr#Xqlyhuvlw|#ri#Wrurqwr#e|#Fodulydwh#Dqdo|wlfv#+XV,#OOF/#vxevfulswlrqv1whfkvwuhhw1frp/#grzqordghg#rq#534<038064#49=3;=64#.3333#e|##Xqlyhuvlw|#ri#Wrurqwr#Xvhu1
PART 2: LOADS & ANALYSIS 81

CODE COMMENTARY
6.6.4.5.4 M2 in Eq. (6.6.4.5.1) shall be at least M2,min calcu- R6.6.4.5.4 In the Code, slenderness is accounted for by
lated according to Eq. (6.6.4.5.4) about each axis separately. magnifying the column end moments. If the factored column
moments are small or zero, the design of slender columns
M2PLQ = Pu(0.6 + 0.03h) (6.6.4.5.4) should be based on the minimum eccentricity provided in Eq.
(6.6.4.5.4). It is not intended that the minimum eccentricity
If M2,min exceeds M2, Cm shall be taken equal to 1.0 or be applied about both axes simultaneously.
calculated based on the ratio of the calculated end moments The factored column end moments from the structural
M1/M2, using Eq. (6.6.4.5.3a). analysis are used in Eq. (6.6.4.5.3a) in determining the
ratio M1/M2 for the column when the design is based on
the minimum eccentricity. This eliminates what would
otherwise be a discontinuity between columns with
calculated eccentricities less than the minimum eccentricity

Analysis
and columns with calculated eccentricities equal to or greater
than the minimum eccentricity.

6.6.4.6 0RPHQWPDJQL¿FDWLRQPHWKRG6ZD\IUDPHV R6.6.4.6 0RPHQWPDJQL¿FDWLRQPHWKRG6ZD\IUDPHV

6
6.6.4.6.1 Moments M1 and M2 at the ends of an individual R6.6.4.6.1 The analysis described in this section deals only
column shall be calculated by (a) and (b). ZLWKSO
ZLWKSODQHIUDPHVVXEMHFWHGWRORDGVFDXVLQJGHÀHFWLRQVLQWKDW
SODQH,IWKHODWHUD
SODQH,IWKHODWHUDOORDGGHÀHFWLRQVLQYROYHVLJQL¿FDQWWRUVLRQDO
(a) M1 = M1nsįsM1s (6.6.4.6.1a) FHPHQWWKHPRP
GLVSODFHPHQWWKHPRPHQWPDJQL¿FDWLRQLQWKHFROXPQVIDUWKHVW
from thee cen ter of twist may
center mayy be
b underestimated by the moment
(b) M2 = M2nsįsM2s (6.6.4.6.1b) HUSUR XUH,QVXFK
PDJQL¿HUSURFHGXUH,QVXFKFDVHVDWKUHHGLPHQVLRQDOVHFRQG
orde analysis
order alys should ld be used.
used

6.6.4.6.2 7KH PRPHQW PDJQL¿HU

QL¿ įs shall be calcucalculated
ated 4.6. 7KUHH GL൵HUH
R6.6.4.6.2 GL൵HUHQW PHWKRGV DUH DOORZHG IRU
by (a), (b), or (c). If įs exceeds 11.5, only
nly (b) or (c) sha
shall be FDOFXO LQJ PHQWPDJ
FDOFXODWLQJWKHPRPHQWPDJQL¿HU7KHVHDSSURDFKHVLQFOXGH
permitted: the Q method,
etho the sum of P concept, and second-order elastic
analysis.
1
(a) δ s = ≥1 (6.6.4.6.2a)
1− Q (a)) Q method:
The iiterative P¨ analysis for second-order moments can
(b) δ s = 1 (6.6.4.6.2b) EH UHSUHVHQWHG E\ DQ LQ¿QLWH VHULHV 7KH VROXWLRQ RI WKLV
≥1
ΣPu series is given by Eq. (6.6.4.6.2a) (MacGregor and Hage
1−
0.75ΣPc 1977). Lai and MacGregor (1983) show that Eq. (6.6.4.6.2a)
closely predicts the second-order moments in a sway frame
(c) Second-order elastic analysis until įs exceeds 1.5.
The P¨ PRPHQW GLDJUDPV IRU GHÀHFWHG FROXPQV DUH
where ™Pu is the summation of all the factored vertical FXUYHGZLWK¨UHODWHGWRWKHGHÀHFWHGVKDSHRIWKHFROXPQV
loads in a story and ™Pc is the summation for all sway- Equation (6.6.4.6.2a) and most commercially available
resisting columns in a story. Pc is calculated using Eq. second-order frame analyses have been derived assuming
(6.6.4.4.2) with k determined for sway members from that the P¨ moments result from equal and opposite forces
6.6.4.4.3 and (EI)HৼIURPZLWKȕds substituted for of P¨Ɛc applied at the bottom and top of the story. These
ȕdns. forces give a straight-line P¨ moment diagram. The curved
P¨ moment diagrams lead to lateral displacements on the
order of 15 percent larger than those from the straight-line
P¨ PRPHQW GLDJUDPV 7KLV H൵HFW FDQ EH LQFOXGHG LQ (T
(6.6.4.6.2a) by writing the denominator as (1 – 1.15Q) rather
than (1 – Q). The 1.15 factor has been omitted from Eq.
(6.6.4.6.2a) for simplicity.
,I GHÀHFWLRQV KDYH EHHQ FDOFXODWHG XVLQJ VHUYLFH ORDGV
Q in Eq. (6.6.4.6.2a) should be calculated in the manner
explained in R6.6.4.3.
The Q IDFWRU DQDO\VLV LV EDVHG RQ GHÀHFWLRQV FDOFXODWHG
using the I values from 6.6.3.1.1, which include the equivalent

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 #Qr#ixuwkhu#uhsurgxfwlrq#ru#glvwulexwlrq#lv#shuplwwhg1
Frs|uljkwhg#pdwhuldo#olfhqvhg#wr#Xqlyhuvlw|#ri#Wrurqwr#e|#Fodulydwh#Dqdo|wlfv#+XV,#OOF/#vxevfulswlrqv1whfkvwuhhw1frp/#grzqordghg#rq#534<038064#49=3;=64#.3333#e|##Xqlyhuvlw|#ri#Wrurqwr#Xvhu1
82 ACI 318-19: BUILDING CODE REQUIREMENTS FOR STRUCTURAL CONCRETE

CODE COMMENTARY
RIDVWL൵QHVVUHGXFWLRQIDFWRUࢥK. These I values lead to a 20
WRSHUFHQWRYHUHVWLPDWLRQRIWKHODWHUDOGHÀHFWLRQVWKDW
FRUUHVSRQGVWRDVWL൵QHVVUHGXFWLRQIDFWRUࢥK between 0.80
and 0.85 on the P¨ moments. As a result, no additional ࢥ
factor is needed. Once the moments are established using Eq.
(6.6.4.6.2a), selection of the cross sections of the columns
involves the strength reduction factors ࢥ from 21.2.2.
(b) Sum of P concept:
7RFKHFNWKHH൵HFWVRIVWRU\VWDELOLW\įs is calculated as an
averaged value for the entire story based on use of ™Pu™Pc.
7KLVUHÀHFWVWKHLQWHUDFWLRQRIDOOVZD\UHVLVWLQJFROXPQVLQ
the story on the P¨H൵HFWVEHFDXVHWKHODWHUDOGHÀHFWLRQRI
all columns in the story should be equal in the absence of
torsional displacements about a vertical axis. In addition, it
is possible that a particularly slender individual column in
DVZD\IUDPHFRXOGKDYHVXEVWDQWLDOPLGKHLJKWGHÀHFWLRQV
HYHQLIDGHTXDWHO\EUDFHGDJDLQVWODWHUDOHQGGHÀHFWLRQVE\
other columns in the story. Such a column is checked using
6.6.4.6.4.
6.6.
The 0.75 in the denominator of Eq. (6.6.4.6.2b) is a
VVUHGXFWLRQID
VWL൵QHVVUHGXFWLRQIDFWRUࢥ ࢥK, as explained in R6.6.4.5.2.
ca culation of (EI)
In the calculation (EI)
I Hৼ, ȕds will normally be zero for
a sway fram cause the llateral loads are generally of short
frame because
GXUD Q6Z ÀHFWLRQVG
GXUDWLRQ6ZD\GHÀHFWLRQVGXHWRVKRUWWHUPORDGVVXFKDV
ZLQGR HDUW DUHDIXQ
ZLQGRUHDUWKTXDNHDUHDIXQFWLRQRIWKHVKRUWWHUPVWL൵QHVV
olum following
of the columns wing a pperiod of sustained gravity load.
)RU WKLV
KLV FFDVH WKH
H GH¿QLWL
GH¿QLWLRQ RI ȕds in 6.6.3.1.1 gives ȕds
= 00. In the unusual al case of a sway frame where the lateral
ned, ȕds will not be zero. This might occur if
loads are sustained,
a building on a slopin
sloping site is subjected to earth pressure on
ide but not on the other.
one side

6.6.4.6.3 Flexural members shall be designed for the total R6.6.4.6.3 The strength of a sway frame is governed
PDJQL¿HGHQGPRPHQWVRIWKHFROXPQVDWWKHMRLQW by stability of the columns and the degree of end restraint
provided by the beams in the frame. If plastic hinges form
in the restraining beam, as the structure approaches a failure
mechanism, its axial strength is drastically reduced. This
VHFWLRQ UHTXLUHV WKH UHVWUDLQLQJ ÀH[XUDO PHPEHUV WR KDYH
HQRXJK VWUHQJWK WR UHVLVW WKH WRWDO PDJQL¿HG FROXPQ HQG
moments at the joint.

6.6.4.6.4 6HFRQGRUGHU H൵HFWV VKDOO EH FRQVLGHUHG DORQJ R6.6.4.6.4 The maximum moment in a compression
the length of columns in sway frames. It shall be permitted member, such as a column, wall, or brace, may occur
WRDFFRXQWIRUWKHVHH൵HFWVXVLQJZKHUHCm is calcu- between its ends. While second-order computer analysis
lated using M1 and M2 from 6.6.4.6.1. SURJUDPV PD\ EH XVHG WR HYDOXDWH PDJQL¿FDWLRQ RI WKH
HQG PRPHQWV PDJQL¿FDWLRQ EHWZHHQ WKH HQGV PD\ QRW
be accounted for unless the member is subdivided along
LWV OHQJWK 7KH PDJQL¿FDWLRQ PD\ EH HYDOXDWHG XVLQJ WKH
procedure outlined in 6.6.4.5.

6.6.5 5HGLVWULEXWLRQ RI PRPHQWV LQ FRQWLQXRXV ÀH[XUDO R6.6.5 5HGLVWULEXWLRQRI PRPHQWV LQ FRQWLQXRXVÀH[XUDO
PHPEHUV PHPEHUV

6.6.5.1 Except where approximate values for moments Redistribution of moments is dependent on adequate
are used in accordance with 6.5, where moments have been ductility in plastic hinge regions. These plastic hinge regions

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 #Qr#ixuwkhu#uhsurgxfwlrq#ru#glvwulexwlrq#lv#shuplwwhg1
Frs|uljkwhg#pdwhuldo#olfhqvhg#wr#Xqlyhuvlw|#ri#Wrurqwr#e|#Fodulydwh#Dqdo|wlfv#+XV,#OOF/#vxevfulswlrqv1whfkvwuhhw1frp/#grzqordghg#rq#534<038064#49=3;=64#.3333#e|##Xqlyhuvlw|#ri#Wrurqwr#Xvhu1
PART 2: LOADS & ANALYSIS 83

CODE COMMENTARY
calculated in accordance with 6.8, or where moments in develop at sections of maximum positive or negative moment
two-way slabs are determined using pattern loading speci- and cause a shift in the elastic moment diagram. The usual
¿HGLQUHGXFWLRQRIPRPHQWVDWVHFWLRQVRIPD[LPXP result is a reduction in the values of maximum negative
negative or maximum positive moment calculated by elastic moments in the support regions and an increase in the values
theory shall be permitted for any assumed loading arrange- of positive moments between supports from those calculated
PHQWLI D DQG E DUHVDWLV¿HG by elastic analysis. However, because negative moments
are typically determined for one loading arrangement and
(a) Flexural members are continuous positive moments for another (6.4.3 provides an exception
(b) İt• at the section at which moment is reduced for certain loading conditions), economies in reinforcement
can sometimes be realized by reducing maximum elastic
6.6.5.2 For prestressed members, moments include those positive moments and increasing negative moments, thus
due to factored loads and those due to reactions induced by narrowing the envelope of maximum negative and positive

Analysis
prestressing. moments at any section in the span (Bondy 2003). Plastic
hinges permit utilization of the full capacity of more cross
6.6.5.3 At the section where the moment is reduced, redis- VHFWLRQVRIDÀH[XUDOPHPEHUDWXOWLPDWHORDGV
tribution shall not exceed the lesser of İt percent and The Code permissible redistribution is shown in Fig.
20 percent. R6.6.5. Using conservative values of limiting concrete

6
strains and lengths of plastic hinges derived from extensive
6.6.5.4 The reduced moment shall be used to calculate late WHVWVÀH[XUDOPHPEHUVZLWKVPDOOURWDWLRQFDSDFLWLHVZHUH
WHVWVÀ
redistributed moments at all other sections within the spans analyzed for redistribution
red of moments up to 20 percent,
ed after rredistribution
such that static equilibrium is maintained depending
ding on the rreinforcement ratio. As shown, the
gemen
of moments for each loading arrangement. permissible redistribution
ible redistributi n percentages are conservative
on
relative to tthe calculated ppercentages available for both fy
acti
6.6.5.5 Shears and support reactions hall be calculat
shall calculated inn = 60
6 ksii and
an 80 ksi. Studie
Studies by Cohn (1965) and Mattock
um considering
accordance with static equilibrium sidering the redi rib-
redistrib- (1959 support
(1959) upp this conclusio
conclusion and indicate that cracking and
uted moments for each loading ar men
arrangement. RQR GHVLJQHG
GHÀHFWLRQRIEHDPVGHVLJQHGIRUUHGLVWULEXWLRQRIPRPHQWV
DUHQR VLJQ \JUHDWHU
DUHQRWVLJQL¿FDQWO\JUHDWHUDWVHUYLFHORDGVWKDQIRUEHDPV
desig d by the distribution
designed istribution of moments according to elastic
hese stud
theory. Also, these studies indicate that adequate rotational
capacity foror the redis
redistribution of moments allowed by the
Codede is available if the members satisfy 6.6.5.1.
The provisions for redistribution of moments apply
equally to prestressed members (Mast 1992).
The elastic deformations caused by a nonconcordant tendon
change the amount of inelastic rotation required to obtain a
given amount of redistribution of moments. Conversely, for
a beam with a given inelastic rotational capacity, the amount
by which the moment at the support may be varied is changed
by an amount equal to the secondary moment at the support
due to prestressing. Thus, the Code requires that secondary
moments caused by reactions generated by prestressing
forces be included in determining design moments.
Redistribution of moments as permitted by 6.6.5 is not
appropriate where approximate values of bending moments
DUHXVHGVXFKDVSURYLGHGE\WKHVLPSOL¿HGPHWKRGRI.
Redistribution of moments is also not appropriate for
two-way slab systems that are analyzed using the pattern
loadings given in 6.4.3.3. These loadings use only 75
percent of the full factored live load, which is based on
considerations of moment redistribution.

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 #Qr#ixuwkhu#uhsurgxfwlrq#ru#glvwulexwlrq#lv#shuplwwhg1
Frs|uljkwhg#pdwhuldo#olfhqvhg#wr#Xqlyhuvlw|#ri#Wrurqwr#e|#Fodulydwh#Dqdo|wlfv#+XV,#OOF/#vxevfulswlrqv1whfkvwuhhw1frp/#grzqordghg#rq#534<038064#49=3;=64#.3333#e|##Xqlyhuvlw|#ri#Wrurqwr#Xvhu1
84 ACI 318-19: BUILDING CODE REQUIREMENTS FOR STRUCTURAL CONCRETE

CODE COMMENTARY
25
ℓ /d = 23

i
i
ks
ks
80
60
b /d = 1/5

=
=
20

fy
fy
Percent change in moment
Calculated
percentage
available
15
Permissible
redistribution
10
allowed by
Minimum 6.6.5.3
permissible
5 net tensile
strain = 0.0075

0
0 0.005 0.010 0.020 0.015 0.025
Net tensile strain, εt
Fig. R6.6.5²3HUPLVVLEOH
R6.6.5²3HUPL
5 UHGLVWULEXWLRQ RI PRPHQWV IRU
PURWDWLRQ FDSD LW\
W\
PLQLPXPURWDWLRQFDSDFLW\

6.7—Linear elastic second-order

-or analysis
nalysis R6. Lin
R6.7—Linear astic sec
elastic second-order analysis
6.7.1 General R6. 1G
R6.7.1 General

In linear
near elastic second-
second-order analyses, the deformed
geom ry oof the structure is included in the equations of
geometry
equilibrium so that P¨H൵HFWVDUHGHWHUPLQHG7KHVWUXFWXUH
P H
P¨
WRUHPDLQ
LVDVVXPHGWRUHPDLQHODVWLFEXWWKHH൵HFWVRIFUDFNLQJDQG
S DUH FRQVLG
FUHHS FRQVLGHUHG E\ XVLQJ DQ H൵HFWLYH VWL൵QHVV EI. In
FRQWUDVW
FRQWUDVWOLQHDUHODVWLF¿UVWRUGHUDQDO\VLVVDWLV¿HVWKHHTXD-
tions of equilibrium using the original undeformed geom-
etry of the structure and estimates P¨H൵HFWVE\PDJQLI\LQJ
the column-end sway moments using Eq. (6.6.4.6.2a) or
(6.6.4.6.2b).

6.7.1.1 A linear elastic second-order analysis shall R6.7.1.17KHVWL൵QHVVHVEI used in an analysis for strength
FRQVLGHU WKH LQÀXHQFH RI D[LDO ORDGV SUHVHQFH RI FUDFNHG GHVLJQ VKRXOG UHSUHVHQW WKH VWL൵QHVVHV RI WKH PHPEHUV
UHJLRQVDORQJWKHOHQJWKRIWKHPHPEHUDQGH൵HFWVRIORDG immediately prior to failure. This is particularly true for a
GXUDWLRQ7KHVHFRQVLGHUDWLRQVDUHVDWLV¿HGXVLQJWKHFURVV VHFRQGRUGHUDQDO\VLVWKDWVKRXOGSUHGLFWWKHODWHUDOGHÀHFWLRQV
VHFWLRQDOSURSHUWLHVGH¿QHGLQ at loads approaching ultimate. The EI values should not be
based solely on the moment-curvature relationship for the
most highly loaded section along the length of each member.
Instead, they should correspond to the moment-end rotation
relationship for a complete member.
To allow for variability in the actual member properties in
the analysis, the member properties used in analysis should
EH PXOWLSOLHG E\ D VWL൵QHVV UHGXFWLRQ IDFWRU ࢥK less than
 7KH FURVVVHFWLRQDO SURSHUWLHV GH¿QHG LQ  DOUHDG\
LQFOXGHWKLVVWL൵QHVVUHGXFWLRQIDFWRU7KHVWL൵QHVVUHGXFWLRQ
factor ࢥK may be taken as 0.875. Note that the overall
VWL൵QHVV LV IXUWKHU UHGXFHG FRQVLGHULQJ WKDW WKH PRGXOXV
of elasticity of the concrete, Ec LV EDVHG RQ WKH VSHFL¿HG
FRQFUHWH FRPSUHVVLYH VWUHQJWK ZKLOH WKH VZD\ GHÀHFWLRQV

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