CALCULATION MODEL OF WELDED CONNECTIONS IN STEEL FRAMES UNDER SEVERE EARTHQUAKES
BEAM-TO-COLUMN CONNECTIONS IN STEEL FRAMES 4.1 Damage evolution equation and model
The damage process was defined with the variable of effective plastic strain. The damage evolution equation for the damage developing process subjected to the extremely low cyclic loading was derived based on the fatigue crack growth formula.
According to Manson- Coffin relationship (Manson, 1953; Coffin, 1954) and Paris formula (Paris, 1963) of fatigue crack growth curve, Solomon (1972) and Krawinkler (1983) proposed the relationship between crack length of the connection welds and the number of cycles under constant amplitude loading in tests.
β = α ∆ε( p)
da a
dN (3)
Crack growth equation was defined as:
β β α ∆ε + = 1.5 ( ) 1 TR EPS e i i a a e (4)
Damage evolution equation of connection for ductile cracking was defined as: =
∫
0 ( , ) b i a x i D dx bt (4)Connections in Steel Structures VII / Timisoara, Romania / May 30 - June 2, 2012 227 228 Connections in Steel Structures VII / Timisoara, Romania / May 30 - June 2, 2012
4.2. Damage parameters calibration
S6~S9 specimens under constant amplitude loadings were used for calibrating the value of β. According to the fitting results, the value of β was classified by two stages based on cyclic amplitudes, as shown in Eq. (6).
⎧ β = ⎨ ⎩ 1.15 1.45 ∆ϕ ≥ ∆ϕ ∆ϕ < ∆ϕ 1.25 1.25 p y p y (6) In which, ∆φp referred to the cyclic tensile or compressive amplitude of connection
plastic rotation.
Various amplitude loadings were used to verify the parameters of damage model, as shown in Figure 9, which indicated the parameter was suitable.
(a) S-2 (b) S-3
Figure 9. Damage curve of connections under various amplitude loads 4.3. Verification and application
Based on the above results of theoretical studies, the crack evolution equations were introduced into the finite element model. Complete decoupling could be used to compute the connection cracking. According to equation (5), the accumulative damage of connection was calculated by calculating the crack of each unit in each cycle (Dassault, 2010). The ABAQUS subroutine was written in Fortran Language. The crack developing process and damage development of connection were introduced into finite element analysis process. The finite element models of the connections tested in this article were established. The value of material parameter αTR = 1/3 was
13.7 in subroutine and β values were determined by the fitting results.
Figure 10. Comparison between FEA taking account of damage and experimental curves
The typical hysteresis curves of connections with damage were obtained as shown in Figure 10 by introducing the damage model into hysteresis curves calculated by the finite element analysis. Proposed model and test results were in good agreement and the accuracy of the model was ensured.
5. ANALYSIS AND APPLICATION OF DAMAGE MODEL OF WELDED BEAM-TO-COLUMN CONNECTIONS
5.1. Damage evolution equation of bilinear model
The relationship of plastic rotation and plastic strain of the beam end was: ϕ =p 2Lpεp
h (7)
In order to establish macro-level quantitative indicators of connection damage, the effective plastic rotation was introduced into the connection crack fatigue formula based on ductility cracking.
β β = α( ) (∆ϕ ) 2 p EPR da h a dN L (8)
The damage evolution equation described by effective plastic rotation was:
β β +1= exp[(2 ) α (∆ϕ , ) ] i i TR EPR i p h D D L (9)
The method to obtain the damage index β was the same as section 4 that constant amplitude loading curves were used for damage curves fitting. The fitting results were taken according to Eq. (10):
⎧ β = ⎨ ⎩ 1 1.3 ∆ϕ ≥ ∆ϕ ∆ϕ < ∆ϕ , , 1.25 1.25 EPR i y EPR i y (10) 5.2. Complete hysteretic curve of connections with damage
The connection moment - rotation curve considering damage was obtained by introducing connection damage evolution equation into the original moment - rotation
Connections in Steel Structures VII / Timisoara, Romania / May 30 - June 2, 2012 229 230 Connections in Steel Structures VII / Timisoara, Romania / May 30 - June 2, 2012
curve.The Di was calculated by Eq. (9) with cyclic accumulation. The connection
moment - rotation curves considering damage subjected to constant amplitude loading and variable amplitude loading were respectively shown in Figure 11.
(1 ) p i K −D (1 ) e i K −D 2 2 (M (1−Di),ϕ) (−My(1−Di),−ϕy) (M3(1−Di),ϕ3) (a) Constant amplitude loading
(1 ) p i K −D (1 ) e i K −D ② ③ ④ ⑤ 2 2 (M′(1−Di),ϕ′) 4 4 (M(1−Di),ϕ′) 2 2 (M(1−Di),ϕ) (−My(1−Di),−ϕy)
(b) Variable amplitude loading
Figure 11. The hysteretic model of connection damage 5.3. Verification of the damage model for connections
In order to validate the proposed connection moment - rotation curve considering damage, the tests resutls were compared with the model results (Figure 12). The calculated damage curve accurately described the phenomenon of connection progressive damage with the loading process.
The cyclic loading tests of connection carried out by Goel and Stojadinovic (1997) and Ricles (2000) in Lehigh University were also selected to validate the proposed simplified calculation model. The numerical simulation using damage model compared well with the test results, as shown in Figures 13 and 14, which could accurately describe the cumulative damage and degradation behaviors of structures.
Figure 12. The comparison results of the tests in this paper 5.4. Application of the damage model in steel frames
The simplified plastic hinge model with damage was applied for dynamic time history analysis of steel frames. A five-storey steel frame model with one span was established. El Centro and Koyna earthquake waves were adopted. In order to study the behaviours of the structure under a severe earthquake, the seismic peak acceleration PGA was adjusted to 0.62g. The frame model was established by B21 beam element. The plastic hinge model with damage was used to consider the effect of connection damage on the steel structure.
Figure 13. The comparison of Goel
connection result Figure 14. The comparison of Ricles connection result
Connections in Steel Structures VII / Timisoara, Romania / May 30 - June 2, 2012 231 232 Connections in Steel Structures VII / Timisoara, Romania / May 30 - June 2, 2012
From Figure 15，before the occurrence of damage and degradation, seismic indexes calculated by two models were in accordance. However, once damage and degradation occurred, the results of the two models were remarkably different. Beam element model without damage overrated the structure deformation capacity, which caused unsafety.
Figure 15. Time history curves of top floor displacement 6. CONCLUSIONS AND GENERAL RECOMMENDATIONS
Based on all the analyses above, some conclusions could be derived from the test and analyses.
1. Tests of 20 local specimens of connection welded regions subjected to monotonic tensile and cyclic loadings were carried out. The effect of material strength, loading patterns and geometric parameters on welded fracture and damage were studied and the fitting analysis of the test results using common damage evolution were carried out. The test results showed that the stress amplitude index of cyclic loading played a decisive role in the weld damage.
2. The experiments of 9 full-scale beam-column connections under cyclic loading were carried out. The damage mechanism of welded connections and the effect of loading amplitude, peak position on connection damage were further studied. The influence of loading history was closely related to the peak displacement position and number of cycles. The sudden strong peak caused the most terrible damage on the connections.
3. The “effective plastic strain” was introduced as the damage development index of metal materials according to the metal ductile fracture theory. By using this index, a damage evolution equation was proposed based on the effective plastic strain. The ABAQUS subroutine to calculate the whole process of connection damage was developed by the damage evolution equation. The model was validated by experimental results.
4. The definition of "effective plastic rotation” of the connection was creatively put forward. The damage evolution equation with the indicator of effective plastic rotation was established. The simplified calculation equation of bilinear model considering the damage for the whole hysteretic curve was derived. The welded connection test results of this article and other researchers were used to prove the accuracy. Finally, the model was applied into the nonlinear dynamic time history analysis of steel frames, which indicated the method had a great value for engineering analysis and design application.
ACKNOWLEDGMENTS
This work described in paper was supported by the National Natural Science Foundation fo China (No. 90815004) and (No. 51038006).
REFERENCES
[1] Bannantine, J.A., Comer, J. J. and Handrock, J.L.(1990), “Fundamentals of metal fatigue analysis”, Prentice-Hall, Englewood Cliffs, N. J.
[2] Coffin, L. F. (1954), “A study of the effects of cyclic thermal stresses on a ductile metal”, Trans. ASME, Vol. 76, (pp. 931–950).
[3] Cosenza, E. and Manfredi, G. (1992), “Seismic Analysis of Degrading Models by means of Damage Functions Concept”, Nonlinear Seismic Analysis and Design of Reinforced Concrete Buildings.
[4] Dassault Systèmes. (2010), “Abaqus Analysis User's Manual”, Dassault Systèmes Simulia Corporation, Providence, RI, U.S.A,.
[5] Kato, B. and Akiyama, H. (1975), “Aseismic Limit Design of Steel Rigid Frames”, Proceeding of Architectural Institute of Japan, No.237.
[6] Kravinkler, H. and Zhorei, M. (1983), “Cumulative Damge in Steel Structures Subjected to Earthquake Ground Motions”. Computers and Structures, Vol. 16(1-4).
[7] Kumar, S. and Usami, T. (1994), “A note on evaluation of damage in steel structures under cyclic loading”, Journal of Structure Engineering, JSCE, Vol. 40A, (pp. 177-178).
[8] Liao, F.F., Wang, W. (2010), “Parameter calibrations of micromechanics-based fracture models of Q345 steel”. Science paper Online, <http://www.paper.edu.cn/ index.php/default/releasepaper/content/201007-457>.
[9] Manson, S.S. (1953), “Behavior of materials under conditions of thermal stress”, Proc., Heat Transfer Symp., University of Michigan Engineering Research Institute, Ann Arbor, Mich., (pp. 9–75).
[10] Park, A.J. and Ang, H.S. (1985), “Mechanistic Seismic Damage Model for Reinforced Concrete”. Journal of Structure Engineering, Proc. ASCE, Vol.111(4). [11] Paris, P. C. and Erdogan, F. (1963), “A critical analysis of crack propagation
laws”, Journal of Basic Engineering, Vol. 85, (pp. 528-534).
[12] Powell, G.H. and Allchabadi, R. (1988), “Seismic damage prediction by deterministic methods: Concept and procedures”, Earthquake Engineering and Structural Dynamics, Vol. 16, (pp. 719-734).
[13] SAC. (1997), “Protocol for Fabrication, Inspection, Testing, and Documentation of Beam-Column Connection Tests and Other Experimental Specimens”, Report No. SAC/BD-97/02, Michigan.
[14] SAC. (2000), “Protocol for Fabrication, Inspection, Testing, and Documentation of Beam-Column Connection Tests and Other Experimental Specimens”, Report No. SAC/BD-97/02, Lehigh University.
[15] Solomon, H.D. (1972), “Low cycle fatigue crack propagation in 1018 steel”, J. Mater., Vol. 7(3), (pp. 299–306).
[16] Kumar S. and Usami T., A Note on Evaluation of Damage in Steel Structures under Cyclic Loading, J. Struct. Eng. JSCE, 1994, 40A: 177-178.
Connections in Steel Structures VII / Timisoara, Romania / May 30 - June 2, 2012 233 234 Connections in Steel Structures VII / Timisoara, Romania / May 30 - June 2, 2012