Lectures 11-4-2 Analysis of Connections Distribution of Forces in Groups of Bolts and Welds

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ESDEP WG 11

CONNECTION DESIGN: STATIC LOADING

Lectures 11.4.2: Analysis of Connections: Distributionof Forces in Groups

of Bolts and Welds

OB!C"#$!%&CO'!

To review the behaviour and the basis for design of local elements in connections.

'(!(!)*#"!&

Lecture 1B.5 !ntroduction to "esign of !ndustrial Buildings

Lecture 1B.# !ntroduction to "esign of $ulti%&tore' Buildings

Lecture (.) *ngineering Pro+erties of $etals

Lecture (., &teel -rades and ualities

Lecture 11.1.( !ntroduction to Connection "esign

Lectures 11.( /elded Connections

Lectures 11.) Bolted Connections

Lecture 11.,.1 0nal'sis of Connections Basic "etermination of orces

(!LA"!D L!C"*(!&

Lecture 11.5 &im+le Connections for Buildings

Lecture 11.2 $oment Connections for Continuous raming

Lecture 11.# Partial &trength Connections for &emi%Continuous raming

&*++A(,

This grou+ of , lectures 311.,.1 % 11.,.,4 ex+lains how the behaviour of local elements in connections ma' beanal'sed so that each com+onent ma' safel' be +ro+ortioned to resist the loads it is reuired to transfer. !t therefore

develo+s the basic conce+ts of force transfer that were +resented in general terms in Lecture 11.1.(.

This second lecture concentrates on the behaviour and design of grou+s of fasteners 3bolts or welds4 as used in the

t'+es of connection described in Lecture 11.1.(. $ethods are +resented for assessing the load on each individual

fastener 3bolt or length of weld4 and for determining the total resistance of the grou+ acting in combination. Thes+ecific to+ics covered include long bolted 6oints7 long welded 6oints7 weld grou+s7 bolt grou+s and welds and bolts

designed to act together in resisting the same a++lied forces.

-O"A"#O-

The notation of *urocode ) 819 has been ado+ted.

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a.Assume the bolts are infinitely stiff and the plates are weak

The bolts do not deform. The' remain straight and +arallel to each other. *ach +iece of +late between a

+air of bolts therefore has the same length7 the same strain and conseuentl' also the same stress. !n theexam+le of igure (7 this means that the forces in the +lates between bolt 1 and bolt ( are >75 7 17>

and >75 . But this also a++lies to the +lates between bolts ( and ) and between bolts ) and ,.

Conclusion the bolts 1 and , transmit the full load . The other bolts are not loaded7 see igure (a.

b.Assume the plates are infinitely stiff and the bolts are weak

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The plates between the bolts do not deform. In other words, every bolt has the same deformation and

therefore is loaded to the same extent. Every bolt carries 0,5 F, i.e. 0,25 F per shear area.

The real distribution of forces is between these two extremes, as is indicated by the solid line !elastic!" in Fi#ure 2c.

The difference between the forces in the outer bolts and the inner bolts is #reater when the stiffness of the plates is

low. This situation occurs when the connection is lon#er more bolts" and the plate thic$ness compared to the boltdiameter is small.

For practical ratios of plate thic$ness to bolt diameter and practical values for the pitch, the followin# approximatedistributions %" of bolt forces apply&

' with four bolts 2('2)'2)'2(

' with six bolts 25')5')0')0')5'25

' with ei#ht bolts 2*')+''5'5'')+'2*

Design recommendation

The part of the connection between the outer bolts must be desi#ned to be as short and stiff as possible, in order to

minimise the differences between the bolt forces.

In practice, however, it is normally permissible to assume an even distribution of forces, owin# to the plastic

deformation capacity of the bolts and plates. -hen a bolt is overloaded, or a plate in bearin# is overloaded, it willdeform plastically. Then, throu#h redistribution of forces, a more even distribution of the forces in the bolts is

obtained.

The amount of deformation capacity that is needed, is #reatly influenced by the len#th of the connection&

/ uniform distribution of forces is assumed if the distance between the outer bolts is not more than )5d, whered is the nominal diameter of the bolt. This means six bolts at a pitch of +d.

For lon#er connections, the desi#n value of the shear force Fv.dper bolt and per shear plane must be reduced

by a factor 1f, see also Fi#ure +.

1f ) ' 13' )5d"4200d )')"

but 1f),0 and 1f0,5.

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The deformation capacity is provided by the bolt bendin# and shear" and4or by the plates yield of net area,

ovaliBation of the bolt hole caused by bearin# stresses".

/nother cause of uneven distribution of forces and thus of a need for deformation capacity is the possibility of

misali#nment of bolt holes. Cecause of fabrication tolerances the diameter of the holes is chosen as the diameter ofbolt plus a clearance. For an D20 bolt, the normal hole diameter is 22 mm. 8ue to this clearance, it is possible that at

low loads elastic deformations" only one bolt in the connection of Fi#ure 2, for example, carries the whole load.


Cecause the deformation capacity of plates is #enerally much bi##er than the deformation capacity of the bolts, it is

recommended that the connection be desi#ned such that yieldin# of the plates in bearin# occurs before yieldin# of thebolts in shear.

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1.1.2 Distribution of shear stresses in long welds

In longitudinally loaded welded connections an uneven load distribution occurs, similar to that just described forbolted connections. The highest stresses occur at the ends of the welds, see Figure 4.

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In this case also a uniform distribution of forces (stresses) may be assumed, provided that the deformation capacity issufficient to allow for the required redistribution of stresses.

In a similar way as for long bolted connections, the differences in the stress distribution depend on the stiffness ratioof the connectors (welds) and the plates; the longer the connection the more uneven the stress distribution.

As with bolted connections, the deformation capacity is provided by the connector (the welds) and/or the adacent

plate material, see !igure "a. #learly the deformation capacity of a thic$ weld is greater than that of a thin weld.

%he plastic &one and the deformation capacity of a weld are proportional to the weld thic$ness. In addition the

ductility of the weld metal and the strength of the weld metal compared with the strength of the plate, have aninfluence on the deformation capacity.

If the yield strength of the weld metal is higher than the yield strength of the plate material, then plasticity occursmainly in the plates. %his is usually the case in common lower grades of steel (up to '*), where +overmatched+

weld metals are applied (as required by urocode -).

In higher strength steels it is sometimes difficult to have an overmatched weld metal with sufficient ductility. %hen an

+undermatched+ weld metal with better ductility properties that is easier to weld may be considered. %he consequence

is that the plastic deformations tend to concentrate in the weld metal. %he deformation capacity depends on (a) the

si&e of the plastic &ones in the weld and the adacent plates and (b) on the ductility of the metal in these &ones. If dueto undermatching weld metal the plastic &ones in the plate are small, then for the same deformation the strains in theweld metal are great. %his means that the ductility requirements for undermatched weld metals should be higher than

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for overmatched weld metals. Therefore, it depends on ductility and yield strength of plate and available weld metals

what situation is favourable: undermatched or overmatched weld metals.

The effect of the length of the weld on the distribution of stresses along the weld has been investigated by means of

finite element calculations. To illustrate this the results of a numerical simulation carried out by Feder [2] are

presented in Figure 5. The stress distribution and the shear deformations in side fillet welds have been determined for

several values of the length l, see Figure 5. For the weld metal, a linear relation between the shear stress and the

relative displacement ! l"a is assumed. For other circumstances #geometry, cross$section of the plates, weld

thic%ness, strength of weld metal and plate metal, etc&, other results will be obtained.

Figure 5 shows that up to a certain limited length of the weld #llim& yielding of the whole weld is possible. 'hen the

length of the weld is e(ual to llim, yis reached in the middle of the weld, at the same time as the ultimate shear stress

uand the rupture displacement uare attained at the ends. Then the average stress, u, at the start of rupture is

obtained by ta%ing the average of a parabola as:

u! #u ) 2y&"* #$2&

For l + llimthe central region of the weld will not have reached yield when the rupture starts at the ends of the weld.

Figure gives results for a side fillet weld in steel -*55 [2]. The members have the same cross$section area ! 2.

The assumed $ diagram is also given in Figure .

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For the rupture strain (u) the value 0,110 is taken. For a weld with 5 mm throat thickness, a displacement of 0,55 mm

is obtained. A further assumption is that the plates do not yield at the ross cross!section when rupture of the weld

occurs. "nder these assumptions, llim#$%0aw, where awis the throat thickness of the weld.

&ith increasin weld lenth the averae shear stress at rupture decreases rapidly. &hen l ' %00a, the stress in the

middle of the weld remains ero

Accordin to *urocode %, the desin resistance of a fillet weld in a lap +oint should be reduced by a factor w.1to

allow for the effects of non! uniform distribution of stress alon its lenth when it is loner than 150a accordin to

the relationship-

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Lw.1= 1,2 - 0,2 (1-3)

where Ljis the overall length of the lap joint in the direction of force transfer. In practice, lap joints with fillet welds

longer than 100a or 10a are seldo! "sed (for a = !!, a length 10 a !eans #0 !!$).

%f co"rse, when the distri&"tion of applied loads on the weld is evenl' distri&"ted along the weld, the a&ove

li!itation does not appl'. a!ples are the welds &etween the we& and the flange of welded &ea!s and theconnection of the we& of a &ea! to a col"!n, see *ig"re #.

1.2 Distribution of Forces in Weld Groups

+he design of weld gro"ps (*ig"re ) is tacled &' considering the strength of the individ"al welds.

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For this, two approaches based on the design assumptions of Clause 6.1.4 of Eurocode 3 can be followed:

1. Calculation based on the stresses in the parent material in the vicinit of the weld.!. Calculation based on the load acting on the welded "oint as a whole.

#he calculation based on the stresses in the parent material is easier and $uic%er than the second method. #he stresses

in the vicinit of the weld can be directl obtained from the structural design calculations. &ecause of the direct lin%between the analsis for the parent material and that of the welds, it is clear that the stresses in the weld are consistent

with those in the parent material. 'f course, it is necessar that in determining the stresses in the parent material, the

stresses must be consistent with other parts in the connection ()ecture 11.4.1*ection !.! and Figure !+. se of the

first method is recommended.

-owever, there are cases where the first method cannot be applied because the stresses in the ad"acent parent material

cannot be simpl determined and the second method must be used. Eamples are:

/ a lap "oint.

/ the connection of a brace to a gusset plate.

1. Calculation based on the stresses in the parent material

0n a double fillet weld shown in Figure , the following stresses act on the throat area (see also )ectures 11.!.!and 11.!.3+.

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= = {0,5tz0,52}/a = [t/(2a2)]z (1-4)

//= (t/2a)zy (1-5)

With the alternative method o !nne" # o $%ro&ode ' it ollo*

(t/2a)[z2/2 + 'z2/2 + 'zy2] %/(#)

or*

(t/2a)[2z2+ 'zy

2] %/(#) (1-)

he e&ond re.%irement %/#i only de&iive i i mall, ie i the re%ltant or&e i oli.%e to thelate

in the arent material only zi reent then it ollo*

(z/%)(#)t/2 (1-3)

or 2'5* 0,31(z/ %)t (1-6)

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For S355: 0,80(z/ fu)t (1-9)

If the theory of plasticity is use for the esi!" of the structure a" the co""ectio" is locate at a poi"t #here a

plastic hi"!e $ay e%elop, the" the $i"i$u$ throat thic&"ess $ust 'e 'ase o" z fy:

his !i%es for S*35: a 0,+ t (1-10)

a" for S355: a 0,55 t (1-11)

his reuire$e"t also applies for statically i"eter$i"ate structures that are esi!"e usi"! the theory ofelasticity. It is i$porta"t to re$e$'er that also i" a" elastic esi!", it is assu$e (i$plicitly) that the $e$'ers

a" the co""ectio"s ha%e sufficie"t efor$atio" capacity to acco$$oate loas a" stresses that usually are

"ot eplicitly ta&e" i"to accou"t i" the esi!" calculatio"s (e.!. stresses ue to u"e%e" settle$e"ts of thesupports te$perature loai"! tolera"ce uri"! fa'ricatio" local o%erloai"! 'y li%e loas, etc.) a" further to

allo# for the approi$atio"s i"here"t i" the esi!" $oels.

herefore, it is "ecessary that the co""ecte parts ca" yiel 'efore rupture of the #els.

he" the esi!" for$ulae of 2urocoe 3 14 are applie #ith z fy, the" the real rupturi"! stre"!th of the#el is at least the real rupturi"! stre"!th of the plate. I" other #ors, actual rupture occurs i" the plate a" "ot

i" the #el.

hus, for the a'o%e reuire$e"t (yiel i" the plate 'efore rupture i" the #el), the esi!" of the #el ca" 'e

'ase o":

z (fyr/fur)fy (1-1*)

#here fyris the $easure yiel stre"!th a" furthe $easure ulti$ate stre"!th of the plate $aterial.

For the esi!" %alues of the yiel stre"!th a" the ulti$ate stre"!th, it follo#s for S355:

z (1-13)

ecause the actual %alue of fyr/furca" 'e hi!her tha" 0,60, it is reuire that:

If efor$atio" capacity is "ecessary, the #els $ust 'e esi!"e to tra"sfer at least 807 of the yiel force i"the (#ea&est) co""ecte $e$'er.

his reuire$e"t !i%es the follo#i"! %alues for the $i"i$u$ throat thic&"ess of a ou'le fillet as prese"te i"Fi!ure 9:

For S*35: a 0,36 t (1-1+)

For S355: a 0,++ t (1-15)

It shoul 'e "ote that, usi"! the $ea" stress $etho accori"! to 2urocoe 3, hapter , !reater throat

thic&"esses are fou" for e" fillet #els. he iffere"ce is a factor of 1,**

hus the applicatio" of the $ea" stress $etho results i" 1,***1,5 ti$es $ore #el $etal tha" "ecessary.

1. Calculation based on the load acting on the welded joint as a whole

his $etho $ust 'e applie if the first $etho is "ot applica'le. For the eter$i"atio" of the stre"!th of a #el!roup, the esi!" %alues for the stre"!th of the separate #els $ay 'e ae, pro%ie that the euili'riu$

reuire$e"ts are fulfille.

his approach is 'ase o" the assu$ptio" that the #els ca" yiel to per$it the reistri'utio" of stresses "ecessary to

acco$$oate local o%erloai"!. I" other #ors, the #els $ust posses sufficie"t efor$atio" capacity.

o !ai" so$e iea of the efor$atio" capacity of #els u"er %arious loai"! co$'i"atio"s, tests 34 ha%e 'ee"carrie out as i"icate i" Fi!ure 10. I" these tests, the #els #ere thi" co$pare #ith the plates i" orer to e"sure

yieli"! i" the #els a" "ot i" the plates. he $easuri"! le"!th lois !i%e" i" Fi!ure 10. he efor$atio" of the

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plates #as su'tracte fro$ the $easure %alues, to o'tai" the efor$atio" of the #el a" the pare"t $aterial i" the

irect %ici"ity of the #el, see Fi!ure +a.

Ds alreay iscusse, it appears that the efor$atio" ($$) at the sa$e stress i" the #el is proportio"al to the throat

thic&"ess. hus, #he" the thic&"ess of a #el is ou'le, "ot o"ly is its stre"!th ou'le, 'ut also its efor$atio"

capacity. his is the reaso" #hy the efor$atio"s are !i%e" as l/a o" the horizo"tal ais i" Fi!ure 11.

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o e$o"strate the i$porta"ce of the a'o%e fi"i"!s, the lap Boi"t of Fi!ure 1* is a"alyze. Suppose that the plates

are i"fi"itely stiff co$pare #ith the stiff"ess of the #els. he" the thic&"esses of the e" fillet #el a" the siefillet #els are a'out the sa$e, the", at the start of rupture (i" the #els), the forces i" all #els are practically eual

to their ulti$ate loa. his ca" easily 'e see" #he" the li"es for 11a" are co$pare. ith asie ae"a" lfor the sie #el a" e" #el a'out the sa$e, l/a is the sa$e for sie #el a" e" #el. herefore, the ulti$atestre"!th of 'oth #els $ay 'e ae. his $ay "ot 'e true if o"e of the #els is %ery s$all, co$pare to the other. It

ca" 'e co"clue therefore, that the ulti$ate stre"!th of the lap Boi"t is eual to the su$ of the ulti$ate stre"!ths ofthe separate #els.

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Now suppose that the throat thickness aendof the end fillet weld is only 10% of the throat thickness asideof the side

fillet welds. At the start of rupture of the end fillet weld, the elongation l=100.10-3. aend=10.10-3aside, see Figure 11.

he stress 11corresponding to l!aside= 10 . 10-3is a"out #30N!$$#, whilst the rupture strength for 11is a"out

30N!$$#.

&n this case, the ulti$ate strength of the lap 'oint is less than the su$ of the ulti$ate strengths of the separate welds.As a result, it is reco$$ended the following design rule is used(


ry to gi)e the end fillet weld and the side fillet weld the sa$e thickness, and ne)er design the end fillet

weld to "e less than 0, ti$es the thickness of the side fillet welds.

he use of a thin weld at the front of the lap 'oint *point A in Figure 1#+, e.g. to pre)ent corrosion, $ust "e a)oided. &f

such a weld is necessary, then it should "e gi)en the sa$e thickness as the other welds. his is particularly i$portant

"ecause the plates are in reality not infinitely stiff co$pared with the welds. he reuired defor$ation capacity

therefore is larger at the front *point A+ than at the "ack of the lap 'oint *point +.

1.3 Non-Linear Distribution of Bolt Forces

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In Section 3.2 of Lecture 11.4.1it was assumed that the bolt forces Riare proportional to the displacement = r .

This assumption is based on linear (elastic beha!iour of the connection. In the same wa" as for the desi#n of weldedconnections$ the theor" of plasticit" ma" be used also for bolted connections$ see %i#ure 13.

&hen the force on the connection increases$ the force on the most hea!il" loaded bolt increases until the "ield force

R"is reached. Then$ with increasin# displacement$ the bolt force sta"s constant until strain hardenin# starts.

'fter the start of "ieldin# at the most hea!il" loaded bolt$ the bolt forces in the bolts which are nearer to the centre ofrotation increase with increasin# rotation. The moment increases until all bolts ha!e reached the "ield force.

%rom tests it ma" be concluded that the plastic moment of such bolted connections is normall" reached atacceptable displacements. Therefore$ in staticall" loaded structures$ the desi#n of such bolted connections ma" be

based on the theor" of plasticit".

In a plastic distribution of bolt forces$ the centre of rotation does not need to be located at the centroid of the bolt

#roup. This can be demonstrated as follows (compare %i#ures 14 and Lecture 11.4.1%i#ure 12.

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The bolts near the centre of rotation have a small moment arm, and therefore do not contribute very much to the

moment resistance of the connection. It is economical therefore to use these bolts for the transfer of shear force V,

and to use the outer bolts to resist the bending moment M.

Several possibilities are shown in Figure 1. The final choice depends on the proportions of the loads which must be

carried! VSdand MSd.

The above design model is based on an elementary principle in the theory of plasticity!

"ny distribution of forces, where the internal forces #bolt forces$ are in e%uilibrium with the e&ternal forces in such away that nowhere is the internal load'carrying resistance #the design resistance of the bolts$ e&ceeded, gives a lower

bound to the design resistance of the connection.

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This principle is only valid if sufficient deformation capacity is available. In bolted connections this capacity can be

assured by designing the bolts such that they are not the controlling item of the strength of the connection:

In shear and bearing: let bearing be decisive, because the deformation capacity in bearing of the plate is much

bigger than the deformation capacity in shear of the bolt.

In tension: let yielding of the plates in bending be decisive rather than rupture of the bolt.

1.4 Combination of Different Types of Fasteners

In general, the load deformation behaviour of different fasteners is such that their design resistances are reached atdifferent deformations, see Figure 16. For this reason the use of more than one type of fastener in the design

calculations is not normally allowed. In this respect the deformation capacity of the types of fastener used is an

important factor.

In the case of welds acting in combination with preloaded bolts that are preloaded after welding is completed, the

design resistances are reached at about the same deformation. Therefore, in this case it is permissible to add thedesign resistances of the preloaded bolts and the welds when determining the design resistance of the connection.

For all other arrangements only one type of fastening may be assumed to be active and all load must be transferredby this e.g. for a connection made originally with bolts that must be strengthened to withstand a higher load the welds

must be designed to carry the whole of the load !not "ust the additional part#.

2. CONCLUDING SUMMA!

$ong connections should be designed in such a way that the forces in the fasteners !bolts and welds# are, as far

as possible, e%ual.

For weld groups, design should, wherever possible, be based on an approach which uses the stress in the parentmaterial in the vicinity of the weld as the controlling parameter.

&olt groups may be designed using a plastic approach providing sufficient deformation capacity to permit fullredistribution of forces is present. 'eformation capacity may be ensured by:

i. for bolts in shear and bearing, ensure that bearing governs.

ii. for bolts in tension, ensure that yield of the plates in tension governs.

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Where more than one type of fastener is used to transfer the same load between the same two components in a

connection, the design should normally assume that all load is transferred by one type of fastener only.

3. REFERENCES

[1] Eurocode 3: !esign of "teel "tructures E#$ 1%%3&1&1: 'art 1.1, (eneral rules and rules for buildings, )E#,

1%%*.

[*] +eder, !. and Werner, (., ns-te ur /raglastberechnung 0on "chweiss0erbindungen des "tahlbaus. "chweissen

und "chneiden, *% 1%22, 4eft 5.

[3] 6igtenberg, +. 7. and $an 8elle, +., 9nderoe naar de 0er0orming 0an statisch belaste hoelassen. 4eron 1*

1%;5 #o. 1 !utch.

Lectures 11-4-2 Analysis of Connections Distribution of Forces in Groups of Bolts and Welds

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Transcript of Lectures 11-4-2 Analysis of Connections Distribution of Forces in Groups of Bolts and Welds