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Computers & Slruchres Vol. 46, No. I, pp. 55-65, 1993 0045-7949/9 3 S6.00 + 0.00Printed in Great Britain. 0 1992 Pergamon Press Ltd

STEEL FRAME ANALYSIS WITH FLEXIBLE JOINTSEXHIBITING A STRAIN-SOFTENING BEHAVIOR

T. H. ALMUSALLAM~and R. M. RICHARDSTDepartment of Civil Engineering, King Saud University, Riyadh, Saudi Arabia$Department of Civil Engineering and Engineering Mechanics, University of Arizona, Tucson, Arizona,U.S.A.

(Receiued 14 October 1991)Abstract-Local buckling of beam and/or connection elements is a source of softening in the mo-ment-rotation behavior in certain connections. This effect can significantly reduce the maximumload-carrying capacity of steel structures. In this study, an elastic-plastic plane frame analysis whichincludes the effect of nonlinear connection behavior is presented to predict the ultimate load-carryingcapacity of steel frames. The structural model comprises members which are assumed to behave elasticallywith nonlinear connection moment-rotation behavior. The main objective of this study is to investigatethe effect of connections which exhibit strain-softening behavior on the ultimate strength and stability offlexible frames. Numerical studies of frames made using the developed computer program are presented.Observations regarding the effects of flexible connections on the strength and deflection of steel framedstructures are discussed. The proposed analyses procedure using flexible joints modeled by the Richardfunction were found to be simple and accurate which compared to elastic-perfectly plastic models.

1. INTRODUCTIONIn the traditional design of steel frameworks, theframes are usually analyzed under the simplificationthat the actual behavior of beam-to-column connec-tions can be idealized to two extreme cases. These arethe pinned-joint connections and the rigid-joint con-nections. In actual structures all connections result ina semi-rigid behavior and neither idealized model iscompletely realistic. An ideally pinned connectionimplies that no moments will be transmitted betweencolumn and beam and the beam and the column willbehave independently. On the other extreme, for thefully rigid connection, the angle between the columnand the beam axes remains unchanged as the framedeforms and no relative rotation occurs between thejoined members. It is generally necessary to incorpor-ate the effect of connection flexibility in the frameanalysis in order to represent the actual behavior ofthe frame. Research on the behavior of semi-rigidconnections started as early as 1917 [l]. During thepast 70 years, many experimental investigations onactual joint behavior have demonstrated that mostconnections should be treated as semi-rigid connec-tions using moment-rotation curves to describe thebehavior of semi-rigid connections. Generally, thesemoment-rotation curves are obtained by performingfull-scale tests or developing analytical models basedon testing the connection segments.Analytical models, based on test data, are: linear,polynomial [2], cubic-B-spline [3], power [4-61 andexponential [7,8] models. Richard et a l . 9, lo] pro-posed a four-parameter model [6] which gives a goodrepresentation of experimentally or analytically

determined moment-rotation curves. In the early196Os, various methods of analysis resulted with thedevelopment of computers. Richard and Gold-berg [l 1, 121 presented a method for analyzing struc-tural systems having nonlinear elements. Moncarzand Gerstle [131analyzed frames with semi-rigid con-nections and considered the response of unbracedmultistory frames under variable load histories. Luiand Chen [7] presented an exponential function torepresent the behavior of connections and used it forthe analysis of flexible frames. Recently, Andersonet al. [14] reviewed nearly 60 published papers withregard to the methods of analysis for flexible framesand evaluated their effects.In the classical limit analysis and design methodthe moment-rotation relationship is assumed to bebilinear in form. With the elastic-perfectly plasticformulation when any section reaches its momentcarrying capacity, it is assumed that under increasedrotation the section will continue to carry the samemaximum moment. However, some steel structuresmay experience instability in some elements such asflange buckling. For example, some connection orbeams may exhibit a distinct maximum moment afterwhich this moment decreases with increasing rotationresulting in strain-softening[lS, 161. This kind ofbehavior affects the rate of redistribution of internalmoments because certain moments in the framedecrease with an increase in rotation. Here, a four-parameter Richard model is used to represent themoment-rotation behavior of semi-rigid connections.These parameters depend on the strength, stiffnessand ductility for a given connection. With the fourparameters determined, the connection model can be

55

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56 T. H. ALMUSALLAMnd R. M. RICHARDused directly for the analysis of frames. A computer whereprogram is developed for the analysis of flexible M = connection moment,frames using this model to represent both strain- 0 = connection rotation,hardening and strain-softening behavior of connec- K = initial or elastic stiffness,tions. K,, = plastic stiffness,MO= reference moment,

2. DI?SCRIPTION OF THE SEMI-RIGID N = curve shape parameter.CONNECTION MODEL

A nonlinear mathematical expression representing Shown in Fig. 1, is the definition of each par-the moment-rotation behavior of semi-rigid connec- ameter. For a given moment-rotation curve, thetions has the following assumptions and require- procedure for determining the four-parameters thatdescribe each M-8 curve is as follows:ments:

1. The rotational deformation of connectionassemblage is small.2. Shear, axial, and torsion deformations on con-nections are neglected.3. The initial elastic stiffness K of the connectionis equal to the slope of the M-0 curve at the origin.4. For any value of 8 the tangent stiffness of aconnection is given by the slope of the M-8expression at that angle.5. As the rotation becomes larger, the slope of thecurve approaches the plastic stiffness Kp (the strain-hardening stiffness or the strain-softening stiffness).

Based on the preceding assumptions and require-ment, a simple and accurate model to representconnection behavior is necessary. The Richardmoment-rotation model was chosen because of itsgenerality and applicability to any type of connec-tion, including those with a strain-softening behavior.The four-parameter Richard model has the form

1. Determine the initial slope of the M-0 curvewhich represents the initial stiffness K which dependson the type and geometry of the connection.2. Determine the plastic stiffness Kp which is theextreme slope of the M-Q curve.3. Compute the shape parameter from theequation

-1n2N=ln( $-&J (2)

where all parameters are defined in the previous steps.Shown in Fig. 2, are some possible deformation

responses that can be obtained with the Richardequation.

3. MOMENT-ROTATION BEHAVIOR OF SPECIFICCORNER CONNECTIONSMoment-rotation curves are the final product of

very complex interactions between the member com-ponents (webs, flanges, angles) and the fasteners(bolt, welds). Therefore, it is very important tounderstand the behavior of single components, as

= Elastic StiRnearKp = Plutic Stlfhkhf. = Reference MomentIV = Curve Shape Parameter

Fig. 1. Richard equation for defining moment-rotation relationships.

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Analysis of steel frames 57

DEFORMATIONFig. 2. Richard curve shapes.

well as of the way in which they interact, as a functionof the geometrical and mechanical factors of thecomplete connection. These moment-rotation curvesare usually nonlinear over most of the range. Thenonlinearity arises mostly from the ductility of somecomponents within the connection. Experimentally, it

has been observed that the flange and web bucklingwill lead to decreasing moment with increasingrotation in some connections causing a significanteffect on the load capacity of the whole structure.Shown in Fig. 3 are welded comer connections whichhave been tested by Beedle [ 151.The analytical modelis used to represent the moment rotation response ofthese types of connections as shown in Figs 4-7 forthese connections.

4. BEAM MODEL FOR THE ANALYSIS OF FRAMESWITH SEMI-RIGID CONNECTIONS

The method of the analysis of flexibly jointedframes are based on the following basic assumptionsand limitations:

1. All members are prismatic and straight.2. The material behavior of the beam is assumedto be linearly elastic.3. The structure behaves linearly except for the

nonlinear moment-rotation characteristics of theconnections.4. Connections are of negligible size. Hence, con-

nection deformation are concentrated at the ends ofthe beams.5. Connections are modeled as a rotational springsacting at the ends of the member.6. The nonlinear behavior of the beam andbeam connections are combined at the center of the

connection.7. Deflections are sufficiently small so that they donot significantly change the geometry of the structureduring loading.

40.25 22 30.25 18

32825 36.125 40.0

~~ ~~

Fig. 3. Some types of comer connections considered in this paper.

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1400T. H. ~~~LL~ and R. M. RICHARD

_ Expeiirnentai-- Richard

-I0 1 2 3 4 5 6 7Rotation (radian) x10-3

Fig. 4. Moment-rotation curve for comer connection G.

8. Overall buckling of individual members or any and is subjected to bending moments Mj and Mjportion of the structure does not occur. applied at its two ends as loads, as shown in Fig. 9.9. Only the moment-rotation behavior of the The total end rotations due to these moment loadsconnection is modeled as a nonlinear function. are f), and t?,, which equal the sum of the beam andThus, axial and shear effects in connections are spring rotations (i.e., 6, = flbi+ t?,). Counter-clock-ignored. wise moments and rotations are taken as positive.The beam element is free of inte~~iate loading and

The stiffness method is developed as a form of has a linear elastic behavior; however, at the ends ofthe slope deflection method. Consider a beam shown the beam, flexible connections which are representedin Fig. 8 with constant moment of inertia and by rotational springs with a nonlinear moment-two rotational spring attached to both ends of tbe rotation relationship are located to account for thebeam. The beam is simply supported at both ends, nonlinear behavior of the beam andJor connection.

0.002 0.004 0.006 0.008 0.01Rotation (radian)

Fig. 5. Moment-rotation curve for comer connection H.

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Analysis of steel frames 59

-Experimencnl- Richard

0.002 0.004 0.006 0.008 0.01Rotation radian)

Fig. 6. Moment-rotation curve for corner connection I.

The moments and the rotations are related by the r 1 1 1(6)following equations M

> = [fl{mL 3)

where Ul, {ml, and (0) are matrices defined below The matrix [fl is the flexibility matrix of the(111)= {Mi M,}T (4) beam-connection model. The incremental vector {0>is defined as the end rotation vector with respect to

and the original axis i - j and {PTI} s the correspondingP> = 0% %I=

incremental end moment vector. KSi and Kti are the(5) tangent stiffnesses of the nonlinear springs at i and j,

6oB- -1.. --._ -

_ ExperimentsI-- Richard

0 1 2 3 4 5 6 7Rotation (mdisa) x10 3

Fig. 7. Moment-rotation curve for corner connection B.

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60 T. H. ALMU~ALLAMnd R. M. RICHARDKR MRE , I A L9-

L t

PRV R

Fig. 8. The beam-connection element model.

respectively. The stiffness of each nonlinear spring iscomputed by differentiating eqn (1)

where all parameters in eqn (7) are defined in Sec. 2.Equation (3) can be written in terms of stiffnessmatrix, simply by taking the inverse of matrix [fl

{ml = [Kl{e>v (f-9where

and

Next, consider the beam shown in Fig. 10(a) witha four degrees of freedom represented by thedisplacement matrix { 3}

{e}r= {Yi ei uj e,>r. (11)As shown in Fig. 10(b), 0; and 0; denote theincremental end rotations of the beam relative to the

beam axis i-j. Therefore, eqn (8) is still valid,provided that 0; and 0; are used instead of f& nd 0,.Thus

where(13)

Fig. 9. The beam model without joint translation.

in which, {0) is defined by eqn (1 l), and [T] is definedas a transformation matrix from i-j coordinates toi - j coordinates and has the form

F-1= (14)

The incremental element stiffness matrix of thebeam shown in Fig. 10(a) will beI = [W-WI VI. (15)

Equation (8) yields the same result as the case ofthe rigid frame analysis when Ksi Ksj 00 and hasthe form of

f 4EI 2EI 1

In the frame analyses herein, the structure stiffnessmatrix is formed by computing the tangent stiffnessmatrix for each element in local coordinates,transforming them to the global system, and theninserting them into the incremental structure stiffnessmatrix. Because of the nonlinearity of the connectionbehavior, the structural stiffness equations weresolved by Euler incremental method or modifiedEuler (second-order Runge-Kutta) method in orderto increase the accuracy of the results with fewerincrements._i,*T

vW

i i(4Fig. 10. Beam with joint translation having absolute endrotation 0 and rotations with respect to beam axis 0.

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Analysis of steel frames 615 NUMERICAL APPLICATIONS FOR THE ANALYSIS OF

FLEXIBLE FRAMESTwo simple frames subjected to different loadings

are considered to study the response of flexiblyconnected frames. In the forthcoming discussion, twoexamples will be presented to illustrate the effectof the connection flexibility with a strain-softeningbehavior on frame behavior. In these examples, aplane frame analysis with elastic hinges will be usedto demonstrate the effect of the connection stiffnesson frame response throughout the entire range ofloading. In the second example, a comparison be-tween the results of five different connections behav-ior will be presented to show how connections withstrain softening behavior effect the load-carryingcapacity of the whole frame.Example 1. Analysis of~ar tial ly restrained rame withstrain softening connectionsFigure 11 shows the geometry and loading of asimple unbraced frame to be analyzed. All connec-tions used here are assigned a strain-softening behav-ior having the same peak moment value as the plasticmoment capacity of the connected members.

Shown for comparison in Fig. 12 are the momentswith the known behavior of the connections rep-resented by the Richard equation for element 3-4 atnodes 3 and 4 with the following parameters:K = 142,307, Kp = 14,230.7, M, = 1087.17 andN = 1.44 (units are in inches and kips). It can be seenfrom Fig. 12 that the moment in the connections withstrain softening behavior decreased rapidly afterpassing the peak in the moment-rotation curve witha rapid increase in the rotation of all the connection.Shown in Fig. I3 is the relationship between the

I20

I_ 1116+:

7 20 -,Fig. 11. Partially restrained frame 1.

applied load and the frame moments and lateraldisplacement at node 2 in nondimensionai form. Thisresult indicates that the moments MM and A&reached their maximum value and then decreasedunder an increase in the load P.The rapid redistribution of moments take placewith the result that the whole structure becomeunstable before attaining a collapse load based upona formation of four plastic hinges. This demonstratesthat the actual ultimate load can be less than theso-called limit state design load. Shown in Fig. I4 isthe relationship between the applied load and theframe strain energy. The highly nonlinear portionis related to the softening region in the moment-rotation relationship.Example 2. Analysis of l exibly jointed fr ame wi thdi erent connection behavior

Loading and geometry for this example frame isshown in Fig. 15. Gravity and lateral loads wereapplied in one stage (proportional loac lin

600 . .. M43- M34

100.\

00 0.01 0.02 0.03 0.64 0.05 0.06 IROTATION, Radian

Fig. 12. Moment-rotation response in frame I for member 3-4.

g). Different

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62 T. H. ALMUSALLAMnd R. M. RICHARD

0.9 . .M43-M340.8 -Lateral Dkp.

0.7 -

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Fig. 13. Moments versus load in frame 1.connection properties were assigned to show theeffect of semi-rigid connections on the behavior of theframe structure. There were two different types ofconnections used in this analysis. One was the real orbeam-to-column connection and the others werepseudo-connections. The pseudo-connections wereused here to simulate the behavior of a plastic hingewith moment-rotation behavior as follows:

for MC,,, < MP, G,, = ~0 (17)for MC,,, 2 MP, L,, = 0, (18)

where M and A&,, are the moment and rotationstiffness of the pseudo-connection, respectively andMp is the plastic moment capacity of the connectedbeam.

The location as a function of the applied load ofpseudo-connections (rigid-perfectly plastic hinges)are usually not known in advance. For frames loadedwith concentrated forces only, all load points shouldbe modeled with this type of connection. Distributedloading can be represented by a series of concentratedforces along each span with short members betweeneach load.

400

1

100 2cMJ 300 400 500strain Energy (kip.ft)

Fig. 14. Load versus strain energy in frame 1.

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Analysis of steel framesP P

63

P/4 -

El

P/4 -

,

El El El3 4 6 9

1 8L A 3A

-15

t-15t-

I_ 30 I 30 -,Column and Roof Girders= W 18x106 Floor Girders= W 27x 46

Fig. 15. Partially restrained frame 2.

The five different cases considered in the analysis ofthe frame shown in Fig. 15 with different Richardparameters used to represent the connection behaviorare listed as follows:

1. All connections are rigidly connected to the beam.2. All beam-to-column connections are consideredto have an elastic-plastic behavior with Kp = 0 ndpseudo-connections are assigned at the ends of eachmember.

3. All beam-to-column connections are assigned astrain-softening behavior with Kp = -K andpseudo-connections at the middle of each beam in thestructure.4. Same as case 3 except Kp = -K/4 were usedinstead of -K for the connection behavior.

5. Same as case 3, but pseudo-connections arereplaced with linear elastic connections.Figure 16 shows all moment-rotation curves forthe analysis. The analyses of all cases were performed

with the maximum moment capacity of the connec-tions the same as the moment capacity of the con-nected beams MP.Shown in Fig. 17 is the moment-rotation curves atnode 7 for member 12 resulting from the analysis ofthe five different cases. From this figure, it can benoted that case 1 has the highest rotation capacitywhereas case 3 shows the lowest rotation capacityamong all other cases. Cases 1 and 2 end with a totalcollapse of the whole frame after the mechanism offailure was reached. For cases 3-5, where the soften-ing behavior was considered in the behavior ofconnections, different types of frame instabilities werefound out wherein a partial collapse of some mem-bers occurred. At that load level the redistribution ofmoments became impossible. In other words when themoment in some connections exceeded the peak value(MP), the connection started to soften decreasingmoment and redistribution took place until the adja-cent connection reached its moment capacity. After

RotationFig. 16. Cases considered for the connections behavior.

CM 46/l E

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64 T. H. LMUS LL M and R. M. RICHARD

._RAgidcmcctiuls- Bhtsdc-Pi&e Corms. (Kp=O). Sofkning Conns. (Kp=-K). . softening CoNls. (Kp=-K/4)

iI* Blastic Besms With (Kp=-K)

0 0.01 0.02 0.03 0.04 0.05 0.06Rotation (radian)

Fig. 17. Moment-rotation curves for member 12 at node 7.

that stage, re~st~bution of moments for that mem-ber were not possible which resulted in partial col-lapse even though a total instability of the framestructure was not attained. The analysis was termi-nated at that level in which the global structuraltangent stiffness matrix developed negative terms onthe diagonal and became indefinite. Figure 18 showsthe load-drift response at node 3 for all cases con-sidered in this study. A summa~ of the results isshown in Table 1.

SUMMARY AND CONCLUSIONS

Experiments carried out over the past decades onsteel beam-to-column connections have demon-strated that the behavior of connections are nonlin-ear. To account for nonlinear behavior of theconnections, the developed computer program usedan incremental analysis procedure that yielded thebending moment and rotational defo~ation in eachconnection as well as a structural displacements. The

_ Rigid Connections- ElasWPla tic Ccmns,(Kp=O)n.... Softening Corms. (Kp=-K)-.-* Softening Corms. (Kp=-K/4)sJ Blascie Beams With (Kp=-K)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Displaeenlern (ft)

Fig. IS. Load-drift response at node 3 for different connections.

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Analysis of steel frames 65Table 1. Summary of the results at ultimate state

CaseNo.Ultimate Maximum Rotationload drift capacity

(kips) (feet) B,_5 radian)1 174.1 0.6 0.042 169.5 0.75 0.033 165.88 0.42 0.0054 168.3 0.58 0.0185 288.0 0.8 0.01

Richard model was selected to represent the inelasticconnection behavior for the proposed analysis offlexible frames.

Three major topics have heen covered in this paper:(1) a simple model to describe the behavior of mosttypes of connections including connections with astrain-softening behavior that can he used directly inthe analysis of a semi-rigid frames; (2) the heammodel for the analysis of frames with semi-rigidconnections (assumptions and stiffness formulation);and (3) numerical application for the analysis offlexible frames considering different types of connec-tions including those with strain softening behavior.

The following conclusions are made based on thesestudies:

1. The four-parameter Richard model represents awide range of connection behavior including theconnections with strain-softening behavior.

2. Tests reported in the literature have shown thatthe moment-rotation relationship of certain semi-rigid connections were sensitive to the local bucklingor distortion of the cross-section which caused asoftening in the moment-rotation behavior. Thislocal buckling was followed almost immediately by asignificant decrease in the moment capacity per unitof load increment and in some cases by almostimmediate collapse to the connection. Also it hadbeen observed experimentally [151 hat elastic-plasticweb buckling led to a drop in the moment-rotationcurve before the moment reached the momentcapacity of the connected beam.3. Connections with softening behavior and littlerotation capacity could not be used safely in plasticdesign unless that hinge were the last to form.

4. Based on the results of the analyses of flexibleframes studied in this paper, it was shown that frameswith strain-softening connections may exhibit partialcollapse if redistribution of load is not possiblebecause of softening behavior.

5. Total collapse of frames with strain-softeningconnections were reached only if the plastic hingeswere the last to form in the location of thoseconnections.

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4.

5.

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7.8.

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11.12.

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16.

RIWERENCES

W. M. Wilson and H. F. Moore, Tests to determine therigidity of riveted joints in steel structures. Bulletin No.104, Engineering Experiment Station, University ofIllinois (1917).M. J. Frye and G. A. Morris, Analysis of flexiblyconnected steel frames. Canad. J. Ciuil Engrs 3,280-291(1975).S. W. Jones, P. A. Kirby and D. A. Nethercot, Model-ing of semi-rigid connection behavior and its influenceon steel column behavior. Proceedings of The Znter-national Conference on J oints in Structural Steel Work,pp. 5.73-5.87 (1981).K. M. Ang and G. A. Morris, Analysis of three-dimen-sional frames with flexible beam-column connections.Canad. J. Civil Engrs 11, 245-254 (1984).N. Krishnamurtby, H. Huang, P. Jeffrey and L. Avery,Analytical M-0 curves for end-plate connections.J. Srruct. Div., AXE 105, 133-145 (1979).R. M. Richard and B. J. Abbott, Versatile elas-tic-plastic stress-strain formula. J. Engng Mech. Diu.,ASCE 101, 511-515 (1975).E. M. Lui and W. F. Chen, Analysis and behavior offlexibly-jointed frames. Engng Struct. 8, 107-l 18 1986).Y. L. Yee and R. E. Melchers, Moment-rotation curvesfor bolted connections. J. Struct. Engng, ASCE 112,615635 (1986).R. M. Richard, P. E. Gillet, J. D. Kriegb and B. A.Lewis, The analysis and design of single plate framingconnections. EnmE Jnl, AZSC 17. 38-52 (1980).R. M. Richard: W. K. Hsia and M. Chmiklowiec,Moment rotation curves for double framing angles.Proceedings of the Sessions at Structures Congress,ASCE, pp. 107-121 (1987).J. E. Goldberg and R. M. Richard, Analysis of nonlin-ear structures. J. Struct. Diu., ASCE 89,333-351 (1963).R. M. Richard and J. E. Goldberg, Analysis of nonlin-ear structures: force method. J. Struct. Div., ASCE 91,33-48 (1965).P. D. Moncarz and K. H. Gerstle, Steel frames withnonlinear connections. J. Strucr. Div., ASCE 107,1427-1441 (1981).D. Anderson, F. Bijlaard, D. Nethrcot and R. Zan-donini, Analysis and design of steel frames with semi-rigid connections. Proposed to IABSE Survey, AISC(1987).L. S. Beedle, Elastic, plastic and collapse characteristicsof structural welded connections. Ph.D. dissertation,Lehigh University (1952).L. S. Beedle, A. A. Topractsoglou and B. G. Johnston,Connections for welded continuous portal frames.Progress Report543-560 (1952). No 4: Part III. The welding Journal