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Correlation between 2- and 3-dimensional finite element analysis of steel bolted end-plate connections
CORRELATION BETWEEN 2- AND 3-DIMENSIONAL FINITE ELEMENT ANALYSIS OF STEEL BOLTED END-PLATE CONNECTIONSt N. KRIS~AMURT~~ Department of Civil Engineering, Vanderbilt University, Nashville, TN 37235,U.S.A.
and DURWOODE. GRADDY§ Research and Development, American BuildingsCompany, Eufaula, AL 36027,U.S.A. (Received 30November 19%) Ah&&-Paper presents a part of a comprehensive research program to investigate the behavior of bolted end plates which are being increasingly used for moment-resist~t connections between steel st~ctural members. The typical configuration studied consisted of a plate welded to the beam cross section and bolted by two rows of pretensioned high-strength bolts at each flange to the adjacent member. While clearly a 3-D finite element analysis would be more suitable for the problem than 2-D analysis, and more meaningful for comparison with test results, it would also be more complex and demanding. A 3-D analysis would also have been infeasible for the exhaustive parameter studies planned. Hence thirteen bench-mark connections, with dimensions spanning values commonly used in the industry, were analyzed by 2-D and 3-D programs, so that their correlation characteristics could be applied for prediction of other 3-D values from corresponding 2-D results. The programs were based respectively on the constant strain triangle and an eight-noded su~~e~~ brick element. The connections were analyzed elastically, under bolt pretension alone (except for two of them which were not pretensioned), and under half and full service loads. Sifrnificant features of the connection modelhng and finite element code modification included the iterative determination of the deformed profile of the end plate under the contact (no tension) support restraints, and the simulation of the bolt pretensiouing and subsequent moment loading processes. The longitudinal displacement (separation) of the back of the end plateand the verticalplatebendingstresses,all
at the beamtensionflange,obtainedfrom2-Dand 3-Danalyseswereevaluated;“correlationfactors,”definedas the ratio of 3-Dvalue to 2-Dvalue,weredeterminedto be 1.4for displacementand rotation, 1.2for average stress, and 1.8 for maximum stress.
~ODU~ON
Bolted end plates are being increasingly used for moment-resistant connections between structural steel members because of their simplicity and economy of fabrication and erection. With joint sponsorship by the American Institute of SteeI Construction (AISC) and the Metal Building Manufactures Association (MBMA), and under the direction of the first author, considerable research has been conducted at Auburn University in Auburn, Alabama during 197l-75, and is being continued at Vanderbilt University, on the behavior of end-plate connections, by finite element analyses and tests. Detailed reports on various phases of the project have been submitted to the sponsors. The ultimate objective of the research was to develop rational design criteria specifically applicable to end-plate connections. The present paper covers an early part of the research&31, in which the second author participated towards his master’s degree thesis 141. The typical configuration studied consisted of a rectangular plate welded to the beam cross section and
tPresented at the Second National Symposium on Computerized Structural Analysis and Design at the School of Engineering and Applied Science, George Washin~on U~versity,-Washington, DC., 29-31 March 1976. ;E;;sF Professor.(FormerlyProfessor,AuburnU~ve~i~.)
bolted to the adjacent member by two rows each of two pretensioned high-strength bolts at each flange of the beam, as shown in Fig. l(a); Figs. lib, c) illustrate the use of the end-plate to connect a beam to a column, and to connect two beams or gable frame members. This type of end-plate connection has not received much attention in the past. Its analysis and design were t~dition~y by classical structural principles on the basis of many s~pl~y~g ass~ptions. The “cracked-section” analogy [Sl, in which the tension is assumed to be carried by the bolts and the compression by a rectangular region of contact between the plate and the adjacent member, does not take into account the bolt pretensioning. The “tee-hanger” analogy[6,7], which forms the basis of the recommended procedure in the AISC Manual [8] and is currently adopted as the standard, was developed from analysis and tests on tee hangers, and is not directly applicable to the end-plate connation. None of the existing methods involve the interaction of the beam with the plate and bolts. Classical approaches are often inadequate to resolve the intricate interaction of the many components of the connection, whose fastener dimensions and spacings and whose component thicknesses are often of the same order of magnitude as the deformation spans of the connected parts. The complexity and expense of fabrication and testing of prototype connections militate against comprehensive testing. To overcome these ditficulties, it was planned to conduct parameter studies on a large number of
381
382
N. KRISHNAMURTHY and DURWOODE. GRADDI
results of subsequent 2-D analyses to predict the more realistic 3-D values. This is a powerful and legitimate technique for the solution of complex problems. It must be emphasized however that any qualitative or quantitative conclusions drawn from such studies strictly apply only to the configuration and conditions investigated and must not be extended or extrapolated to other cases which may be similar in function but not in geometry. CASES
(a)
(cl
(b)
Fig. 1. (a) Typical end-plate connection configuration studied, (b) connection between beam and column, (c) connection between two beam or gable frame members.
connections by the finite element method, to be later evaluated by a limited number of tests. FINITE
ELEMENT
ANALYSIS
While the finite element technique was obviously the best analytical tool for the complicated geometry, irregular loading and varying boundary conditions of the connection problem, it was equally clear that it would take a 3-D analysis rather than Z-D analysis to model the problem adequately. A 3-D analysis would also be more meaningful for comparison with tests. However, 3-D finite element analysis is more complicated than 2-D, and much more demanding than 2-D in time and effort of input preparation and output evaluation, and even more importantly, in time and machine requirements of electronic computation. Thus any exhaustive 3-D investigation was clearly infeasible. It was therefore decided to analyze, by both 2-D and 3-D computer programs, certain “bench-mark” connections considered representative of the fabrication and construction industry, to determine the relationship that might exist between the 2-D and 3-D results. If any correlation could be found, then it could be applied to the
ANALYZED
Thirteen connections with details as listed in Table I. were examined in the study. Their dimensions were chosen on the basis of responses to questionnaires on the typical values and ranges used by representative member companies of the two sponsoring organizations. The letters A and M in the connection designations refer to AISC and MBMA. Connections A06 and MO6 have the minimum dimensions reported by the two groups, and connections A08 and MO8 their maximum dimensions. All other connections represent “average” dimensions. It can be seen that the MBMA beam sections are deeper and generally narrower, and their material thickness much smaller, than AISC values. Connections A04 and MO4 were analyzed as the non-pretensioned counterparts of A01 and Mol. The variables referred to in Table 1 and subsequently. are as follows: E = Modulus of elasticity, ksi. = Beam depth, in. BP = Beam flange width, in. TF = Beam flange thickness, in. Tw = Beam web thickness, in. DS = End-plate depth, in. Bs = End-plate width, in. Ts = End-plate thickness, in. P7 = Pitch from center of flange to outer bolt row, in. Ps = Pitch from center of flange to inner bolt row, in. BD = Diameter of bolt, in. G ,, = Gage distance between the two bolts in a row, in.
Ds
The first three AISC cases and the first four MBMA cases listed were analyzed with a modulus of elasticity of 30,OOO.O ksi, as against 29,000.0ksi for the other cases, with relatively insignificant differences in absolute terms; the difference was of course immaterial in the comparison of 2-D and 3-D values for the same connection. Poisson’s ratio of 0.3 was used throughout. Bolts for connections numbered l-4 in the last digit of the connection designation were chosen on the basis of the questionnaire responses; for subsequent cases, the
Table 1. Details of connections analyzed
1 N.P.
~~‘;
N.P.
N.P. = Nan-pretensioned.
(All dimensions in inches.)
383
Steelboltedend-plateconnections bolt area was standardized as a function of the beam area and material properties, for the four bolts at the beam tension flange to equal in yield capacity the tensile half of the beam, as follows:
(a)
Bn = %2~Aeffp&b,y - Common node nRoller support
where,
(Gap shown around bolt shank far clarity. Bolt head not part of model.)
B.., = Bolt area per row, i.e. for two bolts, AB = Area of beam cross section, VPby= Yield strength of plate and beam material, and e,& = Yield strength of bolt material. For all the cases analyzed, A36 steel and A325 bolts were assumed, and hence U,,by and Ably were taken as 36.0 ksi and 80.0 ksi.
It would have been impossible to model every detail of the connection and adequate portions of the beam and connected members exactly, even if what portions were “adequate” could have been determined. The analysis domain was limited to the end plate and a length of the attached beam equal to the beam depth. Due to symmetry about the beam web, only a vertical half of a connection needed to be idealized in 3-D analysis. For the 2-D model, the connection was analyzed as a plane stress problem with the analysis domain in the symme~y plane through the beam web. All the widths perpendicular to the web were input as the respective thicknesses of the plane elements. The support for the back of the plate was considered rigid. The results could be applied directly to situations where the adjacent component was very thick, or was an identical end-plate and beam combination; for other cases, the deformations of the adjacent component must be evaluated. Bolt heads were omitted to minimize finite element idealization; although later investigations proved bolt heads to be fairly sign&ant in i~uencing plate behavior, their omission in a comparative evaluation of 2-D and 3-D analyses must be considered irrelevant. On the beam tension side, the bolts were modelled with square cross sections providing the same area as the actual bolts. Thus, if Bn was the diameter of the bolt, the side of the equivalent square was, B ti = t/ (7rB$4) = O.@&&,.
(2)
The bolts were attached to the outer surface of the end plate at the common nodes, but passed inde~ndently through the plate, and extended beyond it for two plate thicknesses, as shown in Fig. 2(a, b). This permitted the bolt to be pretensioned from the free rear end, with the front end (which was attached to the plate) being enabled to respond to the interactive movements of the plate under various beam loadings. Except for the common nodes on the plate surface, the plate had separate node numbers even for identical locations on the interface around the bolt shank. The small clearance normally allowed for bolt fit was ignored, While in the 3-D model this did not present any speciat problem, it was intriguing to reconcile the strategy with the planar dom~n of the 2-D model. As su~ested in Fig. 2(c), the two separate regions namely parts of the bolt shank and the end plate were permitted to overlap in the
(b) %li!
-Common node
aRoller support
(Bolt head not part of model)
E F
4ommon
node
CDEA overlaps EFBA
Fig. 2. (a) 3-D bolt idealization, (b) 2-D bolt idealization, (c) detailsof Z-Dmesh overlap. same plane, while having different nodes (except at the common front edge) and different elements. In spite of this topological jugglery, there was no mathematical approximation involved, because after all the structural response was reflected only as the stiffnesses of the appropriate elements-mere numbers that went into their proper slots in the stiffness matrix. In the bolt ideafization adopted for both 2-D and 3-D analysis, the deformed bolt shank and the sides of the bolt hole could, in the models, intrude upon one another. But because of the small slack that would be available and the stress relief at the isolated contact points by local yielding in a real connection, the situation was considered harmless. The bolt pretension on the beam tension side was applied, as in an actual connection, by forces at the end of the bolt as indicated in Fig. 2. As the back of the plate was restrained against re~ward movement, the pretension was transferred to the plate at the common nodes and developed compression in the plate around the bolt shank. On the beam compression side, the bolts themselves were omitted as unnecessary detail, but square holes were left for them in the end plate to reduce its stiffness approp~ately. The pretension transfer was accomplished directly at the bolt hole corners on the plate surface. Figures 3 and 4 depict the 2-D and 3-D idealization used. Further details will be described in succeeding sections. BOUNDARYCONDITIONS
One of the major difficulties in bolted connection analysis, and an insu~o~~ble obstacle in all previous analysis methods, was the unpredictability of the actual support conditions at the back of the end plate. The piate would pull away from the adjacent component around the beam tension flange, and possibly at other regions, to a varying extent depending upon the beam and plate dimensions, bolt sizes and locations, material properties, and load level. The only way to solve this “variable
384
N. KRISHNAMURTHY and DURWOOII E. GR~DDI
l
Roller
(b)
Fig. 3. (a) 2-D
meshwith581nodes. (b) end view and notation, (c) assumed beam stress distribution.
Fig.4. 3-Dmeshfor
AOl: (a)plateeievatjon,~) sideelevation, beamcrosssection,and(d~plan,
boundary value problem” turned out to be by iterative finite element analysis. All the back of plate nodes were initially supported against longitudinal movement, along the beam axis. The connection was analyzed for the desired loading, and the reactions and displacements at the back of plate nodes were then examined for violation of the no-tension contact restraint conditions. If any node had a negative reaction, it meant that the plate tended to pull away from the support; to enable this, the boundary code for the node was changed to permit longitudinal movement in the next cycle. In subsequent cycles, if a previously released node had a negative displacement, it meant that the boundary code for the node was to be reset to prevent longitudinal movement in the next cycle. The analysis, checks, and release and reinstatement procedures were repeated until no change was observed. While in the 2-D analysis no advantage was found in the use of the final boundary conditions of one loading as the initial boundary conditions for the next loading (and on the contrary the
program tended to revert back to the original allsupported condition before working on the new condition), a few cycles were saved in the 3-D analysis by such continuation of boundary conditions. Restraint against vertical movement was provided to simulate the constraint by the adjacent component, by the prevention of vertical movement along one row of nodes in 2-D and one plane of nodes in 3-D, along the projecting portion of the bolt shanks in the beam tension flange region and at the corresponding locations on the plate face in the beam compression region. Addition~ly, in the 3-D analysis, transverse restraint was provided in the vertical symmetry plane and at the restrained bolt shank nodes. Figures 2 and 3 indicate most of the support conditions used. LOADING
CONDITIONS
For all except the two non-pretensioned connections A04 and MO4, three loading stages were considered: (a) Pretension only (zero external load), (b) Pretension and
Steelpeltedend-plateconnections half service load and (c) Pretension and full service load, “service load” being defined as corresponding to a beam extreme fiber stress of 0.6 times the beam yield stress, or 22.0 ksi for the A 36 steel assumed. For A04 and MO4, there was no bolt pretension, and as the half service loading condition was not analyzed, results were obtained only for loading condition fc). To simulate the connection installation as closely to practice as possible and ensure the validity of subsequent analyses under external loading conditions, the following procedure was adopted. The bolt pretension force corresponding to the recommended stress of 0.7 times the bolt yield stress (or 56.0 ksi for the A 325 bolts assumed) was applied as nodal forces at the free ends of the bolt shank in the beam tension region and at the bolt hole comers on the plate face in the beam compression region. The boundary codes for these ‘*pretension nodes” were of course set at this point to permit their free movement under the action of the forces. Even under the bolt pretension alone, the edges and the mid-depth re8ion of the plate tended to lift off of the support and hence cycling was necessary to arrive at the solution. The computed displacements would then correspond to the extension of the bolts induced by the nut tighten&g to achieve the desired pretension in the real connection. The dispfacements at the plate face would likewise correspond to the reduction in plate thickness due to the transfer of bolt pretension. For the beam moment loadings, a linear variation of longitudinal stresses was assumed by the simple beam theory as indicated in Fig. 3(c), and applied as concentrated tensile and compressive nodal forces on the upper and lower halves of the beam at the beam stub end. For these moment loadings, the bolt end conditions were modified, again to simulate the real situation. The boundary codes at the pretension nodes were changed to accept prescribed ~splacements, and the ~splacements at these nodes calculated in stage (a) were imposed at these nodes; thus the pretensioned bolt would respond to the external load very much as the actual tightened nut would permit it. This was all the more important because previous tests[6] had shown that bolt forces changed considerably with increasing loading. COMWTERS PROGRAMS Both the 2-D and 3-D programs were based on the displacement formulation of the finite element method, so popular and common by now that it is already classical and does not need further elaboration. The 2-D program centered round a plane stress algorithm originally developed by Wilson [9], and utilized the constant strain triangle. The input element could be (and in the study was) a qudrilaterai element, internally subdivided into four triangles around the baricenter of the element. The 10 x 10 stiffness matrix of the four triangles was statically condensed into an 8~ 8 matrix for the quadrilateral. The average stresses for the four triangles were output as the element stress. The 3-D program algorithm was developed by Levy [ 101 in acknowledged consultation with Clough of the University of California at Berkeley. It utilized an isoparametric eight-noded brick element, with three degrees of freedom per node. To improve the shear characteristics of the element, nine internal degmes of freedom were added; the 33 X 33 stiffness matrix was condensed to 24 x 24. Although the 2-D program had a bilinear (ehsticPlastic) material capability, only its linear elastic feature
385
was utilized, to match the 3-D linear elastic limitation. At the service load levels considered, it was believed (and generi\lly confirmed later) that any yielding would be purely local, and immaterial to the overall behavior. Both programs were extensively modified and au8mented to suit the connection problem. Reaction routines were added, and so were routines to check and reset conditions at the back of the plate; in the 2-D promo, an ext~~lation routine was added to estimate the surface stresses by least square parabolic fit to the element baricenter stresses. Preprocessor routines for automatic mesh generation, and postprocessor routines for data reduction and tabular display were attached. Capabilities were added to store the calculated values from specified intermediate cycles, so that an incomplete analysis may be continued without loss in the next run. ANALYSIS Meshes were chosen on the basis of limited pilot studies, as compromises between precision and economic feasibility. The 2-D analysis of the tist three connections in each of the two groups was carried out with a mesh of 559 nodes and 488 quadrilateral elements, the plate being divided into four slices. The rest were analyzed with a mesh of 581 nodes and SO8elements, with six divisions in the plate-but the precision of the two meshes were nearly the same. The mesh shown in Fig. 3(a) is the latter. The 3-D analyses of all thirteen connections were carried out with the mesh shown in Fig. 4, with 572 nodes and 287 elements. All the 2-D analyses were carried out on the IBM 360/50 computer at Auburn University, with a 524 K byte total core capacity, of which only about 200 K was used. The 3-D program requiring 376 K free core was too large for the system, and hence a two-step version taking 306K core was developed, in which the stiffness matrix was computed and stored on tape as the first step, and then modified and solved for each particular boundary and loading condition in a separate step. Most of the 3-D analyses were run in this manner at Auburn, but a few were run with the single step version on the much larger IBM 370/155 system at the Rust Research Center of the University of Alabama at Birmingham. A typical 2-D analysis took about live iterative cycles for the pretension stage, and six or seven cycles for subsequent loadings. The pretension run of a 3-D analysis took about ten cycies, and subsequent loadings, with the final boundary conditions of the previous loading used as initial conditions, took about five cycles each. Each 2-D cycle took about 9min, although with some reprogramming it was reduced to about 5 min. Every 3-D cycle took more than 75 min on the IBM 360/50, but only a third as long on the IBM 370/155.
Preli~n~y inspection of the results revealed that maximum deformations and stresses occurred, under moment loadings, in the end plate, especially on its back face around the beam tension flange. Most of the evaluation was therefore focussed on the back of plate. The 3-D transverse displacements u and stresses o;, while of interest and significance in themselves, had no counterpart in the 2-D solutions, and thus could be omitted in the correlation study. The vertical displacements w and the longitudinal stresses 0; were so small relative to other values that their correlation would not have been of much significance either.
N. KRISH~AM~RTHY and DURWOODE. GRAM)\
386
Hence, only the lon~tu~na1 displacements U,that is the plate separation from the support, and the vertical stresses aI, both at the back of the plate, were chosen as the significant quantities for evaluation of the finite element results from the correlation study. The results were first plotted in various graphical forms to reduce the mass of numerical data to visual images of the plate behavior. A set of figures for the connection A 01 under full service load are presented here as typical. Figure 5 shows isometric representations, to distorted scale, of the longitudinal component (u) of the deformed shape of the back of plate by 2-D and 3-D analyses; and Figs. 6-8 show respectively the 3-D contours, 2-D and 3-D variations along vertical sections, and 3-D variations along horizontal sections, for the same quantity u. The lines on the plate surface in Fig. 5 correspond to the finite element grid used. It can be clearly seen that the maximum plate displacement occurs at the tension flange; there is a second maximum near the middepth of the plate, and in a few cases (MO1 for example) this may have a slightly larger value than the flange value. But the connection rotation 0 was defined as (S/OS), where 6 was the u displacement at the back of the end plate at the level of the outer surface of the tension flange, that is at the level A in Fig, 3(a); hence this value 6 was used in the deter~nation of the correlation factor for displacement. Figures 9 and IO represent the 3-D contours, and Z-D and 3-D variations along vertical sections, for the vertical stress oz at the back of the plate. The vertical stress is seen to have three maximum magnitudes, one at the tension flange and one each at each bolt row on either side
---Beam
,111_: :
Fig. 5. Deformedshape (14 com~nent) connectionA01 underfull
of back of plate. service load.
lines
of displacement
for
of the tension flange. All three values were evaluated for stress correlation factors. A comparison of 2-D and 3-D results showed one dominant characteristic: the 3-D model was more flexible than the 2-D counterpart, with larger displacements and stresses. Possibly because of the transverse variations of the effects in the 3-D model, there were a few eases where the 2-D stresses at the first bolt row were slightly larger than 3-D values. As the precision of the two models as measured by convergence properties were nearly the same, the difference in their flexibilities must be attributed
flange o_‘.
and
webcenter
Fig. 6. 3-D contours
*
u of back of plate, for connection
A01 under full service load.
387
Steelbolted end-plateconnections
Fig. 7. Variations of displacement u of back of plate along vertical sections for A01 under full load.
B -Location
[.[[
z[
OO.$.O04in
of bolt
zr
Displacement scale z_[z_[
z~;z.[z=[z~[
zr
Fig. 8. Variations of displacement u of back of plate along horizontal sections for A01 under full load. mainly to the additional constraints imposed on the 2-D
model by the prevention of transverse variation of deformations and stresses, as illustrated in Fig. 5.
ua =
CORRELATIONFACMlRS
The correlation factor Y, was defined as the ratio of the 3-D value to the corresponding 2-D value. As the 3-D model proved to be the more flexible, the factor was almost always greater than unity. To evaluate the displacement correlation factor (q~d),a weighted average of the 3-D displacements across the width of the plate at the chosen level was used, rather than the maximum value, mainly to conform to the definition of connection rotation stated earlier. The weighted average was determined from: n-1 8. = z (ur + u,+db4(2B,) [ _
transverse section were compared with the corresponding 2-D value. This weighted average was determined as:
(3)
in which, ui, is the calculated displacement u at the ith node along the transverse section; b,~, the distance from the ith node to the next, that is the width of the ith element; and n, the number of nodes along the section. In the case of the stresses, both the 3-D maximum value, and a weighted average of all the values along a
1z 1
uibj IBs
(4)
in which, q, is the calculated stress at the center of the back of plate face of the jth element along the transverse section; bj, the width of the gth element; and m, the number of elements along the section. Based on the preceding considerations, the following seven correlation factors could be defined: Vd, for average displacement 6 ; QL, and XL,, for average and maximum u2 at the hrst row of bolts, T’,, and Q,,, for the same at the beam tension flange; and VI:, and V,L, for the same at the second row of bolts. Table 2 gives the seven correlation factors for the thirteen connections analyzed under half and full load conditions, except where data were not available (as with A04 and MO4 under half load), and where data were contradictory but very insign&ant (as with the small az values of opposite signs by 2-D and 3-D, at the first row of bolts). The mean values and standard deviations of the correlation factors are also listed in Table 2. It may be
N. KRISHNAMURTHY and DURWWD E. GRADDY
- - - Beam flange and web center iines
b
Fig. 9. 3-D contours of vertical stress o2 on back of plate for connection A01
Pione onolysis
Three-
emionolonc
< >
iii
ilysi
!i
1
J_ i.ii
Y=1.5?
-~ <
< /
under full service load.
>
Y=275
Fig. 10. Variationsof vertical stress o, on back of plate along vertical sections for A 01 under full load.
noted that the correlation factors for displacement increase somewhat with increasing load, while those for stress generally decrease. The largest correlation factors were for maximum stress at the second row of bolts. From the Table, the critical correlation factors for displacement and vertical stress may be cited as follows: Yllod= 1.42iO.18 *‘,, = 1.21tO.li ‘lir:, = 1.79i 0.39.
CONCLUSION
The values of critical displacements and stresses from 2-D and 3-D finite element analyses of thirteen end-plate connections have been compared, and reasonable correlations between the results have been observed. For practical use in the estimation of 3-D values form 2-D results, correlation factors of 1.4 for displa~ment (and hence rotation), 1.2 for average stress, and 1.8 for maximum stress, may be adopted, for the particular configuration studied. Considerable use of this correlation
389
Steel bolted end-plate connections Table 2. Correlation factors Half Loud Bolt Row 1 I
0.88 0.96 0195
1.61 1.94 1:69
1.06 0.85 0.64
1.98 1.70 1.40
’
Full Load
I
Flange
Bolt-Row 1
Bolt Ron 2
Flange 1
I
1.20 t.43 1:21
1.63 2.T5 1.64 _
1.16 1.29 1.24 _
1.75 2.19 1.88 _
1.21 1.21 1.30
1.52 1.64 1.76
1.07 1.10 0.94
I.73 1.87 1.77
has been made by the first author, in the evaluation of finite element modeb and conclusions from their analyses, on the basis of prototype tests, with satisfactory results. Ac~now~e~e~e~~e-~e financiai support and guidance of the research sponsors AISC and MBMAare gratefully acknowledged. The computing services provided by the Auburn University Computer Center and the University of Alabama (Birmingham) Rust Research Center, are also appreciated. REFgRENCBS 1. N. Krishnamurthy, Finite element analysis of splice-plate connections-a feasibility study. Unpublished Rep., CEAISC/MBMA-1, Dept. of Civil Engng, Auburn University, Alabama (Jan. 1973). 2. N. K~s~arn~hy, Effects of plate thickness and pretensioning in typical bolted end-plate connections. Unpub~~bedRep., C~-AISC~MBMA-2,Dept. of Civil Engng, Auburn University, Alabama (Dec. 1973). 3. N. Krishnamurthy, Correlation between three-dimensional and tw~~mension~ finite element analysis of end-plate
Bolt Row 2 I
1.42
0.67
1.48
1.21
1.57
1.01
1.79
0.18
0.15
0.19
0.11
0.18
0.25
0.39
connections. unpublished Rep., CE-AISCIMBMA-4,Dept. of Civil Engng, Auburn University, Alabama (Oct. 1974). 4. D. E. Graddy, Jr., Feasibility study of finite element analysis of steel splice-plate connections, Unpubiish~ Thesis towards Master’s De8ree in Civil Enginee~ng, Auburn University, Alabama (Mar. 1973). 5. W. McGuire, Sreef Structures, Chap. 6. Prentice-Hall, Englewood Cliffs, New Jersey (1968). 6. R. T. Douty and W. McGuire, High strength bolted moment connections. J. Srnrct. Div., ASCE 91, (SE?), 101-128(Apr. 1%5), 7. R. S. Nair, P. C. Birkemoe and W. H. Munse, High strength bolts subject to tension and prying. J. Struct. Div., ASCE 100, (ST2), 351-372(Feb. 1974). 8. American Institute of Steel Construction, Manual of Steel Construction, 7th Edn, New York (1969). 9. E. L. Wilson, Finite element analysis of two~imension~ structures. Thesis presented to the University of Caliiomia, Berkeley, towards Degree of Doctor of Engineering (1963). 10. S. Levy, 3-D isoparametric finite element pro8ram. General Electric Technical information Series No. 71-C-191, Schenectady, New York (June 1971).
and DURWOODE. GRADDY§ Research and Development, American BuildingsCompany, Eufaula, AL 36027,U.S.A. (Received 30November 19%) Ah&&-Paper presents a part of a comprehensive research program to investigate the behavior of bolted end plates which are being increasingly used for moment-resist~t connections between steel st~ctural members. The typical configuration studied consisted of a plate welded to the beam cross section and bolted by two rows of pretensioned high-strength bolts at each flange to the adjacent member. While clearly a 3-D finite element analysis would be more suitable for the problem than 2-D analysis, and more meaningful for comparison with test results, it would also be more complex and demanding. A 3-D analysis would also have been infeasible for the exhaustive parameter studies planned. Hence thirteen bench-mark connections, with dimensions spanning values commonly used in the industry, were analyzed by 2-D and 3-D programs, so that their correlation characteristics could be applied for prediction of other 3-D values from corresponding 2-D results. The programs were based respectively on the constant strain triangle and an eight-noded su~~e~~ brick element. The connections were analyzed elastically, under bolt pretension alone (except for two of them which were not pretensioned), and under half and full service loads. Sifrnificant features of the connection modelhng and finite element code modification included the iterative determination of the deformed profile of the end plate under the contact (no tension) support restraints, and the simulation of the bolt pretensiouing and subsequent moment loading processes. The longitudinal displacement (separation) of the back of the end plateand the verticalplatebendingstresses,all
at the beamtensionflange,obtainedfrom2-Dand 3-Danalyseswereevaluated;“correlationfactors,”definedas the ratio of 3-Dvalue to 2-Dvalue,weredeterminedto be 1.4for displacementand rotation, 1.2for average stress, and 1.8 for maximum stress.
~ODU~ON
Bolted end plates are being increasingly used for moment-resistant connections between structural steel members because of their simplicity and economy of fabrication and erection. With joint sponsorship by the American Institute of SteeI Construction (AISC) and the Metal Building Manufactures Association (MBMA), and under the direction of the first author, considerable research has been conducted at Auburn University in Auburn, Alabama during 197l-75, and is being continued at Vanderbilt University, on the behavior of end-plate connections, by finite element analyses and tests. Detailed reports on various phases of the project have been submitted to the sponsors. The ultimate objective of the research was to develop rational design criteria specifically applicable to end-plate connections. The present paper covers an early part of the research&31, in which the second author participated towards his master’s degree thesis 141. The typical configuration studied consisted of a rectangular plate welded to the beam cross section and
tPresented at the Second National Symposium on Computerized Structural Analysis and Design at the School of Engineering and Applied Science, George Washin~on U~versity,-Washington, DC., 29-31 March 1976. ;E;;sF Professor.(FormerlyProfessor,AuburnU~ve~i~.)
bolted to the adjacent member by two rows each of two pretensioned high-strength bolts at each flange of the beam, as shown in Fig. l(a); Figs. lib, c) illustrate the use of the end-plate to connect a beam to a column, and to connect two beams or gable frame members. This type of end-plate connection has not received much attention in the past. Its analysis and design were t~dition~y by classical structural principles on the basis of many s~pl~y~g ass~ptions. The “cracked-section” analogy [Sl, in which the tension is assumed to be carried by the bolts and the compression by a rectangular region of contact between the plate and the adjacent member, does not take into account the bolt pretensioning. The “tee-hanger” analogy[6,7], which forms the basis of the recommended procedure in the AISC Manual [8] and is currently adopted as the standard, was developed from analysis and tests on tee hangers, and is not directly applicable to the end-plate connation. None of the existing methods involve the interaction of the beam with the plate and bolts. Classical approaches are often inadequate to resolve the intricate interaction of the many components of the connection, whose fastener dimensions and spacings and whose component thicknesses are often of the same order of magnitude as the deformation spans of the connected parts. The complexity and expense of fabrication and testing of prototype connections militate against comprehensive testing. To overcome these ditficulties, it was planned to conduct parameter studies on a large number of
381
382
N. KRISHNAMURTHY and DURWOODE. GRADDI
results of subsequent 2-D analyses to predict the more realistic 3-D values. This is a powerful and legitimate technique for the solution of complex problems. It must be emphasized however that any qualitative or quantitative conclusions drawn from such studies strictly apply only to the configuration and conditions investigated and must not be extended or extrapolated to other cases which may be similar in function but not in geometry. CASES
(a)
(cl
(b)
Fig. 1. (a) Typical end-plate connection configuration studied, (b) connection between beam and column, (c) connection between two beam or gable frame members.
connections by the finite element method, to be later evaluated by a limited number of tests. FINITE
ELEMENT
ANALYSIS
While the finite element technique was obviously the best analytical tool for the complicated geometry, irregular loading and varying boundary conditions of the connection problem, it was equally clear that it would take a 3-D analysis rather than Z-D analysis to model the problem adequately. A 3-D analysis would also be more meaningful for comparison with tests. However, 3-D finite element analysis is more complicated than 2-D, and much more demanding than 2-D in time and effort of input preparation and output evaluation, and even more importantly, in time and machine requirements of electronic computation. Thus any exhaustive 3-D investigation was clearly infeasible. It was therefore decided to analyze, by both 2-D and 3-D computer programs, certain “bench-mark” connections considered representative of the fabrication and construction industry, to determine the relationship that might exist between the 2-D and 3-D results. If any correlation could be found, then it could be applied to the
ANALYZED
Thirteen connections with details as listed in Table I. were examined in the study. Their dimensions were chosen on the basis of responses to questionnaires on the typical values and ranges used by representative member companies of the two sponsoring organizations. The letters A and M in the connection designations refer to AISC and MBMA. Connections A06 and MO6 have the minimum dimensions reported by the two groups, and connections A08 and MO8 their maximum dimensions. All other connections represent “average” dimensions. It can be seen that the MBMA beam sections are deeper and generally narrower, and their material thickness much smaller, than AISC values. Connections A04 and MO4 were analyzed as the non-pretensioned counterparts of A01 and Mol. The variables referred to in Table 1 and subsequently. are as follows: E = Modulus of elasticity, ksi. = Beam depth, in. BP = Beam flange width, in. TF = Beam flange thickness, in. Tw = Beam web thickness, in. DS = End-plate depth, in. Bs = End-plate width, in. Ts = End-plate thickness, in. P7 = Pitch from center of flange to outer bolt row, in. Ps = Pitch from center of flange to inner bolt row, in. BD = Diameter of bolt, in. G ,, = Gage distance between the two bolts in a row, in.
Ds
The first three AISC cases and the first four MBMA cases listed were analyzed with a modulus of elasticity of 30,OOO.O ksi, as against 29,000.0ksi for the other cases, with relatively insignificant differences in absolute terms; the difference was of course immaterial in the comparison of 2-D and 3-D values for the same connection. Poisson’s ratio of 0.3 was used throughout. Bolts for connections numbered l-4 in the last digit of the connection designation were chosen on the basis of the questionnaire responses; for subsequent cases, the
Table 1. Details of connections analyzed
1 N.P.
~~‘;
N.P.
N.P. = Nan-pretensioned.
(All dimensions in inches.)
383
Steelboltedend-plateconnections bolt area was standardized as a function of the beam area and material properties, for the four bolts at the beam tension flange to equal in yield capacity the tensile half of the beam, as follows:
(a)
Bn = %2~Aeffp&b,y - Common node nRoller support
where,
(Gap shown around bolt shank far clarity. Bolt head not part of model.)
B.., = Bolt area per row, i.e. for two bolts, AB = Area of beam cross section, VPby= Yield strength of plate and beam material, and e,& = Yield strength of bolt material. For all the cases analyzed, A36 steel and A325 bolts were assumed, and hence U,,by and Ably were taken as 36.0 ksi and 80.0 ksi.
It would have been impossible to model every detail of the connection and adequate portions of the beam and connected members exactly, even if what portions were “adequate” could have been determined. The analysis domain was limited to the end plate and a length of the attached beam equal to the beam depth. Due to symmetry about the beam web, only a vertical half of a connection needed to be idealized in 3-D analysis. For the 2-D model, the connection was analyzed as a plane stress problem with the analysis domain in the symme~y plane through the beam web. All the widths perpendicular to the web were input as the respective thicknesses of the plane elements. The support for the back of the plate was considered rigid. The results could be applied directly to situations where the adjacent component was very thick, or was an identical end-plate and beam combination; for other cases, the deformations of the adjacent component must be evaluated. Bolt heads were omitted to minimize finite element idealization; although later investigations proved bolt heads to be fairly sign&ant in i~uencing plate behavior, their omission in a comparative evaluation of 2-D and 3-D analyses must be considered irrelevant. On the beam tension side, the bolts were modelled with square cross sections providing the same area as the actual bolts. Thus, if Bn was the diameter of the bolt, the side of the equivalent square was, B ti = t/ (7rB$4) = O.@&&,.
(2)
The bolts were attached to the outer surface of the end plate at the common nodes, but passed inde~ndently through the plate, and extended beyond it for two plate thicknesses, as shown in Fig. 2(a, b). This permitted the bolt to be pretensioned from the free rear end, with the front end (which was attached to the plate) being enabled to respond to the interactive movements of the plate under various beam loadings. Except for the common nodes on the plate surface, the plate had separate node numbers even for identical locations on the interface around the bolt shank. The small clearance normally allowed for bolt fit was ignored, While in the 3-D model this did not present any speciat problem, it was intriguing to reconcile the strategy with the planar dom~n of the 2-D model. As su~ested in Fig. 2(c), the two separate regions namely parts of the bolt shank and the end plate were permitted to overlap in the
(b) %li!
-Common node
aRoller support
(Bolt head not part of model)
E F
4ommon
node
CDEA overlaps EFBA
Fig. 2. (a) 3-D bolt idealization, (b) 2-D bolt idealization, (c) detailsof Z-Dmesh overlap. same plane, while having different nodes (except at the common front edge) and different elements. In spite of this topological jugglery, there was no mathematical approximation involved, because after all the structural response was reflected only as the stiffnesses of the appropriate elements-mere numbers that went into their proper slots in the stiffness matrix. In the bolt ideafization adopted for both 2-D and 3-D analysis, the deformed bolt shank and the sides of the bolt hole could, in the models, intrude upon one another. But because of the small slack that would be available and the stress relief at the isolated contact points by local yielding in a real connection, the situation was considered harmless. The bolt pretension on the beam tension side was applied, as in an actual connection, by forces at the end of the bolt as indicated in Fig. 2. As the back of the plate was restrained against re~ward movement, the pretension was transferred to the plate at the common nodes and developed compression in the plate around the bolt shank. On the beam compression side, the bolts themselves were omitted as unnecessary detail, but square holes were left for them in the end plate to reduce its stiffness approp~ately. The pretension transfer was accomplished directly at the bolt hole corners on the plate surface. Figures 3 and 4 depict the 2-D and 3-D idealization used. Further details will be described in succeeding sections. BOUNDARYCONDITIONS
One of the major difficulties in bolted connection analysis, and an insu~o~~ble obstacle in all previous analysis methods, was the unpredictability of the actual support conditions at the back of the end plate. The piate would pull away from the adjacent component around the beam tension flange, and possibly at other regions, to a varying extent depending upon the beam and plate dimensions, bolt sizes and locations, material properties, and load level. The only way to solve this “variable
384
N. KRISHNAMURTHY and DURWOOII E. GR~DDI
l
Roller
(b)
Fig. 3. (a) 2-D
meshwith581nodes. (b) end view and notation, (c) assumed beam stress distribution.
Fig.4. 3-Dmeshfor
AOl: (a)plateeievatjon,~) sideelevation, beamcrosssection,and(d~plan,
boundary value problem” turned out to be by iterative finite element analysis. All the back of plate nodes were initially supported against longitudinal movement, along the beam axis. The connection was analyzed for the desired loading, and the reactions and displacements at the back of plate nodes were then examined for violation of the no-tension contact restraint conditions. If any node had a negative reaction, it meant that the plate tended to pull away from the support; to enable this, the boundary code for the node was changed to permit longitudinal movement in the next cycle. In subsequent cycles, if a previously released node had a negative displacement, it meant that the boundary code for the node was to be reset to prevent longitudinal movement in the next cycle. The analysis, checks, and release and reinstatement procedures were repeated until no change was observed. While in the 2-D analysis no advantage was found in the use of the final boundary conditions of one loading as the initial boundary conditions for the next loading (and on the contrary the
program tended to revert back to the original allsupported condition before working on the new condition), a few cycles were saved in the 3-D analysis by such continuation of boundary conditions. Restraint against vertical movement was provided to simulate the constraint by the adjacent component, by the prevention of vertical movement along one row of nodes in 2-D and one plane of nodes in 3-D, along the projecting portion of the bolt shanks in the beam tension flange region and at the corresponding locations on the plate face in the beam compression region. Addition~ly, in the 3-D analysis, transverse restraint was provided in the vertical symmetry plane and at the restrained bolt shank nodes. Figures 2 and 3 indicate most of the support conditions used. LOADING
CONDITIONS
For all except the two non-pretensioned connections A04 and MO4, three loading stages were considered: (a) Pretension only (zero external load), (b) Pretension and
Steelpeltedend-plateconnections half service load and (c) Pretension and full service load, “service load” being defined as corresponding to a beam extreme fiber stress of 0.6 times the beam yield stress, or 22.0 ksi for the A 36 steel assumed. For A04 and MO4, there was no bolt pretension, and as the half service loading condition was not analyzed, results were obtained only for loading condition fc). To simulate the connection installation as closely to practice as possible and ensure the validity of subsequent analyses under external loading conditions, the following procedure was adopted. The bolt pretension force corresponding to the recommended stress of 0.7 times the bolt yield stress (or 56.0 ksi for the A 325 bolts assumed) was applied as nodal forces at the free ends of the bolt shank in the beam tension region and at the bolt hole comers on the plate face in the beam compression region. The boundary codes for these ‘*pretension nodes” were of course set at this point to permit their free movement under the action of the forces. Even under the bolt pretension alone, the edges and the mid-depth re8ion of the plate tended to lift off of the support and hence cycling was necessary to arrive at the solution. The computed displacements would then correspond to the extension of the bolts induced by the nut tighten&g to achieve the desired pretension in the real connection. The dispfacements at the plate face would likewise correspond to the reduction in plate thickness due to the transfer of bolt pretension. For the beam moment loadings, a linear variation of longitudinal stresses was assumed by the simple beam theory as indicated in Fig. 3(c), and applied as concentrated tensile and compressive nodal forces on the upper and lower halves of the beam at the beam stub end. For these moment loadings, the bolt end conditions were modified, again to simulate the real situation. The boundary codes at the pretension nodes were changed to accept prescribed ~splacements, and the ~splacements at these nodes calculated in stage (a) were imposed at these nodes; thus the pretensioned bolt would respond to the external load very much as the actual tightened nut would permit it. This was all the more important because previous tests[6] had shown that bolt forces changed considerably with increasing loading. COMWTERS PROGRAMS Both the 2-D and 3-D programs were based on the displacement formulation of the finite element method, so popular and common by now that it is already classical and does not need further elaboration. The 2-D program centered round a plane stress algorithm originally developed by Wilson [9], and utilized the constant strain triangle. The input element could be (and in the study was) a qudrilaterai element, internally subdivided into four triangles around the baricenter of the element. The 10 x 10 stiffness matrix of the four triangles was statically condensed into an 8~ 8 matrix for the quadrilateral. The average stresses for the four triangles were output as the element stress. The 3-D program algorithm was developed by Levy [ 101 in acknowledged consultation with Clough of the University of California at Berkeley. It utilized an isoparametric eight-noded brick element, with three degrees of freedom per node. To improve the shear characteristics of the element, nine internal degmes of freedom were added; the 33 X 33 stiffness matrix was condensed to 24 x 24. Although the 2-D program had a bilinear (ehsticPlastic) material capability, only its linear elastic feature
385
was utilized, to match the 3-D linear elastic limitation. At the service load levels considered, it was believed (and generi\lly confirmed later) that any yielding would be purely local, and immaterial to the overall behavior. Both programs were extensively modified and au8mented to suit the connection problem. Reaction routines were added, and so were routines to check and reset conditions at the back of the plate; in the 2-D promo, an ext~~lation routine was added to estimate the surface stresses by least square parabolic fit to the element baricenter stresses. Preprocessor routines for automatic mesh generation, and postprocessor routines for data reduction and tabular display were attached. Capabilities were added to store the calculated values from specified intermediate cycles, so that an incomplete analysis may be continued without loss in the next run. ANALYSIS Meshes were chosen on the basis of limited pilot studies, as compromises between precision and economic feasibility. The 2-D analysis of the tist three connections in each of the two groups was carried out with a mesh of 559 nodes and 488 quadrilateral elements, the plate being divided into four slices. The rest were analyzed with a mesh of 581 nodes and SO8elements, with six divisions in the plate-but the precision of the two meshes were nearly the same. The mesh shown in Fig. 3(a) is the latter. The 3-D analyses of all thirteen connections were carried out with the mesh shown in Fig. 4, with 572 nodes and 287 elements. All the 2-D analyses were carried out on the IBM 360/50 computer at Auburn University, with a 524 K byte total core capacity, of which only about 200 K was used. The 3-D program requiring 376 K free core was too large for the system, and hence a two-step version taking 306K core was developed, in which the stiffness matrix was computed and stored on tape as the first step, and then modified and solved for each particular boundary and loading condition in a separate step. Most of the 3-D analyses were run in this manner at Auburn, but a few were run with the single step version on the much larger IBM 370/155 system at the Rust Research Center of the University of Alabama at Birmingham. A typical 2-D analysis took about live iterative cycles for the pretension stage, and six or seven cycles for subsequent loadings. The pretension run of a 3-D analysis took about ten cycies, and subsequent loadings, with the final boundary conditions of the previous loading used as initial conditions, took about five cycles each. Each 2-D cycle took about 9min, although with some reprogramming it was reduced to about 5 min. Every 3-D cycle took more than 75 min on the IBM 360/50, but only a third as long on the IBM 370/155.
Preli~n~y inspection of the results revealed that maximum deformations and stresses occurred, under moment loadings, in the end plate, especially on its back face around the beam tension flange. Most of the evaluation was therefore focussed on the back of plate. The 3-D transverse displacements u and stresses o;, while of interest and significance in themselves, had no counterpart in the 2-D solutions, and thus could be omitted in the correlation study. The vertical displacements w and the longitudinal stresses 0; were so small relative to other values that their correlation would not have been of much significance either.
N. KRISH~AM~RTHY and DURWOODE. GRAM)\
386
Hence, only the lon~tu~na1 displacements U,that is the plate separation from the support, and the vertical stresses aI, both at the back of the plate, were chosen as the significant quantities for evaluation of the finite element results from the correlation study. The results were first plotted in various graphical forms to reduce the mass of numerical data to visual images of the plate behavior. A set of figures for the connection A 01 under full service load are presented here as typical. Figure 5 shows isometric representations, to distorted scale, of the longitudinal component (u) of the deformed shape of the back of plate by 2-D and 3-D analyses; and Figs. 6-8 show respectively the 3-D contours, 2-D and 3-D variations along vertical sections, and 3-D variations along horizontal sections, for the same quantity u. The lines on the plate surface in Fig. 5 correspond to the finite element grid used. It can be clearly seen that the maximum plate displacement occurs at the tension flange; there is a second maximum near the middepth of the plate, and in a few cases (MO1 for example) this may have a slightly larger value than the flange value. But the connection rotation 0 was defined as (S/OS), where 6 was the u displacement at the back of the end plate at the level of the outer surface of the tension flange, that is at the level A in Fig, 3(a); hence this value 6 was used in the deter~nation of the correlation factor for displacement. Figures 9 and IO represent the 3-D contours, and Z-D and 3-D variations along vertical sections, for the vertical stress oz at the back of the plate. The vertical stress is seen to have three maximum magnitudes, one at the tension flange and one each at each bolt row on either side
---Beam
,111_: :
Fig. 5. Deformedshape (14 com~nent) connectionA01 underfull
of back of plate. service load.
lines
of displacement
for
of the tension flange. All three values were evaluated for stress correlation factors. A comparison of 2-D and 3-D results showed one dominant characteristic: the 3-D model was more flexible than the 2-D counterpart, with larger displacements and stresses. Possibly because of the transverse variations of the effects in the 3-D model, there were a few eases where the 2-D stresses at the first bolt row were slightly larger than 3-D values. As the precision of the two models as measured by convergence properties were nearly the same, the difference in their flexibilities must be attributed
flange o_‘.
and
webcenter
Fig. 6. 3-D contours
*
u of back of plate, for connection
A01 under full service load.
387
Steelbolted end-plateconnections
Fig. 7. Variations of displacement u of back of plate along vertical sections for A01 under full load.
B -Location
[.[[
z[
OO.$.O04in
of bolt
zr
Displacement scale z_[z_[
z~;z.[z=[z~[
zr
Fig. 8. Variations of displacement u of back of plate along horizontal sections for A01 under full load. mainly to the additional constraints imposed on the 2-D
model by the prevention of transverse variation of deformations and stresses, as illustrated in Fig. 5.
ua =
CORRELATIONFACMlRS
The correlation factor Y, was defined as the ratio of the 3-D value to the corresponding 2-D value. As the 3-D model proved to be the more flexible, the factor was almost always greater than unity. To evaluate the displacement correlation factor (q~d),a weighted average of the 3-D displacements across the width of the plate at the chosen level was used, rather than the maximum value, mainly to conform to the definition of connection rotation stated earlier. The weighted average was determined from: n-1 8. = z (ur + u,+db4(2B,) [ _
transverse section were compared with the corresponding 2-D value. This weighted average was determined as:
(3)
in which, ui, is the calculated displacement u at the ith node along the transverse section; b,~, the distance from the ith node to the next, that is the width of the ith element; and n, the number of nodes along the section. In the case of the stresses, both the 3-D maximum value, and a weighted average of all the values along a
1z 1
uibj IBs
(4)
in which, q, is the calculated stress at the center of the back of plate face of the jth element along the transverse section; bj, the width of the gth element; and m, the number of elements along the section. Based on the preceding considerations, the following seven correlation factors could be defined: Vd, for average displacement 6 ; QL, and XL,, for average and maximum u2 at the hrst row of bolts, T’,, and Q,,, for the same at the beam tension flange; and VI:, and V,L, for the same at the second row of bolts. Table 2 gives the seven correlation factors for the thirteen connections analyzed under half and full load conditions, except where data were not available (as with A04 and MO4 under half load), and where data were contradictory but very insign&ant (as with the small az values of opposite signs by 2-D and 3-D, at the first row of bolts). The mean values and standard deviations of the correlation factors are also listed in Table 2. It may be
N. KRISHNAMURTHY and DURWWD E. GRADDY
- - - Beam flange and web center iines
b
Fig. 9. 3-D contours of vertical stress o2 on back of plate for connection A01
Pione onolysis
Three-
emionolonc
< >
iii
ilysi
!i
1
J_ i.ii
Y=1.5?
-~ <
< /
under full service load.
>
Y=275
Fig. 10. Variationsof vertical stress o, on back of plate along vertical sections for A 01 under full load.
noted that the correlation factors for displacement increase somewhat with increasing load, while those for stress generally decrease. The largest correlation factors were for maximum stress at the second row of bolts. From the Table, the critical correlation factors for displacement and vertical stress may be cited as follows: Yllod= 1.42iO.18 *‘,, = 1.21tO.li ‘lir:, = 1.79i 0.39.
CONCLUSION
The values of critical displacements and stresses from 2-D and 3-D finite element analyses of thirteen end-plate connections have been compared, and reasonable correlations between the results have been observed. For practical use in the estimation of 3-D values form 2-D results, correlation factors of 1.4 for displa~ment (and hence rotation), 1.2 for average stress, and 1.8 for maximum stress, may be adopted, for the particular configuration studied. Considerable use of this correlation
389
Steel bolted end-plate connections Table 2. Correlation factors Half Loud Bolt Row 1 I
0.88 0.96 0195
1.61 1.94 1:69
1.06 0.85 0.64
1.98 1.70 1.40
’
Full Load
I
Flange
Bolt-Row 1
Bolt Ron 2
Flange 1
I
1.20 t.43 1:21
1.63 2.T5 1.64 _
1.16 1.29 1.24 _
1.75 2.19 1.88 _
1.21 1.21 1.30
1.52 1.64 1.76
1.07 1.10 0.94
I.73 1.87 1.77
has been made by the first author, in the evaluation of finite element modeb and conclusions from their analyses, on the basis of prototype tests, with satisfactory results. Ac~now~e~e~e~~e-~e financiai support and guidance of the research sponsors AISC and MBMAare gratefully acknowledged. The computing services provided by the Auburn University Computer Center and the University of Alabama (Birmingham) Rust Research Center, are also appreciated. REFgRENCBS 1. N. Krishnamurthy, Finite element analysis of splice-plate connections-a feasibility study. Unpublished Rep., CEAISC/MBMA-1, Dept. of Civil Engng, Auburn University, Alabama (Jan. 1973). 2. N. K~s~arn~hy, Effects of plate thickness and pretensioning in typical bolted end-plate connections. Unpub~~bedRep., C~-AISC~MBMA-2,Dept. of Civil Engng, Auburn University, Alabama (Dec. 1973). 3. N. Krishnamurthy, Correlation between three-dimensional and tw~~mension~ finite element analysis of end-plate
Bolt Row 2 I
1.42
0.67
1.48
1.21
1.57
1.01
1.79
0.18
0.15
0.19
0.11
0.18
0.25
0.39
connections. unpublished Rep., CE-AISCIMBMA-4,Dept. of Civil Engng, Auburn University, Alabama (Oct. 1974). 4. D. E. Graddy, Jr., Feasibility study of finite element analysis of steel splice-plate connections, Unpubiish~ Thesis towards Master’s De8ree in Civil Enginee~ng, Auburn University, Alabama (Mar. 1973). 5. W. McGuire, Sreef Structures, Chap. 6. Prentice-Hall, Englewood Cliffs, New Jersey (1968). 6. R. T. Douty and W. McGuire, High strength bolted moment connections. J. Srnrct. Div., ASCE 91, (SE?), 101-128(Apr. 1%5), 7. R. S. Nair, P. C. Birkemoe and W. H. Munse, High strength bolts subject to tension and prying. J. Struct. Div., ASCE 100, (ST2), 351-372(Feb. 1974). 8. American Institute of Steel Construction, Manual of Steel Construction, 7th Edn, New York (1969). 9. E. L. Wilson, Finite element analysis of two~imension~ structures. Thesis presented to the University of Caliiomia, Berkeley, towards Degree of Doctor of Engineering (1963). 10. S. Levy, 3-D isoparametric finite element pro8ram. General Electric Technical information Series No. 71-C-191, Schenectady, New York (June 1971).