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Structural analysis and optimization of steel truss bridge
based on finite element method
Xuran Yang1, 4, Xirui Li2, Puyu Sun3
1College of Transportation Engineering, Nanjing Tech University, Nanjing, 211816,
China
2School of International Education, Jinling Institute of Technology, Nanjing, 211169,
China
3School of Urban Construction, Yangtze University, Jingzhou, 434023, China
4202100141@yangtzeu.edu.cn
Abstract. In view of the defects of the current mainstream structural safety analysis method of
Truss bridge, the safety analysis method at the component level should be further developed.
Therefore, it is necessary to carry out the overall safety analysis and structural optimization at
the structural level. Based on the concrete data of the actual steel truss beam bridge, this paper
uses ABAQUS and MIDAS finite element software to model the truss structure. First of all, the
static analysis is carried out, and then the structural analysis and optimization of the steel truss
beam bridge is carried out with the elastic modulus reduction method. Based on the linear elastic
finite element method, the bearing state of each component is solved, and the ultimate bearing
capacity of the whole structure is calculated iteratively. Thus, the limitation of incremental
nonlinear finite element method is overcome, and higher calculation accuracy and efficiency are
realized.
Keywords: Steel Truss Bridge, Structural Optimization, Elastic Modulus Reduction Method,
Finite Element Analysis, Ultimate Bearing Capacity.
1. Introduction
Steel truss girder bridges have been used in Nanjing Dashengguan Yangtze River Bridge, Qiantang River
Bridge and other mega bridge projects because of their simple construction, high load carrying capacity,
high longitudinal and transverse stiffness and short construction period [1]. At present, bridge structures
are mainly analyzed for structural safety by comparing the cross-sectional internal forces and resistances
of the members based on the consideration of various influence coefficients, which is a safety analysis
method at the member level [2]. Although this method is simple and practical, it cannot grasp the
contribution of each component to the overall safety of the structure from the bearing state and failure
mode. It is difficult to optimize the bearing capacity distribution and material consumption of the
structure [3]. For this reason, it is necessary to carry out the overall safety analysis and structural
optimization at the structural level [4].
The finite element method is the first choice for structural analysis of truss bridges. The finite element
method, since the basic idea and method proposed by Martin, Tuner, Clough, etc. in the middle of the
20th century. And, it has developed into a rigorous theoretical foundation and a wide range of application
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DOI: 10.54254/2755-2721/25/20230750
© 2023 The Authors. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0
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analysis methods [5-8]. This paper combines the advantages and disadvantages of each of the two finite
element software, ABAQUS and MIDAS, for finite element modelling. The elastic modulus reduction
method (EMRM) proposed in recent years gives a new elastic modulus adjustment strategy based on the
conservation principle of element bearing ratio and strain energy. The influence of all internal force
combination effects of the section on member failure can be considered by introducing a generalized
yield criterion. By reducing the elastic modulus of high load-bearing members and iteratively calculating
the linear elastic finite element method, a series of static allowable internal force fields are formed in
the structure. The gradual failure process of components and structures is simulated, and the ultimate
bearing capacity of structures is solved based on it [9].
2. Finite element modeling
The finite element analysis method plays an important role in modern bridge design. The use of finite
element software to analyse the performance of steel truss bridge is not only more convenient and
intuitive, but also more accurate, which needs to be fully used to improve the level of structural design.
The modelling work is the most important part of the whole finite element analysis, and it is necessary
to choose the appropriate software to complete the modelling. However, different finite element analysis
software has been adopted for modelling and analysis of bridges, which have their own advantages and
disadvantages.
2.1. Selection of finite element software
Based on the use and summary of each finite element software, this paper adopts ABAQUS and MIDAS
software for finite element modelling and analyses the characteristics of the software to complement the
advantages and disadvantages of the two software.
MIDAS is more accurate for the whole structure calculation and rough for the node calculation, so it
is more suitable for the whole steel truss bridge modelling analysis. Different from MIDAS, ABAQUS
model and calculation are more accurate and more suitable for modelling and analysis of detailed
structures. However, the boundary conditions are required to be correct and comprehensive. In actual
engineering, it is difficult to extract the forces and moments at the detail structure to meet the pre-
processing data requirements of ABAQUS calculation [10].
2.2. MIDAS finite element modelling
MIDAS software is a data processing software based on structural design finite element analysis, which
has a broad application prospect in the bridge field. A steel truss bridge (Nanjing Dashengguan Yangtze
River Bridge) with a length of 1270m, a width of 34m and a height of 26.6m was constructed by using
MIDAS software with reference to an example steel truss bridge. The material of the bridge was set as
Q420 steel under GB03S standard, and the bridge was set to be constructed by six main members. After
the material and cross-section were defined, the relevant nodes were established first, and then the nodes
were connected sequentially with the set units to improve the connection between the truss structures
and finally form a complete bridge model (Figure 1).
Figure 1. Bridge modeling model and its related parameters.
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The relevant static analysis was performed, and a uniform load of 21kN/m was applied to the full
bridge for the load condition, and the calculations were run and entered the post-processing stage, where
the displacement and stress diagrams were obtained, with the following results (Figure 2 and 3):
Figure 2. Model Stress Diagram.
Figure 3. Model displacement diagram.
It can be seen that the internal force of the joist is mainly axial force, the upper chord is under
compression and the lower chord is under tension, the mid-span position is the maximum stress, and the
stress decreases to both sides, the vertical deflection of the mid-span node is the largest.
Since there is moment transfer between the rods of this steel truss bridge, the MIDAS model using
beam unit is more in line with the engineering reality and can simplify the calculation. The following
load types are added: bridge self-weight, overall warming, overall cooling, dynamic load, The internal
forces of the beam unit after different working conditions are extracted and recorded by MIDAS, and
the extracted data are loaded as part of the boundary conditions for ABAQUS calculation.
2.3. ABAQUS finite element modelling
In steel truss bridge, the load is transferred to the main truss node through each rod, and then to the
bearing and foundation. Therefore, the nodes play a pivotal role in the process of force transmission.
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ABAQUS is used to number the nodes of the steel truss and to build 2D and 3D models of the steel truss
structure in order to optimize the analysis of the truss in detail and accurately.
In this paper, the N-shaped steel truss of 108m side span bridge of Nanjing Dashengguan Yangtze
River Bridge is used as the research object for modelling, as shown in Figure 4. ABAQUS has a rich
library of material models, which can simulate most common engineering materials. The material of the
main truss is Q420qE steel. The material has a yield strength of 420MPa , a density of 7.85×103kg/m3,
a modulus of elasticity of E=2.1×1010N/m2 and a Poisson's ratio of 0.3. The main truss is 16m high with
12m long sections. The bridge main truss consists of upper chord, lower chord and web (diagonal and
vertical bars). The upper and lower chords are connected to the webs through integral nodes. The cross-
sectional area of upper and lower chords is set to 1.2×10-2m2 and the cross-sectional area of webs is set
to 1.4×10-2m2.
Figure 4. Steel truss modeling model.
After completing the modeling, the static analysis is performed first. The restraint is applied at the
two end supports; the load condition is 21kN/m for the whole bridge. From Figure 5, it can be seen that
the internal force of the truss is mainly axial force, the upper chord is under compression and the lower
chord is under tension, the maximum stress is at the upper side of the span and the stress decreases to
both sides, the vertical deflection of the nodes in the middle of the span is the largest. These are consistent
with the theoretical analysis and MIDAS analysis results.
Figure 5. Stress diagram of steel truss.
3. Elastic modulus reduction method
3.1. Stability limit bearing capacity solution method
The elastic modulus reduction method can strategically reduce the elastic modulus of high-bearing
elements in steel truss Bridges. The stiffness damage of steel truss bridge during loading can be
simulated by elastic modulus reduction method, and the stable ultimate bearing capacity of steel truss
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bridge can be calculated by linear elastic iterative analysis. For a steel truss bridge structure subjected
to n loads P1, P2,...Pn, the external load can be represented by the vector P as:
0012(1)
where and are the load reference value and load multiplier, respectively.
The unit load bearing ratio , as an important parameter of the modulus of elasticity reduction
method, characterizes the extent to which the section enters full section yielding. indicates that
the full section yields, while indicates that the section is unstressed. can be defined by using,
is the generalized yield function.
(2)
where k is the iteration step; e is the cell number.
Accordingly, the base load ratio for identifying the dynamic threshold of the high load cell can be
defined as: 0 (3)
(4)
where is the maximum cell load ratio in the structure; is the minimum cell load ratio in the
structure; is the uniformity of load ratio; is the total number of meshed cells.
Where the unit bearing ratio is greater than the base bearing ratio is the high bearing unit of the
iteration step. The elastic modulus reduction method combined with the principle of conservation of unit
deformation energy establishes the elastic modulus adjustment strategy [11-12].
(5)
where and
are the values of the elastic modulus of the cell e during the k and k+1 iterations,
respectively. When k+1 is used, takes the combined modulus of elasticity value .
Based on the results of the linear elastic finite element analysis, the unit load carrying ratio is
obtained. Since the unit load carrying ratio is proportional to the external load, the ultimate load carrying
capacity of the structure at the iteration step can be obtained from the maximum unit load carrying
ratio .
(6)
The above calculation process should be repeat to ensure that the ultimate bearing capacity of the
two adjacent steps meets the following convergence criteria.
(7)
where is the convergence tolerance. As a criterion of convergence, it usually takes the value of
0.001~0.01, and in this paper, it takes 0.001.
If the M iteration satisfies the convergence criterion of equation (7), then the ultimate bearing
capacity PL of the structure is:
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(8)
From modelling to iterative steps, the whole calculation process of elastic modulus reduction method
belongs to linear elastic iterative analysis, which does not involve nonlinear behaviour of materials. The
stability and high efficiency of the iterative calculation process are ensured [13].
3.2. Safety analysis method for truss members
According to the first iteration of the Elastic Modulus Reduction Method (EMRM), equation (2) is used
to obtain the unit bearing ratio of each member of the steel truss bridge under the design load , and
then the member safety factor is obtained as follows:
(9)
where represents the safety factor of the member unit e , which reflects the safety reserve condition
of the member under the combined effect of internal forces.
The structural load carrying capacity safety assessment of the bridge is performed at the member
level according to. When is used, it means that none of the members has failed. On the
contrary, when a member is , the member will enter the load carrying capacity limit state. Usually
the design of the structure requires a certain safety margin, not allowing to be too close to 1 [14].
3.3. Safety analysis method of truss structure
According to the results of the last step of EMRM elastic iterative analysis, the ultimate load of the
structure is obtained from PL, and then combined with equation (8), the overall safety factor of the truss
structure is obtained.
(10)
where KT is the overall structural safety factor or structural safety factor, which can reflect the overall
load bearing state and safety reserve condition of the bridge structure.
The overall safety of the bridge can be assessed at the structural level according to KT . When KT>1 ,
it indicates that the structure will not develop an overall failure mode under the current load bearing
condition. On the contrary, when KT≤1 indicates that the structure enters the load carrying capacity
limit state and will fail. Therefore, it is required that the structural safety factor should not be too close
to 1, so as to guarantee a certain margin of safety for the structure as a whole.
From equation (9), it can be seen that the overall structural safety factor KT can be obtained based on
the member with the largest unit load bearing ratio
in the last iteration of EMRM, according to
which the overall structural safety can be judged. It can be seen that the structural load capacity is closely
related to the member load capacity, and according to the correlation between the two, the structural
design can be optimized based on safety considerations.
4. Model solving analysis
According to the results of the first iteration of EMRM, the bearing ratio of each unit and the
corresponding safety factor can be obtained, and the geometric parameters of the bar sections are listed
in Table 1, and the steel truss model rod numbers are shown as Figure 6. Group 2 is all the lower chords
except those near the supports and Group 3 is the lower chords near the supports at both ends of the
bridge. As shown in Table 2, the results of the first iteration show that the safety factor of all members
of the steel truss bridge under this condition is greater than 2, which indicates that the structural design
meets the load carrying safety and no structural failure will occur under normal conditions.
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Table 1. Geometric parameters of rod section.
Rod category
Rod number
Cross-sectional area /m2
Cross-sectional moment of inertia/m4
Wind-up bar
3,30～37
1.2×10-2
1.44×10-5
Downswing bar
1,5～12
1.2×10-2
1.44×10-5
Ventral rod
2,4,13～29
1.4×10-2
2.04×10-5
Figure 6. Steel truss model rod numbering.
Table 2. Component safety factor.
Group number
Build Unit Number
Component safety factor
1
3, 30 to 37
2.63
2
5 to 11
4.21
3
1,12
7.53
Multiple iterations are carried out for the member unit bearing ratio. As shown in Figure 7, the
horizontal coordinate is the number of iterations and the vertical coordinate is the unit bearing ratio.
According to the last iteration result, the ultimate bearing capacity PL=61.73kN/m of the steel truss beam
bridge can be obtained, so the overall structural safety factor of the steel truss girder bridge is:
(11)
The combination of the member safety factor and the overall structural safety factor shows that the
bridge can maintain a certain safety reserve at both the member and structural levels without local failure
or overall failure.
Figure 7. Iterative process of component unit load ratio.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
12345678
Group 1 Group 2 Group 3
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5. Conclusion
In this paper, a steel truss bridge is constructed, and the bridge model is established by dividing the
discrete elements into truss beams, and the discrete elements are determined. MIDAS takes the whole
of the relevant bridge as the modeling reference object, while ABAQUS takes the relevant side span
bridge truss as the modeling object. After the modeling is completed, comparing the data obtained by
the two-software analysis, it is found that the internal force of the truss, the compression and tension
situation of the upper and lower chord and the maximum stress concentration position are the same, and
are consistent with the theoretical analysis. In structural analysis, EMRM and elastic modulus analysis
were adopted to determine the element bearing ratio and reference bearing ratio, and then the elastic
modulus of the element was reduced, and the ultimate load of the structure was determined by
calculating the specific iteration steps, that is, the rare load that the structural system could bear.
In this paper, the elastic modulus reduction method is used to solve the component safety factor and
the whole safety factor of the bridge structure, and then the safety of the steel truss beam bridge is
analyzed from the two aspects of the component and structure. The generalized yield criterion and elastic
modulus reduction method are introduced into the safety analysis of steel truss girder Bridges. The
influence of combined internal forces on structural safety is considered, and the problem that the
structural safety may be biased when the structural safety is evaluated according to a single internal
force is overcome. The EMRM structure optimization method adopted in the optimization design section
overcomes the defects of the traditional structure optimization design method. According to the variation
of the unit load ratio in the iterative calculation process, the high-load and low-load components of the
bridge structure can be identified. Finally, the structure is optimized by adjusting the sectional strength
of the high-load and low-load members. The optimization by using EMRM method not only makes up
for the defects neglected by traditional methods, which do not consider the influence of internal force
redistribution when the structure enters the elastic-plastic state or even the plastic limit state under rare
loads (that is, the ultimate load), so it cannot meet the strength requirements of the structural system. It
can also improve the utilization rate of materials, thereby reducing the consumption of materials and
maximizing the benefit of resources.
Since simplified measures were adopted in the modeling process of this study, and many factors were
not involved, the next step is to consider and analyze the effects of wind loads, automobile loads and
crowd loads on the structural strength and stiffness of steel truss bridges. In addition, natural disasters
have a great impact on the destruction of buildings, especially earthquakes, which can cause damage
and even collapse of bridges, threatening the life safety of citizens and causing huge losses to the national
economy. Therefore, seismic analysis and seismic response analysis of bridges are particularly important.
It is found through investigation that the seismic resistance of the whole bridge structure can be further
studied through inverse spectrum analysis.
Authors Contribution
All the authors contributed equally and their names were listed in alphabetical order.
References
[1] JTGD60 2004 General Specification for the Design of Highway Bridges and Culverts (Beijing:
China Construction Industry Press)
[2] JTG/TJ21 2011 Highway Bridge Load Capacity Testing and Assessment Procedure (Beijing:
People's Traffic Publishing House)
[3] JTG/TH21 2011 Technical condition assessment standard for highway bridges (Beijing: People's
Traffic Publishing House)
[4] Li Y 1996 Code-based assessment for load-carrying capacity of bridges Bridge Constr. 26(2) 61-
63
[5] Zienkiewicz O C and R.L.Taylor 2005 The Finite Element Method,5thed (Beijing: World Book
Publishing Company)
Proceedings of the 2023 International Conference on Functional Materials and Civil Engineering
DOI: 10.54254/2755-2721/25/20230750
140
[6] Kuichiro Wasutsu 1984 The Variational Method in Elasticity and Plasticity (Beijing: Science
Press)
[7] Feng K and Shi Z C 1981Mathematical Theory of Elastic Structure (Beijing: Science Press)
[8] Qian W C 1980 Variational Method and Finite Element (Beijing: Science Press)
[9] Yang L F, Yu B and Qiao Y P 2009 Limit bearing capacity analysis of frame using elastic modulus
reduction method Acta Mechanica Solida 22(2) 109-115
[10] Zhang Q, Gao J Q and Liu P 2020 Finite element analysis of steel nodes of steel truss bridges
Highway 65(07) 135-140
[11] Yang L F, Yu B and Ju J W 2015 Incorporated Strength Capacity Technique for Limit Load
Evaluation of Trusses and Framed Structures under Constant Loading J. Struct. Eng. 141(11)
4015023
[12] Yang L F, Li Q and Zhang W et al. 2014 Homogeneous Generalized Yield Criterion Based Elastic
Modulus Reduction Method for Limit Analysis of Thin-Walled Structures with Angle Steel
Thin-Walled Struct. 80 (9) 153-158
[13] Xie W W, Ye Z Q and Yang L F. 2018 Iterative linear elastic method for stable ultimate bearing
capacity of steel-tube concrete arch bridge China Railway Sci. 39(01) 39-48
[14] Yang L F, Li Q and Zhang W 2013 Analysis and optimization of two-level load bearing capacity
of steel truss girder bridge structure Civil Constr. Envir. Eng. 35(06) 51-57
Proceedings of the 2023 International Conference on Functional Materials and Civil Engineering
DOI: 10.54254/2755-2721/25/20230750
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