The nonlinear stiffness matrix method was incorporated to investigate the structural performance of steel portal frames with semirigid connections. A portal frame with unstiffened extended endplate connection was designed to demonstrate the adequacy of the proposed method. Besides, the seismic performance of steel portal frames with semirigid connections was investigated through time history analysis where kinematic hysteresis model was assigned to semirigid connections to account for energy dissipation and unloading stiffness. Based on the results of the study, it was found that generally semirigid connections influenced the force distribution which resulted in the decrease in base shear and lighter frame compared to the rigid one. The results also indicated that there was no direct relationship between maximum displacement at the top and connection stiffness in highrise frames.
The structural behaviour of steel portal frame is mainly associated based on its connection’s performance. Accordingly, the accurate modelling of steel portal frame needs to take into account realistic connection modelling if an accurate response is desired to be achieved. It is a usual engineering practice to consider either perfect simple or fully rigid connections between beam and column. Experimental tests, however, have acknowledged the real behaviour of beamtocolumn connections in someplace between these two unrealistic models that possess remarkable flexibility. Based on the majority of design regulations it is only necessary to consider the connection flexibility for the third category; however, it is predicted that majority of beam to column connection types have semirigid performance in some fashion.
The rigidity of beamtocolumn connections have relationship with geometrical factors of the connection components, that is, angle section, bolts size, and endplate. Widespread research projects that consist of numerical and experimental tests have been done so far to demonstrate momentrotation relationship which is suitable to estimate the ideal behaviour of semirigid connections [
It appears from the aforementioned investigations that numerous researches have studied the effects of frames with semirigid connections. However, far too little attention has been paid to the hysteresis modelling of semirigid connections to account for decreasing energy dissipation and unloading stiffness with increasing plastic deformation. The present study explores the influence of semirigid connection on structural performance of steel portal frame through analytical study. The seismic performance of steel portal frames also was investigated through nonlinear time history analysis where kinematic hysteresis model was assigned to semirigid connections to consider energy dissipation capacity and stiffness degradation. The required parameters for the abovementioned hysteresis models were extracted from eight fullscale experimental tests results of unstiffened extended endplate connections conducted at Universiti Teknologi Malaysia.
According to the AISC, the beamtocolumn connection is classified based on the characteristics of momentrotation (
If
Momentrotation curves for connections [
The maximum moment can be carried out by connection introduced as
In Eurocode 3 Part 1–8 [
A supposedly simple connection should be capable of transferring the external forces, without increasing substantial moments that influence adversely members or the structure as a whole. According to Eurocode 3 Part 1–8, the connections are considered as nominally pinned if
The beamtocolumn connections that do not address the criteria for FR connections or a simple connections shall be classified as partially restrained (PR) or semirigid connections. PR connections provide an anticipated deformation between connected members, based on the (
Analysis of semirigid frames requires accurately predicting the connection’s performance. Nonlinear behaviour of connections through momentrotation curve and some of the analytical method are used to predict them. This analytical method is defined as the momentrotation relationship achieved by a decent curve to fullscale experimental test results. One of the most famous methods which is used in this study is suggested by Frye and Morris. This method is characterized through an odd power polynomial as
Chen and Lui [
The main drawback of this formulation is that the tangent connection stiffness may become negative at some value of connection moment
During the analysis of steel portal frame with semirigid beamtocolumn connections, the influence of connection flexibility is considered through assigning rotational joint possessing stiffness
(a) Forces and displacements and (b) rotations in the semirigid frame component.
The nonlinear stiffness matrix of component
In the above equations
By using (
As can be seen in Figure
Assume
Compute the updated tangent matrix
Calculate
Add
Repeat steps (
NewtonRaphson solution, one iteration.
To show the adequacy of suggested method, the moment steel portal frame having semirigid connections was considered as shown in Figure
Unbraced steel portal frame with semirigid connections [
The proposed connections are designed using unstiffened extended endplate as shown in Figure
Unstiffened extended endplate connection as partial strength connections.
To show the effects of semirigid connections on structural performance, the analyses of portal frame with rigid connections were also considered. Both steel frames design was performed in accordance with the 2010 AISC direct analysis method (AISC 36010/IBC 2006). The results of both frames are summarized and given in Table
Structural performance of frame with semirigid and rigid connections [
Component number  Component type  Rigidframe  Semirigid frame 

1  Column  W24 
W21 
2  Column  W21 
W21 
3  Column  W18 
W6 
4  Column  W21 
W24 
5  Column  W16 
W18 
6  Column  W12 
W18 
7  Beam  W18 
W16 


Total weight (kg)  4574.00  3938.10  
Top storey displacement (cm)  0.96  1.20  
Maximum interstorey drift  0.48  0.52 
To evaluate the connection semirigidity effects on steel portal frames subjected to the ground acceleration 5, 10, and 15storey frames were considered as shown in Figure
Sectional properties of 5, 10, and 15storey frames.
Storey  5storey frame  10storey frame  15storey frame 

Column section  


1–3  W12 
W27 
W33 
4–6  W12 
W24 
W30 
7–9  —  W24 
W24 
10–12  —  W14 
W21 
13–15  —  W16 



Beam section  


1–3  W14 
W14 
W14 
4–6  W14 
W14 
W14 
7–9  —  W14 
W14 
10–12  —  W12 
W12 
13–15  —  W10 
5, 10, and 15storey portal frames.
For each analysis, kinematic hysteresis models along with individual stiffness were assigned to the connections. The stiffness was calculated from 6 fullscale experimental tests of flush endplate connection with variable parameters including number of bolt rows, thickness of endplate, and bolts size. A test rig was considered to accommodate a 3 m height column and a 1.3 m span of cantilever beam as shown in Figure
Test rig for fullscale testing.
The momentrotation curves for all the six beamtocolumn connections are shown in Figure
Momentrotation curves of flush endplate connections.
Failure mode at the end of the test.
The results of the initial stiffness, moment capacity, and maximum rotation for all the six beamtocolumn connections are given in Table
Results of all the six beamtocolumn connections.
Specimen  Moment of inertia of beam (cm^{4})  Number of bolts in row  Diameter of slot 
Endplate thickness 
Moment 
Rotation, 
Initial stiffness, 
Max rotation at max load, 

FEP 1  3450  1  20  12 
35.1  11.3  3.1  104.9 
FEP 2  3450  1  24  15 
70.3  12.4  5.6  96.5 
FEP 3  23457  2  20  12 
81.5  6.8  12.0  39.8 
FEP 4  23457  2  20  12 
95.0  6.0  15.8  45.4 
FEP 5  55481  2  24  15 
200.0  6.0  33.0  79.2 
FEP 6  55481  2  24  15 
192.0  5.2  36.9  42.9 
Hysteresis is the process of energy dissipation through deformation (displacement) that affects nonlinear static and nonlinear time history load cases. Several different hysteresis models are available to describe the behaviour of different types of materials. For the most part, these differ in the amount of energy they dissipate in a given cycle of deformation and how the energy dissipation behaviour changes with an increasing amount of deformation. Typical for all models, cyclic loading behaves as follows:
Initial loading in the positive or negative direction follows the backbone curve.
During the reversal deformation, unloading occurs along a different path, usually steeper than the loading path.
After the load level is reduced to zero, continued reversal of deformation causes reverse loading along a path that eventually joins the back bone curve on the opposite side.
Figure
To perform nonlinear dynamic analysis, it is crucial to select earthquake records proportional to the geotechnical properties and soil conditions of the site. In this research the intended frames have been designed on rock beds of soil site class B in compliance with response spectrum of IBC 2009 code. Accordingly, 10 ground motion records were considered from Pacific Earthquake Engineering Research Center (PEER) as given in Table
Different places of ground motions as per PEER records.
Earthquake  Station  PGA (g)  Earthquake  Station  PGA (g) 

Northridge  24087 ArletaNordhoff  0.344  Loma Prieta  47381 Gilroy Array #3  0.555 
Northridge  24278 CastaicOld Ridge  0.217  Victoria, Mexico  6604 Cerro Prieto  0.621 
Northridge  24303 LAHollywood  0.358  Westmorland  5051 Parachute Test Site  0.242 
Northridge  24514 SylmarOlive View  0.535  Kern County  1095 Taft Lincoln School  0.178 
Loma Prieta  1028 Hollister City Hall  0.247  CapeMendocino  89324 Rio Dell Overpass FF  0.385 
Figures
To study the seismic performance of the semirigid and fully rigid frames, the time history analyses were compared with pushover analysis. As can be seen in the Figures
Kinematic hysteresis model.
Base shear versus displacement at the top (5storey frame).
Base shear versus displacement at the top (10storey frame).
Base shear versus displacement at the top (15storey frame).
This study evaluates seismic performance of steel portal frames with semirigid connections. The nonlinear stiffness matrix method was developed to investigate the effects of beamtocolumn connection rigidity and geometric nonlinearity in the seismic response. Besides, three portal frames with different connection stiffness were taken into consideration and their seismic performance was evaluated through time history analysis. The following points emerged from the present investigation:
It was noticed that the semirigid connection modelling produced lighter frames compared to the rigid one.
Beamtocolumn connection flexibility affects the force distribution in the frame and causes decrease in the base shear.
It was concluded that there is no linear relationship between connection stiffness and maximum displacement at the top. The maximum displacement at the top in highrise frame is mainly controlled by frame properties and ground motion level.
Current design code does not take into account adequate design method for frames with semirigid connections for high seismic areas. Specially, for research concern the seismic force distribution and the analysis subjected to the gravity loads need further investigation. Considerably more work will need to be done to determine the cyclic performance of partial strength/semirigid connections.
The authors declare that they have no conflicts of interest.
The authors wish to thank the esteemed technical staff of the Structures and Materials Laboratory, Universiti Teknologi Malaysia (UTM), for their cooperation on and support of this study. Financial support provided by the Universiti Teknologi Malaysia Construction Research Centre (CRC) for conducting the experimental work was invaluable and authors remain obliged.