Download Force Distribution of the System

Chapter 7
Force Distribution in the Structural System
This chapter presents the force transfer mechanisms of the specimen, and
calculates the proportions of the lateral force and overturning moment resisted by major
components of the structural system.
7.1 Lateral Force Distribution
Figure 7.1.1 shows a schematic diagram of the idealized lateral force transfer
paths in the composite steel frame-RC infill wall specimen. It can be seen that the lateral
load is transferred to the base through three mechanisms: shear stud–infill wall
interaction; diagonal compression struts due to the interaction between the steel frame
and infill walls; and deformation of the steel frame alone. The percentage of the lateral
force resisted by each mechanism is determined in this chapter for the purpose of
evaluating the relative importance of these mechanisms at various loading stages.
7.1.1 Lateral Force Transferred by the Headed Studs
The lateral force along the cross sections at interfaces between the middle beam
and the infill wall (sections A-A and B-B in Figure 7.1.1) was resisted by the steel
column, the seat or top angles and the beam headed studs along the interfaces. The lateral
force along the cross section at the bottom interface of the first story (section D-D in
188
P
Vstud
Pstrut
Vstud
A
B
A
B
Vstud
Pstrut
C
C
Vstud
D
D
Figure 7.1.1 Lateral Shear Force Transfer Mechanism of the Specimen
Vn2
Vs2
Pstrut
Vnta
Vna
Vstud_a
Vsta
Tsw
Tnw
Vnb
Vsa
Vnba
Vstud_b
Vsb
Vsba
Pstrut
Vn1
Vs1
Figure 7.1.2 Lateral shear Force Transfer along Sections A-A and B-B
Figure 7.1.1) was resisted by the steel column and the beam headed studs since there
were no top angles there. Figure 7.1.2 shows a schematic diagram of the lateral force
distribution along section A-A and section B-B on both sides of the middle beam. The
189
Possible friction force due to the bearing between the steel beams and the infill wall in
the corner region is neglected. It can be seen that the portion of lateral force transferred
through the studs along the top interface of the beam most likely flowed into the studs
along the bottom interface of the beam directly. The portion of the lateral shear force
transferred through the compression strut was divided into two components: one that
flowed into the panel zone region through shear in the column, and the other that flowed
into the middle beam through tension in the top angle or seat angle of the PR connection.
An equilibrium equation for lateral force is established along section A-A:
P  Vstud _ a  Vna  Vsa  Vnta  Vsta
(7.1.1)
where
P = total applied lateral force, kips
Vstud_a = lateral force resisted by the headed studs along A-A section, kips
Vna = shear force at the bottom of the north column in the second story, kips
Vsa = shear force at the bottom of the south column in the second story, kips
Vnta = axial force in the horizontal leg of the top angle of the north connection,
kips
Vsta = axial force in the horizontal leg of the top angle of the south connection,
kips
As discussed in Section 5.2, the shear forces at the bottom of the steel columns in
the second story, Vna and Vsa, were estimated using measured shear strains at the center of
the column web (see Appendix C). Tables 5.1.2 and 5.1.3 show these two shear forces at
the peak lateral load in the south and north directions, respectively. The axial forces in
the horizontal legs of the top angles, Vnta and Vsta, were assumed to equal the axial forces
in the top flanges of the middle beam and were computed according to Eq. (5.1.7). Table
5.1.4 shows these forces at the peak lateral load in both directions. Therefore, the lateral
force resisted by the interface studs along the middle beam Vstud_a can be written as:
Vstud _ a  P  (Vna  Vsa  Vnta  Vsta )
(7.1.2)
190
The same procedure can be used to calculate the lateral force resisted by the
headed studs along the bottom interface of the middle beam (section B-B), Vstud_b and the
lateral force resisted by the headed studs along the bottom interface of the first story
(section D-D).
Figures 7.1.3 and 7.1.4 show the percentage of the lateral force resisted by the
interface studs along sections A-A, B-B, and D-D, at the peak load of the first cycle of
each group of cycles, but is unrecoverable after the column at the corresponding section
yielded. It can be observed that, initially, the headed studs along the bottom interface of
the second story shared the highest percentage of the lateral load among three interfaces
(approximately 91% for the studs along the bottom interface of the second story, 88%
along the top interface of the first story, and 83% along the bottom interface of the first
story). The lateral force resisted by the studs along the bottom interface of the first story
dropped by 4-5% in next two groups of cycles. The percentage of the lateral force
resisted by the studs along the top interface of the first story also dropped slightly during
the next two groups of cycles, but was maintained at about 80% for loading cycle G4-1
and G5-1. Figures 7.1.3 and 7.1.4 shows that the percentage of the lateral force resisted
by the studs along the bottom interface of the second story dropped a little from cycle
G1-1 to cycle G3-1, but quickly dropped by approximately 30% from the cycle G3-1 to
G4-1.
To further evaluate the strength of the headed studs as a group, Figure 7.1.5
shows a plot of the lateral force resisted by the studs versus the average slip along the top
interface of the first story at the peak lateral load during the first cycle of each group of
cycles. Figure 7.1.6 shows a plot of the lateral force resisted by the studs versus the
average slip along the bottom interface of the second story. The curve linking these peak
values represents a possible envelope of the hysteretic shear force–slip relationship of the
interface headed studs as a group. Figure 7.1.5 shows that, along the envelope, the shear
stiffness of the stud group along the top interface of the first story decreased gradually
before the average slip reached approximately 0.015 inches in cycle G3-1. The stud
group then started to quickly lose its shear stiffness, approaching zero at 0.05 inches of
191
Lateral Force Resisted by Studs (%)
100
1
1
1
2
80
2
4
3
3
2
3
5
60
4
40
20
0
Bottom interface
of story 1
Top interface
of story 1
Bottom interface
of story 2
* 1 – G1-1-A; 2 – G2-1-A; 3 – G3-1-A; 4 – G4-1-A; 5 – G5-1-A
Fig. 7.1.3 Percentage of the Lateral Load Resisted by Interface Studs
at Peak Load in the South Direction
Lateral Force Resisted by Studs (%)
100
1
1
1
80
2
2
3
3
4
2
3
5
60
4
40
20
0
Bottom interface
of story 1
Top interface
of story 1
Bottom interface
of story 2
* 1 – G1-1-C; 2 – G2-1-C; 3 – G3-1-C; 4 – G4-1-C; 5 – G5-1-C
Fig. 7.1.4 Percentage of the Lateral Load Resisted by Interface Studs
at Peak Load in the North Direction
192
Lateral Load Resisted by Studs (kips)
140
G4-1
120
G5-1
100
G3-1
80
G2-1
60
40
G1-1
Loaded in South
20
Loaded in North
0
0.00
0.01
0.02
0.03
0.04
0.05
0.06
Average Slip (inches)
Lateral Load Resisted by Studs (kips)
Fig. 7.1.5 Lateral Load Resisted by Studs versus Average Slip
along the Top Interface of the First Story
140
120
G3-1
G3-2
100
80
G2-1
G3-3
G4-1
60
40
G1-1
Loaded in South
20
Loaded in North
0
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Average Slip (inches)
Fig. 7.1.6 Lateral Load Resisted by Studs versus Average Slip
along the Bottom Interface of the Second Story
193
slip. Therefore, it is reasonable to designate the maximum strength of the stud group
along the top interface of the first story as approximately 130 kips, approximately 1.85
times the design strength of the stud group. Figure 7.1.6 shows that, along the envelope,
the shear stiffness of the stud group along the bottom interface of the second story was
close to that of the stud group along the top interface of the first story until cycle G3-1.
However, the maximum shear strength of the stud group there was achieved at cycle
G3-1 at approximately 100 kips. The shear strength then dropped about 25% during the
next two loading cycles, indicating that some of the studs along this interface failed
during these loading cycles. There was little decrease of the shear strength during the
next cycle, G4-1, although the average slip increased by more than 100% to reach 0.1
inches.
7.1.2 Lateral Force Transferred by the Compression Strut
Figure 5.1.14 shows that, when the specimen was loaded in the south direction,
the horizontal component of the compressive strut along the up-north interface of the first
story can be written as
Fc  Vnb  Vnsa  Tstud  Vnd
(5.1.3)
As discussed in Section 5.1.2, the shear force at the bottom of the north column and
tensile force of the studs along the bottom portion of the north column were negligible.
Therefore, when the specimen was loaded in the south direction, the shear force resisted
by the compressive strut was approximately equal to the sum of the shear at the top of the
north column in the first story, Vnb, and the axial force in the horizontal leg of the seat
angle, Vnsa. Similarly, the lateral force resisted by the compressive strut in the infill wall
of the first story should also approximately equal the shear force at the bottom of the
south column, Vsd, when the specimen was loaded in the south direction. As a result, it is
reasonable to assume that the average of (Vnb+Vnsa) and Vsd is the approximate lateral
load resisted by the diagonal strut in the first story infill wall when the specimen is loaded
in the south direction. Similarly, the average value of sum of the shear at the top of the
south column and in the first story and the axial force in the horizontal leg of the seat
194
angle (Vsb+Vssa), and shear at the bottom of the north column, Vsd, is the approximate
lateral load resisted by the diagonal strut in the first story infill wall when the specimen
was loaded in the north direction. There is no average for the second story since only the
forces along the bottom of the second story were obtained. Figure 7.1.7 and Figure 7.1.8
show the percentage of the lateral load resisted by the diagonal strut in the infill wall of
each story. In the first story, this percentage increased steadily from 10% in the first cycle
to 25% when the specimen reached its maximum strength. In the second story, this
percentage increased rapidly to 50% after cycle G3-1, indicating the diagonal strut
become one of the major mechanisms to transfer the lateral load. As stated above, this is
likely due to the loss of shear strength of the stud group along the bottom interface of the
second story.
7.1.3 Lateral Force Resisted by the “Shear” Deformation of Steel Frame
Figures 7.1.3, 7.1.4, 7.1.7 and 7.1.8 show that, along sections at steel beam-infill
wall interface, the headed studs and the compression strut transferred the majority of the
lateral force. The steel frame was also gaged in the middle of the steel columns of the
first story (section C-C in Figure 7.1.1). The result indicated that the steel columns only
resisted 1-3% of the total lateral force along section C-C. It can be concluded that the
lateral force resisted by the “shear” deformation of steel frame is negligible before the
crushing of concrete occurred.
After the peak load, as the concrete was crushed and the compression strut
deteriorated in the second story, both steel columns in the second story started to resist
the shear force through the shear deformation of steel frame. After the 1.25% cycles, the
maximum lateral load was stabilized to be 70 to 80 kips, which was approximately equal
to the sum of the nominal shear strengths of the two steel columns, 2(0.6FywAw)=83 kips.
7.2. Overturning Moment Distribution
Since the axial force and bending moment of each column were obtained in
Section 5.2, it is feasible to calculate the overturning moment resisted by the steel
195
Lateral Load Resisted
by Diagonal Strut in Infill Wall (%)
100
80
60
4
40
20
1
2
3
4
5
3
1
2
0
Story 1
Story 2
Fig. 7.1.7 Percentage of the Lateral Load Resisted by Diagonal Strut
in Infill Wall at Peak Load in the South Direction
Lateral Load Resisted
by Diagonal Strut in Infill Wall (%)
100
80
60
4
40
4
2
20
5
3
2
1
3
1
0
Story 2
Story 1
Fig. 7.1.8 Percentage of the Lateral Load Resisted by Diagonal Strut
in Infill Wall at Peak Load in the North Direction
196
columns if the neutral axis of each cross section of the entire specimen can be
determined. It is difficult to determine the location of the neutral axis of sections A-A and
B-B (Figure 7.1.1) because of the complicated deformation pattern of the middle beam.
However, for sections C-C and D-D, it is reasonable to assume that the deformation of
these gaged sections is planar, as shown in Figure 7.2.1. As a result, three equations were
established to solve the distances xnc and xsc:
ε nc 
N nc
EAc
(7.2.1)
ε sc 
N sc
EAc
(7.2.2)
xnc  xsc  Lc
(7.2.3)
where
Nnc = axial force in the north column, kips
Nsc = axial force in the south column, kips
nc = axial strain at the center of the north column
sc = axial strain at the center of the south column
Ac = cross section area of the column, inch2
E = modulus of elasticity of the steel column, ksi
xnc = distance from the neutral axis to the centerline of the north column, inches
xsc = distance from the neutral axis to the centerline of the south column, inches
Lc = distance between the centerlines of the two columns, inches
nc
xnc
xsc
Figure 7.2.1 Section Deformation
197
sc
The resulting xnc and xsc are:
xnc 
N nc
Lc
N nc  N sc
(7.2.4)
xsc 
N sc
Lc
N nc  N sc
(7.2.5)
Therefore, the overturning moment resisted by the steel columns was
M sct  N nc xnc  N sc xsc  M nc  M sc
(7.2.6)
Substituting Eq. (7.2.4) and Eq. (7.2.5) into Eq. (7.2.6) yields
M sct
( N nc2  N sc2 )

* Lsc  M nc  M sc
N nc  N sc
where
Mnc = bending moment in the north steel column, kip-inches
Msc = bending moment in the south steel column, kip-inches
The percentages of the overturning moment resisted by combined bending
moment and axial force in the steel columns are shown in Figures 7.2.2 and 7.2.3. These
figures show that, at the bottom of the first story, the percentage of overturning moment
resisted by steel columns was approximately 80%. In the middle of the first story, the
percentage of overturning moment resisted by steel columns varied from about 85% in
cycle G1-1 to nearly 100% in cycle G5-1. It is not a surprise that this calculated
percentage was higher in the middle of the first story than in the bottom, because in
determining the neutral axis of section C-C, the interface slip between the steel columns
and the infill wall was neglected. As a result, the location of the neutral axis was moved
to the compression side of section C-C and the overturning moment was magnified a
little. Therefore, it is reasonable to conclude that in the early loading cycles, the
overturning moment resisted by the steel columns was approximately 80%, with the
remainder resisted by the infill wall.
198
Overturning Moment Resisted
by Steel Columns (%)
120
5
100
1
80
1
2
2
4
3
3
60
40
20
0
Bottom of
Story 1
Middle of
Story 1
* 1 – G1-1-A; 2 – G2-1-A; 3 – G3-1-A; 4 – G4-1-A; 5 – G5-1-A
Figure 7.2.2 Percentage of the Overturning Moment Resisted by Steel Columns
at the Peak Load in the South Direction
Overturning Moment Resisted
by Steel Columns %
120
5
100
80
1
2
1
3
2
3
4
60
40
20
0
Bottom of
Story 1
Middle of
Story 1
Top of
Story 1
Bottom of
Story 2
* 1 – G1-1-C; 2 – G2-1-C; 3 – G3-1-C; 4 – G4-1-C; 5 – G5-1-C
Figure 7.2.3 Percentage of the Overturning Moment Resisted by Steel Columns
at the Peak Load in the North Direction
199
Figures 2.2.2 and 2.2.3 also show that, as the headed studs fail progressively, the
steel frame carried more and more overturning moment. When few studs remained at the
bottom interface of the second story during cycle G5-1, the frame was basically carrying
all the overturning moment.
200