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Capacity analysis of gusset plate connections using the Whitmore, blockshear, global section shear, and finite element methods

Paul L. Rosenstrauch a, Masoud Sanayei a,⇑, Brian R. Brenner a,b

a Department of Civil & Environmental Engineering, Tufts University, Medford, MA 02155, United Statesb Fay, Spofford & Thorndike, Burlington, MA 01803, United States

a r t i c l e i n f o

Article history:Received 12 July 2012Revised 30 August 2012Accepted 31 August 2012

Keywords:Finite element modelGusset plateWhitmore sectionBlock shear failureStress distributionBolted connectionAbaqusLoad ratingTruss bridge

a b s t r a c t

The 2007 collapse of the I-35W Bridge in Minneapolis, Minnesota led to additional load rating require-ments for gusset plates. The Federal Highway Administration provided the direct tension and block shearmethods to assess tensile capacity for use in load rating, as well as a mechanics based method to assessshear capacity on critical planes. However, the three methods provide limited information about ultimatebehavior for complex connections. Using finite elements, the authors modeled the connection from Whit-more’s 1952 study to evaluate the plate capacity and failure mechanism. The analysis showed that thedirect tension method was conservative, but predicted neither capacity nor behavior. Model performancecorresponded well with the critical block shear capacity load in the left diagonal, and shear capacity onthe gross horizontal plane. Yielding on this plane began around critical bolt holes and spread to eitherside of the plate, forming the critical failure mechanism.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The practice of traditional truss analysis has been carried outsuccessfully for decades. Simplifying assumptions include thatmembers carry axial force only, and that connections do not trans-fer moment. These assumptions predate widespread computeranalysis and are appropriate for conservatively assessing the globalbehavior of truss structures. The resulting analysis is suitable fordesign, but is not as useful for understanding in situ structuralbehavior. In reality, members carry more than just axial load, andconnections have some degree of moment restraint. The forcestransmitted by a connection can be significant. Modern computing,however, makes the task of evaluating the connection behaviorfeasible, specifically, the behavior of the gusset plates, which arethe main components of a connection. In the past, gusset plateanalysis was characterized by simplifying assumptions, but the ac-tual behavior is complex. The recent failure of I-35W over the Mis-sissippi River has resulted in new attention and concern on gussetplate analysis.

On August 1st, 2007, the I-35W Bridge over the Mississippi Riv-er in Minneapolis, Minnesota collapsed. According to some analy-

ses, [15] the gusset plates played a major role in the collapsesequence [10]. The event prompted changes to the way in whichgusset plates for bridges are evaluated in the United States. Earlyin 2009, the Federal Highway Administration (FHWA) issued pub-lication FHWA-IF-09-014, which provided guidance and examplesfor the load rating of gusset plates [20]. The calculations for tensilecapacity may also be found in the American Association of StateHighway Transportation Officials (AASHTO) 2010 Design Specifica-tion [3], and they are referenced in the 2011 AASHTO Manual forBridge Evaluation [4]. The calculations are based upon past exper-iments and simple mechanics.

The FHWA method for determining tensile and shear capacity ofgusset plates is straightforward. Three methods for evaluatingcapacity, which assume particular stress distributions, must be ap-plied to the connection. The first is the direct tension method,which evaluates yield or ultimate capacity along the Whitmoresection. The commonly known Whitmore section is found by mul-tiplying plate thickness by the effective width, in turn found byextending two lines from the first row of fasteners in a connectionat thirty degrees away from the line of action to where they inter-sect a line extended through the final row of bolts. This effectivewidth is shown in Fig. 1. The second is the block shear failuremethod, which is characterized by fracture along the tension planebelow a connected member and shear yielding along the connectedmember length as shown in Fig. 2. The third mode is global section

0141-0296/$ - see front matter � 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.engstruct.2012.08.032

⇑ Corresponding author. Tel.: +1 617 627 4116.E-mail addresses: [email protected] (P.L. Rosenstrauch), masoud.

[email protected] (M. Sanayei), [email protected] (B.R. Brenner).

Engineering Structures 48 (2013) 543–557

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shear failure through the critical gross and net sections across theentire plate, as shown in Fig. 3. The mode that predicts the lowestfailure load governs tensile and shear plate capacity. The directtension and block shear failure modes are applied on a member-by-member basis to each connected member in the connection,and the global section shear failure method is applied to each crit-ical plane.

Given that there are approximately 11,400 truss bridges in theUnited States as of 2010 [21], connection behavior at the ultimate

condition in addition to capacity is valuable information forinspectors, engineers, and bridge owners. The finite element (FE)method offers a way in which gusset plate response may be exam-ined as a whole rather than on a member-by-member basis withassumed stress distributions. In this paper, a review of the experi-mental basis for the direct tension and block shear methods is pre-sented. A finite element model (FEM) of the Whitmore experimentwas constructed in the commercial finite element program Abaqus6.11-2, validated with data from that experiment, and is discussed.Calculations for the predicted capacity according to the FHWAmethod follow. The loads in the FEM were then increased until fail-ure was observed and compared to FHWA capacity predictions. Atcritical loading stages, behavior as characterized by stress distribu-tions is discussed.

2. The Whitmore experiment

Richard E. Whitmore studied stress distributions in 1952 bymodeling a connection from a Warren Truss using aluminum forthe members and the two gusset plates [24]. The connection geom-etry is shown in Fig. 4 [24]. Strain gauge rosettes captured defor-mation. Global loading is unknown, but applied member forcesare assumed to be representative of a typical truss analysis assum-ing pinned connections. Output from the Whitmore experiment in-cluded maximum and minimum principal stresses (r1 and r3) andmaximum shear stress (s12) at each strain rosette. Contours ofthese stress distributions were constructed. Because gusset platesare a case of plane stress, the intermediate principal stress (r2) iszero.

Whitmore found that beam theory did not accurately predictthe normal and shear stresses across the critical section identifiedin Fig. 4. He did find that the thirty degree method conservativelypredicted the experimental maximum principal stress. Whitmoreproposed that using this method to find the maximum principalstress may be used in design. The thirty degree method was ‘‘com-monly used’’ in 1952 [24], yet the section bears his name due tothis experiment.

Additional information would have been helpful in evaluatingthe plate’s performance in this test. The stress around the fastenerholes, which may be three times greater than the applied stressesin direct tension [18], was not reported. Furthermore, the vonMises stress (rvm) could have been reported; it provides moreinformation about plate behavior than maximum principal stressbecause it captures three-dimensional behavior and predicts theonset of plasticity. The von Mises stress may be expressed in prin-cipal stresses for plane stress using data from this experiment asshown in the following equation [6],

rvm ¼1ffiffiffi2p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðr2

1Þ þ ðr23Þ þ ðr1 � r3Þ2

qð1Þ

The behavior of the plate at the ultimate condition was notexamined in the Whitmore experiment. The block shear method,however, does take this behavior into account.

3. The block shear experiment

Hardash and Bjorhovde [8] identified the need for a capacitymethod that takes into account the failure mechanism of a connec-tion. To develop this method, 28 plates were tested to failure. Thetests consisted of a reaction member, gusset plate, and a memberloaded in tension as shown in Fig. 5. All plates exhibited the samefailure mechanism; rupture on the net tension plane with yieldingon the gross shear planes as identified in Fig. 2. The degree of shearyielding depended on the connection length. They proposed the

Fig. 3. Critical planes for use in global section shear failure method, from FHWA,2009.

Fig. 1. Effective (Whitmore) width for use in the direct tension method from FHWA2009.

Fig. 2. Critical planes for use in the, block shear method, from FHWA 2009.

544 P.L. Rosenstrauch et al. / Engineering Structures 48 (2013) 543–557


following equations [8] for predicting the block shear capacity of aplate based upon these observations,

Rn ¼ FuSnett þ 1:15Feff lt ð2aÞ

Feff ¼ ð1� ClÞFy þ ClFu ð2bÞ

Cl ¼ 0:95� 0:047l ð2cÞ

in which Rn, Fu, Snet, t, Feff, Cl, Fy, and l are the nominal connectionresistance, ultimate material strength, net tension section length,plate thickness, effective shear strength, connection length factor,material yield strength, and connection length, respectively. Thisequation is for the case where two shear planes exist on either sideof the member, and so a 2 may be factored out from the 1.15 coef-ficient, which may then be expressed approximately as(2) � (0.577). In turn, (0.577) may be expressed as 1=

ffiffiffi3p

. In a gen-

eral coordinate system defined by the mutually perpendicular x, y,and z directions, the von Mises stress may be expressed as in thefollowing equation [6],

rvm ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðrx þ ry þ rzÞ2 � 3ðrxry þ ryrz þ rxrz � s2

xy � s2xz � s2

yzÞq

ð3Þ

For the assumed state of pure shear in the gusset plate along aconnected tension member, Eq. (3) reduces to Eq. (4). From thisequation, the applied shear stress that occurs when the von Misesstress reaches the material’s yield stress, (Fy) is seen to occur at57.7% of the von Mises and yield stresses (Fy) [8],

sxy ¼1ffiffiffi3p rvm ¼ ð0:577ÞFy ð4Þ

The block shear model is a better predictor of the plate’s truebehavior than is the Whitmore method for this particular geome-try. However, the connection geometry is simple in that just oneapplied load is directed through just one connected member. Inan arrangement similar to an in-service truss bridge connectionwith many connected members, this scenario may not controlbehavior.

4. Additional literature

Whitmore’s 1952 experiment and Hardash and Bjorhovde’s1985 experiment represent the basis of the understanding of gus-set plate behavior, and others have expanded their work. TheWhitmore section may be applied to many different types of con-nection geometries [17]. To achieve consistent reliability across allconnection types, it was proposed that the strength used in calcu-lating shear capacity should be the average of the yield and ulti-mate material strengths [7]. The block shear model was shownto be applicable to welded connections, although an increase infracture strength of 25% is required [19]. Load rating gusset platesaccording to the Load and Resistance Factor Rating (LRFR) meth-od’s standards using block shear may identify potentially inade-quate connections, assuming that the original connection wasdesigned optimally according to Allowable Stress Design (ASD)standards [9]. Higgins et al. also explained that mathematically,the differences between the Whitmore and block shear methodsamount to the balance between accounting for ultimate strengthon the net tensile section and yield strength on the gross shearsection.

The finite element method is uniquely suited to capturing localbehavior in large structures [13]. Gusset plate connections of a

Fig. 4. Whitmore experimental setup, from Whitmore, 1952 [24].

Fig. 5. Block shear experimental setup, from Hardash and Bjorhovde, 1985 [8].

P.L. Rosenstrauch et al. / Engineering Structures 48 (2013) 543–557 545


long-span Warren truss were shown to behave as fixed with fullmoment restraint [16]. Strong correlation was found between a fi-nite element model and tests of a braced-frame when memberswere modeled with shell elements, and connected with spring ele-ments [22]. Others were able to match braced frame failure pat-terns to finite element stress distributions [5]. Finite elementanalyses were carried out on the original block shear experiment[12]. By employing element removal once elements reached theirultimate capacity, it was demonstrated that the finite elementmodel was able to predict experimental capacity. Evolutionarystructural optimization techniques were applied to tension con-nections [14], which showed that material demand is concentratedat the base of a connected tension member. End plate connectionshave been modeled with finite elements as well [23]. Full plastifi-cation was observed in the end plate model as a potential failuremode.

Similar to how the block shear experiment characterized capac-ity and failure for gusset plates with one connected member, theanalysis presented in this paper characterizes the failure mecha-nism for a connection with complex geometry as a whole, insteadof on a member-by-member basis. Such an analysis is timely asadditional information may supplement practical applications ofthe FHWA load rating method. The understanding gained from thisassessment may in turn improve other aspects of bridge engineer-ing such as design and inspection.

5. Verification of the Whitmore model

The original Whitmore experiment provides a familiar gussetplate connection for study. The authors constructed a finite ele-ment model based on this connection using Abaqus 6.11-2. Allmembers were modeled with the general purpose S4 linear shellelement. The vertical member was not modeled because it wasnot loaded in the original experiment. However, its bolt holes inthe plate were included. All 37 bolts were modeled with theC3D8 linear solid element with a radius of 0.125 in. A distance of0.3125 in. separates the gusset plate from the connected membersdue to thickness offsets, and so friction between the members wasassumed to be zero. In the actual experiment, friction forces arepresent due to the prestressing forces in the bolts. For this reason,shell-to-solid couplings were applied between the bolts and themembers to provide full connectivity around the hole perimetersrather than applying a contact algorithm. As in [24], the elasticmodulus, E, was 10,000 ksi and the Poisson’s ratio, m, was 0.33.All other dimensions are as shown in Fig. 4.

Horizontal rollers along the left edge of the lower chord and asingle two-dimensional pin, for stability, formed the in-planeboundary conditions. To reduce model size and computation time,Z-symmetry in the X–Y plane (out-of-plane direction) wasemployed. Shell thicknesses were adjusted to account for thehalf-symmetry. One plate was modeled, but the symmetry andthickness adjustments cause the model to behave as if two platestransfer the load. Along the rear symmetry plane of the model,out of plane boundary conditions were provided. The remainingmembers were loaded with distributed edge loads equal in magni-tude to half of their value in the original experiment, also due to

symmetry. The total applied loads are shown in Table 1, and thefront and side views of the completed model are shown in Fig. 6. Adetailed image of the finite element mesh is shown in Fig. 7 wherethe bolt connects the gusset plate in the front to a member beyond.

Stress output was taken at 10 locations which correspond tostrain gauge locations in the original Whitmore experiment.Gauges R-I, R-XI, R-V, and R-VI were selected to capture maximumprincipal stress, R-XVI, R-XVIII, and R-XIII to capture minimumprincipal stress, and R-VIII, R-XV, and R-XII to capture shear stres-ses. Locations are shown in Fig. 8 [24].

6. Discussion of the Whitmore model

Figs. 9–11 show the Whitmore stress contours superimposed onthe FEM color contours. The FEM contours correspond well to thoseconstructed by Whitmore except around the bolt holes due tostress concentrations. This may also be explained by the lack ofexperimental data in these regions. The contours show that themaximum principal and minimum principal stresses are orientedaround the lines of action of the left and right diagonals respectively.

Table 1Finite element model loading.

Member Load (kips)

Left diagonal 3.69Right diagonal �3.265Right lower chord 10.8Left lower chord (reaction from rollers) 6.25

Fig. 6. Elevation and side view of Z-symmetric FEM.

Fig. 7. FE mesh around bolt.



Fig. 8. Gauge locations and orientations from Whitmore, 1952 [24].

Fig. 9. Whitmore contours overlain on FEA output, maximum principal stress r1.

Fig. 10. Whitmore contours overlain on FEA output, minimum principal stress r3.



The shear stress is highest in the central region of the plate wherethe lines of action of the forces meet. The von Mises stress, whichindicates the onset of plasticity, is shown in Fig. 12. It may be inter-preted as a visual combination of the three stress components aswell as an analytical combination.

The author’s post-processed Whitmore’s data according to Eq.(1) to find the expected von Mises stress at each gauge. For eachof the ten gauges, values of stress for maximum principal, mini-mum principal, shear, and von Mises stress are compared inFig. 13a–d. These figures further demonstrate that the stresses inthe FEM closely track those stresses reported in [24]. The averageof the absolute value of the percent errors is 15.3% for the vonMises stress, some of which may be attributed to the fact that fric-tion was not modeled, and to the stress concentrations around thebolt-holes. Considering the complexity of the geometry, smallamount of test data, modeling and measurement error, the modelis considered to be validated in the elastic range based on the testdata.

The largest reported stress from [24] is the maximum principalstress at gauge R–V, 4.045 ksi, located below the left diagonal inFig. 8. This is also the value that led to the validation of the thirtydegree method, because the direct tension method conservativelypredicts 4.31 ksi along this section. Taking into account post pro-

cessed von Mises stress, however, indicates that the highest stressat any of the gauge locations is found at gauge R-XV. It is 5.20 ksi,and occurs in the middle of the plate in the high shear region.

It is apparent in Figs. 9–11 that the bolt holes cause significantstress concentrations. The largest stress is found around bolt holeF, shown in Fig. 14. Other critical holes are identified in Fig. 14for later use. The mesh around these holes is refined to achievean accurate solution. To account for the presence of stress concen-trations, the stresses in the elements on either side of the nodewith the highest stress, and the two elements behind them, wereaveraged. These elements lie within r/2 of the bolt hole perimeter,where r is the bolt-hole radius. This procedure returned a vonMises stress of 35.4 ksi at hole F. Dividing the applied load onthe right lower chord (10.8 kips) by its area (1.5 square inches inthe symmetric model) results in an applied stress of 7.2 ksi. Themaximum stress in the plate is then 4.9 times greater than the ap-plied stress. This ratio is higher than Timoshenko and Young’s [18]expected stress concentration factor of 3, but for this model, manyloads come together in a complex structure. A small degree of localyielding is expected even in existing structures. The high stress istherefore both expected and reasonable.

These discussion points lead to different conclusions than theone reached by Whitmore in 1952. High stresses around hole F

Fig. 11. Whitmore contours overlain on FEA output, shear stress s12.

Fig. 12. FEA von Mises stress contours rvm.



mean that failure is most likely to begin there, and the high stressregion in the middle of the plate may be of more interest than theregion below the left diagonal. However, conclusions on the globalfailure mechanism may not be drawn based on this elastic analysisalone; a capacity analysis is required.

7. FHWA capacity analysis of the Whitmore connection

The authors applied the direct tension, block shear, and globalsection shear methods from the FHWA to the Whitmore modelto find a baseline prediction of plate capacity. Direct tension capac-

ity according to the LRFR method may be calculated according tothe following equations [20],

Pr ¼ /yPny ¼ /yFyAg ð5aÞ

Pr ¼ /uPnu ¼ /uFuAn ð5bÞ

in which Pr, Pny, Pnu, /y, /u, Fy, Fu, Ag and An are the LRFR tensile resis-tance, nominal resistance for yielding, nominal resistance for frac-ture, resistance factor for tension yielding (0.95), resistance factorfor tension fracture (0.80), material yield strength, material ulti-mate strength, gross area along the Whitmore section, and net areaalong the Whitmore section, respectively. If Eq. (5a) predicts a low-er resistance, failure by yielding is implied whereas if Eq. (5b) gov-erns, failure by fracture is implied.

Block shear failure resistance according to the LRFR methodmay be found using the following equation [20],

If Atn P 0:58Avn then

Pr ¼ /bsð0:58FyAvg þ FuAtnÞ

else Pr ¼ /bsð0:58FuAvn þ FyAtgÞ ð6Þ

in which Pr, /bs, Atg, Atn, Avg, and Avn are the LRFR block shear resis-tance, resistance factor for block shear (0.80), area along the grosstensile plane, area along the net tensile plane, area along the grossshear planes, and area along the net shear planes, respectively. Allother variables have been previously defined. The 0.58 coefficientallows the shear yielding strength to be expressed in terms of tensileyielding strength as discussed previously. This method takes into ac-count the possibility that the plate may yield along the gross tensionsection and fracture along the net shear section, although this modeof block shear failure has never been observed experimentally [7].

Fig. 13. Comparison of Whitmore to FEA values of (a) maximum principal stress, (b) minimum principal stress, (c) shear stress, and (d) von Mises stress.

Fig. 14. Critical bolt-hole identification.



Gross and net section shear resistance according to the LRFRmethod may be found by the following equations [20],

Vr ¼ /vyVn ¼ /vy � 0:58 FyAgX ð7aÞ

Vr ¼ /vuVn ¼ /vu � 0:58 FuAn ð7bÞ

in which Vr, Vn, /vy, /vu, Ag, An, and X are the LRFR shear resistance,nominal shear resistance, resistance factor for shear yielding (0.95),resistance factor for shear fracture (0.80), gross area across the hor-izontal and vertical sections of the plate, net area across the hori-zontal and vertical sections of the plate, and shear reductionfactor (1.0 or 0.74), respectively.

In applying the FHWA method to calculate plate capacity, theauthors did not include the resistance factors so that the predictedvalues would reflect the actual failure load. When the resistancefactors are not included, the capacities predicted by the LRFRmethod for each failure mode are equal to those predicted by theLoad Factor Rating (LFR) method when safety coefficients are re-moved. For direct tension, net section resistances are not reportedbecause the gross section governed capacity at all connectedmembers. The block shear resistances according to equations fromHardash and Bjorhovde [8] are less conservative because they pre-dict larger capacities, but are provided for comparison. The blockshear case in which yielding on the gross tension section occursin conjunction with fracture on the net shear planes governed forthe lower chord only, predicting a lower capacity by 0.2 kips. Thecapacity values are reported in Table 2 according to failure modeand connected member. Shear capacity of the gross and net platesections are reported in Table 3.

8. FEM capacity analysis of the Whitmore connection

The authors performed a capacity analysis of the finite elementmodel to evaluate the loads at failure and to observe if the modelwould demonstrate the tensile or shear failure mechanisms pre-dicted by the FHWA. Compression was not considered and is notdiscussed. Bolts were assumed to remain elastic during the analy-sis. For all other members, the material model for aluminum 6061-T6 from Ambriz et al. [2] was used. Ambriz et al. [2] provide truestress and true strain data, reproduced in Fig. 15. For input intoAbaqus, logarithmic plastic strains must be used to define materialbehavior in order to account for the degree of ductility. Eqs. (8a)and (8b) below [1] show the conversion from nominal stressesand strains to true stress and logarithmic plastic strains,

rtrue ¼ rnomð1þ enomÞ ð8aÞ

eplln ¼ lnð1þ enomÞ �

rtrue

Eð8bÞ

in which rtrue, rnom, enom, and eplln are the true stress, nominal stress,

nominal strain, and logarithmic plastic strain, respectively. The truestrains from Ambriz et al. [2] correspond to the logarithmic portionof Eq. (8b), and so this equation was used to process the strain data.In this aluminum model, the yield stress is 38.75 ksi, and the ulti-mate strength is 49.05 ksi. Isotropic strain hardening is assumed,as is the von Mises yield criterion. In the analysis, the curve extends

to a logarithmic strain of 1.00 at a slope of 1% for numericalstability.

The loads in Table 1 that were applied to the original verifica-tion analysis were scaled by a factor of ten, for referencing simplic-ity, so that the model would reach ultimate capacity. Thesymmetric model was used in all analyses, but reported loads arefor the full connection. Analyses were performed with the finitestrain formulation. The initial analysis did not converge becausethe right portion of the lower chord reached capacity before thegusset plate, formed a plastic hinge, and became unstable at 63%of the applied load indicating that the connection is stronger thanits connected members, in accordance with typical connection de-sign. Member thicknesses in the symmetric model were increasedto 0.75 in. to force failure to occur in the plate. The subsequentanalysis did not converge because each bolt deformed differentlyand caused out-of-plane instability in the plate. The plate was thenrestrained in the out-of-plane (Z) direction so that in-plane behav-ior could be assessed. With these changes, the capacity analysisconverged.

9. Damage criteria

A criterion was required in order to determine if material failureoccurred during the analysis. The failure criterion must take intoaccount the onset of fracture, as fracture is included in the calcula-tion of the block shear capacity. An analytical model is proposed in[11] that expresses the equivalent plastic strain at failure as afunction of the stress triaxiality as shown in Eqs. (9) and (10) forductile and shear fracture,

g ¼ 3rm

req¼ r1 þ r2 þ r3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a21 þ a2

2 þ a23 � r1r2 � r2r3 � r1

qr3

ð9Þ

e��eq ¼ e��eqðgÞ ð10Þ

in which g, rm, req, ai, and e��eq are the stress triaxiality, hydrostaticstress, von Mises stress, constants in the approximation of thestress–strain curve, and equivalent plastic strain at fracture, respec-tively, and all other variables have been previously defined. If

Table 2Whitmore connection resistances.

Member Direct tension(kips)

Block shear(kips)

Block shear, 1985(kips)

Left diagonal 64.12 78.92 90.26Right diagonal 52.93 72.18 82.87Lower chord 90.39 100.8 107.99

Table 3Whitmore connection resistances, gross and net plate sections.

Plane Gross section capacity (kips) Net section capacity (kips)

Horizontal 93.27 104.73Vertical 70.80 76.28

Fig. 15. True stress vs. log strain for Al 6061-T6, based on Ambriz et al. [2].



implemented, Eqs. (9) and (10) provide the equivalent strain at frac-ture for a range of possible states of stress. Experimental data forthis method could not be found for aluminum alloy 6061-T6, how-ever. A simpler but less realistic approach is to assume a constantvalue of equivalent plastic strain at failure as stated in [25]. Equiv-alent plastic strain may be calculated as shown in the followingequation,

eeq ¼ffiffiffi23

r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2

1 þ e22 þ e2

3

qð11Þ

in which eeq, and ei where i is 1, 2, and 3, are the equivalent plasticstrain and the principal strains, respectively [25]. According to Xue[26], the critical equivalent plastic strain for failure in aluminum6061-T6 is 0.55. In the finite element model, it is assumed that ele-ments with an equivalent plastic strain greater than 0.55 have beendamaged. As the model does not utilize a damage evolution processthat would model material fracture, once the equivalent plasticstrain exceeds 0.55 anywhere in the model, the model is assumedto have failed, and both the analysis and discussion end.

10. Yielding and fracture

The critical holes represent the regions in the plate where yield-ing is expected to begin. As before, elements around bolt holes areconsidered in groups of four. An element group is consideredyielded when its average von Mises stress exceeds the materialyield strength. In Fig. 16, a local group of elements within r/2. ofthe hole perimeter and a significant group of elements betweenr/2 and r have been identified around the critical holes where r isthe hole radius. Stresses in these regions were monitored in orderto track the behavior of the connection as the applied loads andresulting stresses increase.

Critical holes F and C were the first and last to yield in their localregions at 23% and 32% of the applied load, respectively. Criticalhole E, shown in Fig. 17, was the first to yield in the significant re-gion at 37% of the applied load, while hole C again was the last toyield in this region at 51% of the applied load. The von Mises stressin the plate is shown at 37% and 51% of the applied load in Figs. 18and 19, respectively. Fig. 19 shows that plenty of capacity remainsat 51% of the applied load.

The model was allowed to run to completion. During the anal-ysis, some elements in the local and significant regions exceededtheir ultimate strengths, however, the analysis was allowed to con-tinue as damage was governed by plastic equivalent strain. Whenthe analysis ended, the critical plastic equivalent strain of 0.55had not yet been achieved, thus, the scaled loads were scaled againby a factor of two. At the first increment, the critical plastic equiv-alent strain was achieved. The loads at failure were therefore 10.01times the original Whitmore loads. The model is considered failedat this point and these loads are considered to be the FEA capacityloads.

11. Global plate behavior

As loading increases, the plastic regions around the criticalholes spread and connect in the middle of the plate. The followingpoints document the spread of plasticity using the von Mises stressas the variable of interest at increasing percentages of the appliedload.

� 70%: The plastic regions around holes E and G connect (Fig. 20).� 72%: The plastic region around hole H connects with the region

around holes E and G.� 75%: The plastic regions around hole A and B connect (Fig. 21).� 76%: The plastic region around hole D connects with the region

around holes E, G, and H.� 79%: The plastic region around hole C connects with the region

around holes A and B.� 80%: The plastic region around holes A, B, and C connects with

the region around holes D, E, G, and H.� 85%: The plastic region around hole F connects with the larger

region (Fig. 22).� 86%: The plastic region extends to the right gusset plate edge.� 92%: The plastic region extends to the left gusset plate edge

(Fig. 23).� 97%: The plastic region extends along the upper shear plane in

the gusset plate at the location of the left diagonal.

The joining of the plastic regions around holes A and B at 75% ofthe applied load in Fig. 21 is significant in that it marks the begin-ning of the block shear failure sequence. According to the blockshear method, the net tension plane will fracture between holesA and B. Fig. 21 shows the plastic regions extending towards eachother from holes A and B, at 45� angles from the Whitmore con-tours in Figs. 9 and 10. From mechanics, this illustrates the begin-ning of the failure sequence as plasticity spreads in the orientationof maximum shear, which bisects the angle between principaldirections.

As more load is applied, plasticity first connects the high ten-sion, compression, and shear stress zones in the plate. It then pro-gresses across the gross horizontal section as expected according tothe global section shear failure method, illustrating the governingbehavior for this connection in this analysis. This is shown in thetransition from Figs. 20–24. Because no members cross this sec-tion, it is the most flexible, attracting large deformations and stres-ses. Even in the original elastic analysis, the shear stress contoursin Fig. 11 lead to the expectation that large shearing stresses anddistortion would be present in this region.Fig. 16. Local region (left) and significant region (right).

Fig. 17. Hole E at 37% of applied load.



The von Mises stresses in the plate at failure are shown fromzero to yield in Fig. 24, and from yield to 50 ksi in Fig. 25 usinga different scale. Fig. 26 shows the plastic equivalent strains at

failure. The plastic stresses are higher along the gross horizontalsection than along the left diagonal block shear failure planesand the plastic equivalent strains are oriented mainly along the

Fig. 18. Plate at 37% of applied load.





horizontal section, suggesting that global section shear failure isthe true failure mechanism according to this analysis. If elementremoval for fractured elements was employed at failure at this

time and analysis were to continue, the block shear mechanismmay form in full before the gross horizontal section becomesunstable.






12. FEA to FHWA capacity comparison

According to the direct tension method, the plate should reachelastic capacity when 64.12 kips have been applied to the left diag-onal at 87% of the applied load. Plasticity along the Whitmorewidth, as expected according to the direct tension method, is notthe main mechanism forming in Figs. 21–25. Plasticity formedaround critical holes between 23% and 51% of the applied loadand spread across the gross shear plane between 70% and 92% ofthe applied load. The analysis did not become unstable at 87%.The direct tension method is therefore neither indicative of platebehavior nor of capacity.

Block shear failure around the left diagonal is expected at a loadof 78.92 kips according to the FHWA, or at 90.26 kips according toHardash and Bjorhovde [8]. Neither of these loads had beenreached at failure, but the contours of Figs. 23 and 24 indicate thatblock shear along the left diagonal is a viable failure mechanism forthis plate. Because in the FHWA block shear method it is assumedthat the stresses in the shear planes are constant once they reachthe yield stress, it is expected that the failure load, were the anal-ysis to continue, would be larger 78.92 kips, and possibly closer to90.26 kips.

In the region around the lower chord, failure according to thedirect tension method is expected to occur at a load of 90.39 kips.The two ends of the lower chord experience different loads; at 72%of the applied load, the left portion carries the critical 90.39 kipload and the right portion carries 155.52 kips. Even at the loadsat failure, Figs. 24 and 25 show that plastic stresses are localizedaround bolt holes in the lower chord Whitmore width, not alongthe full width. This can be attributed to continuity, as only a netload of 65.13 kips has been applied at 72% of the applied loadand the plate is not the only structural member transferring theforces. For this connection, the direct tension method again is nei-ther indicative of behavior nor of capacity.

At 81% of the applied load, the left portion of the lower chordreaches its predicted block shear capacity load according to theFHWA (100.80 kips), while the right portion carries 174.96 kips.Again, due to continuity, the plate does not experience stressesin accordance with those expected for block shear failure. If thelower chord were discontinuous, it is possible that the block shearfailure mode would be a relevant failure model.

Above the critical horizontal plane, the net horizontal force act-ing to the left from the diagonals may be resolved. This load is bal-anced by the net load in the lower chord acting to the right, and

Fig. 24. Von Mises stress at failure:zero to yield.

Fig. 25. Von Mises stress at failure: yield to 50 ksi.



together they act to shear the plate across the gross horizontal sec-tion. The global section shear failure method predicts a capacity of91.84 kips for this section, but the analysis indicates that the sec-tion yielded at 92% of the applied load when the horizontal loadwas 83.66 kips. Two reasons explain this behavior. First, von Misesstress was used as an indicator of plasticity. Across the gross hor-izontal section, the shear stresses have indeed not exceeded theshear yielding stress of (0.58)FY as shown in Fig. 27. The scale inFig. 27 has been adjusted to reflect this value. Second, the stressesare not constant along the section as assumed in the global sectionshear failure method. They exceed the shear yield strength in thecentral region and decrease back into the elastic zone as distancefrom the center along the critical section increases. A similar butmore pronounced trend is evident in the shear stress at failure inFig. 28.

Load-to-predicted ratios were calculated between the appliedloads in the model at failure and the predicted FHWA loads. Ratiosare shown in Table 4. It is important to note that the FHWA capac-ity loads used in these calculations are the maximum loads, notthe loads that actually govern. The ratio for the left diagonal is be-low 1 because full block shear capacity was not achieved whenthe model failed. If fracture was modeled fully, a higher loadwould be achieved that reflects the tearing of the plate along

the left diagonal net tension plane, and the higher stresses thatexceed yield along the shear planes. The left and right lowerchords have ratios larger than 1 due to continuity. The ratio forthe horizontal section capacity reinforces the fact that the analysispredicts gross section yield as the main failure mechanism. Ifanalysis were to continue, it is expected that the section woulddevelop stresses in excess of the yield strength, and becomeunstable between the gross section capacity and net section frac-ture capacity. It’s ratio would approach, and would possibly ex-ceed 1.00. Based on the geometry of the connection, the loadapplied to the gross horizontal section is greater than the load ap-plied to the gross vertical section. The vertical section is also stiff-ened by the lower chord, explaining the low ratio of 0.69. Thisratio is not indicative of behavior.

The load case that governs this connection according to theFHWA method is direct tension in the gusset plate at the locationof the left diagonal. This load is 64.12 kips. The load-to-capacity ra-tios are calculated again using the loads in the FEM at failure, andthose member forces which correspond to 87% completion in themodel. In this way, FEM capacity is compared to the FHWA predic-tion for connection capacity. Ratios are shown in Table 5. Whenthese ratios are assessed, it is apparent that the FHWA method isconservative as all ratios are above 1.00.

Fig. 26. Plastic equivalent strains at failure.

Fig. 27. Shear stress in the plate at 92% of applied load.



13. Discussion

Failure patterns and loads observed in this analysis indicate thatthe direct tension method is neither indicative of behavior nor ofcapacity, but this does not call the safety of those existing gussetplates designed by this method into question. The direct tensionmethod predicts lower capacity loads than the block shear methoddoes because block shear capacity is calculated over a larger area.Thus, the direct tension method will govern capacity for load rat-ing, allowable loads will be smaller, and additional capacity isavailable in the ultimate state for connections with similargeometry.

The development of the horizontal plastic section may betracked through the analysis. This region begins as smaller regionsaround critical holes in the central region of the plate where thehighest shear and von Mises stresses are located. When the regionsaround the critical holes connect, they form a ‘‘U’’ shape in the

middle of the plate. As loading increases, the plastic region extendsalong the sides of the diagonals until it reaches the plate edges.Interestingly, Whitmore identified this as the critical section inhis elastic analysis [24], yet discussion focused on maximum prin-cipal stress most likely because this experiment was not carriedout to ultimate conditions. Physically, the most likely failure mech-anism, global section shear failure along the gross horizontal sec-tion, is evident in the way the edges of the plate show lateraldistortion in Figs. 24 and 25.

This analysis shows that the block shear method and global sec-tion shear method are possible in-plane failure mechanisms forgusset plates in bridge connections. The behavior may be evaluatedphysically as well as analytically. Linking physical behavior toloading as demonstrated in this analysis may help engineers toidentify critical planes in a plate and critical plates in a structure,and to carry out effective rehabilitative action on these plates. Amajor result in this research is that the critical areas in the plateare located where there are no connected members, as the mem-bers restrain the plate from deformation. Thus, an engineer mayfind that retrofitting a plate in these areas is less costly thandesigning other expensive measures, which may be difficult toconstruct. Biennial inspections, re-rating, and checking each capac-ity measure help ensure the safety of these structural elements andif aided by analyses similar to the one described in this paper, gus-set plate evaluation and rehabilitation may become more efficientin the future.

14. Conclusions

The connection from Whitmore [24] was modeled in the finiteelement program Abaqus, and validated with data from the origi-nal experiment. The loads were then increased, and load levelsand failure mechanisms were compared to capacity predictionsfound with the FHWA load rating guide. The direct tension methoddid not predict the onset of plasticity, nor did it accurately predictplate behavior. When elements in the model exceeded their ulti-mate strength, block shear and global section shear failure patternswere evident in the plate, and member forces were in accordancewith expected loads indicating a successful analysis. The primaryfailure mechanism for the model was determined to be gross hor-izontal section yielding according to the von Mises stress contoursat failure. Future work is required to gain more understanding intothis problem. Additional understanding will aid engineers in main-taining and rehabilitating the remaining historic truss bridges, andfor the evaluation of gusset plate connections in general.

Fig. 28. Shear stress in the plate at failure.

Table 5FHWA and FEA predicted capacities using governing loading.

Loading Load atfailure (kips)

FHWAcapacity(kips)

FEA-to-FHWAratio

Failuremode

Left diagonal 74.54 64.12 1.16 Direct tensionRight diagonal �65.95 52.93 N/A Direct tensionLeft chord 126.25 90.39 1.40 Direct tensionRight chord 218.16 90.39 2.41 Direct tensionHorizontal section 91.84 76.75 1.20 Gross section yieldVertical section 53.09 91.07 1.25 Gross section yield

Table 4FHWA and FEA predicted capacities using maximum loads.

Loading Load atfailure(kips)

FHWAcapacity(kips)

FEA-to-FHWA ratio

Failuremode

Left diagonal 74.54 78.92 0.94 Block shearRight diagonal �65.95 72.18 N/A Block shearLeft chord 126.25 100.8 1.25 Block shearRight chord 218.16 100.8 2.16 Block shearHorizontal section 91.84 93.27 0.98 Gross section yieldVertical section 53.09 70.8 0.75 Gross section yield



15. Future work

Extensive testing, which was beyond the scope of this research,is required to confirm the yield and fracture patterns observed inthe finite element model. Such tests would capture compression,buckling, and fatigue behavior, which were also beyond the scopebut are important failure mechanisms. The testing of connectionssimilar to the Whitmore connection with a variety of memberthicknesses should be performed to understand how the failuremechanisms change as thickness varies. Further tests should beperformed on a variety of models based on existing truss bridgeconnections. The loading on these models should be realistic andin accordance with the latest AASHTO standards. Using such aloading program would then allow for the determination of test-to-prediction ratios between predicted load-rating and true in-ser-vice behavior under design loading. With more testing and byincorporating an element removal algorithm into the model, anal-ysis and discussion may continue at higher loads. If applied, thesediscussion points would further contribute to the understanding ofgusset plate behavior.

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