Steel Structures 8 (2008) 189-198 www.ijoss.org
Evaluation of Diaphragm Effect for Steel Welded Box Beam
and Circular Column Connections
Young-Pil Kim1 and Won-Sup Hwang2*
1Struatural Department, YOOSHIN Eng. CO., 832-40 Yeoksam-Dong Seoul, Korea2Department of Civil Engineering, Inha University, 253 Younghyun-Dong Incheon, Korea
Abstract
This paper investigates the design equations and the strength behavior of the diaphragm for steel box beams and circularcolumn connections. The strength of the connection is decided by the strength of beam and column as well as connectiondiaphragm, because the connection diaphragm supports the concentration forces from the box beam's lower flange. In previousresearches, however, the calculation procedure of the diaphragm stress from the indeterminate curved-girder model is toocomplicated to apply in regular practice. Irrational assumptions exist in deriving the process of the equation. Moreover, noreasonable design has yet been made because the diaphragm's effect on the strength of the connection has not been considered.Therefore, this study examines the strength behavior of a connection with diaphragm details through non-linear FEM analysisof the connection diaphragm. Finally, the connection strength is evaluated by using rigidity capacity of diaphragm, newdiaphragm design method is proposed.
Keywords: box beam and circular column connection, diaphragm, non-linear analysis, strength, rigidity capacity
1. Introduction
Recently, welded steel piers have been widely applied
for pier structures of urban overpasses and elevated
structures in East Asian countries due to their excellent
earthquake resistance capacity, small space requirements,
and short construction term. At the T-type or framed
beam-to-column connections of box-sectioned steel piers,
it has been widely acknowledged that serious shear lag
and stress concentrations may occur due to abrupt
direction changes in member forces, so it is necessary to
handle these problems properly in the design stage.
Instead of box-sectioned steel piers, circular- sectioned
piers have been introduced lately due to high local-
buckling strength, efficient behavior for changing loading
direction, and fine appearances.
In an early study on welded steel connections, Beedle
et al. (1951) proposed a stress and strength evaluation
method for an H-sectioned beam-to- column connection
by assuming that stresses are uniform in flanges and
webs. Fielding and Huang (1971) indicated that the
strength of the beam-to-column connection of H-
sectioned frame is reduced due to the axial force in the
column. However, they did not recognize the shear lag
phenomenon at the flange of connections. By recognizing
the shear lag phenomenon at box -sectioned beam-to-
column connections of the pier structure, Okumura and
Ishizawa (1968) carried out theoretical and experimental
studies using a simple beam model subjected to a
concentrated mid-span load. Instead of using a simple
beam model, Nakai et al. (1992) suggested an equation
for the shear lag stress from a study using an overhanging
beam model with additional moments due to shear
deformation occurring in the connection. Qi and Mimura
(2002) suggest design and strength evaluation method of
welded beam-column connection. Also, Hwang et al.
(2004) suggest shear lag stress evaluation method for
box-sectioned welded connection using the additional
moment of cantilever beam model.
However, almost all researches, except Okumura and
Ishizawa’s, have been limited to box-sectioned connections
only. Also, most researches do not consider that the
behavior of the box beam and circular column connection
is much more efficient than that of H-sectioned or box-
sectioned connections in steel piers. After Okumura and
Ishizawa, There is no reach result on analysis and
experiment of the box beam and circular column
connection. Basic research is carried out from large scale
test of the box beam-to-circular column connection by
Hwang et al. (2002).
Okumura and Ishizawa (1968) suggest stress equation
Note.-Discussion open until February 1, 2009. This manuscript forthis paper was submitted for review and possible publication on July17, 2008; approved on August 30, 2008
*Corresponding authorTel: +82-32-860-7570; Fax: +82-32-873-7560E-mail:[email protected]
190 Young-Pil Kim and Won-Sup Hwang
from indeterminate curved I-girder model for connection
diaphragm of box beam-to circular column connections
as shown in Fig. 1 and Fig. 2. Their equations are very
complicated form and design variable are not precise.
Then, current design specifications don’t reflect this
equations. Therefore, new design method of connection
diaphragm should be developed by considering practical
diaphragm behavior. In current design specifications and
researches of connection diaphragm, importance of
diaphragm has been neglected, because concentrated
forces of beam lower flange are transferred to circular
column through connection diaphragm. Then, Connection
strength is affected by strength of connection diaphragm.
Also, appropriate reinforcements of diaphragm deficits
area such as working manhole are required. However,
previous research did not consider connection strength
behavior with connection diaphragm shape.
Therefore, to suggest new diaphragm design methods
for the steel box beam and circular column connections.
The strength behavior is examined with the diaphragm
shape from the non-linear FEM analysis. Influence of
diaphragm design parameters is investigated based on
connection strength. Also, required thickness of connection
diaphragm is evaluated with connection diaphragm details.
2. Investigating Existing Researches for Connection Diaphragm
2.1. Calculation of Connection Diaphragm Stress
Previous research (Okumura and Ishizawa, 1968) have
analized stress of connection diaphragm using concept of
stress concentration. Specially, stresses should be calculated
in the point such as ϕ=π, ϕ=π/2, and ϕ=α. Also, normal
and shear stresses of the internal and external fiber are
given as followings.
(1-a)
(1-b)
(1-c)
Where fo and fi is external and internal fiber normal stress.
Also, v is shear stress, Tϕ is axial force, Mϕ is bending
moment, and Qϕ is shear force in any point. In any point,
Tϕ, Mϕ, and Qϕ is given by Table 1. As shown in Table 1.
Okumura’s stress equations are very complicated form.
Then, its practical usage is very difficult in connection
diaphragm design.
2.2. Specifications for connection diaphragm
In current design specifications, the dimension of the
diaphragm should be decided after the review of stress
calculation based on Eq. (1) (KHBDC, 2005; Hanshin
Highway Co., 1985). However, design specifications do
not adopt Eq. (1) due to complicated calculation
procedure. In current stage the stress calculation can be
omitted when following empirical conditions is satisfied.
① Ro≤1500mm, ② tbf≤25mm, ③ r≒Ro/2,
④ td≥b/17, ⑤ tr·br≥250tbf
Where, an each dimension uses a mm unit. Also, a
manhole radius of the diaphragm should be fixed to one
half of column outer radius. A diaphragm thickness tdmust be more thick a flange thickness tbf.
2.3. Investigating of stress equation
Eq. (1) is composed of complicate variables as
described before. Also, assumed condition of deriving
sequence is too finite. Therefore, deriving procedure of
Eq. (1) should be reviewed for normal stress term.
Generally, bending stress of curved girder is given by
Eq. (2) (Ugural and Fenster, 1995).
(2)
Where, Z is variable depending on curvature radius of
curved I-girder. Other variables are defined as Symbols.
For curved girder, variable Z is expressed as followings.
fo
Tϕ
Ar
-----–M
ϕh1
⋅
Are j⋅ ⋅
---------------–=
fi
Tϕ
Ar
-----–M
ϕh2
⋅
Are a⋅ ⋅
---------------+=
νQϕ
Ad
------–=
fTϕ
Ar
-----–M
ϕ
ArR'
---------- 1y
Z R' y+( )-------------------++=
Figure 1. Diaphragm of connections.
Figure 2. Diaphragm details of connections.
Evaluation of Diaphragm Effect for Steel Welded Box Beam and Circular Column Connections 191
(3)
Eccentricity e of Fig. 1 is derived as Eq. (4) from
condition that right term of Eq. (2) equals zero.
(4)
Eq. (4) is rearranged for reciprocal of Z. Eq. (5) is
obtained.
(5)
As Substituting Eq. (5) to Eq. (2), Eq. (6) is obtained.
(6)
When, R' approximate to infinity, Eq. (7) is gained.
(7)
For outer fiber of upper flange Eq. (8) is obtained due
to y=h1 and R+h1=j.
(8)
Precisely, Eq. (8) is same equation with Eq. (1-a),
condition is irrational considering R' is from 1.0 m
to 5.0 m in usual practice. Also, connection diaphragm
sections are excessively decided by using empirical
design method. Excessive diaphragm thickness leads to
abrupt collapse of adjacent member in ultimate state.
Then, beam, column, and diaphragm must maintain
strength balance.
3. Diaphragm Analysis Model
3.1. Outline of analysis model
In this study, to investigate influence of diaphragm
design parameters, 4 analysis groups are produced. They
are LD, DH, DR, and DHR models.
A diaphragm location was moved to the downward
direction by the ld in LD model which is be produced to
examine a strength behavior with a diaphragm location.
LD model is same dimension with specimen NC-50. LD-
00 model is same dimension with NC-50 specimen. LD
model’s details are expressed as shown in Table 1. Where
main parameter of LD model is moved distance-to-
thickness ratio ld over td. The ld was changed based on a
diaphragm thickness td from 0mm to 24mm. At this case,
ld/td is changed from 0.0 to 2.0.
DH model was produced to examine a connection
strength behavior according to Db. DH model which
changes diaphragm web depth-to-radius ratio Db/R. into
0.0, 0.13, 0.25, 0.5, 0.75, and 1.0 as shown in Table 2.
Reinforcing rib is not installed in DH model at
connection diaphragm. Central angle α is established
from 40o to 70o. Also, DH model’s last term of notation
means Db/R values.
DR model is introduced to investigate reinforcing rib
Z1
Ar
-----–y
R' y+----------- Ad
A∫=
eZ
Z 1+----------– R'=
1
Z---
R' e+
e-----------–=
fTϕ
Ar
-----–M
ϕ
Ar
-------1
R'----
1
e---
1
R'----+⎝ ⎠
⎛ ⎞ y
R' y+( )---------------++=
fTϕ
Ar
-----–M
ϕ
Ar
-------1
e---⎝ ⎠⎛ ⎞
–y
R' y+( )---------------+=
fo
Tϕ
Ar
-----–M
ϕh1
⋅A
re j⋅ ⋅
---------------–=
R' ∞≅
Table 1. Member forces in any point
Point Forces Stress Equation
In Case ofϕ=α
Axial Force
Bending Moment
Shear Force
In Case of.ϕ=π/2
Axial Force
Bending Moment
Shear Force
In Case ofϕ=π
Axial Force
Bending Moment
Shear Force
Tϕ
To
αcosF
i1
2------- α
απ--- 1–⎝ ⎠⎛ ⎞
sin–=
Mϕ
Mo
R' To1 αcos–( )
Fi1
2-------
2λµ------ 1 αcos–( ) 1
απ---–
3
4---λ–
1
4---λk+⎝ ⎠
⎛ ⎞ αsin+⟨ ⟩–+=
Qϕ
To
αsinF
i1
2-------
1
π--- αsin α αcos–( ) αcos+⟨ ⟩–=
Tϕ
Fi1
4-------=
Mϕ
Mo
ToR' F
i1R'
λπ---
1
4---
λ8--- 3 k–( ) αsin–+⟨ ⟩+ +=
Qϕ
To
Fi1
2π( )⁄+=
Tϕ
T–o
=
Mϕ
Mo
2ToR' F
i1R'
λπ---
1
4---
λ8--- 3 k–( ) αsin–+⟨ ⟩–+=
Qϕ
0=
192 Young-Pil Kim and Won-Sup Hwang
effect. As shown in Table 3, the width-to-radius ratio br/
R of DR model is changed form 0.25 to 0.75. In current
design specifications (KHBDC, 2005), Db/R od DR
model is fixed to 0.5 because a manhole radius of the
diaphragm should be reserve also the half of a column
outer radius. Also, diaphragm thickness td of DR model is
changed into 12, 15, 18, and 21 mm.
DHR model is similar to DR model. In notation of
DHR model, before hyphen term means Db/R values. Db/
R is defined as 0.25 or 0.75. Also, DR09 and DR12
models mean that each diaphragm thickness td is 9,
12 mm respectively. After hyphen term is central angles
of connections. Last term of notation is value of br/R. br/
R value of DH model and DR model is changed from
0.25 to 0.75.
3.2. Analysis method
In this study, boundary condition of analysis model is
Table 2. Dimensions of LD model
LD modelα
(deg)R
(mm)tbf
(mm)tc
(mm)ld
(mm)tw
(mm)b
(mm)db
(mm)ld/td
LD-00LD-05LD-10LD-15LD-20
50 12 12 12
06121824
12 450 488
0.00.51.01.52.0
Table 3. Dimensions of DH model
DH modelα
(deg)R
(mm)tc
(mm)tbf
(mm)tw
(mm)td
(mm)db
(mm)L
(mm)b
(mm)Db/R br/R
DH12-40series
DH12-4000DH12-4013DH12-4025DH12-4050DH12-4075DH12-4010
40 294 12 12 12 12 448 2000 380
0.000.130.250.500.751.00
0.0
DH09-50series
DH09-5000DH09-5013DH09-5025DH09-5050DH09-5075DH09-5010
50 294 12 12 12 9 448 2000 450
0.000.130.250.500.751.00
0.0
DH12-50series
DH12-5000DH12-5013DH12-5025DH12-5050DH12-5075DH12-5010
50 294 12 12 12 12 448 2000 450
0.000.130.250.500.751.00
0.0
DH12-60 series
DH12-6000DH12-6025DH12-6050DH12-6075DH12-6010
60 294 12 12 12 12 448 2000 510
0.000.250.500.751.00
0.0
DH12-70series
DH12-7000DH12-7025DH12-7050DH12-7075DH12-7010
70 294 12 12 12 12 448 2000 550
0.000.250.500.751.00
0.0
Evaluation of Diaphragm Effect for Steel Welded Box Beam and Circular Column Connections 193
given by Fig. 3(b) based on one of test setup as shown in
Fig. 3(a). Concentrated load P is loaded to the upper roller.
Table 4. Dimensions of reinforcing rib model
DR modelα
(deg)R
(mm)tc
(mm)tbf
(mm)tw
(mm)td
(mm)db
(mm)L
(mm)b
(mm)Db/R br/R
DR09 series
DR09-5025DR09-5050DR09-5075
50 294 12 12 12 9 448 2000 450 0.50.250.500.75
DR12 series
DR12-5025DR12-5050DR12-5075
50 294 12 12 12 12 448 2000 450 0.50.250.500.75
DR15 series
DR15-5025DR15-5050DR15-5075
50 294 12 12 12 15 448 2000 450 0.50.250.500.75
DR18 series
DR18-5025DR18-5050DR18-4075
50 294 12 12 12 18 448 2000 450 0.50.250.500.75
DHR0925series
DHR0925-5025DHR0925-5050DHR0925-5075
50 294 12 12 12 9 448 2000 450 0.250.250.500.75
DHR1225series
DHR1225-5025DHR1225-5050DHR1225-5075
50 294 12 12 12 12 448 2000 450 0.250.250.500.75
DHR0975series
DHR0975-5025DHR0975-5050DHR0975-5075
50 294 12 12 12 9 448 2000 450 0.750.250.500.75
DHR1275series
DHR1275-5025DHR1275-5050DHR1275-5075
50 294 12 12 12 12 448 2000 450 0.750.250.500.75
Figure 3. Boundary condition of test and anlaysis.
Figure 4. Residual stress distribution.
194 Young-Pil Kim and Won-Sup Hwang
Nishimura (1998)’s material constitutive laws were
applied to non-linear F.E.M analysis in this study. Initial
imperfections such as initial deflection and residual stress
are considered. Specially, residual stress of connection
member is applied to box beam and circular column as
shown in Fig. 4(a) and Fig. 4(b). Compression residual
stress is assumed by (-)0.4 fy or (-)0.5fy. Tensile residual
stress is assumed by (+)1.0fy. Figure 5 shows diaphragm
mesh division of diaphragm model and reinforcing rib
model case. Also, diaphragm web and rib details are
shown in Fig. 6.
3.3. Verification of analysis results
To verify analysis method, analysis results of NC-40,
NC-50, NC-60, and NC-70 model compare with test
result. Table 5 shows dimension of test specimens.
Where, notation of test model is indicated under Table 5.
Table 6 shows mechanical properties of materials. SS400
steel class is applied to analysis and test model. Elastic
modulus E is 1.99×106 MPa and yield strength fy is 286
MPa. Figure 7 shows comparison of between analysis
results and test results. P-δ relationship of analysis agree
well with one of test less than about 100 mm displacement
that bursting of welding part is occurred. After bursting of
welding part, connection strength is reduced rapidly in
test results. However, two results have good agreement in
view of yield load. Therefore, it is judged that propriety
of non-linear FEM analysis procedure is ensured in this
study.
Figure 5. Mesh division of analysis model.
Figure 6. Analysis model detail of connections.
Figure 7. Comparison of analysis results and test Results.
Table 5. Dimensions of connection specimens
Modelá
(deg.)R
(mm)tc
(mm)tfb
(mm)tw
(mm)db
(mm)d2
(mm)L
(mm)B
(mm)
NC-40NC-50NC-60NC-70
40516069
290286292293
11.8311.9111.8611.88
11.8311.9111.8611.88
11.8311.9111.8611.88
485484486488
444359293212
1997199419951995
377445504547
Table 6. Mechanical properties of materials
fy (MPa) fu (MPa) E (×105 MPa) fy/fu Steel Class
286 464 1.9992 0.62 SS400
Evaluation of Diaphragm Effect for Steel Welded Box Beam and Circular Column Connections 195
4. Investigation of Design Parameter
4.1. Influence of diaphragm location
Figure 8 shows strength ratio and displacement relationship.
The strength ratio is defined by P-to-Po(NC-50) ratio. Where
Po(NC-50) means yield load of test specimen NC-50 (Lee
and Lu, 1989). As ld increasing, strength ratio decrease.
When ld is 24 mm, strength ratio reduced about 20%.
Strength of NC-50 model is located between strength of
LD-15 and LD-20. Therefore, NC-50 could be analogized
to relocation of diaphragm about 20 mm. At this case, ld/
td equals 1.67. Figure 9 shows ld/td and yield strength ratio
P-to-Po(NC-50) relationship. Connection strength is linearly
increased with increment of ld/td. When ld/td is 2.0, 20%
strength reduction is occurred comparing with LD-00.
4.2. Influence of diaphragm web depth
Figures 10 and 11 show strength ratio DH model with
diaphragm web depth-to-radius ratio Db/R. Strength ratio
Po/Po(NC-40) is defined by Po-to-Po(NC-40) ratio. However,
Po(NC-50) value is selected as yield load of DH-50100
model instead of NC-50 model yield strength due to
diaphragm location effect. As shown in Figure 10,
strength of NC-40 agrees with one of connection which
diaphragm is not installed. Also, Strength ratio Po/Po(NC-40)
of connection model which Db/R is 0.75 is obtained
sufficiently comparing NC-40 model. However, Po/Po(NC-
40) is reduces gradually when Db/R is smaller than 0.75.
Then, strength ratio Po/Po(NC-40) of DH12-4000 model
becomes 0.45. Similarly, Db/R effect of 50o models has
almost same trend comparing 40o models. Specially,
strength ratio Po/Po(NC-50) of DH12-5000 model becomes
0.60. strength ratio shows upward tendency with increase
of central angle due to reinforcement effect of beam web.
Therefore, Figure 12 shows Db/R vs strength ratio with
central angle. Strength ratio develop gradually with
increase of central angle. However, strength ratio of
connection is enough to use without reinforcement when
Db/R is more than 0.75.
4.3. Influence of Influence of rib
DR12 model is br/R is changed as diaphragm depth Db
is fixed to 0.5R and td=tbf. In Db/R=0.5, strength ratio of
rib installed connection is 0.96. strength ratio of non-rib
Figure 8. P-δ relationship of LD model.
Figure 9. Yield load ratio with diaphragm location.
Figure 10. Influence of diaphragm web depth (α=40o).
Figure 11. Influence of diaphragm web depth (α=50o).
196 Young-Pil Kim and Won-Sup Hwang
installed connection is 0.86. Then, strength ratio of DR12
models is smaller than one of DH12-50100 which
diaphragm manhole is not installed as shown in Figure
13. However, increasing of yield strength is not occurred
with increase br/R as shown in Figure 14. Figure 15
shows strength ratio with diaphragm web thickness. The
diaphragm thickness td is thicker, the connection strength
is stronger. Diaphragm web thickness td must be increased
instead of reinforcing rib width br to develop diaphragm
strength.
5. Design Equation of connection Diaphragm
Okumura’s diaphragm stress equation based on curved
girder model is very complicated and difficult to apply
usual practice. Also, strength of diaphragm web and rib is
unreasonably evaluated. Then, more reasonable design
method of connection diaphragm is required. This study
introduces moment inertia of diaphragm from circular
column axis to consider diaphragm rigidity. Id/I could be
introduced by using diaphragm rigidity Id in connection
which manhole is not installed. Also, connection strength
is expressed by P/Po(DH-50100), Where Po(DH-50100) is yield
strength of connection which manhole is not installed.
Analysis results of 50o models are selected based on
central angle range of design specification, i.e. 45-55o.
Analysis results are rearranged by Id/I and P/Po(DH-50100)
relationship as shown Fig. 16. As Id/I is more than 1.0,
connection yield strength ratio is reduced rapidly.
However, connection yield strength ratio is more than 1.0
when Id/I is smaller than 1.0. This trend is expressed as
followings.
for
(9)
for
Id/I 1.0 condition should be confirmed to maintain
diaphragm strength because strength ratio decrease
remarkably as Id/I is more than 1.0. When Id/I equals 1.0,
P
Po
----- 1.0= IdI⁄ 1.0≥
1.0
IdI⁄( )
1.25-------------------= I
dI⁄ 1.0<
Figure 12. Db/R and strength ratio relationship.
Figure 13. Influence of reinforcing rib width (DR12).
Figure 14. Influence of reinforcing rib width (DR21).
Figure 15. Strength ratio with diaphragm web thickness.
Evaluation of Diaphragm Effect for Steel Welded Box Beam and Circular Column Connections 197
Required diaphragm thickness treq with Db/R is obtained
by numerical solution as shown in Fig. 17. When Db/
R=1.0, treq/td=1.0. treq/td equals 1.1 at Db/R=0.5. Also,
treq/td increases by geometric progression when Db/R is
smaller than 0.25. It is recommended that Db/R of
connection diaphragm should be more than 0.25. Also,
numerical solution such as Fig. 17 is difficult to apply
practical usage. Then, treq is suggested by function of
fourth order Db/R as followings.
(10)
Figure 18 shows comparison of numerical solution and
Eq. (10) that is proposed. Eq. (10) agrees well with
numerical solution of Db/R versus treq/td. Where Eq.(10) is
rearranged by considering flange yielding condition td≥tbf
.
(11)
As a result, required diaphragm thickness-to-flange
thickness ratio treq/tbf could be expressed by fourth order
Db/R as shown in Eq. (11). Required diaphragm thickness
treq could be calculated from Eq. (11) with diaphragm web
depth Db. It is simpler than Okumura’s stress equation.
Also, more reasonable design of connection diaphragm
could be possible.
6. Conclusions
This paper investigates problems of existing design
specification and proposes a newly diaphragm design
method in order to develop the design method of steel
piers. From this study, the following conclusions can be
drawn:
(1) Connection strength reduces linearly with increase
of diaphragm moved length ld. When ld equals two times
of diaphragm thickness td, connection strength reduced by
20%.
(2) Diaphragm web depth Db should be designed over
0.75R in case of diaphragm model in order to confirm
connection strength.
(3) Variation of connection strength is negligible with
the reinforcing rib width.
(4) In view of connection strength, It is judged that
diaphragm web depth is the most important variable.
(5) Equation of required diaphragm web thickness treq
is proposed based on diaphragm web depth Db.
Acknowledgments
This work is a part of a research project supported by
Korea Ministry of Construction & Transportation (MOCT)
through Korea Bridge Design & Engineering Research
Center at Seoul National University. The authors wish to
express their gratitude for the financial support.
References
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treq
td
------- 1.8Db
R------ 1–⎝ ⎠⎛ ⎞
4
1Db
R------+ 0.25≥ ≥
treq
tbf
------- 1.8Db
R------ 1–⎝ ⎠⎛ ⎞
4
1Db
R------+ 0.25≥ ≥
Figure 16. Relationship of diaphragm rigidity capacityratio versus strength ratio.
Figure 17. Relationship of Db/R versus treq/td.
Figure 18. Comparison between treq/td and Eq. (10).
198 Young-Pil Kim and Won-Sup Hwang
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Symbols
Stresses
fo: external fiber normal stress
fi: internal fiber normal stress
v: shear stress
Forces
Tϕ: axial force in any point.
Mϕ: bending moment in any point.
Qϕ: shear force in any point.
To: indeterminate axial force
Mo: indeterminate bending moment
Structural dimension
br: reinforcing rib width
Dd: diaphragm web depth
dd: net diaphragm web depth
td: diaphragm web thickness
tc: circular column thickness
tr: diaphragm rib thickness
tr: diaphragm rib thickness
R: radius of circular column
R': distance between axis of circular column and
neutral axis of diaphragm
Ro: external radius of circular column
Sectional property
b3: effective width of circular column (=td+1.56 )
r: distance between the central axis of circular
column and neutral axis of curved girder
(=Ar/[brln(d/a)+tdln(g/d)+b3ln(j/g)])
Ar: sectional area of diaphragm (=brtr+tddd+b3tc)
Ad: sectional area of diaphragm web (=tddd)
e: eccentricity (=R'−r)
k: non-dimensional variable considering shear lag
stress (=(2fi1−fs1)/(2fi1))
λ: ratio of j and R' (=R/R'=j/R')
Id: moment inertia of connection diaphragm for the
central axis of circular column. Where, diaphragm
has not manhole like test specimen.
I: moment inertia of connection diaphragm for the
central axis of circular column. Where, manhole is
installed in diaphragm. Reinforcing rib is neglected
in calculation of moment inertia)
Rtc