The purpose of this research was to investigate the effect of geometric variables on the bolt load distributions of a coldformed steel bolt connection. The study was conducted using an experimental test, finite element analysis, and an analytical method. The experimental study was performed using singlelap shear testing of a concentrically loaded bolt connection fabricated from G550 coldformed steel. Finite element analysis with shell elements was used to model the coldformed steel plate while solid elements were used to model the bolt fastener for the purpose of studying the structural behavior of the bolt connections. Material nonlinearities, contact problems, and a geometric nonlinearity procedure were used to predict the failure behavior of the bolt connections. The analytical method was generated using the spring model. The boltplate interaction stiffness was newly proposed which was verified by the experiment and finite element model. It was applied to examine the effect of geometric variables on the singlecolumn multibolt connection. The effects were studied of varying bolt diameter, plate thickness, and the plate thickness ratio (
A bolt connection is generally used in coldformed steel structures. It can result in a highstress concentration at contact regions, especially with thin coldformed steel members. The shearing of coldformed steel, bearing on material in front of the bolt, tearing of the coldformed steel in the net section, and shearing of the bolt are the main failure modes of a single bolt connection. In multibolt connections, the failure modes are identified by numerical and analytical methods, with the results showing combined bearing failure and netsection failure modes [
Stages of force distribution in bolt connections [
In the first step, the connection evolves elastic behavior with the outer bolts being more loaded than the inner ones. In the second step, the outer bolt rows reach their elastic limit strength but the inner bolt rows are still in the elastic stage. In the third step, the plastic behavior is distributed to the inner bolt rows. In the fourth step, the plastic behavior is developed in all the bolt rows. However, the limitation of the coldformed ductility between the boltplate in bearing may be reached before the connection develops full plastic strength. These failures result in a lower average bearing stress value per bolt hole of the multibolt connection. In this case, the connection would not develop the ultimate bearing capacity as recommended by the design method [
This study describes the method of analysis for determining the load distribution among bolts in a singlecolumn multibolt connection. A boltplate interaction stiffness equation is proposed which is verified by the experimental test and finite element analysis (FEA). The experimental investigation was performed using lap shear bolt connection testing and the connections were fabricated from coldformed steel.
The laboratory test was performed using G550 coldformed steel with a plate thickness of 1.0 mm for the bolt connection test. G550 is a hotdipped zinccoated structural steel and has a minimum yield strength of 550 MPa. The material properties for G550 steels were specified by following ASTM Standards A37007 [
Material properties of coldformed steel, G5501T.
Test number  SD  COV  

CT1  CT2  CT3  CT4  CT5  Mean  

1.045  1.045  1.044  1.045  1.045  1.04  0.0004  0.04 

12.64  12.65  12.59  12.58  12.52  12.59  0.0522  0.41 

608.47  605.77  605.90  617.50  606.46  608.82  4.9715  0.81 

627.00  622.50  625.50  630.00  620.00  625.00  3.8890  0.62 

213.51  213.49  213.49  213.58  213.58  213.53  44.1757  0.02 

1.03  1.02  1.03  1.02  1.02  1.02  0.0054  0.53 
Material properties of bolts.
Test number  SD  COV  

BT1  BT2  BT3  BT4  BT5  Mean  

4.93  4.97  4.97  4.97  4.94  4.95  0.019  0.393 

930.00  949.00  927.00  976.00  872.00  930.80  38.232  4.107 

963  930  963  997  909  952.40  33.908  3.560 

204.08  204.08  204.09  204.10  204.09  204.08  7.385  0.003 
The bolt connection specimen and the hole size are shown in Figure
Bolt connection specimens.
Bolt connection geometries
Hole size
The bolt connection specimens were tested using a universal testing machine (UTM) in the structural laboratory of Kasetsart University, Bangkok, Thailand, as shown in Figure
Bolted connection test setup.
Test apparatus
Test specimen
In the present study, the ANSYS finite element package [
Geometry, element type, and material model of FEA.
Geometry model
SHELL281
SOLID186
Contact element
Pure penalty algorithm
Mesh and boundary condition
Actual stress and plastic strain
Threedimensional, 20node solid elements (SOLID186) with three degrees of freedom at each node were used to model the steel bolts and the washers, as shown in Figure
The load capacity and failure mode of single bolt connection tests are shown in Figure
Summary of the lap shear connection results.
Test  Loading capacity, 
Failure mode 


S1  10,812  Bearing  0.97 
S2  10,660  Bearing  0.95 
S3  10,707  Bearing  0.96 
S4  10,800  Bearing  0.97 
S5  10,758  Bearing  0.96 
Mean  10,747  0.96  
FEA  11,167  Bearing 
Loaddeformation curves and failure mode of single bolted connection.
Loaddeformation curves
Failure mode
In order to identify the connection failure modes, the VonMises stress contour was normalized by the ultimate stress of the coldformed steel material as shown in Figure
Stress normalization and strain distribution of bolt connection.
HartSmith [
In multibolt connections, the bolts in different rows carry different amounts of the load which depend on the bolt diameter, plate stiffness, and other geometry configurations [
Deformation of single bolt connection.
An analytical beam theory with the shear deformation and rotational bending effects included is the theory known as Timoshenko beam theory [
Forces in bolt.
The bolt tolerated the bearing force, shearing force, and bending moment initiated by an eccentric load from the steel plates. The statically indeterminate system was divided in Figure
Statically indeterminate system.
A cantilever beam with partially uniform loads and end moments idealizes the load components of a steel bolt. The superposition analysis method was used to analyze the bending deformation and shear deformation for a cantilever beam carrying a uniform load of intensity over part of the span as shown in Figures
Bending deformation of cantilever beam.
Shear deformation of cantilever beam.
The bending moment was introduced by eccentric force from the plate. It was derived by sum of the rotation of the neutral axis at the end of the bolt as shown in Figure
Rotation of the bolt.
The bending rotation and shear rotation components can be written as (
Stiffness components of single bolt connection.
The boltplate interaction stiffness (
In the bending stiffness component, the bolt deformation was analyzed using (
The flexural deformation of the bolt was calculated by sum of deformation at point
Flexural deformation of the bolt.
The flexural deformations of the bolt at point
The bolt bending flexibility (
The Timoshenko beam theory [
Shear deformation of the bolt.
The deformations of the bolt at point
Schematic diagram of bearing deformation in the plate and bolt.
Bearing deformation in the plate
Bearing deformation in the bolt
A common method for modeling a connection stiffness association of the single bolt connection is represented by the spring model as shown in Figure
The spring model of single bolt connection.
Connection model
Spring model
Plate stiffness
The matrix equation represented a single bolt connection which can be written as
The plate stiffness was calculated in two zones as shown in Figure
Plate stiffness.
Stiffness  Stiffness equation  Value (N/mm) 



1/63692 


1/63692 

Equation ( 
1/50660 
The stiffness, load vector, and the node displacement matrix have been implemented in the Maplesoft program [
Comparison of the loaddeformation curve of the proposed spring model.
Comparison with tests and FEA
Comparison with other equations
The Swift equation [
An application of boltplate interaction stiffness analysis for the multibolt connection is presented by the triple bolt lap connection as in Figure
Multibolt connection analysis.
The matrix equation representing the multibolt spring model in Figure
Calculation of stiffness.
Stiffness equation  Stiffness component 














Equation ( 
The multibolt connection stiffness.
Multibolt connection spring model
Plate stiffness
Connection load and load distribution of analytical model and FEA.
Finite element model of triple bolt connection
Connection load
Load distribution
The analytical spring model as in the previous section was used to examine the effects of various connection parameters. The parametric study calculated the load distribution of the bolt connection with variations in the bolt diameters, plate thickness (
Concept of the sensitivity analysis.
Variation in the bolt diameter introduced efficiency of load distribution shown in Table
Bolt load distribution for variation in bolt diameter.

Bolt load distribution (%)  Efficiency, 


Bolt 1  Bolt 2  Bolt 3  
4  34.99  30.03  34.99  90.09 
6  34.53  30.95  34.53  92.84 
8  34.27  31.46  34.27  94.37 
10  34.11  31.78  34.11  95.33 
12  34.00  32.00  34.00  95.99 
Bolt load distribution and sensitivity analysis results of varying bolt diameters.
Bolt load distribution
Parameter sensitivity
The efficiency of bolt load distribution had small changes with increased plate thickness as shown in Table
Bolt load distribution for variation in plate thickness.

Bolt load distribution (%)  Efficiency, 


Bolt 1  Bolt 2  Bolt 3  
0.40  34.75  30.51  34.75  91.52 
0.60  34.72  30.56  34.72  91.69 
1.00  34.67  30.67  34.67  92.01 
1.40  34.59  30.83  34.44  92.62 
1.60  34.48  31.05  34.48  93.14 
Bolt load distribution and sensitivity analysis results of varying plate thickness.
Bolt load distribution
Parameter sensitivity
The efficiency of load distributions was changed suddenly with changes to the
Bolt load distribution for variation in plate thickness ratio (

Bolt load distribution (%)  Efficiency, 


Bolt 1  Bolt 2  Bolt 3  
1  34.72  30.56  34.72  91.69 
2  38.14  30.65  31.21  91.95 
3  39.67  30.75  29.59  88.76 
4  40.39  30.86  28.75  86.25 
5  40.69  30.98  28.33  84.98 
Bolt load distribution and sensitivity analysis results of the plate thickness ratio (
Bolt load distribution
Parameter sensitivity
The efficiency of load distribution decreased when the
Effect of the
From the parametric study results, it was clear that varying the geometries influenced the connection stiffness, which significantly affected the load distribution. The sensitivity analysis results of the multibolt connection from variation in the bolt diameter, plate thickness, and
The bolt load distribution efficiency of a coldformed steel, multibolt connection was presented. The analytical precedence with a spring model was used to simplify the stiffness of the bolt connection. A new boltplate interaction stiffness equation was proposed which was verified by the FEA and experimental tests. The spring model with boltplate interaction stiffness was used to estimate the bolt load distribution of the multibolt connection. The results showed that the analytical model was accurate and had good correlation with the FEA. Furthermore, the analytical procedure was used to examine the efficiency of load distribution for the multibolt connection and these results showed an uneven sharing of the load among bolts with one of them carrying the major component of the load. Thus, the stress distribution of the major load carrying hole resulted in highstress concentration which caused weakness in the mechanically fastened connection. Moreover, increasing
The authors declare that they have no conflicts of interest.
The authors wish to acknowledge the Kasetsart University Research and Development Institute (KURDI) for providing funding support and BlueScope Lysaght (Thailand) Limited for material support.