Example 1: A Simply-Supported Beam
The first example is a simply-supported beam (see Fig. 11), which was tested by Lay and Galambos (1964). The section is W130x28, with the measured properties being A=3606 mm2, T=11.16×106 mm4, S=168.8×103 mm3, Z=190.1×103 mm3, d=132 mm, Fy=247 MPa, and Fr = 19.8 MPa. Mp = ZxFy = 47 kN-m, and load capacity P=71 kN. Equations 14 to 17 are used to establish 0pu-K relation for this beam. w2=0.9, w3=0.308 and w4=0.21.
The middle portion of this beam is a region of constant moment. This entire region begins to yield when its moment exceeds the first-yield moment. Therefore, this beam is of particular interest for this study since the wide spread of plasticity both through the depth of the section and along the length of the beam took place. Such a structure with a large plastic zone is generally regarded as being difficult for a plastic-hinge method to model.
First, the beam was analysed by using four elements, as shown in Fig. 12b where the digit inside a rectangular box represents the element number. The middle portion is represented by two identical elements 2 and 3. The moment-deflection curve at midspan section is drawn in Fig. 11. Compared with the test curve, the errors of the theoretical analysis are within 5%. Next, the beam was reanalysed using a 3-element discretization scheme (see Fig. 12a). Table 1 records the Qpu value for hinge-springs of different discretization schemes. The deflection at the one-third point at the incipience of collapse is also recorded in Table 1. The plasticity-factors under 90% of the collapse load are depicted in Fig. 12, where an oval represents a hinge-spring and the number inscribed in it is the corresponding plasticity-factor p.
Table 1 Analysis of the simply-supported beam using different discretization schemes
For the 3-element scheme, Table 1 shows that the Bpu value for the hinge-springs of element 2 is
9.2 times as large as that for the hinge-springs of elements 1 and 3. Therefore, the hinge-springs of element 2 have a significantly smaller p value (i. e., p1=p2=0.18) than the hinge-springs of elements 1 (p2=0.67) and 3 (pi=0.67) as illustrated in Fig. 12a, even though all these hinge-springs are subjected
to an identical moment. This substantially smaller p value for element 2 is translated into a much smaller stiffness for the middle portion of the beam, which is in keeping with the fact that the entire middle portion is a plastic zone. Thus, the parameter Bpu is capable of imitating the spread of plasticity along the length of a member.
It is noted from Table 1 that the two analyses obtain identical deflection. It is further noted from Figs. 12a and 12b that the plasticity-factors are identical for these two discretization schemes. The two discretization schemes have identical beam stiffness at every loading step. This can be explained by inspecting Eqs. 7 and 8 and Bpu values in the table. The Bpu value for elements 2 and 3 of the 4- element scheme is half of that for element 2 of the 3-element scheme (0pu is proportional to element length L. The elements in the middle portion have k= -1). Hence, the product RmL in Eq. 8 remains unchanged whether the middle portion of the beam is represented by one or two (or more) elements.
Example 2: A 2-Story Moment Frame
Consider the steel frame subjected to the gravity loads shown in Fig. 13. The structure supports specified loads of 110 kN/m on floor beams B1 and B2 and 51 kN/m on roof beams B3 and B4, where the factored loads are 1.4 times the corresponding specified loads. Fy=248 MPa and Fr=0.3Fy.
For the nonlinear analysis of this example, each column is represented by two elements while each beam is represented by four elements. The distributed loads are lumped at nodal points. Equations 18 to 21 are used to establish the 0pu-K relationship. The analysis terminated when the frame failed at load ratio X=1.09. The lateral displacement of the top right corner of the frame during the loading history is described by the continuous line in Fig. 14. The analysis results are in close agreement with those from other researchers.
This paper presents a nonlinear analysis method for steel frames using a new plastic-hinge model. Two separate parameters are used in the hinge model to mimic the spread of plasticity both through section depth and along member length respectively. The proposed plastic-hinge model is general and applicable to all kinds of beam-columns, while specific numerical examples were conducted for steel frames with wide flange I-sections. The method is simple to implement since it only involves the
modification of a conventional elastic matrix displacement procedure. Numerical examples demonstrated the accuracy of the proposed analysis method.
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