Chapter 9

Plastic Mechanism Analysis

This chapter evaluates the plastic analysis mechanisms for one-story infilled frames proposed by other researchers by using experimental data obtained from the literature, and establishes a plastic mechanism for analysis of the two-story composite steel frame-reinforced concrete infill wall specimen test conducted in the present research.

9.1 Evaluation of Existing Plastic Mechanism Models

As discussed in Section 1.3.2, different approaches have been tried to predict the maximum strength of infilled steel frames subjected to lateral load since the 1960’s. One of these approaches is plastic mechanism analysis, in which the collapse modes and collapse loads for an infilled steel frame are dependent on the bending strength of the steel members and crushing stress of the infill wall material. Among all the researchers, only Liauw and Kwan (1983a, 1983b) have established plastic mechanisms for infilled steel frames both with and without interface connectors, with the backing of static lateral loading tests and finite element analysis. This section evaluates the plastic mechanisms proposed by Liauw and Kwan (1983a, 1983b) by using experimental results from cyclic loading tests on one-story, one-bay RC infilled steel frames obtained from the literature.

The plastic mechanisms defined by Liauw and Kwan (1983b) for a one-story, one-bay infilled steel frame having interface connectors are shown in Figure 9.1.1. They include the following aspects of behavior at the limit of useful response to lateral loading: 1) Corner crushing with yielding of the infill/beam connectors and plastic hinges forming at two joints and in the columns (mode 1, Figure 9.1.1.(a)); 2) Corner crushing with yielding of the infill/column connectors and plastic hinges forming at two joints and in the beams (mode 2, Figure 9.1.1.(b)); 3) Diagonal crushing with yielding of the

Figure 9.1.1 Failure Modes of a Single Story Infilled Steel Frame Having Interface Connectors [after Liauw and Kwan (1983)]

infill/beam connectors and plastic hinges forming at four joints (mode 3, Figure 9.1.1.(c)); 4) Diagonal crushing with yielding of the infill/column connectors and plastic hinges forming at four joints (mode 4, Figure 9.1.1.(d)).

These plastic mechanisms were primarily based on experimental observations of small models, having a scale approximately equal to 1:10 to 1:12. The steel frames in these test models comprised 0.87 inches ´ 0.87 inches solid square steel bars, which have different section characteristics if compared to rolled wide flange sections. Furthermore, the interface connectors were J hooks or U hooks bent from wire, instead of using headed studs. Because these parameters differ from the test conducted in this research, and also to further confirm the validity of equations of Liauw and Kwan (1983a, 1983b), these mechanisms are evaluated in the present study by using additional experimental results of larger-scale one-story specimens with headed studs as interface connectors.

The parameters of the tests used for verification are listed in Tables 9.1.1 and 9.1.2. These tests have been conducted by two groups of researchers in Japan since 1980 (Makino et al., 1980; Makino, 1984; Hayashi and Yoshinaga, 1985, 1986, 1987, 1994). These tabulated specimens are all one-story and one-bay, comprising steel members with wide flange sections and reinforced concrete infill walls. The reinforcement ratio of the RC infill walls was approximately 0.5%. The specimens have either the strong axis of the steel columns or the weak axis of the steel columns oriented in the plane of the infill wall. The stud spacing varied from approximately 4.5 inches to 12 inches. In each test, each column was subjected to a static gravity load equal to one-third of its axial yielding strength, while cyclic lateral loading was added on the top joint. In Tables 9.1.1 and 9.1.2, l = center-to-center spacing of columns; h = story height (center-to-center spacing of beams); t = thickness of the RC infill wall; fc’ = compressive strength of the concrete; Fy = yield strength of the frame steel; Qsn = nominal shear strength of a single headed stud; d = distance between adjacent studs.

The nominal shear strength of a single headed stud in these specimens was calculated according to the following formula:

(9.1.1)

where

t = wall thickness, inches

Equation (9.1.1) was established by Makino (1984) according to the shear tests on the same studs as those used in their infilled steel frame specimens. Because the studs used in these specimens were not confined by reinforcement cages, the strength reduction factor, f, was used to take into account the limited confinement provided by the concrete alone.

The analytically predicted maximum lateral loads of the four different failure modes of Figure 9.1.1 for each of the specimens are tabulated in Table 9.1.3. In the table, the error of the analytical prediction is calculated as follows:

(9.1.2)

Comparison of the results of the four failure modes with the experimental peak load shows that mode 3 is the predicted governing failure mode for all of the Japanese tests except for specimen D8, for which the mode 1 and mode 3 had almost the same maximum lateral load. The predicted failure modes were consistent with the observed modes in the tests. Figure 9.1.2 shows the error of the analytically predicted maximum lateral load for the specimens having the strong axis of the steel columns oriented in the plane of the infill wall, and Figure 9.1.3 shows the error of the analytically predicted maximum lateral load for the specimens having the weak axis of the steel columns oriented in the plane of the infill wall. Figure 9.1.2 shows that the plastic mechanisms proposed by Liauw and Kwan (1983b) overestimate the maximum lateral load of the specimens having the strong axis of the steel columns oriented in the plane of the infill wall by approximately 20% on average. Figure 9.1.3 shows that the plastic mechanisms proposed by Liauw and Kwan (1983b) underestimate the maximum lateral load of the

Table 9.1.1 Parameters for Steel Frame-RC Infill Wall Specimens from Japan

Reference / Specimen
Designation / Specimen
Number / Interface Connectors / Number of Connectors / Frame*1
section / Column*2
Orientation
Beam / Column
Makino
(1984) / A2 / 1 / Studs / 3 / 2 / a / X
B2 / 2 / Studs / 3 / 2 / a / Y
C2 / 3 / None / - / - / b / X
C4 / 4 / Studs / 3 / 2 / b / X
C6 / 5 / Studs / 5 / 3 / b / X
C8 / 6 / Studs / 3 / 2 / b / X
C9 / 7 / Studs / 3 / 2 / b / X
C10 / 8 / Studs / 9 / 5 / b / X
C11 / 9 / None / - / - / b / X
C12 / 10 / Studs / 3 / 2 / b / X
C13 / 11 / Studs / 3 / 2 / b / X
D2 / 12 / None / - / - / b / Y
D3 / 13 / Studs / 3 / 2 / b / Y
D4 / 14 / Studs / 5 / 3 / b / Y
D5 / 15 / Studs / 3 / 2 / b / Y
D6 / 16 / Studs / 9 / 5 / b / Y
D7 / 17 / None / - / - / b / Y
D8 / 18 / Studs / 3 / 2 / b / Y
Hayashi et al.
(1985) / SRCV1-2X / 19 / Welded reinforcing bar / - / - / c / X
Hayashi et al.
(1986) / SRCV1-7X / 20 / Studs / 5 / 3 / d / X
Hayashi et al.
(1987) / SRCV2-7X / 21 / Studs / 5 / 2 / d / X
SRCV1-2Y / 22 / Studs / 3 / 3 / c / Y
Hayash et al.
(1994) / SRCV2-7X / 23 / Studs / 6 / 3 / d / X

1: a = both the columns and beams are fabricated from H-4.92´4.92´0.26´0.35 (wide flange section - depth ´ width ´ web thickness ´ flange thickness, in inches)

b = both the columns and beams are fabricated from H-4.92´4.92´0.18´0.24

c = both the columns and beams are fabricated from H-3.94´3.94´0.24´0.32

d = the columns are fabricated from H-3.94´3.94´0.24´0.32, and the beams are fabricated from H-5.90´2.95´0.20´0.28

2: X = Strong axis of the steel column is oriented in the plane of the infill wall

Y = Weak axis of the steel column is oriented in the plane of the infill wall

Table 9.1.2 Parameters for Steel Frame-RC Infill Wall Specimens from Japan

Specimen
Designation / Specimen
Number / Frame Dimension (inches) / fc’
(ksi) / Fy (ksi) / Qsn
(kips) / d
(inches) / Mp(kip-ft)
l / h / t / Column / Beam / Column / Beam
A2 / 1 / 47.2 / 39.4 / 2.4 / 1.9 / 46.7 / 46.7 / 3.1 / 11.8 / 35.1 / 35.1
B2 / 2 / 47.2 / 39.4 / 2.4 / 2.6 / 46.7 / 46.7 / 4.0 / 11.8 / 16.7 / 35.1
C2 / 3 / 47.2 / 39.4 / 2.4 / 3.1 / 46.7 / 46.7 / - / - / 24.9 / 24.9
C4 / 4 / 47.2 / 39.4 / 2.4 / 2.6 / 46.7 / 46.7 / 4.0 / 11.8 / 24.9 / 24.9
C6 / 5 / 47.2 / 39.4 / 2.4 / 2.4 / 46.7 / 46.7 / 3.8 / 7.9 / 24.9 / 24.9
C8 / 6 / 47.2 / 39.4 / 2.4 / 3.3 / 46.7 / 46.7 / 4.7 / 11.8 / 24.9 / 24.9
C9 / 7 / 47.2 / 39.4 / 2.4 / 2.8 / 46.7 / 46.7 / 4.2 / 11.8 / 24.9 / 24.9
C10 / 8 / 47.2 / 39.4 / 2.4 / 3.2 / 46.7 / 46.7 / 1.0 / 4.7 / 24.9 / 24.9
C11 / 9 / 47.2 / 39.4 / 2.4 / 3.3 / 46.7 / 46.7 / - / - / 24.9 / 24.9
C12 / 10 / 47.2 / 39.4 / 2.4 / 3.2 / 46.7 / 46.7 / 4.6 / 11.8 / 24.9 / 24.9
C13 / 11 / 47.2 / 39.4 / 2.4 / 3.1 / 46.7 / 46.7 / 5.9 / 11.8 / 24.9 / 24.9
D2 / 12 / 47.2 / 39.4 / 2.4 / 2.3 / 46.7 / 46.7 / - / - / 11.4 / 24.9
D3 / 13 / 47.2 / 39.4 / 2.4 / 2.3 / 46.7 / 46.7 / 3.6 / 11.8 / 11.4 / 24.9
D4 / 14 / 47.2 / 39.4 / 2.4 / 2.3 / 46.7 / 46.7 / 3.6 / 7.9 / 11.4 / 24.9
D5 / 15 / 47.2 / 39.4 / 2.4 / 2.6 / 46.7 / 46.7 / 4.0 / 11.8 / 11.4 / 24.9
D6 / 16 / 47.2 / 39.4 / 2.4 / 2.7 / 46.7 / 46.7 / 0.9 / 4.7 / 11.4 / 24.9
D7 / 17 / 47.2 / 39.4 / 2.4 / 2.7 / 46.7 / 46.7 / - / - / 11.4 / 24.9
D8 / 18 / 47.2 / 39.4 / 3.1 / 2.7 / 46.7 / 46.7 / 4.7 / 11.8 / 11.4 / 24.9
SRCV1-2X / 19 / 43.3 / 43.3 / 2.0 / 3.7 / 47.1 / 47.1 / - / - / 20.5 / 20.5
SRCV1-7X / 20 / 63.0 / 45.3 / 2.0 / 2.7 / 52.6 / 47.6 / 3.7 / 9.8 / 22.9 / 24.1
SRCV2-7X / 21 / 63.0 / 45.3 / 2.0 / 3.5 / 46.0 / 52.5 / 4.6 / 9.8 / 20.0 / 26.5
SRCV1-2Y / 22 / 43.3 / 43.3 / 2.0 / 2.4 / 50.6 / 50.6 / 3.4 / 9.8 / 10.7 / 22.0
SRCV2-7X / 23 / 82.6 / 45.3 / 2.0 / 3.7 / 50.7 / 45.7 / 4.8 / 11.2 / 22.0 / 23.1

Table 9.1.3 Comparison of the Analytical Maximum Lateral Load with the Test Results

Specimen
Designation / Specimen
Number / Maximum Lateral Load (kips) / Error (%)
Plastic Analysis Theory / Test
Mode 1 / Mode 2 / Mode 3 / Mode 4 / Minimum
A2 / 1 / 93.3 / 109.5 / 78.9 / 90.5 / 78.9 / 76.8 / 2.7
B2*1 / 2 / 79.6 / 112.9 / 70.0 / 86.4 / 70.0 / 69.0 / 1.5
C2 / 3 / 94.2 / 113.0 / 79.0 / - / 79.0 / 65.4 / 20.8
C4 / 4 / 95.9 / 111.8 / 80.8 / 97.4 / 80.8 / 72.6 / 11.3
C6 / 5 / 96.3 / 111.0 / 81.5 / 96.2 / 81.5 / 78.0 / 4.5
C8 / 6 / 107.4 / 125.1 / 92.3 / 113.1 / 92.3 / 69.7 / 32.4
C9 / 7 / 99.1 / 115.5 / 83.9 / 101.7 / 83.9 / 69.9 / 20.0
C10 / 8 / 100.2 / 118.2 / 85.0 / 105.6 / 85.0 / 77.8 / 9.2
C11 / 9 / 97.0 / 116.3 / 81.9 / - / 81.9 / 72.2 / 13.4
C12 / 10 / 105.1 / 122.5 / 90.0 / 109.9 / 90.0 / 69.0 / 30.4
C13 / 11 / 135.4 / 157.7 / 125.2 / 158.8 / 125.2 / 100.5 / 24.6
D2*1 / 12 / 54.0 / 81.7 / 48.8 / - / 48.8 / 65.6 / -25.6
D3*1 / 13 / 63.0 / 89.8 / 57.9 / 72.3 / 57.9 / 73.5 / -21.2
D4*1 / 14 / 66.7 / 92.7 / 61.5 / 75.2 / 61.5 / 71.5 / -13.9
D5*1 / 15 / 83.3 / 111.8 / 80.8 / 97.4 / 80.8 / 80.0 / 1.0
D6*1 / 16 / 63.7 / 93.1 / 60.1 / 77.5 / 60.1 / 79.1 / -24.0
D7*1 / 17 / 58.7 / 88.8 / 55.1 / - / 55.1 / 77.8 / -29.2
D8*1 / 18 / 79.2 / 112.6 / 80.4 / 103.2 / 79.2 / 97.3 / -18.7
SRCV1-2X / 19 / 88.8 / 88.8 / 79.3 / 79.3 / 79.3 / 74.3 / 6.8
SRCV1-7X / 20 / 90.9 / 118.6 / 78.9 / 112.4 / 78.9 / 61.9 / 27.4
SRCV2-7X / 21 / 102.9 / 180.0 / 88.7 / 179.8 / 88.7 / 66.1 / 34.2
SRCV1-2Y*1 / 22 / 56.5 / 68.1 / 53.1 / 53.1 / 53.1 / 59.3 / -10.4
SRCV2-7X / 23 / 113.6 / 180.2 / 104.2 / 225.2 / 104.2 / 71.6 / 45.5

1: the specimen has column weak axes in the plane of the walls

Fig. 9.1.2 Error of Predicted Maximum Lateral Load for Japanese Infilled Steel Frames with Strong Axis of Steel Columns Oriented in the Plane of the Infill Wall

Fig. 9.1.3 Error of Predicted Maximum Lateral Load for Japanese Infilled Steel Frames with Weak Axis of Steel Columns Oriented in the Plane of the Infill Wall

specimens having the weak axis of the steel columns oriented in the plane of the infill wall by approximately 15% on average. It can be concluded that the plastic mechanism analysis is not conservative for infilled steel frames having the strong axis of the steel columns oriented in the plane of the infill wall.

There are several possible reasons that might contribute to the overestimation of the maximum lateral load of the specimens having the strong axis of steel columns oriented in the plane of the infill wall: 1) Liauw and Kwan (1983a, 1983b) assumed that the crushing of the concrete initiated when the maximum stress in the corner region reached the compressive strength of the concrete. It was recommended by Macgregor (1997) that 0.85fc’ is a more appropriate value for the crushing strength of concrete in a nodal region bounded by compressive struts and bearing plates. 2) The reduction of the plastic moment capacity of the frame members due to the shear force and, particularly, the axial force, was not recognized by Liauw and Kwan (1983a, 1983b). The effect of shear force on the plastic moment capacity of a steel cross section can usually be neglected if the shear force is less than the nominal yielding strength of the steel column (ASCE-WRC, 1971). However, for strong axis bending of steel members, the influence of axial force should not be neglected if the axial force is larger than 15% of the nominal compressive yielding strength of this cross section (ASCE-WRC, 1971). 3) The depth of the steel members was neglected so that the infill wall was slightly oversized in the plastic mechanism calculations of Liauw and Kwan (1983b). In the next section, the above three factors will be considered in establishing the plastic mechanism for the two-story composite steel frame-RC infill wall specimen tested in the present project.