This paper presents a refined analysis for evaluating lowcycle fatigue crack initiation life of welded beamtocolumn connections of steel frame structures under strong earthquake excitation. To consider different length scales between typical beam and column components as well as a few crucial beamtocolumn welded connections, a multiscale finite element (FE) model having three different length scales is formulated. The model can accurately analyze the inelastic seismic response of a steel frame and then obtain in detail elastoplastic stress and strain field near the welded zone of the connections. It is found that the welded zone is subjected to multiaxial nonproportional loading during strong ground motion and the elastoplastic stressstrain field of the welded zone is threedimensional. Then, using the correlation of the FatemiSocie (FS) parameter versus fatigue life obtained by the experimental crack initiation fatigue data of the structural steel weldment subjected to multiaxial loading, the refined evaluation approach of fatigue crack initiation life is developed based on the equivalent plastic strain at fatigue critical position of beam end seams of crucial welded connections when the steel frame is subjected to the strong earthquake excitation.
The brittle fracture of traditional beamtocolumn welded connections of steel frames is one of the most major causes for the failure of the steel structures under strong earthquake excitation. Seismic damage investigations after the Northridge and Kobe earthquakes show that fatigue crack initiation and propagation at or near the welded connections may cause serious seismic damage, which can endanger the safety of steel frame structures and even cause the collapse of steel frame structures [
Fracture of beamtocolumn welded connections.
Ibeam to Icolumn
Ibeam to boxcolumn
The fatigue cracks may initiate at the weld of the bottom flange of the beams near the connection, or sometimes at the weld of the top flange [
The current studies on fatigue damage of beamtocolumn welded connections are commonly based on the global seismic response at the ends of columns and beams of a steel frame structure, and the hysteretic characteristics between the internal forces and the displacement at the ends of component are used to assess the fatigue life of connections. Krawinkler and Zohrei [
Although some alternative connections, such as strengthening connection by adding cover plates and reduced beam section connection, are developed to prevent weld from lowcycle fatigue cracking by shifting maximum plastic strain position away from the column flange. However, when the steel frame structure with improved beamtocolumn connections is subjected to severe ground motion, it is still necessary to investigate the occurrence probability of plastic hinge at the beams ends and further fatigue failure risk analysis of the connections, which implies that thorough understanding of the local stress and strain fields is required.
A refined numerical analysis is proposed for fatigue crack initiation of beamtocolumn welded connections in this paper. A multiscale finite element (FE) model is formulated for the inelastic analysis of steel frames, where the model contains three different length scales elements to describe, respectively, beamcolumn components, selected critical beamtocolumn welded connections based on earthquake resistant conceptual design of steel frame structure and seismic fortification standard of “no damage under weak earthquake” and “no collapse under strong earthquake,” and welded seams at the connections. It is explored that the stain status of the fatigue critical points at weld seams of the crucial connections is threedimensional and their loading pattern is multiaxial nonproportional under strong earthquake excitation. Then taking the experimental relationship between the strain amplitude and the cyclic numbers prior to crack initiation of welded structural steel specimen into consideration, a refined analysis approach is proposed to evaluate fatigue crack initiation life of the welded connections based on the equivalent plastic strain of fatigue critical points.
The equations of motion for a multistory and highrise steel frame structure subjected to strong ground motion can be expressed as
It is known that under strong earthquake excitation, columntobeam welded connections and the partial regions of beams and columns near the critical connections of a steel frame may be in an elasticplastic stress state, while the remaining parts of the beams and columns are still in the linear elastic range. Although the elastoplastic analysis of the steel frame using fullscale threedimensional FE model is accurate, it is inefficient and time consuming when the steel frame structure is large. Therefore, the multiscale FE modeling is a proper choice for the refined local elasticplastic seismic analysis of the welded connections.
The connections are classified as critical ones and normal ones. The particular attention will be paid to the accurate elasticplastic stress and strain fields at the critical connections and at the beams and columns which are connected with these connections. The multiscale FE model consists of elasticplastic threedimensional beam and solid elements. The solid elements with millimeter scale are used for the welded seam zone of the critical connections; the solid elements with centimeter scale are used for the panel zone of critical welded connections and the regions of the beams and columns adjacent to the connections, while the beam elements with meter scale are used for the simulation of the other regions of the beams and columns far away from the key connections and other beams and columns. The multiscale elements of the FE model can be assembled together by assigning the coupling interactions between the elements boundary.
As an example, a fourstorey steel frame shown in Figure
Configurations and dimensions of the beam and column sections.
Component  Location  Dimensions 

Beam  Outer beams in 
H450 × 250 × 12 × 20 
Inner beams in 
H450 × 200 × 12 × 20  
Outer beams in 
H450 × 250 × 12 × 20  
Inner beams in 
H500 × 300 × 14 × 25  


Column  Square columns  □400 × 400× 20 × 20 
Numerical example of a 4story steel frame.
The welding method for the connection is manual arc welding with filler metal of E4301. All welds are single V groove weld. The construction materials for beams and columns are assumed to be Q235B. The details of welded connections are shown in Figure
Details of welded connections.
The commercial FE package ABAQUS [
The floor slabs, which play an important role in a steel frame to resist external loads effectively, are modeled by means of shell element S31 in the multiscale FE formulation. The seismic action is transferred by adding the rigid link interaction at common boundaries between the shell elements of slab and the solid element (or the beam element). The typical local multiscale FE model of the frame is shown in Figure
Multiscale FE model of the beamtocolumn welded connection.
To conduct the refined analysis, it is important to clarify whether the welding residual stresses play important roles in plastic strain at welded connections of steel frames under the strong ground motion excitation. Song [
The weldment specimens, whose base metal is structural steel of grade Q345 with yield strength of 425 MPa, with longitudinal seam at the midspan, are cyclic loaded with constant amplitude by fatigue test machine and the residual stress distribution along transversal cross section of the specimens is measured as shown in Figure
Transversal residual stress distribution of the weldment specimen subjected to cyclic loading.
When a steel frame structure is subjected to the strong earthquake motion, the welded connections may behave plastically and the stresses at the weld zone may exceed the yield strength of the base metal. Therefore, the welding residual stresses can be fully relieved according to the residual stress relief law, and then welding residual stresses are not taken into consideration in the refined seismic plastic response analysis in the present study.
When the frame is subjected to a strong earthquake excitation, the stresses of partial regions of beams and columns linked to the connection may enter elastoplastic range while the balance may keep in the elastic state. To ensure that the refined solid elements C3D8R can reflect the elastoplastic behavior of the connection and the beams elements B31 can model the elastic behavior of the beams and columns adequately, the coupling boundaries between beam elements and solid ones should be properly determined.
If an elastic beam component is subjected to bending moment, its deformation obeys the Euler Bernoulli hypotheses. Accordingly, the interfaces between the beam element and solid elements can be divided into two parts. Firstly, the refined modeling region is formulated to contain the beam and column components with sufficient lengths to ensure the elastic response of the coupling constraint sections; for example, the positions at the quarter length of the beam and column near the connection can be taken as the coupling sections. The kinematic coupling technique is employed for establishing coupling interactions between the solid elements C3D8R for connection regions and the beam elements B31. Secondly, sections of demarcation between the elastic region and the plastic region in the refined welded connection modeling at the maximum response time are used as the coupling sections to rebuild a multiscale FE model.
For the fourstorey steel frame shown in Figure
PEMAG of the bottom flange.
It can be seen in Figure
Plastic strain distribution of beam flange.
A typical welded connection of the Ibeam and boxcolumn is shown in Figure
Weld of the beamtocolumn connection.
It is assumed that large deformations of the welded seam occur along the longitudinal direction of the beam. In this case, the transverse deformations of the welded seam due to Poisson’s ratio should be restricted by the column flange, and then the local stress and strain states of the welded zones are threedimensional, which cannot be treated by the onedimensional classical beam theory. Therefore the multiscale FE model is adopted for the threedimensional inelastic analysis.
In the analysis, based on the experimental results the elastoplastic constitutive model is used to simulate the mechanical properties of the Q235B steel as shown in Figure
Constitutive model of Q235B steel.
When the steel frame is subjected to the earthquake of intensity 9, Figure
Plastic strain magnitude of the bottom flange weld at
The six stress time histories of the weld toe, that is, the fatigue critical position, are shown in Figure
Stress time history of the weld toe.
The normal stress distribution at the time
Normal stress distribution along the beam axial direction.
In order to select a proper approach for fatigue crack initiation analyses of beamtocolumn welded connection, the primary task is to evaluate the threedimensional strain state at the critical point, which may be produced either by proportional loading or by nonproportional loading. Based on multiaxial fatigue theory and experimental study [
Since the mechanism of the fatigue crack initiation for the structural steel satisfies the metal crystal slip theory, fatigue crack initiation occurs on the plane with the maximum shear strain, which can be defined as the critical plane according to the multiaxial fatigue theory [
The strain tensor at a material point can be written as
A point on the inclined plane can be selected as the origin of an
Strain acting on a plane in a threedimensional coordinate system.
Therefore the critical plane of maximum shear strain at a point of the weld zone can be located by a pair of angles
The FE results for the time histories of angles
Timevarying angle of maximum shear strain at the fatigue critical point in the case of rare earthquake.
Intensity of 6
Intensity of 9
To realize the refined analysis for fatigue crack initiation life of the welded connections of the steel frame under a strong earthquake excitation, experiments for determining fatigue lives under uniaxial and multiaxial loading conditions are conducted on the Q235B steel weldment.
The weldment has a tubular geometry with the outside and inside diameters of 18 mm and 14 mm, respectively. The wall thickness in the gage section is 2 mm. The total length of the weldment is 185 mm, and an 18 mm long welded zone is at the center of the gage length. The geometry of the weldment is displayed in Figure
Geometry of the welded metal specimen (unit: mm).
Multiaxial fatigue tests are carried out on a MTS809 tensiontorsion servohydraulic testing system under straincontrolled loading using a tensiontorsion strain extensometer, which is mounted at the center of the outside of the specimen gage section to measure the strain responses. The uniaxial, inphase, and 90° outofphase loading are sinusoidal wave with constant amplitudes and the three test strain paths are displayed in Figure
Fatigue test loading paths: (a) uniaxial (UA), (b) inphase (IP), and (c) 90° outofphase (OP).
The cyclic mechanical properties of metal materials can be expressed by RambergOsgood model as follows [
The cyclic stressstrain curves of Q235B weldment under different loading paths.
It can be seen from Figure
In the uniaxial fatigue analysis, the MansonCoffin equation in terms of the equivalent strain parameter is widely applied for fatigue life estimation.
Fatigue properties of Q235B welded specimens.
Fatigue properties  Uniaxial tests  Inphase tests  90° outofphase tests 


481.14  508.27  519.35 

0.0375  0.0654  0.0202 

−0.0755  −0.0708  −0.3555 

−0.3154  −0.0624  −0.3181 
Figure
Correlation of the equivalent strain parameters versus fatigue life.
Fatemi and Socie [
Correlation of the FS parameters versus fatigue life.
It can be seen from Figure
From Figure
The weighted mean value of maximum damage critical plane of the fatigue sensitive point is defined as
The normal strain and shear strain time histories of critical plane.
Based on the cyclic rainflow counting algorithm, the cycle counting number for shear strain range and the normal strain are obtained according to the normal strain and shear strain time histories of critical plane, as shown in Figure
Cyclic rainflow counting results of shear strain range and normal strain of critical plane.
Maximum shear strain range
Maximum normal strain
Substituting the cyclic rainflow counting results of shear strain range and normal strain during earthquake excitation time of 20 seconds into FS parameter, fatigue life
The paper presented a refined analysis of elastoplastic behavior of beamtocolumns welded connection of the steel frame under strong earthquake by using the multiscale FE modeling technique. The major findings are listed as follows:
The authors declare that they have no conflicts of interest.
The authors are grateful for the financial support from the National Natural Science Foundation of China under Grant no. 51438002 and no. 51378409.