[Technical Field]
5 [0001]
The present invention relates to a reinforced concrete (RC) member joining
structure.
Priority is claimed on Japanese Patent Application No. 2015-198054, filed
October 5, 2015, the content of which is incorporated herein by reference.
10 [Background Art]
[0002]
Conventionally, a technology for shortening a construction period by increasing
precast concrete (PCa) construction ratio and reducing cast-in-place concrete has been
developed and implemented.
15 [0003]
For example, there is a construction method in which a plurality of sleeves
(sheath pipes) of mechanical joints are arranged in an upper end portion of a stub, and a
precast concrete column projecting a plurality of main reinforcement from a lower end
portion toward the lower side is installed and joined on the stub.
20 [0004]
In this construction method, each of the plurality of main reinforcement
projecting to the lower side of the PCa column is plugged inside a sleeve of the
mechanical joint of the stub. Then, a grout material is injected into inside sleeves of the
plurality of mechanical joints of the stub and a joint portion between PCa columns
25 vertically adjacent to each other, and the stub and the PCa columns are integrally joined
3
(for example, see Patent Document 1).
[Citation List]
[Patent Literature]
[0005]
5 [Patent Document 1]
Japanese Unexamined Patent Application, First Publication No. 2012-77547
[Summary of Invention]
[Technical Problem]
[0006]
10 Meanwhile, while a design technique for connecting main reinforcement of one
pair of RC members using a mechanical joint has been established, a design technique
for connecting main reinforcement using non-contact lap splice overlapping the main
reinforcement while having a gap formed between the main reinforcement has not been
sufficiently established.
15 [0007]
The present invention is in consideration of the situation described above, and an
object thereof is to provide an RC member joining structure enabling RC members to be
appropriately joined using a non-contact lap splice.
[Solution to Problem]
20 [0008]
In order to solve the problems described above, the present invention employs
the following aspects.
[0009]
(1) In an RC member joining structure according to one aspect of the present
25 invention, one RC member is formed by extending main reinforcement from a joining
4
end face to the outer side, and the other RC member is a column and is formed by having
sheath pipes buried to open to the joining end face in parallel with the main
reinforcement, the main reinforcement of the one RC member is inserted into the sheath
pipe, a grout material is filled inside the sheath pipe, the main reinforcement of the one
RC member and a main reinforcement of 5 the other RC member are connected by a
non-contact lap splice with a gap therebetween, and the required lap splice length Lb of
the non-contact lap splice is set in accordance with Equations (1) to (6) below.
[0010]
[Formula. 1]
10
[0011]
[Formula. 2]
[0012]
15 [Formula 3]
[0013]
5
[Formula 4]
[0014]
[Formula 5]
5
[0015]
[Formula 6]
[0016]
10 Here, Ld is a lap splice length (mm), and db is the nominal diameter (mm) of the
main reinforcement of the column. Lb is the ineffective lap splice length (mm) at
bottom end and is acquired by Equation (4). σyf is the yield strength (N/mm2) of the
main reinforcement of the column.
[0017]
15 H is the shear span (mm) of the column. hb is a grout thickness at the joint
(mm). Ds is the sheath pipe outer diameter (mm). τbmax is the bond strength (N/mm2)
of the main reinforcement of the column and is acquired by Equation (5). τsbmax is the
bond strength (N/mm2) of the sheath pipe and is acquired by Equation (6).
[0018]
20 γt is the ratio of the joint upper ineffective length to the lap splice length. Rd is
6
the design limit member angle (%). Ry is the yield member angle (%). dcy is the
effective depth (mm) of the column and is the distance from a compressive stress fiber of
the cross-section of a pedestal portion to a centroid position of the steel reinforcement in
tension (joint reinforcement).
5 [0019]
α is the bond strength reduction coefficient. fc is the concrete design reference
strength (N/mm2).
[0020]
(2) In the RC member joining structure described in “(1)” above, a yield moment MY at
10 the cross-section based on arrangement of main reinforcement of the other RC member
arranged on the inner side of the sheath pipe may be above an acting bending moment
MLE at the joint end point, and the required lap splice length Ld may be set so as not to
yield in flexure at a cross-section of the joint end point of the non-contact lap splice by
satisfying a condition of Equation (7) below.
15 [0021]
[Formula 7]
[0022]
Here, MY is the yield moment (kNm) at the cross-section according to the
20 column main reinforcement at the joint end point and is acquired by Equation (8) below.
MLE is the moment (kNm) acting on the joint end point. Mcy is the yield moment (kNm)
at the cross-section of the pedestal and is acquired by Equation (9) below. φe is the
coefficient of bending moment and is 1.0.
7
[0023]
As is total the cross-sectional area (mm2) of the steel reinforcement in tension.
dY is the effective depth (mm) of the column and is a distance from the compressive fiber
stress of the joint end point cross-section to the centroid position of the steel
reinforcement in tension. a 5 is the stress block depth (mm) of concrete and is calculated
by Equation (10) from the balance at the cross-section. Fc is the concrete design
reference strength (N/mm2), fy is the specified yield strength (N/mm2) of the main
reinforcement of the column, and b is a member width (mm).
[0024]
10 [Formula 8]
[0025]
[Formula 9]
15 [0026]
[Formula 10]
[0027]
(3) In the RC member joining structure described in “(1)” or “(2)” described
20 above, the required transverse reinforcement amount p wd in a lap splice section may be
8
set in accordance with Equation (11) and Equation (12).
[0028]
[Formula 11]
5 [0029]
[Formula 12]
[0030]
Here, Cs is the smaller value of a gap (mm) between sheath pipes and twice the
10 minimum covering thickness (mm) of the sheath pipe. Ds is the sheath pipe outer
diameter (mm), and pwd is the required transverse reinforcement ratio. fwy is the
specified yield strength (N/mm2) of the transverse reinforcement.
[0031]
σtd is the design splitting stress intensity (N/mm2). σt0 is the reference splitting
strength (N/mm2). σtav is the average stress intensity (N/mm215 ) acting on a splitting face
when concrete is failed in splitting and is acquired by Equation (13).
[0032]
α is the bond reduction coefficient. τb is the working bond stress (N/mm2) of
the main reinforcement in a column and is acquired by Equation (14) below. τsb is the
working bond stress (N/mm220 ) of the sheath pipe and is acquired by Equation (15) below.
[0033]
9
ft is a pulling strength (N/mm2) of concrete and is acquired by Equation (17)
below.
[0034]
[Formula 13]
5
[0035]
[Formula 14]
[0036]
10 [Formula 15]
[0037]
[Formula 16]
15 [0038]
[Formula 17]
10
[Advantageous Effects of Invention]
[0039]
According to an RC member joining structure of one aspect of the present
invention, the structure of a joining portion having hig 5 h reliability can be realized by
using the non-contact lap splice of which a design method has not been conventionally
established.
[Brief Description of Drawings]
[0040]
10 Fig. 1 is the diagram showing an RC member joining structure according to one
embodiment of the present invention.
Fig. 2 is the arrow view taken along line X1-X1 shown in Fig. 1.
Fig. 3 is the arrow view taken along line X2-X2 shown in Fig. 1.
Fig. 4 is the arrow view taken along line X3-X3 shown in Fig. 1.
15 Fig. 5 is the diagram showing a steel reinforcement stress distribution of an RC
member joining structure according to one embodiment of the present invention.
Fig. 6 is the diagram showing one example of dimension settings when
calculating a required transverse reinforcement ratio of the RC member joining structure
according to one embodiment of the present invention.
20 Fig. 7 is the diagram showing a cross-sectional strength determination position
of a joint end point of the RC member joining structure according to one embodiment of
the present invention.
Fig. 8 is the diagram showing a stress block method according to American
11
Concrete Institute (ACI) 318.
Fig. 9 is the diagram showing the dimensional shape and the state of the
arranged reinforcement of a test specimen.
[Description of Embodiments]
5 [0041]
Hereinafter, an RC member joining structure according to one embodiment of
the present invention will be described with reference to Figs. 1 to 9.
[0042]
First, in this embodiment, as shown in Figs. 1 to 4, on a stub 1 consisting of one
10 RC member, a PCa column 2 of the other RC member is installed, and the stub 1 and the
PCa column 2 are joined together to erect the PCa column 2.
[0043]
The stub 1 is formed with a plurality of main reinforcement 3 sticking out of an
upper end face (joining end face) thereof to the upper side by a predetermined length.
15 [0044]
On the bottom end portion of the PCa column 2, a plurality of sheath pipes 4
having an approximately bottomed cylinder shape are integrally buried in a vertical
direction along the axis, and a grouting gap is installed as joining end face. In addition,
the PCa column 2 according to this embodiment includes a squeezing part 6 bending
20 portions near the lower end portion of the plurality of main reinforcement 5 at a gradient
angle of 1/6 to be near the center, the main reinforcement 5 is set so as to approach the
sheath pipes 4 by this squeezing part 6, and one pair of sheath pipes 4 and the main
reinforcement 5 are buried in concrete in parallel with a predetermined gap therebetween.
[0045]
25 In an RC member joining structure A according to this embodiment, inside the
12
sheath pipe 4 of the PCa column 2, the main reinforcement 3 projecting to the lower side
of the stub 1 is plugged to have a gap with the main reinforcement 5 of the PCa column 2
in the horizontal direction, and a grout material is injected into the inside of each sheath
pipe 4 and the joint portion 7. Accordingly, the main reinforcement 3 and 5 of the stub
1 and the PCa column 2 are connected 5 by a non-contact lap splice 8 through the sheath
pipes 4, and the stub 1 and the PCa column 2 are integrally jointed together.
[0046]
Here, a design method of a case in which the main reinforcement 2 and 5 of the
stub 1 and the joining portion of the PCa column 2 are configured as the non-contact lap
10 splice 8 as described above will be described.
[0047]
In addition, a joining portion targeted by this design method, for example, is
preferably applied to a building that is constructed in an earthquake region (regions
corresponding to Seismic zones 2 to 4 in U.S. UBC 1997 section 1653) and satisfies the
15 following conditions.
[0048]
(Restriction on column to be applied)
1-1) The axial force NU at the ultimate state is Ag·fc/10 or less.
1-2) The shear span ratio MU/(QU·D) at the ultimate state is 5.0 or more.
20 1-3) The designed limit member angle of the column at the ultimate state with plastic
deformation considered is 2.0% or less.
[0049]
Here, D is the depth (mm) of the column, NU is the axial force (kN) exerted in
the column at the ultimate state, MU is the bending moment (kNm) exerted in the column
25 at the ultimate state, QU is the shear force (kN) exerted in the column at the ultimate
13
state, Ag is the cross-sectional area (mm2) of the column, and fc is the compressive
strength (N/mm2) of cylinders.
[0050]
(Frame system)
The frame system is a frame struc 5 ture or a frame structure used also as a bearing
wall.
[0051]
(Basic items)
Other basic items are as follows.
10 2-1) The seismic load is according to standards applied in the country or the region.
2-2) The calculation of the cross-section for the axial-direction force, the bending and the
shear forces is performed on the basis of “Steel reinforcement concrete structure
calculation standard and commentary” edited by the Architectural Institute of Japan”
(hereinafter, referred to as the RC standard) or ACI 318 for a stress analysis. The
15 calculation thereof may be acquired on the basis of standards applied in the
corresponding country or region.
2-3) The design of the joint is performed on a basis that will be described later (Design of
joint).
[0052]
20 (Used materials)
(Concrete)
3-1) The concrete used is normal concrete.
3-2) The specified strength is fc = 21 to 30 N/mm2 (cylinder strength), or fcu = 25 to 40
N/mm2 (cube strength) or more and is converted into a cylinder strength using fc = 0.8
25 fcu. The conversion factor may be a numerical value according to the applied standard.
14
[0053]
(Steel)
4-1) The steel reinforcement is a deformed bar having a diameter of 25 mm or less.
4-2) The specified yield strength of the main reinforcement is 500 N/mm2 or less.
5 [0054]
(Sheath pipe)
A Japanese Industrial Standard (JIS) G3302 hot-dip galvanized steel sheet or a
steel bar or a material equivalent thereto is used.
[0055]
10 (Grout material)
5-1) The grout material is applied to the inside of the sheath pipe and a PCa pedestal.
5-2) The compressive strength is 60 N/mm2 or more (cylinder strength).
[0056]
(Design of PCa column)
15 Next, the design of a PCa column will be described.
[0057]
(Calculation of axial-direction force and bending moment)
First, for the axial-direction force and the bending moment, the followings are
applied.
20 6-1) The stress intensity of the inside of the cross-section is calculated on the basis of the
RC standard or the ACI 318, and an allowable bending moment or an ultimate bending
moment is acquired. Here, the calculation may be on the basis of standards applied to
the corresponding country or region.
6-2) A minimum main reinforcement amount of the column follows the rules of the
25 design reference to be applied.
15
[0058]
(Calculation of shear strength)
The calculation of the shearing is performed as bellows.
7-1) An allowable shear strength or an ultimate shear strength is acquired on the basis of
the RC standard or ACI 318. 5 In addition, those may be acquired on the basis of
standards applied to the corresponding country or region.
7-2) The minimum shear reinforcement ratio follows the rules of the applied design
standard.
7-3) The shear force is calculated at the pedestal and a joint end point.
10 7-4) At a small gap of the pedestal, the shear friction strength for grout is reviewed.
[0059]
(Design of Joint)
Next, the joint (lap splice) of the pedestal portion is designed in accordance with
the following sequences (a) to (c).
15 (a) A lap splice length Ld according to attachment is calculated.
(b) Next, it is checked that bending yield does not occur at the cross-section of the joint
end point.
(c) Then, the required transverse reinforcement amount pwd of the joint section is
calculated.
20 [0060]
(a) Calculation of lap splice length Ld according to attachment
The required lap splice length Ld of the lap splice portion is calculated using the
following Equations (18) to (23). In addition, please refer to Fig. 5 for symbols
included in the equations.
25 [0061]
16
[Formula 18]
[0062]
[Formula 19]
5
[0063]
[Formula 20]
[0064]
10 [Formula 21]
[0065]
[Formula 22]
15 [0066]
[Formula 23]
17
[0067]
Here, Ld is a lap splice length (mm), db is the nominal diameter (mm) of the
main reinforcement of the column. Lb is the joint lower ineffective length (mm) and is
acquired by Equation (21). σyf is the 5 yield strength (N/mm2) of the main reinforcement
of the column and is 1.1 times the specified yield strength.
[0068]
H is the shear span (mm) of the column. hb is the grout thickness at the precast
joint (mm), and it is assumed that hb = 0 in the review for safety. Ds is the sheath pipe
outer diameter (mm). τbmax is the bond strength (N/mm210 ) of the column main
reinforcement and is acquired by Equation (22). τsbmax is the bond strength (N/mm2) of
the sheath pipe and is acquired by Equation (23).
[0069]
γt is rhe ratio of the joint upper ineffective length to the lap splice length and is
15 0.13. Rd is the design limit member angle (%) and is 2.0%. Ry is the yield member
angle (%) and is 0.75%. dcy is the effective depth (mm) of the column and is the
distance from the compressive fiber stress of the cross-section of the pedestal portion to
the centroid of the steel reinforcement (joint reinforcement) in tension.
[0070]
20 α is the bond reduction coefficient and is assumed to be 0.8 when the column is
manufactured through concrete laying down casting and is assumed to be 1.0 otherwise.
fc is the concrete design reference strength (N/mm2).
[0071]
18
(b) Checking no yield at the cross-section of the joint end point
It is checked that the yield moment MY based on the PCa column main
reinforcement arranged on the inner side of the sheath pipe is above an acting bending
moment MLE at the joint end point using the following Equation (24). In a case where
Equation (24) is not satisfied, it is predicted that a 5 crack is expected to initiate at the joint
end point, and accordingly, the required lap splice length Ld is increased so as to satisfy
Equation (24).
[0072]
Here, MY is the yield moment (kNm) at the cross-section according to the
10 column main reinforcement at the joint end point and is acquired by Equation (25) below.
MLE is the moment (kNm) acting on the joint end point. Mcy is the yield moment
strength (kNm) at the cross-section of the pedestal and is acquired by Equation (26)
below. As described above, H is the shear span length (mm) of the column. hb is the
grout thickness at the joint portion (mm) and is assumed to be hb = 0 in reviewing the
15 safety. Ld is the lap splice length (mm). Lb is the joint lower ineffective length (mm)
and is acquired by Equation (21). φe is the coefficient of the bending moment and is
assumed to be 1.0.
[0073]
In addition, As is the cross-sectional area (mm2) of the steel reinforcement in
20 tension. dY is the effective depth (mm) of the column and is the distance from the
compressive fiber stress of the joint end point cross-section to the centroid position of the
steel reinforcement in tension. dcy is the effective depth (mm) of the column and is the
distance from the compressive fiber stress of the cross-section of the pedestal up to the
centroid of the steel reinforcement (joint reinforcement) in tension. a is the stress block
25 depth (mm) of concrete and is calculated by Equation (27) from the balance of the
19
cross-section. Fc is the concrete specified strength (N/mm2), fy is the specified yield
strength (N/mm2) of the main reinforcement of the column, and b is the member width
(mm).
[0074]
5 [Formula 24]
[0075]
[Formula 25]
10 [0076]
[Formula 26]
[0077]
[Formula 27]
15
[0078]
(c) Calculation of the required transverse reinforcement amount pwd at the joint section
The required transverse reinforcement amount pwd is calculated by the following
20
Equations (28) and (29).
[0079]
Here, Cs is the smaller value of a gap (mm) between sheath pipes and twice the
minimum covering thickness (mm) of the sheath pipe. Ds is the sheath pipe outer
diameter (mm), and pwd is the required transverse 5 reinforcement ratio. fwy is a specified
yield strength (N/mm2) of the transverse reinforcement and is 390 N/mm2 or less.
[0080]
σtd is the design splitting stress intensity (N/mm2). σt0 is the reference splitting
strength (N/mm2) and is assumed to be 1.0. σtav is the average stress intensity (N/mm2)
10 acting on the splitting face when concrete is failed in splitting and is acquired by
Equation (30).
[0081]
α is the bond reduction coefficient and is assumed to be 0.8 when manufacturing
the column through concrete laying down casting and is assumed to be 1.0 otherwise.
τb is the working bond stress (N/mm215 ) of the main reinforcement of the column and is
acquired by Equation (31) below. τsb is the working bond stress (N/mm2) of the sheath
pipe and is acquired by Equation (32) below.
[0082]
db is the nominal diameter (mm) of the main reinforcement of the column. σy
is the yield strength (N/mm220 ) of the main reinforcement of the column and is 1.1 times
the specified yield strength. H is the shear span (mm) of the column. hb is the grout
thickness at the joint portion (mm) and is assumed to be hb = 0 in reviewing the safety.
Ld is the lap splice length (mm). Lt is the joint upper ineffective length (mm) and is
acquired by Equation (33) below.
25 As described above, Lb is the joint lower ineffective length (mm) and is acquired
21
by Equation (21). DS is the sheath pipe outer diameter (mm), and fc is the concrete
specified strength (N/mm2). ft is the aplitting strength (N/mm2) of concrete and is
acquired by Equation (34) below.
[0083]
5 [Formula 28]
[0084]
[Formula 29]
10 [0085]
[Formula 30]
[0086]
[Formula 31]
15
22
[0087]
[Formula 32]
[0088]
5 [Formula 33]
[0089]
[Formula 34]
10 [0090]
The design method described above is applied under the following conditions
8-1) to 8-9).
8-1) The diameters, the numbers, and the materials of the main reinforcement of the
column inside the sheath pipe and the main reinforcement of the column inside the PCa
15 are assumed to be the same.
8-2) A slope of the inclined main reinforcement in the squeezing part of PCa column is
1/6 or less. At a bending point of the main reinforcement of the column, an additional
transverse reinforcement is arranged to bear a transverse component of a force.
8-3) The minimum lap splice length is 40 times the diameter of the main reinforcement
23
or more.
8-4) The lap splice length is set in consideration of a construction error together with the
calculated required length.
8-5) The diameter of the transverse reinforcement of the joint section is 10 mm or more,
5 and the gap is 100 mm or less.
The transverse reinforcement of the upper portion of the joint section follows the
rules of the design standard to be applied.
8-6) In order to prevent splitting failure, all the the sheath pipes are constrained by the
transverse reinforcement (core reinforcement).
10 8-7) Hooks of the hoop and the core reinforcement are 135 degrees or 180 degrees with
an additional length of 6d or more.
8-8) Clearance of the sheath pipe and the main reinforcement in the column except for a
combination of the sheath pipe and the main reinforcement of the column forming the
joint is 4/3 times the coarse aggregate and is the diameter of the main reinforcement or
15 more.
8-9) A gap between the main reinforcement of the stub and the main reinforcement of the
PCa column at the non-contact lap splice is defined to be 1/5 or the non-contact lap splice
length and is 150 mm or less.
[0091]
20 (Stress state assumed in this design)
As the stress state assumed in this design, the stress state of the steel
reinforcement of the joining portion when the bending yield moment is reduced at the
end portion of the column is assumed to be that shown in Fig. 5.
In addition, the split line at the bond splitting failure and various dimensions are
25 assumed to be those shown in Fig. 6. In order to check that the joint end point does not
24
yield in flexure, a distribution of bending moments shown in Fig. 7 is assumed. In
addition, when the cross-section of the joint end point yields in flexure, the upper joint
ineffective length Lt shown in Fig. 5 becomes longer than the assumed length, and, as a
result, the effective lap splice length becomes shorter than the assumed length, and then it
is predicted that slipping-5 out of the main reinforcement of the column from the joining
portion occurs. For this reason, the cross-section of the end point of the joint is planned
not so as to yield in flexure.
[0092]
(Setting of design limit member angle Rd)
10 The design limit member angle Rd used for calculating the joint lower
ineffective length Lb is set on the basis of a story drift angle determination condition
2.0% of a general building in US ASCE 7-05.
In this design, a design equation based on the concept that the joining portion
may not be failed up to the design limit member angle Rd in advance is formed. Based
15 on the appropriate evaluation in accordance with the design policy of a building, the
numerical value of Rd can be appropriately changed. Here, the upper limit value of Rd
is 2.5%, a value that has been determined through experiment. When the numerical
value of Rd is decreased, the joint lower ineffective length Lb increases, and the required
lap splice length Lb decreases in accordance with Equation (18).
20 [0093]
(Yield strength for reviewing no yield at the joint end point)
The yield moment My at the cross-section of the joint end point and the yield
moment Mcy at the cross-section of the pedestal are calculated on the basis of the
following Equations (35) and (36) acquiring an ultimate strength using the concrete stress
25 block method (Fig. 8) represented in ACI 318. Since a technique for acquiring a yield
25
strength in simplified calculation is not represented in ACI 318, here, based on these
equations, Equations (25) and (26) having a strength reduction coefficient φ of 1.0 are
used for the yield strength for reviewing no yield at the cross-section of the joint end
point.

An RC member joining structure in which one RC member is formed by
extending main reinforcement from a joining end face to the outer side, and the other
RC member is a column and is formed by having sheath pipes buried to open to the
joining end face in parallel with the main reinforcement,
wherein the main reinforcement of the one RC member is inserted into the
sheath pipe, a grout material is filled inside the sheath pipe, and the main reinforcement
of the one RC member and a main reinforcement of the other RC member are connected
by a non-contact lap splice with each other, having a gap therebetween, and
wherein the required lap splice length [[Lb]] Ld of the non-contact lap splice is
set in accordance with Equations (1) to (6) below:
[Claim 2]
The RC member joining structure according to claim 1,
wherein a yield moment MY at the cross-section based on arrangement of main
reinforcement of the other RC member arranged on the inner side of the sheath pipe is
above an acting bending moment MLE at the joint end point, and the required lap splice
35
length Ld is set so as not to yield in failure at the cross-section of the joint end point of
the non-contact lap splice by satisfying a condition of Equation (7) below;
[Formula 7]
wherein MY is the yield moment ([[kNm]]kN m) at the cross-section according
to the column main reinforcement at the joint end point and is acquired by Equation (8)
below, MLE is the moment (kNm) acting on the joint end point, Mcy is the yield moment
(kNm) at the cross-section of the pedestal and is acquired by Equation (9) below, and φe
is the coefficient of bending moment and is 1.0; and
wherein As is the total cross-sectional area (mm2) of the steel reinforcement in
tension, dY is the effective depth (mm) of the column and is the distance from the
compressive fiber stress of the joint end point cross-section to the centroid of the steel
reinforcement in tension, a is the stress block depth (mm) of concrete and is calculated
by Equation (10) from the balance at the cross-section, and [[Fc]] fc is the concrete
specified strength (N/mm2), fy is the specified yield strength (N/mm2) of the main
reinforcement of the column, and b is member width (mm).
[Formula[Claim 3]
The RC member joining structure according to claim 1 or 2,
wherein the required transverse reinforcement amount pwd in a lap splice
section is set in accordance with Equation (11) and Equation (12);
[Formula 11]