- Email: [email protected]
Elastic-plastic modeling of structural fasteners for steel bracing connections
J. Construct. Steel Research 30 (1994) 13-38
© 1994 Elsevier Science Limited Printed in Malta. All fights reserved 0143-974X/94/$7.00
ELSEVIER
Elastic-Plastic Modeling of Structural Fasteners for Steel Bracing Connections O. S. Bursi Department of Structural Mechanics and Design Automation, University of Trento, Via Mesiano 77, 38050 Trento, Italy and Department of Civil, Environmental and Architectural Engineering,University of Colorado, Boulder, Colorado 80309-0428, USA (Received 17 April 1992; revised version received 30 September 1992; accepted 4 March 1993)
ABSTRACT This paper presents part of the results of a study devoted to the analysis of heavy steel bracing connections, and to the effects of those connections on the behavior of braced flames subjected to static loads. In particular, the paper addresses techniques for modelino the stiffness and strength behavior of structural fasteners within the framework of hardenino plasticity. Through macroscopic considerations, an incremental constitutive law which relates forces to relative displacements in an interface element is defined, which can be implemented infinite element software in a consistent and straightforward manner. Owino to the complex behavior offasteners such as bolted clip angles, fillet welds and bolts subjected to shear loading, the constitutive laws are based on stiffness and strength values obtained from physical tests.
1 INTRODUCTION 1.1 Purpose of the study Braced steel frames such as the braced structure shown in Fig. 1, subjected to static vertical and lateral loads, are often analyzed as ideal pin-jointed trusses. 1 Actually, the framing members are often connected by gusset plate connections such as shown in Fig. 2, which may be bolted or welded to the adjacent members. These bracing connections behave quite differently from the postulated ideal pin-joints. Stiffness and strength characteristics of these connections may affect the frame behavior in several principal ways: 1. The extensional flexibility of the bracing connection, disregarded in 13
14
O. S. Bursi
/ / /
T- N
ma
M
I
Fig. 1. Concentrically braced frame.
N
N
1F
Fig. 2. Beam web-to-column and bracing connections.
pin-jointed as well as rigid-jointed analysis, may contribute appreciably to frame sway under lateral loads. 2. The rotational rigidity of the bracing connection may affect both stiffness and strength of the structure. The beam-to-column joint behavior may approach that of a rigid joint, thus inducing secondary bending, which may be beneficial in cases of distributed member loads, or harmful in the case of axially loaded columns. These stiffness changes also affect the dynamic response of the structure. It is the purpose of this study to develop rational approaches to the prediction of both bracing connection and braced structure behavior, to verify them by physical tests and t o draw conclusions for design practice which may lead to more functional and economical braced steel frames. The work is presented in three papers, of which the first, current one describes the behavior of bolted and welded fasteners and its formulation for use in bracing connection analysis. The second paper 2 concerns itself with connection behavior and its prediction, and finally the third paper 3 considers the structural effects of bracing connections on the response of braced frames subjected to static lateral loads. 1.2 State of the art
Current US practice is probably best summarized by the works of Thornton. 4'5 The three design procedures suggested for bracing connections under tensile monotonic loading are based on the upper-bound
Structural fasteners for steel bracing connections
15
theorem of limit analysis and only satisfy equilibrium. In particular, Method 34,s is capable of producing uniform force distributions on all structural fasteners and therefore will yield the most economical bracing connection. Because these methods only satisfy equilibrium from an upper-bound limit analysis point of view they are unable to account for deformations. They cannot predict the actual distribution of forces in the fasteners, but can consider effects of eccentricity in an approximate fashion. A more sophisticated approach is that of Williams et al., 6 who used a finite element analysis to arrive at distribution of forces among individual fasteners. These analyses resulted in design guidelines for bracing connections under tensile loading. No conclusion was drawn about bracing connection deformations, but of importance was the statement that force distributions among the fasteners depend on the connection-structure member interaction (frame action). Gross 7 investigated the above phenomenon as well as eccentricity and column orientation effects on bracing connection strength by means of a series of ¼-scale tests on braced frame subassemblages. That experimental study verified that force distributions among the fasteners depend on frame action, and that the distribution of the moment between the beam and the column introduced by an eccentric brace loading depends on the beam-to-column connection rigidity. No rational procedure was proposed for predicting these phenomena. Nevertheless, gusset tearout capacity was predicted very closely adopting Hardash and Bjorhovde's method, 8 while the gusset buckling capacity was predicted conservatively using the AISC procedure. 9 Other studies on the behavior of bracing connections under compressive monotonic loading were published based on tests 1° and finite element analyses. 1~ Only guidelines for the design of the gusset plate with fixed boundary conditions were given, owing to the difficulty of simulating the partial restraint offered by structural fasteners. It appears that, at present, no accurate method is available to predict both the stiffness and strength of bracing connections and the effects of connection behavior on the response of full-size braced frames. This will be the purpose of the current study.
1.3 Approach of the study In order to characterize stiffness and strength of bracing connections, the finite element methodology is used (Fig. 3). Figure 3(a) shows the different kinds of elements used in the analysis: elements 1 and 2 are the interface or fastener element used to characterize the bolted double clip angle fastener; element 3 models the behavior of a unit length of Weld; element 4
O. $. Bursi
16 A C T U A L NODE
TL
EL. N.
ISOLATED ELEMENT
2D-MODEL r
i
1,2
terra
;_-_~_, I #__J
I I
(a)
{b)
Fig. 3. Beam web-to-column and bracing connections: (a) FEM idealization; (b) fastener and framing member models.
models the bolt behavior in shear; and element 5 simulates the plate behavior. Figure 3(b) shows how these elements are modeled either as nodal interface elements or, in the case of No. 5, following standard plane-stress theory. All of these elements have an elastic range, followed by inelastic behavior. This first, present paper presents the formulation of a nodal 2D interface element, developed within the framework of rate-independent workhardening plasticity. Such an element is characterized by two nonsymmetric stiffness matrices K, a yield surface F and a flow function G.* As a result, the mechanical behavior of different types of fastener used in bracing connections, including bolted clip angles, fillet welds and high strength bolts subjected to shear loading, can be simulated in a versatile way. Because these interface elements must simulate combined nonlinear problems, such as contact and sliding surface, plasticity and membrane effects, the relevant constitutive laws were obtained through physical tests. 12-17 For this reason, results of tests using one type of bolted double clip angle with several bolt rows and under various loading combinations are highlighted. Furthermore, load-displacement curves for fillet weld fasteners and for high strength bolts subjected to shear found in the literature 1.q7 are presented. Finally, the methodology by which such results are incorporated into the interface elements is shown. *Notation: Dots over symbols denote time derivatives. Matrices are indicated by bold-face symbols, transpose by a superscript T. Parentheses enclose arguments.
Structural fasteners for steel bracing connections
17
The second paper 2 describes the finite element analysis of bracing connections using the fastener (interface) element developed in the first paper. The results of these analyses are then verified by means of an extensive test program. One might note that these analyses could be a valuable tool for bracing connection design; this, however, is not within the scope of the current study. Lastly, the third pape r3 uses the analytically described bracing connection behavior to compute relevant connection flexibilities, which serve for the definition of a simplified model able to simulate bracing connection behavior and capable of being incorporated in standard frame analysis programs. With that connection model the effects of bracing connections on structure behavior are observed, including sways, eccentricity effects and secondary bending.
2 A T W O - D I M E N S I O N A L N O D A L INTERFACE ELEMENT FOR STRUCTURAL FASTENERS The behavior of structural fasteners in bracing connections can be modeled in the context of plane finite element analysis through a twodimensional (2D) interface element. These elements can be generally classified as: (1) continuous 1s'19 interface elements; or (2) nodal or point 2°-22 interface elements. The former elements start from the notion of a continuous relative displacement field, whereas the latter (Fig. 4(a)) lump relative displacement 2D-MODEL
ISOLATED ELEMENT
Y, v n~ /
GUSSET PLATE
//-- Kn' n' Kn.t F Kt't' Kt'n
$1 ~ 3 u 3 2 "2 S2 x,~u (a)
COLUMN (b)
Fig. 4. Structural fastener: (a) nodal interface element;(b) bolted double clip angle unit.
18
O. S. Bursi
to the nodes. The continuous interface elements have been suggested to be superior, 21 but Schellekens22 has shown that in the case of large stiffness values those elements can cause instability and therefore oscillations in the response. Because in the case of bolted clip angles (Fig. 4(b)) the relative forces are transmitted through bolts with finite spacing length, and because high values of stiffness are associated with fillet welds, interface elements with lumped properties were chosen to simulate the mechanical behavior of these structural fasteners. The theoretical description of the behavior of the fasteners is based on a 2D nodal interface element in which both the microscopic (static and dynamic friction, inherent anisotropy, localized plasticity, contact) and macroscopic (slide surface, spreading plasticity and membrane effects) phenomena are included in the macroscopic characteristics of the element and reproduced as continuous nonlinearities. Therefore the adoption of only macroscopic considerations allowed the incremental relationships between the generalized forces and the generalized displacement components of the element to be determined. The formulation of the element requires: (1) the mathematical description of the stress and strain state of the element; (2) the definition of a criterion which defines the stress state which corresponds to the beginning of inelastic deformations; and (3) the determination of a relationship between the stress state and the inelastic deformations. Because these requirements are similar to the requirements which have to be met in the theory of rate-independent plasticity, 23 this theory was adopted for the formulation of incremental relationships of the interface element. In particular, item (1) requires the definition of the elastic stiffness matrix K of the interface element (Fig. 4(a)) which relates increments of stress resultants ~; and increments of relative displacements d c. Because of different fastener responses, the terms of the stiffness matrix K will assume different values if the interface element is subjected to tension-shear loading or compression-shear loading. Item (2) implies the determination of a yield function F(S, h) (Fig. 5), defined such that negative values of F correspond to macroscopic elastic force states in the interface element, zero values of F correspond to plastic force states, and positive values of F are not possible. The vector h represents the variables of the plastic process.
Structural fasteners for steel bracing connections
19
Fn+.=O .... ~ . ~ L j ~
/ t
~ Si
S.+~=St -a~K
3 8 .+a
Fig. 5. Incremental consistency of the elastic predictor-plastic corrector approach.
Item (3) requires the definition of a flow function G(S, h), whose gradient gives the direction of the incremental plastic displacements dp (Fig. 5). Because the above directions are functions of complex microscopic and macroscopic phenomena, the flow rule is 'non-associated', i.e. F ~ G. The formulation of the incremental relationships of the interface element as well as the integration of the constitutive laws is reported in the Appendix. Combined nonlinear problems, such as contact and slide surfaces, localized and spread plasticity as well as membrane effects, have up to now prevented a rigorous analytical approach to the determination of stiffness and strength properties of structural fasteners. Therefore, the constitutive matrix K, the yield function F and the potential flow G of fasteners, such as bolted double clip angles, fillet welds and high strength bolts subjected to shear, were obtained through fitting of experimental data. 12-t 7
3 PHYSICAL MODELING OF BOLTED DOUBLE CLIP ANGLES 3.1 T e s t p r o g r a m
In Fig. 3 the bolted clip angle fasteners are characterized by an element consisting of only one row of bolts; obviously, such single-bolt units behave differently from multiple-bolt units as would actually be present in real connections. For this reason, two- and three-bolt units were also tested, and since their behavior is presumed closer to that of the multi-row clip angles found in bracing connections, the presentation which follows centers on the results of these two- and three-bolt unit tests. In the test program undertaken, 29 tests to failure were carded out ta using one type of bolted double clip angle with one, two and three bolts.
O. S. Bursi
20
These specimens were subjected to pure tension, pure shear, pure compression, combined tension-shear and compression-shear. Tables 1 and 2 summarize the test program for two- and three-bolt units. A typical test specimen consisted of bolted double clip angles fabricated from angles 4 in x 3 in x ¼in (100 mm x 76 mm x 6 mm). Clip angle dimensions in the case of two-bolt units are shown in Fig. 6, while dimensions relative to three-bolt units can be obtained by adding one bolt unit to the two-bolt unit specimen. Because several tests were carried out, angles characterized by two different yield stresses (angle materials 1 and 2, Tables 1 and 2), i.e. fy = 42.8 and 47.7ksi (295 and 329 MPa), were adopted. The clip angles were connected by one, two or three rows of A325 high strength tensioned bolts (yield stress fy,b = 102 ksi (703 MPa)) with TABLE 1
Summary of Calibration Tests with Two-Bolt Units
Test
TI-2 TS301-2 TS451-2
TS452-2 TS453-2 TS601-2 TS602-2 S1-2 $2-2 $3-2 CS451-2 C1-2
Angle material 1 2 1 1 1 2 2 1 1 1 1 1
Load type
•
Testino equipment
Pure tension
0 30 ° 45 °
A B30 A
Tension-shear Tension-shear
1345 Tension-shear
60 °
Pure shear
90 °
Compression-shear
135 ° 180 °
Pure compression
1345 B60 B60 A C C A A
TABLE 2
Summary of Calibration Tests with Three-Bolt Units
Test
TS301-3 TS451-3 TS452-3 TS601-3 TS602-3 S1-3 $2-3
Anole material
Load type
•
Testing equipment
2 2 2 2 2 2 2
Tension-shear Tension-shear
30 ° 45 °
Tension-shear
60 °
Pure shear
90 °
B30 B45 1545 B60 B60 C C
21
Structural fasteners for steel bracing connections
~C-S ~T,-S (;4
~C-S
~T-S S rl
P6*, I ,,a,,,
u
, ,-~- , i
/'\
I
G6 G8
G5 G3
G_L4x3xl/4
G16 G18
G15 G13
3/4 A-SZ5 Bolts
Ril0at clip angle unit
Fig. 6. Two-bolt unit test specimens and linear and rosette strain gage locations: shear (S), combined tension-shear ( T - S ) or compression-shear ( C - S ) loading. Dimensions in inches (1 in=25.4 mm).
¼in (19 mm) diameter, in ~ i n (20.6ram) punched holes. Several testing devices (see Tables 1 and 2) were fab'ricated in order to apply different load types to the specimens according to the displacement approach. For the sake of brevity, only two types of test rig are presented.X2 The first one, labeled A in Tables 1 and 2, is shown in Fig. 7: the connecting plate between the two clip angles was fabricated 1 in (25.4 ram) thick in order to concentrate all deformations in the clip angles. It allowed different load types to be applied to specimens with one or two bolt rows: (a) pure tension, • = 0 °, (b) pure compression, ~ = 180 °, (c) combined tension-shear, ® = 45 °, (d) combined compression-shear, • = 135 °, and ¢)=0"
tT
~) =180"
;c
0=45"
(/)=135"
tT-S
C-S
0=90" Is
LVDTs LVDTs~/''~
C-S
a)
b)
c)
d)
Fig. 7. Testing equipment A and LVDT locations.
~S
o~
22
O. S. Bursi
finally (e) pure shear, • = 90 °. The angle tan • defines the direction between the applied load and the tension axis (Fig. 7). The second type of test rig, labeled C in Tables 1 and 2, is shown in Fig. 8. The welded plate assembly allowed pure shear loading to be applied to up to three-bolt units. In order to monitor local strains in the specimens and for comparison with future FEM predictions, linear and rosette strain gages were attached to the vertical and horizontal legs of the clip angles according to the layout shown in Fig. 6. Furthermore, for measurement of displacements, symmetrically placed LVDT transducers on opposing sides (Figs 7 and 8) were used to average the displacements. The values of displacement components, measured by transducers (Figs 7 and 8) in the normal and shear (n, t) directions (Fig. 4(a)), allowed the behavior of bolted double clip angles to be characterized in terms of force-displacement relations. 3.2 Test results
Because in actual bracing connections only bolted clip angles with multiple rows are used, results obtained from two- and three-bolt unit tests were adopted for the physical modeling.
~
~ =90'
S
SEC. A-A
JS
"~B
ts
..+
ts SEC.
B-B
Fig. 8. Testing equipment C and LVDT locations.
Structural fasteners for steel bracin0 connections
23
For the sake of brevity, only the test results obtained for the two-bolt units are presented in Fig. 9. Figure 9(a) plots tensile force component applied per bolt versus tensile displacement component (along axis n in Fig. 4(a)); Fig. 9(b) plots shear force component per bolt versus shear displacement (along axis t in Fig. 4(a)). Both curves highlight the anisotropic behavior of bolted clip angles as well as the interaction effect between tension and shear. Similar behavior was observed in one- and three-bolt unit tests, and in compression-shear tests. ~2 These curves lend themselves to representation by bilinear elastic-strain hardening relations.
3.3 Modeling Force-displacement relationships relative to directions n (Fig. 9(a)) and t (Fig. 9(b))--see Fig. 4--were represented according to bilinear laws of elastic-plastic hardening type (see, for example, Fig. 10). As a result, the characteristic value of both the elastic stiffness K e and the hardening stiffness K p were obtained by linear regression techniques. 12 These values allowed each plastic failure load S p to be identified. Analyses carried out by Sigfusdottir and GerstlC 3 on isolated bracing connections indicate that a bilinear representation compared to a multilincar one leads to errors in the structural response assessment that are within engineering accuracy
(4%). Experimental elastic stiffness values scaled to one bolt clip angle unit and relative to loading conditions of tension and shear as a function of tan are reported in Figs 11(a) and (b) respectively. The trend of variation of 25
25
gO" ~20 -
~15~
~=o' 45"
10-
20. v W15. O nfl~lO-
5-
5.
o.;o o,~o o.~o oAo " o. TENSION DISPLACEMENT (inches)
(a)
o.lo o.lo o.~ oAo SHEAR DISPLACEMENT (inches)
(b)
Fig. 9. Bolted double clip angle unit responses: (a) tension component; (b) shear component.
24
O. S. Bursi 25P U R E TENSION TEST W / O - B O L T UNITS ~20 -
~15-
zlo-
_o z
030
l
o.Jo
o.,.~
oAo
o.
TENSION DISPLACEMENT ( i n c h e s )
Fig. 10. Bilinear model of a bolted clip angle test (T1-2) force-displacement relationship. 5000 o
5000 ¸ rl-
Z~---
I ~ ' 0 - B O L T UNIT TEST THREE-BOLT UNIT TEST ANALYTICAL SIMULATION
~4ooo LO b_ z 3000LL
A
__2000"
(~
r'l&---
u) ~.~4ooo • v
'PWO-BOLT UNIT TEST THREE-BOLT UNIT TEST ANALYTICAL SIMULATION
~ 3000Ll(,3 ~..~2000
Z~
A
¢/3
O
[] D
0.~
rY 1 0 0 0
(2
~.6o
2.60 ton~
(a)
~.oo
o.oo
2.6o
1 .oo
~.oo
( t a n ~ ) -1
(b)
Fig. 11. Experimental and analytical elastic stiffness variation of a bolted double clip angle unit: (a) tension component; (b) shear component. stiffness values indicates that the tension stiffness decreases owing to the presence of larger shear components (Fig. l l(a)) while the shear stiffness decreases owing to the presence of larger tension components (Fig. 1 l(b)). To compute the (2 x 2) constitutive stiffness matrix K of the interface element (Fig. 4(a)) from the tensile and shear stiffnesses obtained experimentally, and plotted i n Fig. 11 as functions of the applied load direction tan ~, the following load-displacement relation is considered:
{AST = FKnn Kntl AT
ASsJ LK,. K.J(AsJ
(1)
Structural fasteners for steel bracino connections
25
in which AT, As are the measured tension and shear displacements, ASr and ASs are the imposed tension and shear forces, and KM and K , are the measured stiffnesses under pure tension and shear. The coupling stiffnesses K,t and Ktn are unknown. To find these, eqns (1) are written in the form:
KT@ (2)
Ks¢ = Kn +
' ~,KT®J tan •
in which KTo(=AST/AT) and Kso(= ASs/As) are the measured stiffnesses under load applied at an angle • with the tension direction, where tan is equal to ASs/AST. Using the values of Kr® and Kso obtained under load at angles ~ = 30 °, 45 ° and 60 °, whose mean values were approximated by eqns (2) (solid lines in Fig. 11), the desired coupling stiffnesses can be determined. For the case of combined tension and shear in the range prior to yielding (Fig. l l(a)), the resulting fastener stiffness matrix in kips/in is:
I163 4
K=-501
- 541 1297
(3)
For the case of the inelastic range and the compression and shear, as well as for the fastener weld units, similar calculations led to the element stiffness matrices needed in the analysis. Plastic load values S p scaled to one bolt clip angle unit and relative to force-displacement component curves (Fig. 9) were plotted within a plane (Fig. 12). In such a plane the horizontal axis represents the shear force Ss, while the vertical axis in the positive direction represents the tension force ST and in the negative direction the compression force Sc. As expected, bolted double clip angles have more strength when subjected to compression-shear loading. Two elliptic interaction curves were fitted through experimental data points in order to define the yield function F (Fig. 12). Plastic stiffnesses of force--displacement curves in the (n, t) directions allowed both components and ratios of plastic displacement increments Ad p to be evaluated. As a result, two elliptic functions G were calibrated (Fig. 13), such that their gradient represents the directions of plastic displacement increments Adp.
26
O. S. Bursi
rl_ 'TWO-BOLT UNff 11[$1"
~Sr,d.
s~
Z~- THREE-BOLT UNIT TEST
+ ~,iT3.o/ =t
/
oo~\
/~nn
/
,L
r I
Ts~
/
\
n\
/ \....
[]
/
Forces are In kips 1 kipl~4.45 kN
\
I
Forces are in kips
- ~ ( sc ~ +(s__,~ =l \-~e.T/
/
]
Ss, d t
"
\ (
Sc '~
i" Ss ~=
~--~.7/ + ~,~'~.o) :
\~Ls/
OC
Sc,d.
Fig. 12. Yield surface F of a bolted double clip angle unit.
'
Fig. 13. Potential flow surface G of a bolted double clip angle unit.
4 PHYSICAL MODELING
OF FILLET WELDS
4.1 Load-displacement response The global behavior of fillet welds connecting planar members (Fig. 14(a)) and loaded centrally or eccentrically is usually based on a linear isotropic
!
I
(a) A
(b)
B
C
(c)
(d)
Fig. 14. Fillet weld test specimen (Ref. 14) and decompositions: (a) displacement AF and loading angle ~; (b) case A; (c) case B; (d) case C.
Structural fasteners for steel bracing connections
27
relationship between toe displacement and load strength of the weld element. Results of physical tests ~4'15 show that stiffness and strength characteristics of fillet welds depend both on the direction of the applied load and on the connection geometry: fillet welds loaded perpendicular to their axis are stronger and less ductile than welds loaded parallel to their axis. Such behavior is mainly attributed to two factors: (i) the restraint to deformation of the fillet weld by connecting elements, which is maximum for transverse welds; and (ii) the change in the angle of the failure plane which increases the weld strength. ~* The complexity of the above behavior and the lack of a rational analytical model able to simulate the anisotropic behavior of fillet welds suggested use of test data of Kennedy and co-workers. 14'15 They carried out 42 tests to failure on specimens made of symmetrically connected elements (Fig. 14~a)) by fillet welds of ¼in (6 mm) and ] i n (9 mm) leg, respectively. The yield stress fy,w of the weld metal was equal to 67 ksi (465 MPa). Test specimens were subjected to different combined loads of tensionshear (~ = 0 ° (pure tension), 15°, 30 °, ..., 90 ° (pure shear)) in order to duplicate realistic loading conditions. Figure 14(a) shows the geometry of a characteristic test specimen ~* and the relative displacement component AF of fillet weld edges along the loading direction that was measured and the loading angle ~. In the tests of Ref. 14, only one component of displacement, A~ (Fig. 14(a)), was measured; with only this one component, it was not possible to determine the necessary two displacement components Ar and As (see eqn (1)). This made it necessary to assume a relationship between these components. To this end, three decompositions of the relative displacements of the fillet weld along the (n, t) directions were considered (Figs 14(b)-(d)). In the first one labeled A (Fig. 14(b) the transverse component was assumed zero; in the second (B) and third (C) a normal component (A~, Fig. 14(c); At, Fig. 14(d)) equal to an arbitrary value, 15% of AF, was considered in opposite directions respectively. In all three cases the same load F (Fig. 14(a)) was considered and decomposed. Load--displacement Curves obtained according to these cases are reported in Fig. 15(a) for the normal (n) and in Fig. 15(b) for the shear (t) direction, respectively, relative to the particular case of a • = 45 ° loading angle. Figure 15 shows that both the stiffness and strength variations are not significant even in the case of other loading angles ~. Numerical simulations performed on isolated bracing connections with boltedwelded fasteners, in which stiffness and strength properties for fillet welds obtained from all three decompositions were adopted (Fig. 14), indicate variations in the response within engineering accuracy (+4.5%).
28
O. S. Bursi IILI~.
~:45
~T) = 4 5 °
°
w ¢dl tv.
Z ~
W
4.00
"B
/,
I I 1 ~.
"r
o.dl
TENSION DISPLACEMENT(inches)
(a)
o.,
0.~
o.61
SHEAR DISPLACEMENT(inches)
(b)
Fig. 15. Fillet weld unit force-displacement relations according to cases A, B and C: (a) tension component;(b) shear component.
The location of fillet welds in the tests with a loading angle of @= 45 °, 60 ° and 75 °14 shows clearly that a structure was tested because two mirror-image fillet weld elements were considered instead of a single one. As a result, the normal component of the displacement (Ai~, Fig. 14(c), or A[, Fig. 14{d)) surely present in a single fillet weld element canceled out. Therefore, further tests should be carried out in order to measure distinctly both the tensile and the shear component displacement of one single element of fillet weld under various loading combinations. Owing to the results of the above sensitivity analysis and the lack of experimental data, only the relative displacement AF (case A, Fig. 14(b)) was considered for the related decomposition. As a result, load-displacement relationships were obtained for the normal (n) and shear (t) directions. Figure 16 shows in the case of tension-shear loading the force-displacement component relationships of a fillet weld of ¼in (6 mm) leg and 1 in (25.4 mm) length. Both diagrams highlight clearly the anisotropic stiffness and strength behavior of fillet weld elements in the tension (Fig. 16(a)) and the shear (Fig. 16(b)) direction. The experimental behavior of fillet welds for compression-shear loading conditions was also obtained from Refs 14 and 15.
4.2 Modeling Similarly to bolted clip angles, the load-displacement response of welded fasteners was modeled by a linearized law of bilinear type. Therefore, values of both the elastic K e and hardening stiffness K p were computed, as well as the plastic load S p.
Structural fasteners for steel bracing connections
29
16
A .~_Q.12°
~.a
0~ 0.000
0 2 5 0.00~ TENSION DISPLACEMENT (inches)
(
o.o~
0.0% o.o',5
o.o~o
SHEAR DISPLACEMENT (inches)
(a)
0.02.j
(b)
Fig. 16. Fillet weld unit responses: (a} tension component; (b) shear component.
The fitting process of experimental elastic stiffnesses allowed the elastic constitutive matrix K of the interface element to be determined for the loading cases of tension-shear and compression-shear. The plastic load vector S p relative to a fillet weld of ¼in (6 ram) leg and 1 in (25.4 mm) length is represented in the tension-shear interaction domain Ss--ST (Fig. 17). In the compression-shear domain $s-Sc a value of the plastic load of pure compression S~ equal to 0.891 times the value of the plastic load in pure tension S~ was assumed according to experimental data. x4 As a result, two interaction curves were fitted for the definition of the yield function F (Fig. 17). , ST, d n /s,~,,
A-
(s,~,
;2,s/ - ~l~.s/ = ;
TEST
_./
/TX
/ } I
÷ \
.
4
F o r c e s a r e in k i p s 1 k i p s = 4.45 kN
--T~.o) + \ Fo/ = I SC
Fig. 17. Yield surface F of a fillet weld unit.
I \ Forces
Ire
/ /
x in kips
( Sc ~
[$l ~'
~--~.3/ + u-~.s) = 1
Sc, dn
Fig. 18. Potential flow surface G of a fillet \ weld unit.
30
O. S. Bursi
Similarly to bolted clip angle fasteners, two other functions (Fig. 18) were calibrated on plastic displacement increments Adp, in order to represent the potential flow function G.
5 PHYSICAL MODELING OF HIGH STRENGTH BOLTS S U B J E C T E D T O S H E A R
5.1 Load-displacement response The connection between the gusset plate and the bracing member (Fig. 2) is often made with high strength bolts subjected to single or double shear. Several shear load versus displacement relationships are available in the literature, 16'~7 relative to tests by subjecting bolts to shear induced by plates either in tension or compression. Figure 19 shows a typical experimental response of an A325 bolt (yield stress fy,b = 92 ksi (635 MPa)) with a ¼in (19 mm) diameter subjected to single shear by plates A36 (fy = 36 ksi (248 MPa)) in tension. 17 Relative displacements were measured by two dial gages clamped on test plates. Similar plots were obtained for other plate thicknesses as well as with varying bolt diameters. 35 ¸
25 ¸
~20 =
n
o15
=0.5
K = 12400 k / i n Kp = 0 k / i n
10
K I = 12400 k / i n
5
0
o
o os
OlO
o'15
oi0
0.~s o:3o
D e f o r m a t i o n (inches)
Fig. 19. Load-displacement response of a bolt subjected to single shear (Ref. 17).
Structural fasteners for steel bracino connections
31
5.2 Modeling Since relative displacements were measured through test plates, x7 the elastic response of the connected plates was subtracted to averaged readings. In this way the load--displacement relationship shown in Fig. 19 represents both the elastic and inelastic deformations of the test bolt as well as the inelastic deformations of the connected plates. The shear spring of the 2D interface element was the only one employed to describe analytically the bolt fastener-plate behavior (element 4, Fig. 3). Bilinear laws obtained through fitting of curves similar to that shown in Fig. 19 were used to simulate the shear force-displacement relationships of bolted shear fasteners. 6 CONCLUSIONS The results of the experimental investigation carried out by the author on bolted double clip angles and data found in the literature for fillet weld connections and high strength bolts subjected to shear allowed the behavior of these structural fasteners to be characterized. Microscopic phenomena (static and dynamic friction, contact and localized plasticity) as well as macroscopic phenomena (slide surface, spread plasticity and membrane effects) were represented through the constitutive law of a 2D nodal interface element as continuous nonlinearities for each fastener type examined. As a result, a finite element model of heavy bracing connections using the interface element and plane stress elements with three and four nodes can be developed. The validity of this approach is shown in a companion paper, 2 where the response of a full-scale bracing connection with bolted-bolted and bolted-welded fasteners was numerically predicted and experimentally verified.
ACKNOWLEDGMENTS This work was carried out under a grant from the National Research Council of Italy (CNR) while visiting at the University of Colorado at Boulder. Thanks are also due to the Department of Civil, Environmental and Architectural Engineering (Boulder) for general support. Finally, special thanks are extended to the technical staff of the Department of Structural Mechanics and Design Automation (Trento) for their assistance
32
o. s. Bursi
throughout the testing program. However, the opinions expressed in this paper are those of the writer, and do not necessarily reflect those of the sponsors.
REFERENCES 1. Salmon, C. G. & Johnson, J. E., Steel Structures--Design and Behavior, Emphasizing Load and Resistance Factor Design (3rd edn). Harper & Row, New York, 1990. 2. Bursi, O. S., Gerstle, K. H., Sigfusdottir, A. & Zitur, J., Behavior and analysis of bracing connections of steel frames. J. Construct. Steel Res., 30 (1994) 39-60. 3. Bursi, O. S. & Gerstle, K. H., Analysis of flexibly connected braced steel frames. J. Construct. Steel Res. 30 (1994) 61-83. 4. Thornton, W. A., A cost comparison of some methods for design of bracing connections. In Proc. 2nd Int. Workshop Connections in Steel Structures: Behavior, Strength and Design, Pittsburgh, PA, April 1991. 5. Thornton, W. A., On the analysis and design of bracing connections. In Proc. 1991 AISC National Steel Construction Conf., Washington, DC, June 1991, pp. 26-1-33. 6. Williams, G. C. & Richard, R. M., Steel connection designs based on inelastic finite element analysis. Design report prepared for American Institute for Steel Construction, Department of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, AZ, May, 1986. 7. Gross, J. L., Experimental study of gusseted connections. In Proc. 1989 AISC National Steel Construction Conf., Nashville, TN, June 1989 pp. 11-1-22. 8. Hardash, S. G . & Bjorhovde, R., New design criteria for gusset plates in tension. Engn9 J. AISC, 21(3) (1985) 139-48. 9. AISC, Engineering for Steel Construction. American Institute of Steel Construction, Chicago, IL, 1984. 10. Hu, S. Z. & Cheng, J. J. R., Compressive behavior of gusset plate connections. Structural Engineering Report No. 153, University of Alberta, Canada, 1987. 11. Chakrabarti, S. K. & Richard, R. M., Inelastic buckling of gusset plates. Struct. Engng Rev., 2 (1990) 13-29. 12. Bursi, O. S., Tests and modeling of bolted double clip angles of steel bracing connections. Internal Report, SMDA Department, University of Trento, Italy, Oct. 1990. 13. Sigfusdottir, A. & Gerstle, K. H., Analysis and tests of bolted bracing connections. In Ninth Structures Congress, ASCE, 29 April-1 May 1991, pp. 430-3. 14. Miazga, G. S. & Kennedy, D. J. L., Behaviour of fillet welds as a function of the angle of loading. Can. J. Civ. Engng, 16 (1989) 583-99. 15. Lcsik, D. F. & Kennedy, D. J. L., Ultimate strength of fillet welded connections loaded in plane. Can. J. Civ. Engng, 17 (1990) 55-67. 16. Wallaert, J. J. & Fisher, J. W., Shear strength of high strength bolt. J. Struct. Div. ASCE, 91(ST3) (1965) 1003-18. 17. Richard, R. M., Gillett, P. E., Kriegh, J. D. & Brett, A. L., The analysis and design of single plate framing connections. Engng J. AISC, 2 (1980) 38-51.
Structural fasteners for steel bracing connections
33
18. Schafer, H., A contribution to the solution of contact problems with the aid of bond elements. Comp. Meth. Appl. Mech. Enong, 6 (1975) 335-54. 19. Rots, J. G., Bond-slip simulations using smeared cracks and/or interface elements. Res. Report 85-01, Struct. Mech. Dept. of Civil Engng, Delft Univ. of Tech., The Netherlands, 1985. 20. Ngo, D. & Scordelis, A. C., Finite element analysis of reinforced concrete. J. Am. Concrete Inst., No. 64-14 (1967) 152-63. 21. Nilson, A. H. et al. (eds), Finite element analysis of reinforced concrete. State-of-the-Art Report, ASCE, New York, 1982. 22. Schellekens, J. C. J., Interface elements in finite element analysis. Res. Report 25-2-90-5-17, Struct. Mech. Dept. of Civil Engng, Delft Univ. of Tech., The Netherlands, Oct. 1990. 23. Curnier, A., A theory of friction. Int. J. Solids Struct., 20(7) (1984) 637-47. 24. Simo, J. C. & Taylor, R. L., Consistent tangent operators for rate independent elastoplasticity. Comp. Meth. Appl. Mech. Engng, 48 (1985) 101-18. 25. Ortiz, M. & Popov, E. P., Accuracy and stability of integration algorithms for elastoplastic constitutive relations. Int. J. Num. Meth. Engng, 21 (1985) 1561-76. 26. Reich, R. & Montgomery, K., MICROFEM-88. Internal Report, CEAE Department, University of Colorado, Boulder, CO, Oct. 1988. APPENDIX: A CONSTITUTIVE LAW FOR A 2D N O D A L I N T E R F A C E E L E M E N T In the following subsections the constitutive law of a 2D nodal or point interface element is developed in incremental form and an explicit relationship between increments of forces and displacements is obtained. Theoretical as well as computational aspects are given. A.I Elastic stiffness matrix
A two-dimensional nodal interface element characterized by lumped properties with coupling, constituted by two linear springs S1 and $2, is considered (Fig. 4(a)). It models a structural fastener unit between the gusset plate and framing members, such as a bolted double clip angle unit (Fig. 4(b)). External nodes (1 and 3; Fig. 4(a)) of that element are attached to the gusset plate, while the common node (2) is attached to the column flange. Nodes 2 and 3, which have the same location in the finite element model, are shown separated (Figs 3(b) and 4(a)) for clarity. A local coordinate system, with axes normal and tangent to the column flange (Fig. 4(a)), is indicated by (n, t). These axes are assumed to be principal axes of the element. The kinematic variables to be used in the constitutive law are the relative displacements d between the gusset plate
34
O. S. Bursi
and the column flange in the normal and tangential direction, which are defined as:
d=(u2 -ul)e
(A1)
where ul defines the displacements of the element SI (Fig. 4(a)), while e is a unit vector in the coordinate directions (n, 0. The stress resultants that the interface element supports are denoted by S and the convention that compressive stresses are negative is used. In the following, an explicit relation between increments of forces and displacements is developed. Because the relation is incremental, it is valid for arbitrary load and displacement histories that can involve unloaded and subsequent reloading in the element. According to the theory of elasto-plasticity and with the restriction of small displacement gradients, the relative displacements d are additively decomposed into: d = de + dp
(A2)
where superscripts e a n d p denote the elastic part of the displacement. Because the forces ~hat-the interface element supports must be reversible, they can be related only to the elastic part of eqn (A2). Therefore, the relation between increments of forces and displacements is expressed by the linear equation: ~;=Kd e
(A3)
where K represents the elastic stiffness matrix of the element. In greater detail, Kn.n defines the normal stiffness, Ks.s defines the shear stiffness, while the constants Kn,t and Kt.~ represent the coupling stiffnesses (Fig. 4(a)). Because the increment of the normal force S~ must assume the same value for positive or negative increments in elastic shear displacements dT, the sign of the constant K,.t must be direction-dependent. Finally, owing to the different properties of fasteners, the constants of the stiffness matrix K assume different values if the interface element is subjected to combined tension-shear or compression-shear loading. A.2 Yield surface and potential flow
It is necessary to postulate a rule for plastic displacements d p. In particular, it is assumed: 1. There is a scalar yield function F(S, h), defined such that negative values of F correspond to elastic stress states in the interface element,
Structural fasteners for steel bracing connections
35
zero values of F correspond to plastic stress states and positive values of F are not possible; the vector h represents the internal variables of the plastic process. 2. Increments of forces are linearly related to increments of displacements. 3. A suitable relation analogous to the flow rule employed in plasticity:
tiP
if F < 0 or /~ < 0 (unloading)
f°
• t~G
if F =/" = 0 (loading)
(A4)
where G(S, h) is a scalar-valued flow function, whose gradient gives the direction of the incremental plastic displacements d p, while ,~ is a nonnegative scalar which defines the magnitude of the increments (Fig. 5). Because the above directions are functions of complex microscopic and macroscopic phenomena, the flow rule is 'non-associated', i.e. F ~: G (see Figs 12--13 and 17-18).
A.3 Integration of constitutive relations During the loading process of the interface element, the stress state must satisfy the yield criterion and remain on the relative surface, i.e. F = 0 and dF = 0, which can be written: OF=
~F
dF d S + -=-. Oh=0 dh
(A5)
Because the interface element presents directional properties (anisotropy), the internal variables h which describe the hardening of the yield surface F are expressed as a function of the normal plastic work dW]---S~r dd~ and the shear plastic work d Wtp = Stx dd,v. Substituting eqns (A2)-(A4) into eqn (A5), the following incremental constitutive law is obtained: ~=K~pd
(A6)
where for purely elastic response or unloading (F < 0 or/~ <0):
Kep = K
(A7)
O. S. Bursi
36
and for imminent plastic p h e n o m e n a (F = P = 0): OG OFT
Kep
~
K----K 0S 0S S
--
OFT
--K OS
OG
OFTOG
OS
Oh dS
(A8)
The plastic multiplier )[ can therefore be explicitly expressed as: OF T
--Kd 0S ~'=OFT_ OG OFT OG K as
(A9)
Classical integration schemes employ a tangent stiffness formulation and are traditionally explicit in nature• The current development is, however, oriented towards an 'elastic predictor-plastic corrector' strategy, based on implicit integration. In fact, implicit schemes are more versatile and robust in an environment where the nature of loading can be highly variable. Let F . = F ( S . , h . ) = 0 represent the yield condition at the time t = t . of the element subjected to a finite load increment AS, in which the displacement increment Ad is given, and let F. + x = F . + ~(S. + ~, h. + 1) = 0 represent the yield condition of the element satisfied by the state of stress resultants S. + ~ = S. + AS and internal variables h.+ 1 = h. + Ah at the time t = t.+ x. The expansion of the yield condition in a Taylor series: F,, + 1 = F. + AF' + 0"5A 2 F" + . . . + HO = 0
(AIO)
opens mainly two possibilities to enforce the yield condition F, + x = 0 and therefore to determine the plastic multiplier A2. The first approach is based on truncation of the Taylor expansion after the first term on the right-hand side of eqn (A10), which leads to the traditional linearized form of consistency, i.e. AF = 0. This linearization results in an explicit format of the plastic multiplier A2 (eqn (A9)) and of the elastic-plastic tangent operator (eqn (A8)), but this implies errors in the case of nonlinear internal variables h and yield functions F.
Structural fasteners for steel bracin0 connections
37
The second and more direct approach, adopted here, is based on the fulfilment of the yield condition at the end of each load increment: F.+ :(S, +AS, h. + Ah) = 0
(All)
which results in greater reliability and allows errors in the plastic return step to be minimized.24 Projecting the trial stress resultant S, on to a subsequent loading surface F , + ~ = 0 in a single step (Fig. 5), two unknowns need to be identified, i.e. the magnitude and the direction of the plastic return step. Such changes can be expressed explicitly in terms of the plastic multiplier A2 and the direction of the plastic flow: d~ s,.k,
(A12)
By using the implicit integration scheme 'generalized midpoint rule', 25 the resultant stress state S, and the plastic internal variables h. adopted for the computation of the plastic flow gradient (eqn (A12)) can be evaluated through the equations: S. =(1 - ct)S, + atSn+ x
(A13)
h , = (1 - ~)h,, + ~h,,+ 1
(A14)
at an intermediate point (Fig. 5) defined by the parameter ct (0~
(A15)
which can be solved for the unknown plastic multiplier A2. The integration scheme of the constitutive relations of the nodal interface element was implemented in a nonlinear finite element computer program. 26 Such a program uses an incremental-iterative solution procedure for the determination of each different structure equilibrium state,
38
O. S. Bursi
based on the initial stress method. Because both the elastic constitutive matrix K (eqn (A3)) and the elastic-plastic tangent operator of the interface element are nonsymmetric, only the symmetric part of the global structure stiffness matrix K s , i.e. Ksym 8 _- ½(Kg -~ K I )
(A16)
was used in the iterative algorithm for the solution of the nonlinear equation system. The algorithm guarantees a limited iteration number for an assigned iteration tolerance.
© 1994 Elsevier Science Limited Printed in Malta. All fights reserved 0143-974X/94/$7.00
ELSEVIER
Elastic-Plastic Modeling of Structural Fasteners for Steel Bracing Connections O. S. Bursi Department of Structural Mechanics and Design Automation, University of Trento, Via Mesiano 77, 38050 Trento, Italy and Department of Civil, Environmental and Architectural Engineering,University of Colorado, Boulder, Colorado 80309-0428, USA (Received 17 April 1992; revised version received 30 September 1992; accepted 4 March 1993)
ABSTRACT This paper presents part of the results of a study devoted to the analysis of heavy steel bracing connections, and to the effects of those connections on the behavior of braced flames subjected to static loads. In particular, the paper addresses techniques for modelino the stiffness and strength behavior of structural fasteners within the framework of hardenino plasticity. Through macroscopic considerations, an incremental constitutive law which relates forces to relative displacements in an interface element is defined, which can be implemented infinite element software in a consistent and straightforward manner. Owino to the complex behavior offasteners such as bolted clip angles, fillet welds and bolts subjected to shear loading, the constitutive laws are based on stiffness and strength values obtained from physical tests.
1 INTRODUCTION 1.1 Purpose of the study Braced steel frames such as the braced structure shown in Fig. 1, subjected to static vertical and lateral loads, are often analyzed as ideal pin-jointed trusses. 1 Actually, the framing members are often connected by gusset plate connections such as shown in Fig. 2, which may be bolted or welded to the adjacent members. These bracing connections behave quite differently from the postulated ideal pin-joints. Stiffness and strength characteristics of these connections may affect the frame behavior in several principal ways: 1. The extensional flexibility of the bracing connection, disregarded in 13
14
O. S. Bursi
/ / /
T- N
ma
M
I
Fig. 1. Concentrically braced frame.
N
N
1F
Fig. 2. Beam web-to-column and bracing connections.
pin-jointed as well as rigid-jointed analysis, may contribute appreciably to frame sway under lateral loads. 2. The rotational rigidity of the bracing connection may affect both stiffness and strength of the structure. The beam-to-column joint behavior may approach that of a rigid joint, thus inducing secondary bending, which may be beneficial in cases of distributed member loads, or harmful in the case of axially loaded columns. These stiffness changes also affect the dynamic response of the structure. It is the purpose of this study to develop rational approaches to the prediction of both bracing connection and braced structure behavior, to verify them by physical tests and t o draw conclusions for design practice which may lead to more functional and economical braced steel frames. The work is presented in three papers, of which the first, current one describes the behavior of bolted and welded fasteners and its formulation for use in bracing connection analysis. The second paper 2 concerns itself with connection behavior and its prediction, and finally the third paper 3 considers the structural effects of bracing connections on the response of braced frames subjected to static lateral loads. 1.2 State of the art
Current US practice is probably best summarized by the works of Thornton. 4'5 The three design procedures suggested for bracing connections under tensile monotonic loading are based on the upper-bound
Structural fasteners for steel bracing connections
15
theorem of limit analysis and only satisfy equilibrium. In particular, Method 34,s is capable of producing uniform force distributions on all structural fasteners and therefore will yield the most economical bracing connection. Because these methods only satisfy equilibrium from an upper-bound limit analysis point of view they are unable to account for deformations. They cannot predict the actual distribution of forces in the fasteners, but can consider effects of eccentricity in an approximate fashion. A more sophisticated approach is that of Williams et al., 6 who used a finite element analysis to arrive at distribution of forces among individual fasteners. These analyses resulted in design guidelines for bracing connections under tensile loading. No conclusion was drawn about bracing connection deformations, but of importance was the statement that force distributions among the fasteners depend on the connection-structure member interaction (frame action). Gross 7 investigated the above phenomenon as well as eccentricity and column orientation effects on bracing connection strength by means of a series of ¼-scale tests on braced frame subassemblages. That experimental study verified that force distributions among the fasteners depend on frame action, and that the distribution of the moment between the beam and the column introduced by an eccentric brace loading depends on the beam-to-column connection rigidity. No rational procedure was proposed for predicting these phenomena. Nevertheless, gusset tearout capacity was predicted very closely adopting Hardash and Bjorhovde's method, 8 while the gusset buckling capacity was predicted conservatively using the AISC procedure. 9 Other studies on the behavior of bracing connections under compressive monotonic loading were published based on tests 1° and finite element analyses. 1~ Only guidelines for the design of the gusset plate with fixed boundary conditions were given, owing to the difficulty of simulating the partial restraint offered by structural fasteners. It appears that, at present, no accurate method is available to predict both the stiffness and strength of bracing connections and the effects of connection behavior on the response of full-size braced frames. This will be the purpose of the current study.
1.3 Approach of the study In order to characterize stiffness and strength of bracing connections, the finite element methodology is used (Fig. 3). Figure 3(a) shows the different kinds of elements used in the analysis: elements 1 and 2 are the interface or fastener element used to characterize the bolted double clip angle fastener; element 3 models the behavior of a unit length of Weld; element 4
O. $. Bursi
16 A C T U A L NODE
TL
EL. N.
ISOLATED ELEMENT
2D-MODEL r
i
1,2
terra
;_-_~_, I #__J
I I
(a)
{b)
Fig. 3. Beam web-to-column and bracing connections: (a) FEM idealization; (b) fastener and framing member models.
models the bolt behavior in shear; and element 5 simulates the plate behavior. Figure 3(b) shows how these elements are modeled either as nodal interface elements or, in the case of No. 5, following standard plane-stress theory. All of these elements have an elastic range, followed by inelastic behavior. This first, present paper presents the formulation of a nodal 2D interface element, developed within the framework of rate-independent workhardening plasticity. Such an element is characterized by two nonsymmetric stiffness matrices K, a yield surface F and a flow function G.* As a result, the mechanical behavior of different types of fastener used in bracing connections, including bolted clip angles, fillet welds and high strength bolts subjected to shear loading, can be simulated in a versatile way. Because these interface elements must simulate combined nonlinear problems, such as contact and sliding surface, plasticity and membrane effects, the relevant constitutive laws were obtained through physical tests. 12-17 For this reason, results of tests using one type of bolted double clip angle with several bolt rows and under various loading combinations are highlighted. Furthermore, load-displacement curves for fillet weld fasteners and for high strength bolts subjected to shear found in the literature 1.q7 are presented. Finally, the methodology by which such results are incorporated into the interface elements is shown. *Notation: Dots over symbols denote time derivatives. Matrices are indicated by bold-face symbols, transpose by a superscript T. Parentheses enclose arguments.
Structural fasteners for steel bracing connections
17
The second paper 2 describes the finite element analysis of bracing connections using the fastener (interface) element developed in the first paper. The results of these analyses are then verified by means of an extensive test program. One might note that these analyses could be a valuable tool for bracing connection design; this, however, is not within the scope of the current study. Lastly, the third pape r3 uses the analytically described bracing connection behavior to compute relevant connection flexibilities, which serve for the definition of a simplified model able to simulate bracing connection behavior and capable of being incorporated in standard frame analysis programs. With that connection model the effects of bracing connections on structure behavior are observed, including sways, eccentricity effects and secondary bending.
2 A T W O - D I M E N S I O N A L N O D A L INTERFACE ELEMENT FOR STRUCTURAL FASTENERS The behavior of structural fasteners in bracing connections can be modeled in the context of plane finite element analysis through a twodimensional (2D) interface element. These elements can be generally classified as: (1) continuous 1s'19 interface elements; or (2) nodal or point 2°-22 interface elements. The former elements start from the notion of a continuous relative displacement field, whereas the latter (Fig. 4(a)) lump relative displacement 2D-MODEL
ISOLATED ELEMENT
Y, v n~ /
GUSSET PLATE
//-- Kn' n' Kn.t F Kt't' Kt'n
$1 ~ 3 u 3 2 "2 S2 x,~u (a)
COLUMN (b)
Fig. 4. Structural fastener: (a) nodal interface element;(b) bolted double clip angle unit.
18
O. S. Bursi
to the nodes. The continuous interface elements have been suggested to be superior, 21 but Schellekens22 has shown that in the case of large stiffness values those elements can cause instability and therefore oscillations in the response. Because in the case of bolted clip angles (Fig. 4(b)) the relative forces are transmitted through bolts with finite spacing length, and because high values of stiffness are associated with fillet welds, interface elements with lumped properties were chosen to simulate the mechanical behavior of these structural fasteners. The theoretical description of the behavior of the fasteners is based on a 2D nodal interface element in which both the microscopic (static and dynamic friction, inherent anisotropy, localized plasticity, contact) and macroscopic (slide surface, spreading plasticity and membrane effects) phenomena are included in the macroscopic characteristics of the element and reproduced as continuous nonlinearities. Therefore the adoption of only macroscopic considerations allowed the incremental relationships between the generalized forces and the generalized displacement components of the element to be determined. The formulation of the element requires: (1) the mathematical description of the stress and strain state of the element; (2) the definition of a criterion which defines the stress state which corresponds to the beginning of inelastic deformations; and (3) the determination of a relationship between the stress state and the inelastic deformations. Because these requirements are similar to the requirements which have to be met in the theory of rate-independent plasticity, 23 this theory was adopted for the formulation of incremental relationships of the interface element. In particular, item (1) requires the definition of the elastic stiffness matrix K of the interface element (Fig. 4(a)) which relates increments of stress resultants ~; and increments of relative displacements d c. Because of different fastener responses, the terms of the stiffness matrix K will assume different values if the interface element is subjected to tension-shear loading or compression-shear loading. Item (2) implies the determination of a yield function F(S, h) (Fig. 5), defined such that negative values of F correspond to macroscopic elastic force states in the interface element, zero values of F correspond to plastic force states, and positive values of F are not possible. The vector h represents the variables of the plastic process.
Structural fasteners for steel bracing connections
19
Fn+.=O .... ~ . ~ L j ~
/ t
~ Si
S.+~=St -a~K
3 8 .+a
Fig. 5. Incremental consistency of the elastic predictor-plastic corrector approach.
Item (3) requires the definition of a flow function G(S, h), whose gradient gives the direction of the incremental plastic displacements dp (Fig. 5). Because the above directions are functions of complex microscopic and macroscopic phenomena, the flow rule is 'non-associated', i.e. F ~ G. The formulation of the incremental relationships of the interface element as well as the integration of the constitutive laws is reported in the Appendix. Combined nonlinear problems, such as contact and slide surfaces, localized and spread plasticity as well as membrane effects, have up to now prevented a rigorous analytical approach to the determination of stiffness and strength properties of structural fasteners. Therefore, the constitutive matrix K, the yield function F and the potential flow G of fasteners, such as bolted double clip angles, fillet welds and high strength bolts subjected to shear, were obtained through fitting of experimental data. 12-t 7
3 PHYSICAL MODELING OF BOLTED DOUBLE CLIP ANGLES 3.1 T e s t p r o g r a m
In Fig. 3 the bolted clip angle fasteners are characterized by an element consisting of only one row of bolts; obviously, such single-bolt units behave differently from multiple-bolt units as would actually be present in real connections. For this reason, two- and three-bolt units were also tested, and since their behavior is presumed closer to that of the multi-row clip angles found in bracing connections, the presentation which follows centers on the results of these two- and three-bolt unit tests. In the test program undertaken, 29 tests to failure were carded out ta using one type of bolted double clip angle with one, two and three bolts.
O. S. Bursi
20
These specimens were subjected to pure tension, pure shear, pure compression, combined tension-shear and compression-shear. Tables 1 and 2 summarize the test program for two- and three-bolt units. A typical test specimen consisted of bolted double clip angles fabricated from angles 4 in x 3 in x ¼in (100 mm x 76 mm x 6 mm). Clip angle dimensions in the case of two-bolt units are shown in Fig. 6, while dimensions relative to three-bolt units can be obtained by adding one bolt unit to the two-bolt unit specimen. Because several tests were carried out, angles characterized by two different yield stresses (angle materials 1 and 2, Tables 1 and 2), i.e. fy = 42.8 and 47.7ksi (295 and 329 MPa), were adopted. The clip angles were connected by one, two or three rows of A325 high strength tensioned bolts (yield stress fy,b = 102 ksi (703 MPa)) with TABLE 1
Summary of Calibration Tests with Two-Bolt Units
Test
TI-2 TS301-2 TS451-2
TS452-2 TS453-2 TS601-2 TS602-2 S1-2 $2-2 $3-2 CS451-2 C1-2
Angle material 1 2 1 1 1 2 2 1 1 1 1 1
Load type
•
Testino equipment
Pure tension
0 30 ° 45 °
A B30 A
Tension-shear Tension-shear
1345 Tension-shear
60 °
Pure shear
90 °
Compression-shear
135 ° 180 °
Pure compression
1345 B60 B60 A C C A A
TABLE 2
Summary of Calibration Tests with Three-Bolt Units
Test
TS301-3 TS451-3 TS452-3 TS601-3 TS602-3 S1-3 $2-3
Anole material
Load type
•
Testing equipment
2 2 2 2 2 2 2
Tension-shear Tension-shear
30 ° 45 °
Tension-shear
60 °
Pure shear
90 °
B30 B45 1545 B60 B60 C C
21
Structural fasteners for steel bracing connections
~C-S ~T,-S (;4
~C-S
~T-S S rl
P6*, I ,,a,,,
u
, ,-~- , i
/'\
I
G6 G8
G5 G3
G_L4x3xl/4
G16 G18
G15 G13
3/4 A-SZ5 Bolts
Ril0at clip angle unit
Fig. 6. Two-bolt unit test specimens and linear and rosette strain gage locations: shear (S), combined tension-shear ( T - S ) or compression-shear ( C - S ) loading. Dimensions in inches (1 in=25.4 mm).
¼in (19 mm) diameter, in ~ i n (20.6ram) punched holes. Several testing devices (see Tables 1 and 2) were fab'ricated in order to apply different load types to the specimens according to the displacement approach. For the sake of brevity, only two types of test rig are presented.X2 The first one, labeled A in Tables 1 and 2, is shown in Fig. 7: the connecting plate between the two clip angles was fabricated 1 in (25.4 ram) thick in order to concentrate all deformations in the clip angles. It allowed different load types to be applied to specimens with one or two bolt rows: (a) pure tension, • = 0 °, (b) pure compression, ~ = 180 °, (c) combined tension-shear, ® = 45 °, (d) combined compression-shear, • = 135 °, and ¢)=0"
tT
~) =180"
;c
0=45"
(/)=135"
tT-S
C-S
0=90" Is
LVDTs LVDTs~/''~
C-S
a)
b)
c)
d)
Fig. 7. Testing equipment A and LVDT locations.
~S
o~
22
O. S. Bursi
finally (e) pure shear, • = 90 °. The angle tan • defines the direction between the applied load and the tension axis (Fig. 7). The second type of test rig, labeled C in Tables 1 and 2, is shown in Fig. 8. The welded plate assembly allowed pure shear loading to be applied to up to three-bolt units. In order to monitor local strains in the specimens and for comparison with future FEM predictions, linear and rosette strain gages were attached to the vertical and horizontal legs of the clip angles according to the layout shown in Fig. 6. Furthermore, for measurement of displacements, symmetrically placed LVDT transducers on opposing sides (Figs 7 and 8) were used to average the displacements. The values of displacement components, measured by transducers (Figs 7 and 8) in the normal and shear (n, t) directions (Fig. 4(a)), allowed the behavior of bolted double clip angles to be characterized in terms of force-displacement relations. 3.2 Test results
Because in actual bracing connections only bolted clip angles with multiple rows are used, results obtained from two- and three-bolt unit tests were adopted for the physical modeling.
~
~ =90'
S
SEC. A-A
JS
"~B
ts
..+
ts SEC.
B-B
Fig. 8. Testing equipment C and LVDT locations.
Structural fasteners for steel bracin0 connections
23
For the sake of brevity, only the test results obtained for the two-bolt units are presented in Fig. 9. Figure 9(a) plots tensile force component applied per bolt versus tensile displacement component (along axis n in Fig. 4(a)); Fig. 9(b) plots shear force component per bolt versus shear displacement (along axis t in Fig. 4(a)). Both curves highlight the anisotropic behavior of bolted clip angles as well as the interaction effect between tension and shear. Similar behavior was observed in one- and three-bolt unit tests, and in compression-shear tests. ~2 These curves lend themselves to representation by bilinear elastic-strain hardening relations.
3.3 Modeling Force-displacement relationships relative to directions n (Fig. 9(a)) and t (Fig. 9(b))--see Fig. 4--were represented according to bilinear laws of elastic-plastic hardening type (see, for example, Fig. 10). As a result, the characteristic value of both the elastic stiffness K e and the hardening stiffness K p were obtained by linear regression techniques. 12 These values allowed each plastic failure load S p to be identified. Analyses carried out by Sigfusdottir and GerstlC 3 on isolated bracing connections indicate that a bilinear representation compared to a multilincar one leads to errors in the structural response assessment that are within engineering accuracy
(4%). Experimental elastic stiffness values scaled to one bolt clip angle unit and relative to loading conditions of tension and shear as a function of tan are reported in Figs 11(a) and (b) respectively. The trend of variation of 25
25
gO" ~20 -
~15~
~=o' 45"
10-
20. v W15. O nfl~lO-
5-
5.
o.;o o,~o o.~o oAo " o. TENSION DISPLACEMENT (inches)
(a)
o.lo o.lo o.~ oAo SHEAR DISPLACEMENT (inches)
(b)
Fig. 9. Bolted double clip angle unit responses: (a) tension component; (b) shear component.
24
O. S. Bursi 25P U R E TENSION TEST W / O - B O L T UNITS ~20 -
~15-
zlo-
_o z
030
l
o.Jo
o.,.~
oAo
o.
TENSION DISPLACEMENT ( i n c h e s )
Fig. 10. Bilinear model of a bolted clip angle test (T1-2) force-displacement relationship. 5000 o
5000 ¸ rl-
Z~---
I ~ ' 0 - B O L T UNIT TEST THREE-BOLT UNIT TEST ANALYTICAL SIMULATION
~4ooo LO b_ z 3000LL
A
__2000"
(~
r'l&---
u) ~.~4ooo • v
'PWO-BOLT UNIT TEST THREE-BOLT UNIT TEST ANALYTICAL SIMULATION
~ 3000Ll(,3 ~..~2000
Z~
A
¢/3
O
[] D
0.~
rY 1 0 0 0
(2
~.6o
2.60 ton~
(a)
~.oo
o.oo
2.6o
1 .oo
~.oo
( t a n ~ ) -1
(b)
Fig. 11. Experimental and analytical elastic stiffness variation of a bolted double clip angle unit: (a) tension component; (b) shear component. stiffness values indicates that the tension stiffness decreases owing to the presence of larger shear components (Fig. l l(a)) while the shear stiffness decreases owing to the presence of larger tension components (Fig. 1 l(b)). To compute the (2 x 2) constitutive stiffness matrix K of the interface element (Fig. 4(a)) from the tensile and shear stiffnesses obtained experimentally, and plotted i n Fig. 11 as functions of the applied load direction tan ~, the following load-displacement relation is considered:
{AST = FKnn Kntl AT
ASsJ LK,. K.J(AsJ
(1)
Structural fasteners for steel bracino connections
25
in which AT, As are the measured tension and shear displacements, ASr and ASs are the imposed tension and shear forces, and KM and K , are the measured stiffnesses under pure tension and shear. The coupling stiffnesses K,t and Ktn are unknown. To find these, eqns (1) are written in the form:
KT@ (2)
Ks¢ = Kn +
' ~,KT®J tan •
in which KTo(=AST/AT) and Kso(= ASs/As) are the measured stiffnesses under load applied at an angle • with the tension direction, where tan is equal to ASs/AST. Using the values of Kr® and Kso obtained under load at angles ~ = 30 °, 45 ° and 60 °, whose mean values were approximated by eqns (2) (solid lines in Fig. 11), the desired coupling stiffnesses can be determined. For the case of combined tension and shear in the range prior to yielding (Fig. l l(a)), the resulting fastener stiffness matrix in kips/in is:
I163 4
K=-501
- 541 1297
(3)
For the case of the inelastic range and the compression and shear, as well as for the fastener weld units, similar calculations led to the element stiffness matrices needed in the analysis. Plastic load values S p scaled to one bolt clip angle unit and relative to force-displacement component curves (Fig. 9) were plotted within a plane (Fig. 12). In such a plane the horizontal axis represents the shear force Ss, while the vertical axis in the positive direction represents the tension force ST and in the negative direction the compression force Sc. As expected, bolted double clip angles have more strength when subjected to compression-shear loading. Two elliptic interaction curves were fitted through experimental data points in order to define the yield function F (Fig. 12). Plastic stiffnesses of force--displacement curves in the (n, t) directions allowed both components and ratios of plastic displacement increments Ad p to be evaluated. As a result, two elliptic functions G were calibrated (Fig. 13), such that their gradient represents the directions of plastic displacement increments Adp.
26
O. S. Bursi
rl_ 'TWO-BOLT UNff 11[$1"
~Sr,d.
s~
Z~- THREE-BOLT UNIT TEST
+ ~,iT3.o/ =t
/
oo~\
/~nn
/
,L
r I
Ts~
/
\
n\
/ \....
[]
/
Forces are In kips 1 kipl~4.45 kN
\
I
Forces are in kips
- ~ ( sc ~ +(s__,~ =l \-~e.T/
/
]
Ss, d t
"
\ (
Sc '~
i" Ss ~=
~--~.7/ + ~,~'~.o) :
\~Ls/
OC
Sc,d.
Fig. 12. Yield surface F of a bolted double clip angle unit.
'
Fig. 13. Potential flow surface G of a bolted double clip angle unit.
4 PHYSICAL MODELING
OF FILLET WELDS
4.1 Load-displacement response The global behavior of fillet welds connecting planar members (Fig. 14(a)) and loaded centrally or eccentrically is usually based on a linear isotropic
!
I
(a) A
(b)
B
C
(c)
(d)
Fig. 14. Fillet weld test specimen (Ref. 14) and decompositions: (a) displacement AF and loading angle ~; (b) case A; (c) case B; (d) case C.
Structural fasteners for steel bracing connections
27
relationship between toe displacement and load strength of the weld element. Results of physical tests ~4'15 show that stiffness and strength characteristics of fillet welds depend both on the direction of the applied load and on the connection geometry: fillet welds loaded perpendicular to their axis are stronger and less ductile than welds loaded parallel to their axis. Such behavior is mainly attributed to two factors: (i) the restraint to deformation of the fillet weld by connecting elements, which is maximum for transverse welds; and (ii) the change in the angle of the failure plane which increases the weld strength. ~* The complexity of the above behavior and the lack of a rational analytical model able to simulate the anisotropic behavior of fillet welds suggested use of test data of Kennedy and co-workers. 14'15 They carried out 42 tests to failure on specimens made of symmetrically connected elements (Fig. 14~a)) by fillet welds of ¼in (6 mm) and ] i n (9 mm) leg, respectively. The yield stress fy,w of the weld metal was equal to 67 ksi (465 MPa). Test specimens were subjected to different combined loads of tensionshear (~ = 0 ° (pure tension), 15°, 30 °, ..., 90 ° (pure shear)) in order to duplicate realistic loading conditions. Figure 14(a) shows the geometry of a characteristic test specimen ~* and the relative displacement component AF of fillet weld edges along the loading direction that was measured and the loading angle ~. In the tests of Ref. 14, only one component of displacement, A~ (Fig. 14(a)), was measured; with only this one component, it was not possible to determine the necessary two displacement components Ar and As (see eqn (1)). This made it necessary to assume a relationship between these components. To this end, three decompositions of the relative displacements of the fillet weld along the (n, t) directions were considered (Figs 14(b)-(d)). In the first one labeled A (Fig. 14(b) the transverse component was assumed zero; in the second (B) and third (C) a normal component (A~, Fig. 14(c); At, Fig. 14(d)) equal to an arbitrary value, 15% of AF, was considered in opposite directions respectively. In all three cases the same load F (Fig. 14(a)) was considered and decomposed. Load--displacement Curves obtained according to these cases are reported in Fig. 15(a) for the normal (n) and in Fig. 15(b) for the shear (t) direction, respectively, relative to the particular case of a • = 45 ° loading angle. Figure 15 shows that both the stiffness and strength variations are not significant even in the case of other loading angles ~. Numerical simulations performed on isolated bracing connections with boltedwelded fasteners, in which stiffness and strength properties for fillet welds obtained from all three decompositions were adopted (Fig. 14), indicate variations in the response within engineering accuracy (+4.5%).
28
O. S. Bursi IILI~.
~:45
~T) = 4 5 °
°
w ¢dl tv.
Z ~
W
4.00
"B
/,
I I 1 ~.
"r
o.dl
TENSION DISPLACEMENT(inches)
(a)
o.,
0.~
o.61
SHEAR DISPLACEMENT(inches)
(b)
Fig. 15. Fillet weld unit force-displacement relations according to cases A, B and C: (a) tension component;(b) shear component.
The location of fillet welds in the tests with a loading angle of @= 45 °, 60 ° and 75 °14 shows clearly that a structure was tested because two mirror-image fillet weld elements were considered instead of a single one. As a result, the normal component of the displacement (Ai~, Fig. 14(c), or A[, Fig. 14{d)) surely present in a single fillet weld element canceled out. Therefore, further tests should be carried out in order to measure distinctly both the tensile and the shear component displacement of one single element of fillet weld under various loading combinations. Owing to the results of the above sensitivity analysis and the lack of experimental data, only the relative displacement AF (case A, Fig. 14(b)) was considered for the related decomposition. As a result, load-displacement relationships were obtained for the normal (n) and shear (t) directions. Figure 16 shows in the case of tension-shear loading the force-displacement component relationships of a fillet weld of ¼in (6 mm) leg and 1 in (25.4 mm) length. Both diagrams highlight clearly the anisotropic stiffness and strength behavior of fillet weld elements in the tension (Fig. 16(a)) and the shear (Fig. 16(b)) direction. The experimental behavior of fillet welds for compression-shear loading conditions was also obtained from Refs 14 and 15.
4.2 Modeling Similarly to bolted clip angles, the load-displacement response of welded fasteners was modeled by a linearized law of bilinear type. Therefore, values of both the elastic K e and hardening stiffness K p were computed, as well as the plastic load S p.
Structural fasteners for steel bracing connections
29
16
A .~_Q.12°
~.a
0~ 0.000
0 2 5 0.00~ TENSION DISPLACEMENT (inches)
(
o.o~
0.0% o.o',5
o.o~o
SHEAR DISPLACEMENT (inches)
(a)
0.02.j
(b)
Fig. 16. Fillet weld unit responses: (a} tension component; (b) shear component.
The fitting process of experimental elastic stiffnesses allowed the elastic constitutive matrix K of the interface element to be determined for the loading cases of tension-shear and compression-shear. The plastic load vector S p relative to a fillet weld of ¼in (6 ram) leg and 1 in (25.4 mm) length is represented in the tension-shear interaction domain Ss--ST (Fig. 17). In the compression-shear domain $s-Sc a value of the plastic load of pure compression S~ equal to 0.891 times the value of the plastic load in pure tension S~ was assumed according to experimental data. x4 As a result, two interaction curves were fitted for the definition of the yield function F (Fig. 17). , ST, d n /s,~,,
A-
(s,~,
;2,s/ - ~l~.s/ = ;
TEST
_./
/TX
/ } I
÷ \
.
4
F o r c e s a r e in k i p s 1 k i p s = 4.45 kN
--T~.o) + \ Fo/ = I SC
Fig. 17. Yield surface F of a fillet weld unit.
I \ Forces
Ire
/ /
x in kips
( Sc ~
[$l ~'
~--~.3/ + u-~.s) = 1
Sc, dn
Fig. 18. Potential flow surface G of a fillet \ weld unit.
30
O. S. Bursi
Similarly to bolted clip angle fasteners, two other functions (Fig. 18) were calibrated on plastic displacement increments Adp, in order to represent the potential flow function G.
5 PHYSICAL MODELING OF HIGH STRENGTH BOLTS S U B J E C T E D T O S H E A R
5.1 Load-displacement response The connection between the gusset plate and the bracing member (Fig. 2) is often made with high strength bolts subjected to single or double shear. Several shear load versus displacement relationships are available in the literature, 16'~7 relative to tests by subjecting bolts to shear induced by plates either in tension or compression. Figure 19 shows a typical experimental response of an A325 bolt (yield stress fy,b = 92 ksi (635 MPa)) with a ¼in (19 mm) diameter subjected to single shear by plates A36 (fy = 36 ksi (248 MPa)) in tension. 17 Relative displacements were measured by two dial gages clamped on test plates. Similar plots were obtained for other plate thicknesses as well as with varying bolt diameters. 35 ¸
25 ¸
~20 =
n
o15
=0.5
K = 12400 k / i n Kp = 0 k / i n
10
K I = 12400 k / i n
5
0
o
o os
OlO
o'15
oi0
0.~s o:3o
D e f o r m a t i o n (inches)
Fig. 19. Load-displacement response of a bolt subjected to single shear (Ref. 17).
Structural fasteners for steel bracino connections
31
5.2 Modeling Since relative displacements were measured through test plates, x7 the elastic response of the connected plates was subtracted to averaged readings. In this way the load--displacement relationship shown in Fig. 19 represents both the elastic and inelastic deformations of the test bolt as well as the inelastic deformations of the connected plates. The shear spring of the 2D interface element was the only one employed to describe analytically the bolt fastener-plate behavior (element 4, Fig. 3). Bilinear laws obtained through fitting of curves similar to that shown in Fig. 19 were used to simulate the shear force-displacement relationships of bolted shear fasteners. 6 CONCLUSIONS The results of the experimental investigation carried out by the author on bolted double clip angles and data found in the literature for fillet weld connections and high strength bolts subjected to shear allowed the behavior of these structural fasteners to be characterized. Microscopic phenomena (static and dynamic friction, contact and localized plasticity) as well as macroscopic phenomena (slide surface, spread plasticity and membrane effects) were represented through the constitutive law of a 2D nodal interface element as continuous nonlinearities for each fastener type examined. As a result, a finite element model of heavy bracing connections using the interface element and plane stress elements with three and four nodes can be developed. The validity of this approach is shown in a companion paper, 2 where the response of a full-scale bracing connection with bolted-bolted and bolted-welded fasteners was numerically predicted and experimentally verified.
ACKNOWLEDGMENTS This work was carried out under a grant from the National Research Council of Italy (CNR) while visiting at the University of Colorado at Boulder. Thanks are also due to the Department of Civil, Environmental and Architectural Engineering (Boulder) for general support. Finally, special thanks are extended to the technical staff of the Department of Structural Mechanics and Design Automation (Trento) for their assistance
32
o. s. Bursi
throughout the testing program. However, the opinions expressed in this paper are those of the writer, and do not necessarily reflect those of the sponsors.
REFERENCES 1. Salmon, C. G. & Johnson, J. E., Steel Structures--Design and Behavior, Emphasizing Load and Resistance Factor Design (3rd edn). Harper & Row, New York, 1990. 2. Bursi, O. S., Gerstle, K. H., Sigfusdottir, A. & Zitur, J., Behavior and analysis of bracing connections of steel frames. J. Construct. Steel Res., 30 (1994) 39-60. 3. Bursi, O. S. & Gerstle, K. H., Analysis of flexibly connected braced steel frames. J. Construct. Steel Res. 30 (1994) 61-83. 4. Thornton, W. A., A cost comparison of some methods for design of bracing connections. In Proc. 2nd Int. Workshop Connections in Steel Structures: Behavior, Strength and Design, Pittsburgh, PA, April 1991. 5. Thornton, W. A., On the analysis and design of bracing connections. In Proc. 1991 AISC National Steel Construction Conf., Washington, DC, June 1991, pp. 26-1-33. 6. Williams, G. C. & Richard, R. M., Steel connection designs based on inelastic finite element analysis. Design report prepared for American Institute for Steel Construction, Department of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, AZ, May, 1986. 7. Gross, J. L., Experimental study of gusseted connections. In Proc. 1989 AISC National Steel Construction Conf., Nashville, TN, June 1989 pp. 11-1-22. 8. Hardash, S. G . & Bjorhovde, R., New design criteria for gusset plates in tension. Engn9 J. AISC, 21(3) (1985) 139-48. 9. AISC, Engineering for Steel Construction. American Institute of Steel Construction, Chicago, IL, 1984. 10. Hu, S. Z. & Cheng, J. J. R., Compressive behavior of gusset plate connections. Structural Engineering Report No. 153, University of Alberta, Canada, 1987. 11. Chakrabarti, S. K. & Richard, R. M., Inelastic buckling of gusset plates. Struct. Engng Rev., 2 (1990) 13-29. 12. Bursi, O. S., Tests and modeling of bolted double clip angles of steel bracing connections. Internal Report, SMDA Department, University of Trento, Italy, Oct. 1990. 13. Sigfusdottir, A. & Gerstle, K. H., Analysis and tests of bolted bracing connections. In Ninth Structures Congress, ASCE, 29 April-1 May 1991, pp. 430-3. 14. Miazga, G. S. & Kennedy, D. J. L., Behaviour of fillet welds as a function of the angle of loading. Can. J. Civ. Engng, 16 (1989) 583-99. 15. Lcsik, D. F. & Kennedy, D. J. L., Ultimate strength of fillet welded connections loaded in plane. Can. J. Civ. Engng, 17 (1990) 55-67. 16. Wallaert, J. J. & Fisher, J. W., Shear strength of high strength bolt. J. Struct. Div. ASCE, 91(ST3) (1965) 1003-18. 17. Richard, R. M., Gillett, P. E., Kriegh, J. D. & Brett, A. L., The analysis and design of single plate framing connections. Engng J. AISC, 2 (1980) 38-51.
Structural fasteners for steel bracing connections
33
18. Schafer, H., A contribution to the solution of contact problems with the aid of bond elements. Comp. Meth. Appl. Mech. Enong, 6 (1975) 335-54. 19. Rots, J. G., Bond-slip simulations using smeared cracks and/or interface elements. Res. Report 85-01, Struct. Mech. Dept. of Civil Engng, Delft Univ. of Tech., The Netherlands, 1985. 20. Ngo, D. & Scordelis, A. C., Finite element analysis of reinforced concrete. J. Am. Concrete Inst., No. 64-14 (1967) 152-63. 21. Nilson, A. H. et al. (eds), Finite element analysis of reinforced concrete. State-of-the-Art Report, ASCE, New York, 1982. 22. Schellekens, J. C. J., Interface elements in finite element analysis. Res. Report 25-2-90-5-17, Struct. Mech. Dept. of Civil Engng, Delft Univ. of Tech., The Netherlands, Oct. 1990. 23. Curnier, A., A theory of friction. Int. J. Solids Struct., 20(7) (1984) 637-47. 24. Simo, J. C. & Taylor, R. L., Consistent tangent operators for rate independent elastoplasticity. Comp. Meth. Appl. Mech. Engng, 48 (1985) 101-18. 25. Ortiz, M. & Popov, E. P., Accuracy and stability of integration algorithms for elastoplastic constitutive relations. Int. J. Num. Meth. Engng, 21 (1985) 1561-76. 26. Reich, R. & Montgomery, K., MICROFEM-88. Internal Report, CEAE Department, University of Colorado, Boulder, CO, Oct. 1988. APPENDIX: A CONSTITUTIVE LAW FOR A 2D N O D A L I N T E R F A C E E L E M E N T In the following subsections the constitutive law of a 2D nodal or point interface element is developed in incremental form and an explicit relationship between increments of forces and displacements is obtained. Theoretical as well as computational aspects are given. A.I Elastic stiffness matrix
A two-dimensional nodal interface element characterized by lumped properties with coupling, constituted by two linear springs S1 and $2, is considered (Fig. 4(a)). It models a structural fastener unit between the gusset plate and framing members, such as a bolted double clip angle unit (Fig. 4(b)). External nodes (1 and 3; Fig. 4(a)) of that element are attached to the gusset plate, while the common node (2) is attached to the column flange. Nodes 2 and 3, which have the same location in the finite element model, are shown separated (Figs 3(b) and 4(a)) for clarity. A local coordinate system, with axes normal and tangent to the column flange (Fig. 4(a)), is indicated by (n, t). These axes are assumed to be principal axes of the element. The kinematic variables to be used in the constitutive law are the relative displacements d between the gusset plate
34
O. S. Bursi
and the column flange in the normal and tangential direction, which are defined as:
d=(u2 -ul)e
(A1)
where ul defines the displacements of the element SI (Fig. 4(a)), while e is a unit vector in the coordinate directions (n, 0. The stress resultants that the interface element supports are denoted by S and the convention that compressive stresses are negative is used. In the following, an explicit relation between increments of forces and displacements is developed. Because the relation is incremental, it is valid for arbitrary load and displacement histories that can involve unloaded and subsequent reloading in the element. According to the theory of elasto-plasticity and with the restriction of small displacement gradients, the relative displacements d are additively decomposed into: d = de + dp
(A2)
where superscripts e a n d p denote the elastic part of the displacement. Because the forces ~hat-the interface element supports must be reversible, they can be related only to the elastic part of eqn (A2). Therefore, the relation between increments of forces and displacements is expressed by the linear equation: ~;=Kd e
(A3)
where K represents the elastic stiffness matrix of the element. In greater detail, Kn.n defines the normal stiffness, Ks.s defines the shear stiffness, while the constants Kn,t and Kt.~ represent the coupling stiffnesses (Fig. 4(a)). Because the increment of the normal force S~ must assume the same value for positive or negative increments in elastic shear displacements dT, the sign of the constant K,.t must be direction-dependent. Finally, owing to the different properties of fasteners, the constants of the stiffness matrix K assume different values if the interface element is subjected to combined tension-shear or compression-shear loading. A.2 Yield surface and potential flow
It is necessary to postulate a rule for plastic displacements d p. In particular, it is assumed: 1. There is a scalar yield function F(S, h), defined such that negative values of F correspond to elastic stress states in the interface element,
Structural fasteners for steel bracing connections
35
zero values of F correspond to plastic stress states and positive values of F are not possible; the vector h represents the internal variables of the plastic process. 2. Increments of forces are linearly related to increments of displacements. 3. A suitable relation analogous to the flow rule employed in plasticity:
tiP
if F < 0 or /~ < 0 (unloading)
f°
• t~G
if F =/" = 0 (loading)
(A4)
where G(S, h) is a scalar-valued flow function, whose gradient gives the direction of the incremental plastic displacements d p, while ,~ is a nonnegative scalar which defines the magnitude of the increments (Fig. 5). Because the above directions are functions of complex microscopic and macroscopic phenomena, the flow rule is 'non-associated', i.e. F ~: G (see Figs 12--13 and 17-18).
A.3 Integration of constitutive relations During the loading process of the interface element, the stress state must satisfy the yield criterion and remain on the relative surface, i.e. F = 0 and dF = 0, which can be written: OF=
~F
dF d S + -=-. Oh=0 dh
(A5)
Because the interface element presents directional properties (anisotropy), the internal variables h which describe the hardening of the yield surface F are expressed as a function of the normal plastic work dW]---S~r dd~ and the shear plastic work d Wtp = Stx dd,v. Substituting eqns (A2)-(A4) into eqn (A5), the following incremental constitutive law is obtained: ~=K~pd
(A6)
where for purely elastic response or unloading (F < 0 or/~ <0):
Kep = K
(A7)
O. S. Bursi
36
and for imminent plastic p h e n o m e n a (F = P = 0): OG OFT
Kep
~
K----K 0S 0S S
--
OFT
--K OS
OG
OFTOG
OS
Oh dS
(A8)
The plastic multiplier )[ can therefore be explicitly expressed as: OF T
--Kd 0S ~'=OFT_ OG OFT OG K as
(A9)
Classical integration schemes employ a tangent stiffness formulation and are traditionally explicit in nature• The current development is, however, oriented towards an 'elastic predictor-plastic corrector' strategy, based on implicit integration. In fact, implicit schemes are more versatile and robust in an environment where the nature of loading can be highly variable. Let F . = F ( S . , h . ) = 0 represent the yield condition at the time t = t . of the element subjected to a finite load increment AS, in which the displacement increment Ad is given, and let F. + x = F . + ~(S. + ~, h. + 1) = 0 represent the yield condition of the element satisfied by the state of stress resultants S. + ~ = S. + AS and internal variables h.+ 1 = h. + Ah at the time t = t.+ x. The expansion of the yield condition in a Taylor series: F,, + 1 = F. + AF' + 0"5A 2 F" + . . . + HO = 0
(AIO)
opens mainly two possibilities to enforce the yield condition F, + x = 0 and therefore to determine the plastic multiplier A2. The first approach is based on truncation of the Taylor expansion after the first term on the right-hand side of eqn (A10), which leads to the traditional linearized form of consistency, i.e. AF = 0. This linearization results in an explicit format of the plastic multiplier A2 (eqn (A9)) and of the elastic-plastic tangent operator (eqn (A8)), but this implies errors in the case of nonlinear internal variables h and yield functions F.
Structural fasteners for steel bracin0 connections
37
The second and more direct approach, adopted here, is based on the fulfilment of the yield condition at the end of each load increment: F.+ :(S, +AS, h. + Ah) = 0
(All)
which results in greater reliability and allows errors in the plastic return step to be minimized.24 Projecting the trial stress resultant S, on to a subsequent loading surface F , + ~ = 0 in a single step (Fig. 5), two unknowns need to be identified, i.e. the magnitude and the direction of the plastic return step. Such changes can be expressed explicitly in terms of the plastic multiplier A2 and the direction of the plastic flow: d~ s,.k,
(A12)
By using the implicit integration scheme 'generalized midpoint rule', 25 the resultant stress state S, and the plastic internal variables h. adopted for the computation of the plastic flow gradient (eqn (A12)) can be evaluated through the equations: S. =(1 - ct)S, + atSn+ x
(A13)
h , = (1 - ~)h,, + ~h,,+ 1
(A14)
at an intermediate point (Fig. 5) defined by the parameter ct (0~
(A15)
which can be solved for the unknown plastic multiplier A2. The integration scheme of the constitutive relations of the nodal interface element was implemented in a nonlinear finite element computer program. 26 Such a program uses an incremental-iterative solution procedure for the determination of each different structure equilibrium state,
38
O. S. Bursi
based on the initial stress method. Because both the elastic constitutive matrix K (eqn (A3)) and the elastic-plastic tangent operator of the interface element are nonsymmetric, only the symmetric part of the global structure stiffness matrix K s , i.e. Ksym 8 _- ½(Kg -~ K I )
(A16)
was used in the iterative algorithm for the solution of the nonlinear equation system. The algorithm guarantees a limited iteration number for an assigned iteration tolerance.