Second-order inelastic dynamic analysis of 3-D steel frames

Transcription

1 International Journal of Solids and Structures 43 (26) Second-order inelastic dynamic analysis of 3-D steel frames Seung-Eock Kim *, Cuong Ngo-Huu, Dong-Ho Lee Deartment of Civil and Environmental Engineering/Construction Tech. Research Institute, Sejong University, 98 Kunja-dong, Kwangjin-ku, Seoul, , South Korea Received 3 November 24; received in revised form 19 May 25 Available online 2 July 25 Abstract This aer resents a reliable numerical rocedure for nonlinear time-history analysis of three-dimensional steel frames subjected to dynamic loads. Geometric nonlinearities of member (P-d) and frame (P-D) are taken into account by the use of stability functions in framed stiffness matrix formulation. The gradual yielding along the member length and over the cross-section is included by using a tangent modulus concet and a softening lastic hinge model based on a modified version of Orbison yield surface. A comuter rogram utilizing the average acceleration method for the integration scheme is develoed to numerically solve the equation of motion of framed structure formulated in an incremental form. The results of several numerical examles are comared with those derived from using beam element model of ABAQUS rogram to illustrate the accuracy and the comutational efficiency of the roosed rocedure. Ó 25 Elsevier Ltd. All rights reserved. Keywords: Dynamic analysis; Geometric and material nonlinearities; Plastic hinges; Stability functions; Three-dimensional steel frames 1. Introduction The second-order inelastic static analysis of three-dimensional steel frames has been studied extensively in recent years together with the raid develoment of comuter technology. The finite element method using the interolation functions and the fiber aroach for reresenting the second-order effect and the sread of lasticity is erformed by Izzuddin and Smith (1996), Teh and Clark (1999), and Jiang et al. (22). Although it can include the interaction between normal and shear stresses and its solution is considered to be accurate, it has not been alied widely for daily use in office engineering design because of its * Corresonding author. Tel.: ; fax: address: sekim@sejong.ac.kr (S.-E. Kim) /$ - see front matter Ó 25 Elsevier Ltd. All rights reserved. doi:1.116/j.ijsolstr

2 1694 S.-E. Kim et al. / International Journal of Solids and Structures 43 (26) highly comutational cost. A more simle and efficient method is the beam-column method using stability functions and refined lastic-hinge aroach roosed by Liew et al. (2) and Kim et al. (22, 23, 21). The benefits of using this method are that they enable only one or two elements to relatively accurately redict the nonlinear resonse of each framed member and, hence, to save comutational time. The studies on the second-order inelastic dynamic time-history analysis of three-dimensional steel frames are relatively few comared with those of static analysis. Porter and Powell (1971) used a single yield surface to model the abrut yielding from comletely elastic to comletely erfectly lastic state of the member ends, and geometric nonlinearity was ignored in the dynamic analysis of three-dimensional ie frames. Cambell (1994) develoed a three-dimensional fiber lastic hinge beam-column element for the secondorder inelastic dynamic analysis of framed structures. The geometric nonlinearity caused by axial force was included, but that caused by the interaction between axial force and bending moments was neglected. This method overestimates the strength and stiffness of the member subjected to significant axial force. Chan (1996) used an udated Lagrangian formulation for the large deflection dynamic analysis of sace frames, but did not consider the yielding of the material. Al-Bermani and Zhu (1996) also used an udated Lagrangian formulation and a bounding-surface kinematic hardening material model in conjunction with the lumed lasticity assumtion. However, this elastolastic method overredicts the caacity of stocky members since it neglects to consider the gradual reduction of stiffness as yielding rogresses through and along the member. Chi et al. (1998) and El-Tawil and Deierlein (21) resented a comuter rogram for dynamic analysis of mixed frame structures comrised of steel, reinforced concrete, and/or comosite members. Three-dimensional beam-columns were modeled using a flexibility-based distributed lasticity formulation that utilized a bounding surface to model inelastic member cross-section resonse. The geometric nonlinear behavior was modeled through an udated Lagrangian geometric stiffness aroach. Recently, the OenSees finite element oen source software has develoed by McKenna et al. (25) to simulate the resonse of structural and geotechnical systems subjected to earthquake loading. The threedimensional nonlinear beam-column elements formulated by force- and dislacement-based aroaches include both concentrated and distributed lasticity tyes using the numerical integration method. Geometric nonlinearity effect is included by the use of the corotational coordinate transformation technique. For the last two studies mentioned, the members of structure need to be divided into many elements to cature the second-order effect accurately, and the numerical integration rocedure is relatively time-consuming, so the analysis time is relatively long. Therefore, it is not convenient to aly them in a daily ractical design. The urose of this aer is to extend the alication of the stability functions and the refined lastichinge aroach for second-order inelastic dynamic time-history analysis of three-dimensional frames. Lateral-torsional buckling is assumed to be revented by adequate lateral bracing. The section of members is assumed to be comact so that it should be able to develo full lastic moment caacity without local buckling; waring torsion is ignored. The material model used is elastic erfectly lastic. The reduction of torsional stiffness is not considered in lastic hinge. The strain reversal effect is treated by the alication of double modulus theory. A comuter rogram utilizing the average acceleration method for the integration scheme is develoed to numerically solve the equation of motion of framed structure formulated in an incremental form. Several examles are resented to rove the robustness of the roosed numerical rocedure in redicting the dynamic resonse of three-dimensional framed structures. 2. Formulation 2.1. Stability functions accounting for second-order effects Stability functions are used to cature the second-order effects since they can account for the effect of the axial force on the bending stiffness reduction of a member. As resented by Kim et al. (21), the

3 S.-E. Kim et al. / International Journal of Solids and Structures 43 (26) force dislacement equation using stability functions may be written for three-dimensional beam-column element as EA P 8 9 L d EI M ya S y EI 1 S y L 2 L h ya >< M yb >= EI S y EI 2 S y ¼ L 1 >< h L yb >= M EI za S z EI 3 S z ð1þ L 4 h za L M EI zb 6 S z EI 4 4 S z L 3 7 L 5 h zb >: >; >: >; T GJ / L where P, M ya, M yb, M za, M zb, and T are axial force, end moments with resect to y and z axes, and torsion, resectively; d, h ya, h yb, h za, h zb,and/ are the axial dislacement, the joint rotations, and the angle of twist; A, I y, I z, and L are area, moment of inertia with resect to y and z axes, and length of beam-column element; E, G, and J are elastic modulus, shear modulus, and torsional constant of material; S 1, S 2, S 3, and S 4 are the stability functions with resect to y and z axes, resectively, and are resented as 8 ffiffiffiffi ffiffiffiffi q y sin q y 2 q cos ffiffiffiffi q y 2 2cos ffiffiffiffi q y ffiffiffiffi ffiffiffiffi if P < >< q y sin q y S 1 ¼ 2 q y cosh ffiffiffiffi q y ffiffiffiffi ð2aþ ffiffiffiffi q y sinh q y 2 2cosh ffiffiffiffi q y þ ffiffiffiffi ffiffiffiffi if P > >: q y sinh q y 8 2 q y ffiffiffiffi ffiffiffiffi q y sin q y 2 2cos ffiffiffiffi q y ffiffiffiffi ffiffiffiffi if P < >< q y sin q y S 2 ¼ ffiffiffiffi ð2bþ ffiffiffiffi q y sinh q y 2 q y 2 2cosh ffiffiffiffi q y þ ffiffiffiffi ffiffiffiffi if P > >: q y sin q y 8 ffiffiffiffi ffiffiffiffi q z sin q z 2 q cos ffiffiffiffi q z >< 2 2cos ffiffiffiffi ffiffiffiffi ffiffiffiffi if P < q z q z sin q z S 3 ¼ 2 q cosh ffiffiffiffi ffiffiffiffi ffiffiffiffi ð2cþ q z q z sinh q >: z 2 2cosh ffiffiffiffi ffiffiffiffi ffiffiffiffi if P > q z þ q z sinh q z 8 2 q z ffiffiffiffi ffiffiffiffi q z sin q z >< 2 2cos ffiffiffiffi ffiffiffiffi ffiffiffiffi if P < q z q z sin q z S 4 ¼ ffiffiffiffi ffiffiffiffi ð2dþ q z sinh q z 2 q z >: 2 2cosh ffiffiffiffi ffiffiffiffi ffiffiffiffi if P > þ sin q z q z where q y = P/( 2 EI y /L 2 ), q z = P/( 2 EI z /L 2 ), and P is ositive for tension CRC tangent modulus model associated with residual stresses q z The CRC tangent modulus concet is used to account for gradual yielding (due to residual stresses) along the length of axially loaded members between lastic hinges. The elastic modulus E (instead of moment of inertia I) is reduced to account for the reduction of the elastic ortion of the cross-section since the

4 1696 S.-E. Kim et al. / International Journal of Solids and Structures 43 (26) reduction of the elastic modulus is easier to imlement than a new moment of inertia for every different section. From Chen and Lui (1987), the CRC tangent modulus E t is written as E t ¼ 1.E for P 6.5P y ð3aþ E t ¼ 4 P E 1 P for P >.5P y ð3bþ P y P y 2.3. Parabolic function for gradual yielding due to flexure The tangent modulus model is suitable for the member subjected to axial force, but not adequate for cases of both axial force and bending moment. A gradual stiffness degradation model for a lastic hinge is required to reresent the artial lastification effects associated with bending. The arabolic function is used to reresent the transition from elastic to zero stiffness associated with a develoing hinge. From Kim et al. (21), when the arabolic function for a gradual yielding is active at both ends of an element, the sloe deflection equation may be exressed as E P ta 38 9 d L M ya k iiy k ijy h ya >< M yb >= k ijy k jjy >< h ¼ yb >= M za k iiz k ijz ð4þ h za M 6 zb 4 k ijz k jjz 7 5 h zb >: >; >: >; GJ T / L where k iiy ¼ g A S 1 S2 2 Et I y ð1 g S B Þ ð5aþ 1 L E t I y k ijy ¼ g A g B S 2 L k jjy ¼ g B S 1 S2 2 Et I y ð1 g S A Þ 1 L k iiz ¼ g A S 3 S2 4 Et I z ð1 g S B Þ 3 L ð5bþ ð5cþ ð5dþ E t I z k ijz ¼ g A g B S 4 ð5eþ L k jjz ¼ g B S 3 S2 4 Et I z ð1 g S A Þ ð5fþ 3 L The terms g A and g B is a scalar arameter that allows for gradual inelastic stiffness reduction of the element associated with lastification at end A and B. This term is equal to 1. when the element is elastic, and zero when a lastic hinge is formed. The arameter g is assumed to vary according to the arabolic function g ¼ 1. for a 6.5 ð6aþ g ¼ 4að1 aþ for a >.5 ð6bþ

5 S.-E. Kim et al. / International Journal of Solids and Structures 43 (26) where a is a force-state arameter that measures the magnitude of axial force and bending moment at the element end. The term a in this study is exressed in a modified version of Orbison full lastification surface of cross-section, as resented by McGuire et al. (2), as follows: a ¼ 2 þ m 2 z þ m4 y þ 3.52 m 2 z þ 3.6 m 2 y þ 4.5m4 z m2 y ð7þ where = P/P y, m z = M z /M z (strong-axis), m y = M y /M y (weak-axis). If the member forces violate yield condition, says a > 1, the member forces will be corrected to return the yield surface along a ath assing though the origin by the alication of the bi-section method. To treat the strain reversal effect in the hinge due to the abrut change in alied direction of dynamic load, the scalar arameter g, which allows for gradual inelastic stiffness reduction of the element associated with lastification at member end as resented in Eq. (5), is modified based on the double modulus theory in Chen and Lui (1987) as follows: g d ¼ ffiffiffiffiffiffiffiffiffiffiffi g dy g dz where 4g g dy ¼ 1 þ ffiffiffiffi 2 for weak axis ð9aþ g g dz ¼ 2g 1 þ g for strong axis ð9bþ ð8þ 2.4. Shear deformation effects As resented by Kim et al. (21), to account for transverse shear deformation effects in a beam-column element, the stiffness matrix may be modified as 8 >< >: P M ya M yb M za M zb T E ta 8 9 d L C iiy C ijy h ya >= C ijy C jjy >< h yb >= ¼ C iiz C ijz h za 6 4 C ijz C jjz 7 5 h zb >; >: >; GJ / L ð1þ in which C iiy ¼ k iiyk jjy k 2 ijy þ k iiya sz GL k iiy þ k jjy þ 2k ijy þ A sz GL C ijy ¼ k iiyk jjy þ k 2 ijy þ k ijya sz GL k iiy þ k jjy þ 2k ijy þ A sz GL C jjy ¼ k iiyk jjy k 2 ijy þ k jjya sz GL k iiy þ k jjy þ 2k ijy þ A sz GL ð11aþ ð11bþ ð11cþ

6 1698 S.-E. Kim et al. / International Journal of Solids and Structures 43 (26) C iiz ¼ k iizk jjz k 2 ijz þ k iiza sy GL k iiz þ k jjz þ 2k ijz þ A sy GL C ijz ¼ k iizk jjz þ k 2 ijz þ k ijza sy GL k iiz þ k jjz þ 2k ijz þ A sy GL C jjz ¼ k iizk jjz k 2 ijz þ k jjza sy GL k iiz þ k jjz þ 2k ijz þ A sy GL ð11dþ ð11eþ ð11fþ where A sy and A sz are the shear areas with resect to y and z axes, resectively Vibration analysis From Chora (21), in order to obtain the natural frequencies and vibration modes of three-dimensional steel frame, the following eigenroblem needs to be solved ½K Šu ¼ x 2 ½MŠu ð12þ where [K ] is the initial stiffness matrix, [M] is the common lumed mass matrix, u is a mode shae, and x is the circular frequency corresonding to u The equation of motion The incremental form of the equation of motion for frames is given by ½MŠ½D ušþ½cš½d_ušþ½kš½duš ¼½DFŠ ð13þ in which [K] is the stiffness matrix as mentioned earlier and [C] =a[m]+b[k ] is the viscous daming matrix, where a and b are mass- and stiffness-roortional daming factors, resectively; ½D uš; ½D_uŠ, [Du], and [DF] are the incremental acceleration, velocity, dislacement, and exciting force vectors, resectively, over a time increment of Dt. With the adotion of the Newmark method that was resented in Chora (21) for ste-by-ste solution of Eq. (13), the following equations are used ½ tþdt _uš ¼½ t _ušþð1 cþdt½ t ušþcdt½ tþdt uš ½ tþdt uš¼½ t ušþdt½ t _ušþð.5 bþðdtþ 2 ½ t ušþbðdtþ 2 ½ tþdt uš ð14aþ ð14bþ in which ½ t uš; ½ t _uš, and[ t u] are the total acceleration, velocity, and dislacement vectors at time t. The arameters b and c define the variation of acceleration over a time ste and determine the stability and accuracy characteristics of the method. Here b and c are taken as 1/4 and 1/2 corresond to the assumtion of the average acceleration method. Finally, the incremental equation of motion (13) can be exressed as " # ½KŠþ c bdt ½CŠþ 1 bðdtþ ½MŠ ½ t DuŠ ¼½DFŠþ 1 2 bdt ½MŠþc b ½CŠ ½ t _ušþ 1 2b ½MŠþDt Once [ t Du] is known, the remaining unknown vectors are comuted as follows: c 2b 1 ½CŠ ½ t uš ð15þ

7 S.-E. Kim et al. / International Journal of Solids and Structures 43 (26) ½ t D_uŠ ¼ c bdt ½t DuŠ c b ½t _ušþdt 1 c ½ t uš 2b ð16aþ ½ t D uš ¼ 1 bðdtþ 2 ½t DuŠ 1 bdt ½t _uš 1 2b ½t uš ½ tþdt uš¼½ t ušþ½duš ½ tþdt _uš ¼½ t _ušþ½d_uš ½ tþdt uš ¼½ t ušþ½d uš ½ tþdt FŠ¼½ t FŠþ½DFŠ ð16bþ ð16cþ ð16dþ ð16eþ ð16fþ The rocedure resented in equations (15) and (16) is reeated for the next time stes until the considered frame is collased or desired time duration ends. 3. Verifications A comuter rogram PAAP-Dyna is develoed based on the above-mentioned formulation to redict vibration behavior of three-dimensional framed structures as well as its nonlinear resonse under earthquake loading. It is verified for validity using the results generated by ABAQUS rogram through three numerical examles. Although the use of shell element of ABAQUS can lead to extremely accurate results, it is not alied here because it is too time-consuming for such a nonlinear dynamic analysis. Therefore, the B33 beam element with 13 numerical integration oints (five oints in web, five in each flange) of ABAQUS is used to model framed structures herein. Four earthquake records of the El-Centro 194, the Loma Prieta 1989, the Northridge 1994, and the San Fernando 1971, as resented in Fig. 1, are used as ground motion inut data. Their eak ground accelerations and time stes are listed in Table 1. For each roblem, the static load due to the weight of the mass and dead loading (if any) is alied first to the structure by a static analysis, and then the earthquake loading is alied by a dynamic time-history analysis. The mass- and stiffness-roortional daming factors are chosen based on first two modes of frame so that the equivalent viscous daming ratio is equal to 5% Two-storey three-dimensional frame for second-order effect verification Fig. 2 shows the geometric and material roerties of a two-storey three-dimensional frame with masses lumed at the framed nodes. The geometry of this frame is symmetric, but the mass distribution is not. The vertical concentrated loads alied at all framed nodes are chosen to be very large in order to show the second-order effect clearly. In numerical modeling, each framed element is divided into two equal elements. After erforming the vibration analysis, first two natural eriods along the alied earthquake direction and corresonding mode shaes of the frame are obtained and comared in Table 2 and Fig. 3. It can be seen that a strong agreement of dynamic roerties of the study frame generated by ABAQUS and roosed rograms is obtained. The roof dislacement resonses of frame and a comarison of their eak values and corresonding times of the linear elastic (LE) and second-order elastic (SE) analyses are shown in Fig. 4 and Table 3. As can be observed from those, the difference of dislacement resonse of LE and SE analyses in each case

8 17 S.-E. Kim et al. / International Journal of Solids and Structures 43 (26) Acceleration (g) Acceleration (g) a -1.2 b Acceleration (g) Acceleration (g) c -1.2 d -1.2 Fig. 1. Earthquake records: (a) El Centro (194); (b) Loma Prieta (1989); (c) Northridge (1994) and (d) San Fernando (1971). Table 1 Peak ground acceleration and its corresonding time ste of the earthquake records Earthquake PGA (g) Time ste (s) El Centro (194) (Array, #9, USGS Station 117) Loma Prieta (1989) (Caitola,, CDMG Station 47125) Northridge (1994) (Simi Valley-Katherine, 9, USC Station 955).64.1 San Fernando (1971) (Pacoima Dam, 254, CDMG Station 279) is distinct, and all results obtained by ABAQUS and the roosed rograms are nearly the same, which rove the accuracy of the roosed rogram in redicting the second-order effect Verification examles of inelastic effect Two-storey three-dimensional frame The same structure, as resented in the revious examle, is used here for verification, excet for the alication of vertical concentrated loads at framed nodes (Fig. 5). The roof dislacement resonses of the frame obtained from the second-order elastic (SE) and second-order inelastic (SI) analyses are shown in Fig. 6. A comarison of the eak roof dislacement and its corresonding time is shown in Table 4. Itis

9 S.-E. Kim et al. / International Journal of Solids and Structures 43 (26) Fig. 2. Two-storey frame for second-order effect verification. Table 2 Comarison of first two natural eriods along the alied earthquake direction of two-storey frame Mode Period (s) Error (%) ABAQUS PAAP-Dyna (roosed) First Second ABAQUS PAAP-Dyna st mode nd mode Fig. 3. Comarison of first two mode shaes along the alied earthquake direction of two-storey frame.

10 172 S.-E. Kim et al. / International Journal of Solids and Structures 43 (26) ABAQUS (LE) PAAP-Dyna (LE) 4 ABAQUS (SE) PAAP-Dyna (SE) Dislacement (mm) Dislacement (mm) a -6 Dislacement (mm) ABAQUS (LE) PAAP-Dyna (LE) 4 Dislacement (mm) t -4-6 ABAQUS (SE) PAAP-Dyna (SE) 4-8 b -8 Dislacement (mm) ABAQUS (LE) PAAP-Dyna (LE) Dislacement (mm) ABAQUS (SE) PAAP-Dyna (SE) -9-9 Dislacement (mm) ABAQUS (LE) PAAP-Dyna (LE) c Dislacement (mm) ABAQUS (SE) PAAP-Dyna (SE) d -16 Fig. 4. Roof dislacements of two-storey frame for second effect verification: (a) El Centro earthquake; (b) Loma Prieta earthquake; (c) Northridge earthquake and (d) San Fernando earthquake.

11 S.-E. Kim et al. / International Journal of Solids and Structures 43 (26) Table 3 Comarison of eak roof dislacement and its corresonding time of two-storey frame for second-order effect verification Eq. tye Min/max Analysis tye ABAQUS PAAP-Dyna (roosed) Dis. Error (%) Dis. (mm) Time (s) Dis. (mm) Time (s) El Centro Max LE SE Min LE SE Loma Prieta Max LE SE Min LE SE Northridge Max LE SE Min LE SE San Fernando Max LE SE Min LE SE Fig. 5. Two-storey frame for inelastic effect verification. observed that the ABAQUS and roosed rograms give nearly identical results in all cases, including the slight ermanent shifts in dislacement due to inelastic behavior for SI analysis cases under Loma Prieta, Northridge, and San Fernando earthquakes. As is evident from the figure, the difference in dislacement resonse of SE and SI analyses in these cases is relatively clear. For the case of El Centro earthquake having smallest PGA, it is noted that the dislacement resonses in SE and SI cases are almost identical, because the behavior of the framed structure is almost in elastic range in this case.

12 174 S.-E. Kim et al. / International Journal of Solids and Structures 43 (26) Dislacement (mm) Dislacement (mm) Dislacement (mm) ABAQUS (SI) PAAP-Dyna (SE) ABAQUS (SE) PAAP-Dyna (SE) ABAQUS (SE) PAAP-Dyna (SE) 3 Dislacement (mm) Dislacement (mm) Dislacement (mm) a b c ABAQUS (SI) PAAP-Dyna (SI) ABAQUS (SI) PAAP-Dyna (SI) ABAQUS (SI) PAAP-Dyna (SI) ABAQUS (SE) PAAP-Dyna (SE) 14 1 ABAQUS (SI) PAAP-Dyna (SI) Dislacement (mm) Dislacement (mm) d Fig. 6. Roof dislacements of two-storey frame for inelastic effect verification: (a) El Centro earthquake; (b) Loma Prieta earthquake; (c) Northridge earthquake and (d) San Fernando earthquake.

13 S.-E. Kim et al. / International Journal of Solids and Structures 43 (26) Table 4 Comarison of eak roof dislacement and its corresonding time of two-storey frame for inelastic effect verification Eq. tye Min/max Analysis tye ABAQUS PAAP-Dyna (roosed) Dis. Error (%) Dis. (mm) Time (s) Dis. (mm) Time (s) El Centro Max SE SI Min SE SI Loma Prieta Max SE SI Min SE SI Northridge Max SE SI Min SE SI San Fernando Max SE SI Min SE SI Four-storey three-dimensional frame This four-storey, two by two bay steel frame was resented by Cambell (1994). The framed geometry and members are ket to be same to the original frame, but the in-lane rigid slabs are neglected in modeling. Besides, because the original masses were too small, they are changed into bigger ones to be more realistic and to be caable of showing inelastic behavior clearly under earthquakes. The geometry, mass distribution, and material roerties of the modified frame are shown in Fig. 7. Each member of the frame is only modeled by one element in numerical modeling. A comarison of first three natural eriods along the alied earthquake direction and their corresonding mode shaes of the four-storey frame obtained by vibration analysis of ABAQUS and the roosed rograms are shown in Table 5 and Fig. 8. It can be seen that a very good correlation is found. The roof dislacement resonses along X-axis of node A of the frame obtained by ABAQUS and roosed rograms are shown in Fig. 9 for four different earthquakes. Their eak dislacements and corresonding times are comared in Table 6. It is observed that excet for the similar resonse in the cases of the frame subjected to the El Centro earthquake with the smallest PGA, the difference of dislacement resonse of SE and SI analyses in the other cases is clear, and the obtained results correlate very well, including the ermanent drifts of dislacement in SI analysis cases. As in the revious examle, this one also indicates that the roosed rogram is able to accurately redict dislacements, which is an imortant index for a erformance-based seismic design. The locations of lastic hinges and their corresonding times at the most severe yielding state of this study frame subjected to Northridge and San Fernando earthquakes are shown in Fig. 1. It can be seen that the results obtained by ABAQUS and PAAP-Dyna are similar. With using Intel Pentium IV 3.2 GHz, 2 GB RAM comuter, the comutational times of the ABAQUS and PAAP-Dyna rograms for four-storey frame subjected to Loma-Prieta earthquake, which is the roblem having the longest analysis time among four cases, are 12 h and 5 min, resectively. This result roves the high comutational efficiency of the roosed comuter rogram.

14 176 S.-E. Kim et al. / International Journal of Solids and Structures 43 (26) Fig. 7. Four-storey frame. Table 5 Comarison of first three natural eriods along the alied earthquake direction of four-storey frame (X direction) Mode Period (s) Error (%) ABAQUS PAAP-Dyna (roosed) First Second Third ABAQUS PAAP-Dyna st mode nd mode rd mode Fig. 8. Comarison of first three mode shaes along the alied earthquake direction of four-storey frame (X-direction).

15 S.-E. Kim et al. / International Journal of Solids and Structures 43 (26) a b c d Fig. 9. Roof dislacements at node A of four-storey frame: (a) El Centro earthquake; (b) Loma Prieta earthquake; (c) Northridge earthquake and (d) San Fernando earthquake.

16 178 S.-E. Kim et al. / International Journal of Solids and Structures 43 (26) Table 6 Comarison of eak roof dislacement at node A and its corresonding time of four-storey frame Eq. tye Min/max Analysis tye ABAQUS PAAP-Dyna (roosed) Dis. error (%) Dis. (in) Time (s) Dis. (in) Time (s) El Centro Max SE SI Min SE SI Loma Prieta Max SE SI Min SE SI Northridge Max SE SI Min SE SI San Fernando Max SE SI Min SE SI Fig. 1. Plastic hinge formation at the most severe yielding state of four-storey frame: (a) Northridge earthquake ABAQUS: t = 5.5 s; PAAP-Dyna, t = 5.5 s and (b) San Fernando earthquake: ABAQUS: t = 3.72 s; PAAP-Dyna, t = 3.71 s. 4. Conclusions A simle and effective numerical rocedure for the nonlinear dynamic time-history analysis of the threedimensional steel frames considering both geometric and material nonlinearities has been resented. The comuter rogram develoed for this research is verified for accuracy and comutational efficiency through

17 S.-E. Kim et al. / International Journal of Solids and Structures 43 (26) three numerical examles with four different earthquake loadings. It is also caable of accurately redicting natural eriods and vibration mode shaes of framed structures. The good results obtained in a short analysis time rove that this comuter rogram can effectively be used for office design in redicting nonlinear behavior of steel framed structures subjected to static and dynamic load instead of using the time-consuming and costly commercial structural software. Acknowledgement The work resented in this aer was suorted by funds of National Research Laboratory Program (Grant No. M J--83-1) from Ministry of Science and Technology in Korea. Authors wish to areciate this financial suort. References Al-Bermani, F.G.A., Zhu, K., Nonlinear elastolastic analysis of satial structures under dynamic loading using kinematic hardening models. Engineering Structures 18 (8), Cambell, S.D., Nonlinear elements for three dimensional frame analysis. Ph.D. Thesis, University of California at Berkeley, CA. Chan, S.L., Large deflection dynamic analysis of sace frames. Comuter and Structures 58 (2), Chen, W.F., Lui, E.M., Structural Stability: Theory and Imlementation. Elsevier, New York. Chi, W.M., El-Tawil, S., Deierlein, G.G., Abel, J.F., Inelastic analyses of a 17-storey steel framed building damaged during Northridge. Engineering Structures 2 (4 6), Chora, A.K., 21. Dynamics of Structures: Theory and Alications to Earthquake Engineering. Prentice-Hall, New Jersey. El-Tawil, S., Deierlein, G.G., 21. Nonlinear analysis of mixed steel concrete frames, Parts I and II. Journal of Structural Engineering 127 (6), Izzuddin, B.A., Smith, D.L., Large-dislacement analysis of elastolastic thin-walled frames, art I and II. Journal of Structural Engineering 122 (8), Jiang, X.M., Chen, H., Liew, J.Y.R., 22. Sread-of-lasticity analysis of three-dimensional steel frames. Journal of Constructional Steel Research 58 (2), Kim, S.E., Lee, J., Park, J.S., D second-order lastic-hinge analysis accounting for lateral torsional buckling. International Journal of Solids and Structures 39 (8), Kim, S.E., Lee, J., Park, J.S., D second-order lastic-hinge analysis accounting for local buckling. Engineering Structures 25 (1), Kim, S.E., Park, M.H., Choi, S.H., 21. Direct design of three-dimensional frames using ractical advanced analysis. Engineering Structures 23 (11), Liew, J.Y.R., Chen, H., Shanmugam, N.E., Chen, W.F., 2. Imroved nonlinear lastic hinge analysis of sace frame structures. Engineering Structures 22 (1), McGuire, W., Gallagher, R.H., Ziemian, R.D., 2. Matrix Structural Analysis. John Wiley & Son, Inc., New York. McKenna, F., Fenves, G.L., et al., 25. Oen system for earthquake engineering simulation (OenSees). Available from: <htt://oensees.berkeley.edu>. Pacific Earthquake Engineering Research Center, University of California, Berkeley. Porter, F.L., Powell, G.H., Static and dynamic analysis of inelastic frame structures. Reort No. EERC 71-3, University of California, Berkeley. Teh, L.H., Clark, M.J., Plastic-zone analysis of 3D steel frames using beam elements. Journal of Structural Engineering 125 (11),