**3.2. Two story space frame**

A two-story space subjected to combined action of gravity load and lateral load is depicted in Fig. 13 with its geometric dimension. The Young modulus, Poisson ratio, and yield stress of material are 19,613 *E* MPa, 0.3 , and 98 *<sup>y</sup>* MPa, respectively. This frame was previously analyzed by De Souza (2000) using the force-based method with fiber model. De Souza (2000) used one element per member in the modeling. The B23 element of ABAQUS is also employed to model this frame. Each framed member is modeled by one present element. The aim of this example is to demonstrate capability of the present element in capturing the effects of both geometric and material nonlinearities.

The ultimate loads of the frame obtained by different methods are presented in Table 1. The load-displacement responses of the frame are also plotted in Fig. 14. It can be seen that the results of the present element are well compared with those of De Souza (2000) using the force-based method. It should be noted that only one element per member is used in present study and De Souza (2000). The B23 element of ABAQUS overestimates ultimate strength of this frame if each framed member is modeled by less than fifty B23 elements. The difference between B23 element and present element is negligible when more than fifty B32 elements are used, and the ultimate strength and load-displacement curve obtained by ABAQUS and present study are then close each other.

**Figure 12.** Load-displacement curves of steel columns

**Figure 11.** Steel columns

**3.2. Two story space frame** 

of material are 19,613 *E* MPa, 0.3

present study are then close each other.

when the member is divided into many elements, the *P*

P

0.002P

W8x31

5 m

effect, and hence, the results of cubic element are close to the obtained results.

SAP2000, which is based on the cubic interpolation functions, overpredicts the buckling loads by 18% and 16% for the cantilever column and simply supported column, respectively, when the columns are modeled by one element per member. The load-displacement curves shown in Fig. 12 indicate that SAP2000 requires more than five cubic elements per member in modeling to match the results predicted by the present element. This is due to the fact that

A two-story space subjected to combined action of gravity load and lateral load is depicted in Fig. 13 with its geometric dimension. The Young modulus, Poisson ratio, and yield stress

previously analyzed by De Souza (2000) using the force-based method with fiber model. De Souza (2000) used one element per member in the modeling. The B23 element of ABAQUS is also employed to model this frame. Each framed member is modeled by one present element. The aim of this example is to demonstrate capability of the present element in

The ultimate loads of the frame obtained by different methods are presented in Table 1. The load-displacement responses of the frame are also plotted in Fig. 14. It can be seen that the results of the present element are well compared with those of De Souza (2000) using the force-based method. It should be noted that only one element per member is used in present study and De Souza (2000). The B23 element of ABAQUS overestimates ultimate strength of this frame if each framed member is modeled by less than fifty B23 elements. The difference between B23 element and present element is negligible when more than fifty B32 elements are used, and the ultimate strength and load-displacement curve obtained by ABAQUS and

 , and 98 *<sup>y</sup>* 

(a) Cantilever column (b) Simply supported column

capturing the effects of both geometric and material nonlinearities.

P

W8x31

2.5 m 2.5 m

0.004P

effect is transformed to the *P*

MPa, respectively. This frame was

Advanced Analysis of Space Steel Frames 87

Method Ultimate load (kN) Difference (%)

The last example is a large scale twenty-story space steel frame as shown in Fig. 15. The aim of this example is to demonstrate the capability of two proposed methods in predicting the strength and behavior of large-scale structures. A50 steel with yield stress of 344.8 Mpa, Young's modulus of 200 Gpa, and Poisson's ratio of 0.3 is used for all sections. The load applied to the structure consists of gravity loads of 4.8 kN/m2 and wind loads of 0.96 kN/m2 acting in the Y-direction. These loads are converted into concentrated loads applied at the beam-column joints. The obtained results are also compared with those generated by Jiang

Jiang et al. (2002) used both the plastic hinge and spread-of-plasticity elements to model this structure to shorten the computational time because the use of a full spread-of-plasticity analysis is very computationally intensive. When a member modeling by one plastic hinge element detected yielding to occur between the two ends, it was divided into eight spreadof-plasticity elements to accurately capture the inelastic behavior. In this study, each framed member is modeled by only one proposed element. The load-displacement curves of node A at the roof of the frame obtained by the present elements and mixed element of Jiang et al. (2002) are shown in Fig. 16. The ultimate load factor of the frame is also given in Table 2. A

Method Ultimate load factor Difference (%)

Jiang et al. (2002) 1.000 -

Present (refined plastic hinge model) 1.021 2.10

Present (fiber model) 1.0002 0.02

De Souza (2000) 128.05 -

**Table 1.** Comparison of ultimate load of two-story space frame

**3.3. Twenty-story space frame** 

et al. (2002) using the mixed element method.

very good agreement between the results is seen.

**Table 2.** TAnalysis result of twenty-story space frame

ABAQUS (5 element/member) 140.26 9.53 ABAQUS (20 element/member) 132.19 3.23 ABAQUS (50 element/member) 130.74 2.10 Present (refined plastic hinge model) 128.50 0.35 Present (fiber model) 128.82 0.60

**Figure 13.** Two-story space frame

**Figure 14.** Load- displacement curves of two-story space frame


**Table 1.** Comparison of ultimate load of two-story space frame
