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Case Studies in Structural Engineering 6 (2016) 53–62

Contents lists available at ScienceDirect

Case Studies in Structural Engineering

j ourna l h om epa ge: w ww.elsev ier .com/ locate /csse

n approximate method of dynamic amplification factor forlternate load path in redundancy and progressive collapseinear static analysis for steel truss bridges

oang Trong Khuyena, Eiji Iwasakib,∗

Graduate School of Engineering, Nagaoka University of Technology, 1603-1 Kamitomioka Nagaoka, Niigata, JapanDepartment of Civil and Environmental Engineering, Nagaoka University of Technology, 1603-1 Kamitomioka Nagaoka, Niigata, Japan

r t i c l e i n f o

rticle history:eceived 22 March 2016eceived in revised form 30 May 2016ccepted 6 June 2016vailable online 7 June 2016

eywords:ynamic amplification factoredundancyrogressive collapseteel truss bridgeslternate load path

a b s t r a c t

Linear static analysis with an alternate load path using dynamic amplification factor (DAF)is often used for redundancy and progressive collapse analysis of steel truss bridges toavoid using the more time-consuming dynamic analysis. This study presents an empiricalequation to calculate the DAF for this type of analysis against the initial sudden memberfracture. Currently, this analysis employs an approximate model with a single degree offreedom to calculate the DAF. With a 5% damping ratio, the constant DAF of 1.854 is usedfor all types of steel truss bridges. However, this approach is inaccurate because the DAFvaries between bridges and with the location of the fractured members as well. Consideringsome of the approaches developed for building structures but adapting them to steel trussbridges, this paper proposes an empirical equation that allows for the computation of theDAF from the maximum norm stress �is/�iy in static linear elastic analysis of the damagedmodel with a member removal. A total of 30 illustrative cases for two typical steel trussbridges are investigated to obtain the data points for the empirical equation. The proposedempirical equation is the enveloped line offset from the best fit line for the data points inillustrative cases.© 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC

BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

. Introduction

Progressive collapse is the spread of an initial local failure from element to element, member to member, eventuallyesulting in the collapse of a part, entire structures or a disproportionately large part [1]. A sudden member failure is aynamic event in which the structural motion is initiated by energy released by the sudden loss of a load-carrying member.our methods, including linear static analysis, nonlinear static, linear dynamic and nonlinear dynamic methodologies, arevailable for redundancy and progressive collapse analysis of structures for the sudden fracture of a member or component2,3]. The event of a sudden member fracture relates to both the primary loading, which causes the initial fracture, and

mpact loading, which causes structural motions after the initial fracture. The dynamic method is a direct solution to addressmpact loading and the dissipation procedure of the energy induced by the initial member fracture. This approach is accurate,ut it requires much intensive computation with time-history transient analysis. Static analysis with an alternate load path,

∗ Corresponding author.E-mail addresses: s115081@stn.nagaokaut.ac.jp (H.T. Khuyen), iwas@nagaokaut.ac.jp (E. Iwasaki).

http://dx.doi.org/10.1016/j.csse.2016.06.001214-3998/© 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/

icenses/by-nc-nd/4.0/).

54 H.T. Khuyen, E. Iwasaki / Case Studies in Structural Engineering 6 (2016) 53–62

Fig. 1. Illustration of the procedure to obtain data points of DAF versus max normalized stress and how to find the empirical DAF formula.

which amplifies the primary loading with a dynamic amplification factor (DAF) to form the impact loading, is an alternativeapproach for analysis without using dynamic analysis.

Currently, linear redundancy and progressive collapse linear static analysis of steel truss bridges have employed a singledegree of freedom (SDOF) model to conventionally calculate the DAF [4,5]. With a 5% damping ratio, the conventional DAF is1.854, constant for all bridges. This approach is conservative because the bridge system acts as multiple degrees of freedominstead of a single degree of freedom. The DAF varies between bridges and with the location of the fractured members, aswell.

To consider a model with multiple degrees of freedom, Goto et al. propose the root mean square mode combinationmethod to approximate the DAF [6]. This approach is moderately accurate and requires some correction factors. Althoughno other studies have yet been published about the approximation of the DAF for bridge systems, such approaches by Liu[7], McKay et al. [8], DoD, U.S. [9], and Stevens et al. [10] that approximate DAF in a building system are valuable. McKayet al., DoD, U.S., and Stevens et al. propose different linear functions of norm rotation, which is the ratio of the total memberrotation to the member-yield rotation, to compute the DAF of steel buildings. On the other hand, Liu computes the DAF byusing the function of max(Mu/Mp), where the max operator is applied to all affected beams that are directly adjacent toand above the removed column. Mu and Mp are the factored moment demand under the original unamplified static gravityloads and the factored plastic moment capacity, respectively, of an affected beam. These approaches may be limited to onlyone building system because the norm rotation and Mu/Mp are critical parameters for the behavior of a building system. Ina steel truss bridge system, when a member fractures, in addition to axial force, the members are also subject to bendingmoments. Considering this behavior, this study proposed the DAF as a function of the maximum norm stress �is/�iy, where�is and �iy are stress in a static analysis and the yield stress of bridge members. In this paper, a total of 30 illustrative casesare investigated in 3D models. The empirical equation to calculate the DAF was defined as the enveloped line offset fromthe best fit line for the data points from illustrative cases.

2. New DAF calculation method and analysis procedure

The empirical equation to calculate the DAF is defined as a function of the maximum norm stress �is/�iy, where �is and�iy are the stress in static analysis and yield stress of the ith bridge member. For a given member fracture scenario, the DAF isobtained by the stress DAF and is then confirmed by an alternate static analysis with amplified loading using the calculatedamplification factor to ensure that the structural responses best match those from the linear dynamic analysis. The processof computing the DAF in a given damaged scenario undertakes the following procedure, as in Fig. 1.

Step 1: Statically apply the primary load GL, as defined in Section 3.3, to the damaged bridge, remove the member thatis being fractured and perform the static linear elastic analysis. Then, measure the norm stress�is/�iy of the members andthe maximum �is/�iy, where �is and �iy are stresses in the static analysis and yield stress of the ith bridge member.

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H.T. Khuyen, E. Iwasaki / Case Studies in Structural Engineering 6 (2016) 53–62 55

Step 2: Carry out the linear dynamic analysis to obtain the position of maximum vertical deflection during time history.he time loading is shown in Fig. 7. To do so, the analysis involves a linear elastic analysis of the intact bridge with therimary loading GL to measure the end forces F of the to-be-removed member. The analysis then statically applies therimary loading GL and end forces in the damaged bridge model with a member removed to make sure that the deflectionf the model is the same as the intact bridge just before the damage. In this step, the physical function of the to-be-removedember is replaced by its end forces. Then, the end force F is suddenly forced down to zero to make a sudden fracture of theember. The next step is to record the chord deflection and demand-capacity ratio R. In the standard definition, the DAF is

he ratio of deflections or stresses in the dynamic analysis of deflections or stresses in the static analysis [11]. Hence, in thistudy, the DAF is calculated through Eq. (1). Normalizing the stress by the yielded stress, Eq. (1) can be rewritten as Eq. (2).he system DAF is the slope of the best fit line of the data points of �idm/�iy and �is/�iy. This is the trial DAF, which needso be confirmed in Step 3.

DAFi = �idm

�is(1)

DAFi = �idm/�iy

�is/�iy= Nidm/Ni0 + Mxidm/Mxi0 + Myidm/Myi0

Nis/Ni0 + Mxis/Mxi0 + Myis/Myi0(2)

here �idm, Nidm, Mxidm, and Myidm are the maximum dynamic stress of a member, dynamic axial force, and bendingoments of the ith member, respectively, after member failure in the dynamic analysis at the time step when vertical

isplacement reaches the peak. At this time step, the stress reaches the maximum value as well. �is, Nis, Mxis, and Myis arehe corresponding static stress, static corresponding axial force, and moments of the ith member, respectively, after the

ember failure in the linear static analysis. �iy is the yield stress of the ith member. Ni0, Mxi0, and Myi0 are the initial yieldapacity of the section of the ith member.

Step 3: Alternate static analysis is executed with an amplified load, which is multiplied by the trial DAF in Step 2 to confirmhether the deflection and demand-capacity ratio, R, are the same as in the linear dynamic analysis. The R is calculated by

qs. (3) and (4) [3,4].

R = NNP

+ Mx

Mpx+ My

Mpy(3)

R = PPu

+ 11 − P/Pex

Meqx

Mpx+ 1

1 − P/Pey

Meqy

Mpy(4)

n the above equations, N and P are the tension and compression forces, respectively, and Mx and My are bending momentsround the strong and weak axis, respectively. Np, Mpx and Mpy are the plastic axial strength and full plastic momenttrengths around the strong and weak axis, respectively. Pe is the Euler buckling load. Pu is the ultimate compressiontrength associated with the global buckling. The equivalent uniform moments, Meqx, and Meqy are used to convert theinear distributed moment state into a uniform moment condition.

. Illustrative application to steel truss bridges

.1. Geometry of illustrative bridges

In this study, the DAF calculation method is illustrated using two steel truss bridges. To apply it to real bridge conditions,wo original bridge designs in Japan were investigated. The first bridge, Bridge A, is a steel deck-through truss type. Theridge has a single span that is 90.0 m in length and 9.0 m in width. The concrete deck thickness is 200 mm. The memberetails are presented in Fig. 2, Fig. 4 and Table 1. The second bridge, Bridge B, is a steel upper deck truss type. The bridgepan is 73.8 m, and the deck width is 8.30 m. The details of Bridge B are shown in Fig. 3, Fig. 4 and Table 2. The concrete deckhickness is 200 mm. In both bridges, the structural steel is Japanese steel SM490A with yield stress �y = 315 MPa.

.2. Structural modeling

The illustrative bridges were modeled on a 3D model by finite element analysis software Diana 9.4.3 to obtain the actualehavior of the structures when a member is fractured. Fig. 5 presents the original models of the illustrative bridges beforeemoving the member. The structural steel was modeled using 3D beam elements with 3 nodes and six degrees of freedomt each node. The truss members are designed to have pin connection at the truss joints. However, with the existence of largeusset plates and numerous bolts and rivets, those truss joints can retain bending moments. Hence, a beam element with

rigid truss joint better reflects the structural response to loading [12]. The concrete deck was modeled using curved shelllements with 8 nodes per element. The eccentricities among the structural members were addressed by rigid beams to

rovide an accurate structural layout as well as the deformation relationship of the members. E and �y are the elastic modulusnd yield stress, respectively, of the structural steel. This study assumed structures with elastic behaviors to calculate theAF, which is used in the linear static analysis with the alternate load path. Rayleigh damping was used, with the massamping coefficient �̄ forced to zero. The stiffness damping coefficient �̄ was defined by assuming a 5% damping ratio for

56 H.T. Khuyen, E. Iwasaki / Case Studies in Structural Engineering 6 (2016) 53–62

U3

UL

L2

UL

F2

CB

D2

4250

S2

CB

D4

F1

CB

1100

0

LL38500

8500

U4

UL

(a) Upper laterals

L3

UL

S1

D3

2800

F1

D4

F2

LL3

U1

UL

L1

UL

F1

(b) Main truss

L4

EP

F1

CB

D4

2800 F2

CB

D4

S2

(c) Stringers and dia phragms

F2

LL2

U2

UL

L1

UL

F1

D1

4250

F2

D5

S2

D4

S2

(d) Floor beams and lower lat eral s

LL3

9000 0=10000x 9

LL1

Fig. 2. Geometry of Bridge A.

LL2

U1

UL1

U1

ST2

SW1

U3

6000

V1

F2

(c) Main truss

D5

F2

LL1 LL 2

L1

UL1

ST2

U1

ST2

SW1

V1

6000

D2

F2

(a) Upp er laterals

73800=1230 0x6

D6

F2

LL1

UL1

LL2

L2

UL1

ST2

U2

SW1

V1

F1

D3

F2

(b) Floor bea ms and string ers

7000

ST1

LL2

D1

UL1

L3

UL1

ST2

SW2

U2

6000

D4

F2

(d) Lower laterals

Fig. 3. Geometry of Bridge B.

Web

Lower Flg.

35

Web

Upper Flg.Upper Flg.

Web

Upper Flg.

Lower Flg.

Web

Lower Flg.

Upper Flg.

Web

Upper Flg.

Web

Lower Flg.Lower Flg.

Web

Web3

Web 35

Lower Flg.

Box-1 shape Box-2 shape I, H shape C shape T shapeBox-3 shape

Fig. 4. Member shape configuration.

the first vibration mode. A 5% damping ratio is the value used in the SDOF model to derive the conventional DAF of 1.854. A5% damping is also the most acceptable damping ratio for the bridge structural system.

3.3. Loadings

The event of a member fracture relates two types of loadings, including primary loading GL and impact loading. Theprimary loading is used to cause the initial member fracture. It may be overweight trucks, traffic collision, corrosion orfatigue cracks. The impact loading, on the other hand, is the generated loading due to structural motion, which is in turn

H.T. Khuyen, E. Iwasaki / Case Studies in Structural Engineering 6 (2016) 53–62 57

Table 1Details of Bridge A.

Member Shape Web (mm) Upper Flg. (mm) Lower Flg. (mm)

U1 Box-1 575 × 14 570 × 10 500 × 10U2 Box-1 575 × 14 570 × 18 500 × 18U3 Box-1 575 × 22 570 × 18 500 × 18U4 Box-1 575 × 22 570 × 22 500 × 22

L1 Box-2 540 × 14 500 × 10 570 × 10L2 Box-2 540 × 14 500 × 14 570 × 14L3 Box-2 540 × 15 500 × 18 570 × 18L4 Box-2 540 × 15 500 × 19 570 × 19

D1 Box-1 575 × 14 570 × 10 500 × 10D2 H 472 × 14 445 × 14 445 × 14D3 Box-3 478 × 10 445 × 11 445 × 11D4 H 472 × 9 445 × 14 445 × 14D5 Box-3 480 × 9 445 × 10 445 × 10

EP I 1200 × 9 300 × 14 300 × 14CB I 1300 × 9 350 × 15 350 × 12UL I 176 × 9 200 × 12 200 × 12

LL1 T 226 × 11 – 300 × 15LL2 T 179 × 10 – 300 × 16LL2 I 163 × 10 300 × 16 300 × 16

S1 I 800 × 9 240 × 15 240 × 15S2 I 800 × 9 240 × 13 240 × 13

F1 C 174 × 8 90 × 13 90 × 13F2 C 274 × 8 90 × 13 90 × 13

Table 2Details of Bridge B.

Member Shape Web (mm) Upper Flg. (mm) Lower Flg. (mm)

U1 Box-1 650 × 13 470 × 16 390 × 13U2 Box-1 650 × 15 470 × 22 390 × 22U3 Box-1 650 × 16 470 × 25 390 × 25

L1 Box-2 400 × 9 390 × 9 470 × 11L2 Box-2 400 × 13 390 × 16 470 × 19L3 Box-2 400 × 16 390 × 22 470 × 25

D1 Box-3 390 × 16 330 × 16 330 × 16D2 H 390 × 13 300 × 13 300 × 13D3 Box-3 390 × 16 300 × 16 300 × 16D4 H 358 × 9 280 × 13 280 × 13D5 Box-3 390 × 10 280 × 9 280 × 9D6 H 370 × 9 280 × 9 280 × 9

V1 Box-3 390 × 10 280 × 9 280 × 9

F1 I 800 × 9 200 × 12 200 × 12F2 I 800 × 9 230 × 12 200 × 12

UL1 T 176 × 8 – 110 × 8LL1 T 178 × 8 – 110 × 10LL2 T 176 × 8 – 110 × 8SW1 T 204 × 10 – 134 × 12SW2 I 176 × 9 200 × 12 200 × 12

S1 I 650 × 9 240 × 11 230 × 10S2 I 650 × 9 240 × 11 200 × 10

dtfoLt

SW3 T 204 × 10 – 134 × 12SW4 T 176 × 8 – 110 × 8

ue to instantaneous geometrical change after the initial fracture. In static analysis, the impact loading is addressed byhe alternate load path using a dynamic amplification factor. This study does not consider the required primary loadingsor redundancy and progressive analysis of truss bridges. Instead, the load primary loading is assumed to be at some level

f standard design load GL = 1.0D + 0.5(L + I); 1.0D + 0.75(L + I); 1.0D + 1.0(L + I); 1.0D + 1.25(L + I); or 1.0D + 1.5(L + I), where D,, and I represent the dead load, live load and live load dynamic allowance, respectively. The loading levels are selectedo ensure that the Eigen-mode patterns of the bridge are the same when a member fractures. The loading in this study,

58 H.T. Khuyen, E. Iwasaki / Case Studies in Structural Engineering 6 (2016) 53–62

Fig. 5. Original models of illustrative bridges: (a) Bridge A; (b) Bridge B.

(a) (b)

(c) (d)

Notes: p1=10 kN/m2; p2=3.4k N/m2

(p1+p2)/2 or p2/2p1+p2 or p2

Damaged truss

20005500 750

p1+p2 or p2

silewalk

750

p2

silewalk

5500

Notes: p1=10k N/m2; p2=3.5kN/m2

(p1+p2)/2 or p2/2

Damaged truss

500

p+p1 2p

Target member8500

10000x9 =90 000

2

1000 0

p2

37300

p2

Target member

26500

p1+p2

10000

Fig. 6. Live load distribution: (a) Bridge A from the transverse direction; (a) Bridge B from the transverse direction; (c) Bridge A from the longitudinaldirection; (d) Bridge B from the longitudinal direction.

Fig. 7. Member fractures and time loadings.

as shown in Fig. 6, is prescribed by the Japanese Standard for Highway Bridges because illustrative bridges were designedby this specification [13]. In dynamic analysis, the model mass includes the self-weight of the steel structures, which arerepresented by the mass density, and the equivalent mass of the live load level, including its dynamic allowance and theweight of the concrete slab converted to stringer joints, as shown in Fig. 5 [6]. The time loading is included as in Fig. 7. Thesectional force F is suddenly forced down to zero at fracture time �t, while the vertical load GL is kept constant to expressthe sudden fracture. An additional series of dynamic analyses with various values of �t showed that when �t is less than

0.01 s, the results of dynamic analysis are the same as those shown in Fig. 8. In this study, the time �t was set to 0.0001 sfor all study cases.

H.T. Khuyen, E. Iwasaki / Case Studies in Structural Engineering 6 (2016) 53–62 59

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

00 0.5 1 1.5 2 2.5 3

Ver

tial

Dis

plac

emen

t Y

(m

)

Time (sec)

0.5s

0.2s

0.1s

0.0 1s

0.0 01s

0.0001s

0.0 0001sSame results

Used in analyses

Fractured member

Fig. 8. Effect of fractured time �t.

Case1 Case2

Fracture member

Case3 Case4 Case5 Case6

Bridge A Bridge B

Fig. 9. Member fracture scenarios.

-0.4

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

00 3 6 9 12 15

Ver

tial

dis

plac

emen

t Y

(m

)

Linea r dynamic 5%

Stati c

max. displacement (time step 19)

3

ohicFc

3

p2w

Time (sec)

Fig. 10. Deflection in linear dynamic analysis for Case 1 at GL = 1.0 D + 1.0(L + I).

.4. Member fracture scenarios

In steel truss bridges, members are fractured due to overloading, traffic collision, fatigue cracks, corrosion, etc. Becausef these causes of damage, the compression members buckle and lose their capacity. The buckling is not a sudden failure;ence, the case of a sudden loss of the capacity of a compression member does not occur. However, when a tension member

nitiates the cracks, then the cracks may be prolonged instantly, leading the entire section to fracture and suddenly lose itsapacity. Hence, in this study, six tension members that represent the typically damaged scenarios were investigated as inig. 9. The fracture was assumed to occur at five primary loading states as in Section 3.3. The DAF of a total of 30 illustrativeases was calculated.

.5. Results and discussion

The results from Case 1 at loading GL = 1.0D + 1.0(L + I) are proposed to illustrate the above calculation process. Fig. 10lots the time history of the vertical displacement of the mid-span damaged truss side in the linear dynamic analysis in Step. The time history of the vertical displacement attained its peak at time step 19. Hence, the sectional force at time-step 19as used to define the norm stress in dynamic analysis�idm/�iy. Combining with �is/�iy in the static analysis in Step 1, the

60 H.T. Khuyen, E. Iwasaki / Case Studies in Structural Engineering 6 (2016) 53–62

y = 1. 3232xR² = 0.9983

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5 1 1.5

DAFtrial=1.3232

σis/σiy

σ idm

/σiy

Fig. 11. DAF for Case 1 at GL = 1.0D + 1.0(L + I).

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 10 20 30 40 50 60 70 80

Dis

plac

emen

t(m

)

Distance (m)

Dynamic Y Dyn amic Z

Alternate Y Alternate Z

Fig. 12. Deflection of upper chords in intact side for Case 1 at GL = 1.0 D + 1.0(L + I).

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 10 20 30 40 50 60 70 80

Dis

plac

emen

t(m

)

Dynamic Y

Dyn amic Z

Alternate Y

Alternate Z

Distance (m)

Fig. 13. Deflection of upper chords in damaged site for Case 1 at GL = 1.0 D + 1.0(L + I).

data points of the end sections of all truss members are plotted in Fig. 11. It can be seen that the data points are locatedin a linear line. This result indicates that even though a model of multiple degrees of freedom was used, the DAF of thebridge system can be expressed with a single value. This value is defined as the DAFtrial, which is shown in Fig. 1. In thisillustrative case, the value of the DAFtrial is 1.3232, which is the slope of the best fit line of the data points. Figs. 12 and 13depict the comparison of the deflections of the upper chord between the linear static analysis with the alternate load path

in Step 3 compared and the dynamic analysis in Step 2. Fig. 14 shows the comparison of R between the alternate load pathanalysis and dynamic analysis. We find that the deflection and R in the alternate load path analysis are similar to dynamicanalysis. Hence, the final DAF is decided to be equal to the DAFtrial. A similar process was applied to a total of 30 cases in twoillustrative bridges. Fig. 15 displays the data points of the final DAF and the max �is/�iy of all illustrative cases. The proposed

H.T. Khuyen, E. Iwasaki / Case Studies in Structural Engineering 6 (2016) 53–62 61

1.59 1.18

Intact truss sideFractured member

Damaged truss side

1.62 1.16 1.161.04

1.41 1.05

dynamic

1.171.05

1.42 1.05

1.011.01

alternate

Fig. 14. Demand-capacity ratio R for Case 1 at GL = 1.0 D + 1.0(L + I).

y = 0.2548x + 0.96 09R² = 0.9361

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

0.3 0.6 0.9 1.2 1.5 1.8

DIF

Proposed: y = 0.255x + 1.00

Bridge A

Bridge B

Data fit

max σis/σiy

Fig. 15. Approximation of dynamic amplification factor.

Table 3Comparison between the accurate DAF and approximate DAF for study cases at GL = 1.0D+ 1.0(L + I).

Cases Exact DAF (dynamic) Empirical equation

max �is/�iy DAF (empirical equation) Error

Case 1 1.325 1.3079 1.327 0.8%Case 2 1.216 0.9539 1.238 1.4%Case 3 1.083 0.5724 1.143 4.2%

ebt

4

bfTaa

sitii

Case 4 1.150 0.8122 1.203 5.0%Case 5 1.272 1.2738 1.318 4.1%Case 6 1.325 1.3040 1.326 0.5%

mpirical equation is the enveloped line offset from the best fit line of the data points. Table 3 compares the DAF calculatedy the proposed empirical equation to the exact DAF defined by dynamic analysis at GL = 1.0D + 1.0(L + I). It is shown thathe empirical equation derived a good approximation to the exact DAF from the dynamic analysis.

. Conclusions

Data points of dynamic amplification factor DAF versus max �is/�iy of a total of 30 illustrative cases in two typical trussridges showed that the DAF can be approximated by a function of max �is/�iy. The empirical equation to calculate the DAFor redundancy and progressive collapse analysis of steel truss bridges against the initial fracture of a member was proposed.he empirical equation means an advance to the field of providing an approximation, which could help in the analysis of

dynamic event. With this method, the analyses can all be static, becoming an easy problem that is quick to solve withccuracy.

It is also noted that, although the proposed empirical equation for DAF calculation is expected to be applicable to anyteel truss bridge, the conclusions and the findings of the empirical equation derived in this paper are solely based on the

llustrative cases of only two types of steel truss bridges. Hence, the validation of the empirical equation to general steelruss bridges may not be immediately filled; therefore, further research is needed. Additionally, the value of max �is/�iyn the data points for the empirical equation ranged from 0.4 to 1.8. Although this value rarely extends beyond this rangen modern bridge conditions, this situation needs further investigation. However, when the norm stress is large, the linear

[

[[[

62 H.T. Khuyen, E. Iwasaki / Case Studies in Structural Engineering 6 (2016) 53–62

analysis may be inaccurate. In this case, it is recommended to use a nonlinear analysis with a dynamic amplified factorderived from nonlinear dynamics to analyze the structure.

References

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