3.

Shear Lag in Wide Flanges

ln a box girder (Fig. 3.1a), the web and ftange plates are interconnected so thatrelative displacements cannot occur. Therefore, at the junction of the web with

the ftange the longitudinal strain in the web (ex,w) must be equal to that in theftange (ex,r). A shear ftow develops between the web and the ftange which causes

(a)

(tJ)

Fig. 3.1. Shear lag effects: (a) distribution of longitudinal normal stressesacross ftange widths, (b) warping of the c:ross-section.

87

Shear Lag in Wide F/anges

shear deformation of the flange plate. The longitudinal displacements in the

parts of the flange remote from the webs lag behind those nearer the webs (Fig.3.1b). This effect leads to a non-uniform distribution of the longitudinal normalstresses across the flange width (Fig. 3.1a). The effect is particularly pronouncedin wide flanges and in flanges with longitudinal stiffeners.

This phenomenon, termed shear lag, results in a considerable increase of thelongitudinal stresses (J in the regions of the flange clo se to the webs in com­

parison with those given by the elementary theory of bending (Fig. 3.1a). Thusneglect of shear lag would lead to an underestimation of the stresses developedin the flange plates at positions adjacent to the webs, and hence to an unsafedesign. The shear lag may also significantly influence the girder deflections.

It is an accepted practice in structural engineering to represent the effect of shearlag by adopting an effective breadth concept. The actual width of the flange plate bis replaced by a reduced width bef over which the longitudinal stresses may beconsidered uniformly distributed, and the application of the elementary theory ofbending to the transformed girder cross-section gives the correct value of a maxim­

um longitudinal stress (Je (Fig. 3.1a). A similar procedure may also be carried out fordeflections. However, when the structure is subjected to large concentrated loads, theconcept of effective breadth gives reliable information only in those parts of thestructure that are not very close to the point of application of the load or the supportreaction. In the immediate neighbourhood of a point load, the actual stress state candiffer rather substantially from that resulting from any simplified analysis, includingthe effective breadth concept.

3.1 Methods of AnalysisExtensive analysis of the shear lag effect has been carried out during the last

few decades. An analytical method has been given by Girkmann [3.1]. Morerecently many analytical models and methods have been developed. Amongthese are numerical solutions based on finite element or finite difference met­

hods, exact and approximate methods based on folded plate theory, and ap­proximate methods based on simplified structural behaviour.

3.1.1 The Finite Element Method

The finite element method has become practically universal for the solutionof mechanics problems in recent years. The continuum is replaced by an assem­bly of finite elements interconnected at nodal points. Stiffness matrices aredeveloped for the finite elements based on assumed displacement patterns, andthen an analysis based on the direct stiffness method may be performed todetermine nodal point displacements and, subsequently, the internal stresses in

88

Methods oj Ana/ysis

the finite elements. As this method is well doeumented in the literature, noattempt is made to review it here in detail.

Moffat and Dowling [3.2] produeed a eomprehensive parametrie study oftheshear lag effeet in box girders; this study was based on the use of the finiteelement method. They found that while only one mesh division over the depthof a girder was suffieient,finemesh divisions had to be used over the girder widthand length, partieularly in the region of a point load or a support.

Although a finite element solution is eapable of giving a eomprehensive andadequate pieture of the stress distribution, it requires the use of large eomputersand is too eostly, partieularly if repeated analyses are required at the preliminarydesign stage.

3.1.2 The Folded Plate Theory

Steel box girders are usually of eonstant eross-seetion and, henee, the foldedplate theory is ideally suited to predieting shear lag effeets. This theory, des­eribed fully above in Chapter 2, takes advantage of harmonie analysis and maybe applied to a variety of support eonditions. Use of the folded plate theoryresults in a eonsiderable saving of eomputer time over the finite element method.

3.1.3 The Finite Strip Method

A direet applieation of the theory of elastieity to determine the stiffnessmatrixof eurved folded plate elements beeomes exceedingly eomplex. A theory knownas the finite strip method [3.3], [3.4], [3.5] may be used in these eases. Thismethod may be eonsidered as a speeial form of the finite element method. ftapproximates the behaviour of eaeh plate by an assembly of longitudinal finitestrips for whieh seleeted displaeement patterns, varying as harmonies lon­gitudinally and as polynomials in the transverse direction, are assumed torepresent the behaviour of the strip in the total strueture. With this assumption,the displaeement at any point in the strip ean be expressed in terms of eightnodal point displaeements and, hence, the element stiffness matrix determined.The remaining proeedure is similar to that used in the folded plate method.

3.1.4 Harmonie Analysis of Shear Lag in Flanges withClosely-Spaeed Stiffeners and in Composite Flanges

This Seetion deseribes a simple method, employing harmonie analysis, whiehenables shear lag effeets in wide flanges to be predieted from hand ealeulations[3.6]. This approaeh is presented here in more depth than the preeeding general

89

Shear Log in Wide Flanges

methods, which are well documented elsewhere. The method is suitable for theanalysis of girders with flanges which are not stiffened, as in the case of concretegirders or flanges of composite girders, and of girders where the stiffeners are soclo sely spaced that it is reasonable to assume the stiffener properties to be spread

evenly (or smeared) over the flange width. This is indeed the case for manybridge and aircraft girders. The method may be applied to multi-cellular girders,as well as to girders with inclined web plates and with overhanging or can­tilevered flanges.

Although the method is suitable for hand calculations, it has been program­med for a personal computer for added convenience; a suitable program isincluded in Table 3.1.

Table 3.1. Basic Listing oj a Program Jor a Personal Computer Jor Harmonie Analysis oj ShearLog in Flanges with Closely Spaeed Stiffeners and in Composite Flanges

10 LET Wl = O20 INPUT U,R,E,B,T,I,L,P,Y,X,Y,J930 FOR J = 1 TO J9 STEP 2

40 LET F = (U-R)tO.550 LET K = J.PhF/L60 LET A = K.Y70 LET O = K.B/280 LET C = (EXP A + EXP(-A)).0.590 LET S = (EXP 0- EXP (-0)).0.5

100 LET Q = 2.P.L.T.B.E.SIN(J.PhV/(2.L)) .SIN(J.PI/2)/(V.Jt2.Plt2.1)110 LET W = -F.Q.SIN(J.PI.X/L).C/(S.T)120 LET Wl = Wl + W130 NEXT J140 PRlNT "HARMONIC ANALYSIS OF SHEAR LAG IN STlFFENEO STEELFLANGES WITH A CONCRETE LAYER"

150 PRlNT "t(ax).E/t(sh).G ="; U160 PRlNT "coefficientr ="; R170 PRlNT "distance from neutral axis to fiange ="; E180 PRlNT "breadth of fiange = ft; B190 PRlNT "modified fiange thickness ="; T200 PRlNT "second moment ofarea ="; I210 PRlNT "span="; L220 PRlNT "load ="; P230 PRlNT "loaded length ="; V240 PRlNT "x coordinate at which the stress is desired ="; X250 PRlNT "y coordinate at which the stress is desired =" ; Y260 PRlNT "STRESS ="; Wl

90

Methods oj Ana/ysis

IpI~

Imtm

·[t-----9JI: x .I

W~

IFonn of Input Data

l

I. ó .1

Notation

No.Input Quantity Ref.in theExample

ProgramI

quantity TE/t*G Bq. 3.17U3.0784

2

eoefficien t r Bq. 3.10R0.24277

3

distance e from neutralaxis of whole eross-section

Bq. 3.22to flange mid-thiekness

Fig. 3.3b, eE27.37

4

breadth b of the flangeFig.3.3.

B300

5

modified thiekness TEq.3.1

of the flangeFig.3.3dT8.987

6

second moment I of wholeeross-section ineI. stiffeners

Bq. 3.22I8.044 62 x 106

and modified layer7

span length L L1000

8

magnitude of the loadp

p6000

9

loaded length a v25

10

position of preseribed eross-section for results x

Fig.3.3eX475

II

transverse coordinate y for re-sults

Fig.3.3.eY150

12

number of hannoniesused

J939

Result of the example: STRESS = 7.257

91

Shear Lag in Wide F/anges

A continuous girder with various support conditions can be approximated byan assembly of simply supported beams and cantilevers, as shown in Fig. 3.2.The bending moment and shear force diagram s are established from continuousbeam analysis in the fi.rstinstance, as in Fig. 3.2b, thereby defining the points atwhich the bending moment becomes zero and the shear force is known. Theshear lag analysis is then carried out for each individual portion of the beam, asin Fig. 3.2c.

I"", .... ""'''''''é /o~r;""".." ...,,""~ ..,,·..~ a) ContinfJOusgirderI I I I:: I I I II : : ::

1\ I :\J I II\....,~ : ~ AI;'--: IJ)Bending moment: :: I : '-""'" i I : diagram1 I I I I I II I I I I I I

~ II II I II I I I I I: I I I I II I I I I I

pIIIIIIIIlI I I I ITT' I I I I: : : I:

, ...•bm I!I I I IJ"" II II 111111111. 1 :

I I

~ c)tguira/ent system

Fig. 3.2. Representation of a continuous girder by an assemblyof simply supported beams and cantilevers.

This process is slightly approximate, since the additional flexibility due toshear lag itself alters the overall bending moment and shear force diagrams instatically indeterminate structures. However, for girders of practical propor­tions, this effect is insignificant and is neglected in this simple method.

Harmonie analysis can be applied directly to the case of simply supportedgirders. However, cantilevers may also be analysed according to the idea presen­ted in Section 2.4, Le. by first establishing a substitute beam, as shown in Fig.2.24.

ln addition to unstiffened flanges (Fig. 3.3a), this method may be applied tothe analysis of stiffened flanges provided the stiffeners are sufficiently doselyspaced to allow their properties to be smeared over the flange width (Fig. 3.3b),and to analysis of a quite general case of flanges with both stiffeners and aconcrete layer (Fig, 3.3c) as described in Section 3.2.7.

It is assumed that the shear effects are transmitted by the composite flangeplateitself, but the longitudinal axial forces are carried by the stiffeners, the steel

92

a arT1,t. __ ._ :4$ _._.

Methods oj Ana/ysis

. a) 6ird4r withunstJflMet/ (Iangu

Ó) 6irder wllhsflflenet/ (Ionges

q,[t

C) 6irder with .

stJflened flangesani} a concrefe laJer

d) 6irt/er with

e.iuiyotent unsflffenedf/anges

x

y

Fig. 3.3. Idealization or ftange plate.

93

Shear Log in Wide Flanges

flange sheet and the concrete layer. It is, therefore, necessary to introduce amodified flange thickness t(Fig. 3.3d) for the axial force-carrying action, definedby:

(3.1)

where As is the cross-sectional area of each stiffener and a is the stiffener spacing,as in Fig. 3.3b, te is the thickness of the concrete layer (Fig. 3.3c), and E, Eerepresent Young's moduli of steel and concrete, respectively.

A shear flow q and a normal force nx per unit width act on a typical elementof the flange, as in Fig. 3.3e. The equation governing the equilibrium in thelongitudinal direction is

onx oq- + - = O.ox oy

The direct strains in the longitudinal direction are given by

(3.2)

(3.3)

If the small transverse forces in the flange are neglected and if it is assumedthat the transverse direct strains of the steel sheet and the concrete layer mustbe the same, then

(3.4)

in which snx and enx are the portions of the normal force in the flange transmittedby the steel and concrete components respectively, and vand ve are Poisson'sratios.

Since

(3.5)

we have

(3.6)

94

Methods oj AnaJysis

from which

v Ve-+­tsE teEe

Considering the above relations, we get

(3.7)

snx nx

8y = -v tsE = -VVe ( As)ve t + ~ E + vt~e

which can be written in the following simple form:

(3.8)

(3.9)

in which

(3.10)r=

ve(t + ~s) E + vt~e

Note that for a steel flange (without the concrete layer) r = v, while for a possibleconcrete flange itself r = ve'

The shear strain in the composite flange may be approximated as

qy=----,tG + teGe

which can be written in the form

(3.11 )

q--,Y - t*G

(3.12)

where the modified ,thickness of the composite flange corresponding to the shearaction is

Ge

t* = t + teCi' (3.13)

95

Shear Log in Wide F/anges

It is seen that whilst the modified flange thickness t, which includes thestiffener contribution, has been used in the expressions for direct strains in Eq.(3.3), the shear strain is dependent upon the thickness t*. This is because thestiffeners cannot participate in the shear-carrying action of the flange.

The condition of compatibility may be expressed as

(3.14)

Substituting the strains from Eqs. (3.3), (3.9) and (3.12),

a2n a2n t1\ a2qx x--r-=---,al ax2 t*G ax ay

and substituting for the shear flow q from Eq. (3.2)

(3.15)

It was noted earlier, and is shown in Fig. 3.2, that a girder with varioussupport conditions can be idealized as an assembly of simply supported spans.To satisfy the corresponding boundary conditions, the normal force nx at anypoint may be expressed by the following Fourier series

(3.16)

where L is the length of the simply supported span.Byconsidering thej-th term ofthe series only and substituting into the partial

differential equation (3.15), the following ordinary differential equation is ob·tained

(3.17)

j1t~-1\where ,. = - - - r.

J L t*G

Note that for an unstiffened steel flange (i.e. t = t* = t), 'i = (i1t/ L) ~ .

96

Methods oj Analysis

The general solution of Eq. (3.17) gives the amplitude of the normallongitudi­nal fotce as

(3.18)

where C1j and C2j are constants to be evaluated from the appropriate boundaryand loading conditions.

Shear lag analysis is carried out for loads which are placed symmetrically onthe girder cross-section. Take the origin of the transverse coordinate y to be atthe mid-width of the flange, i.e. at the axis of symmetry as in Fig. 3.3e. Thenfrom symmetry,

C2j = O,

so that, from Eq. (3.18), the distribution of the normal force across the flangewidth is given by

(3.19)

The value of the remaining constant C1j can be determined from the shearloading condition at the edge of the flange. Combining Eqs. (3.2), (3.16) and(3.19),

aq anx 00 j1t j1tX 00 j1t j1tX

- = - - = - L Niy) - cos - = - L - C1j cosh 'jY cos - ,ay ax j=1 L L j=1 L L

so that by integrating with respect to y and substituting for 'j from Eq. (3.17)the shear flow at any point may be expressed as

( t-E )-1/2 00 •

J1tX

q(x, y) = - - - r L C1j sinh 'jY cos-.t*G j=1 L

(3.20)

(3.22)

At the edge of the flange, where y = b12, the shear flow will be denoted by qe(x).Therefore, from Eq. (3.20)

- (b) . ( fE ) -1/2 00 b j1tXq X,- = qe(x) = - - - r L C1jsinh'j-cos-. (3.21)

2 t*G j=1 2 L

From simple beam theory, the shear flow qe(x) transmitted from the web tothe edge of the flange can be approximated as

tbe

qe(x) = V(x) -.21

97

Shear Log in Wide F/anges

(Note that for a double-symmetrical cross-section, we have simply qe(x) == V(x) (tbd/4I)). There V(x) is the total shear force acting on the beamcross-section at position x, I is the second moment of area of the completecross-section, induding stiffeners and concrete parts (reduced by the ratio EjE,see Fig. 3.3), d is the depth of the web and e is the vertical distance from thecross-section neutral axis to the mid-thickness of the flange (Fig. 3.3b, c).

The shear flow transmitted at the edge can also be expressed in the form ofa Fourier series. For the case of simply supported ends, the series takes the form

where

2 f.L j1tX tbe f.L j1tXQej = - qe(x) cos - dx = - V(x) cos - dx.

L o L IL o L

(3.23)

(3.24)

The values of the coefficient Qej' evaluated according to this equation, are listedin the first column of Table 3.2 for three typical cases.

Table 3.2. Va/ues oj (hl' Coetficil'l/I Qc.i fÓr Differcnt Types or Loading.

Type 0/ loadlngae,/

SiCiiOn J.!.II (ég .li2~SSSeet/on.1. f.§

lJntlorm/yd/strilJutea'Lw loe

Lw (to r IAs) e/ood/ng w .2.2W

jl JrlJ jl JrIIA. ~ ~I • L I

{oreeP d/str/IJuted oyerlengtll tf (symmetrico/ PL(torIAs)e . jr . jJró

obout mk/-spon) pa oe . jJr . jrO-

~.2 Ó/JrlJ Sin2 SIn-U.2 Ó.l 2J SIn-SIn--/ Jr .2 t?L.. tLI

.A-

IL.1

force P d/str/IJuted oysr.2 pabe . jJr'l' j,,5

PL(tb +IAs)e . jr,! . jJrolengtl1 tf (ot generol posil/on)

ó/r2J Sin L Stn 2L.2 ÓJlJrlJ SlnTStnuP

~

.li. (for : ~ 7l sL - : )L

I

98

(3.25)

Methods oj Ana/ysis

By equating the two expressions for qe(x) from Eqs. (3.21) and (3.23), theremaining eonstant C1'; is obtained as

( tE )1/2 Q.CI'; = - t*G - r eJ bsinh 'j­

2

Having thus determined the two eonstants of integration, the amplitude of thenormallongitudinal force for any partieular harmonie ean be obtained from Bq.

(3.19) as

)1/2 eosh 'jY-E .t -r Qej b

Nj(y) ~ - (,'G sinh Cj:2

(3.26)

(3.27)

The magnitude of this force varies across the width of the flange; its peakvalue occurs at the edge, Le. where Y = bl2

(b) ( tE )1/2 bNj - = - - - r Qej cotanh 'j - .2 ~G 2

Knowing the amplitude Nj, the value of the longitudinal normal foree per unitwidth, nx(x, y), may be determined from Eq. (3.16) for any position on theflange. Also, the shear flow at any point q(x, y) may be determined from Eq.(3.20) to complete the solution.

The evaluation of the amplitude Nj can be simplified by introducing a newterm k defined as:

(3.28)

and k is then simply caleulated as the produet of the number of the harmoniej and the quantity

7tb ~2L~ t*0- r,

which is a constant for any partieular girder. Substituting for k in Eq. (3.27), theamplitude Nj is obtained as

_ 2LQej kj cotanh kj •j7tb

(3.29)

99

Shear Lag in Wide F/anges

The values of the product k cotanh k have been calculated for a large 'range ofvalues of k and the results are listed in Table 3.3. It may be noted that for valuesof k in excess of 2, cotanh k is approximately unity; and advantage can be takenof this when evaluating the contribution of terms involving higher values of j.

Table 3.3 Va/ues oj the Product k cotanh k - see Eq. (3.29)

k k cotanh kkk cotanh kkk cotanh kkk cotanh kkk cotanh k

0.50

1.082 O1.001.313 O1.501.65722.02.07464.54.501 10.52

1.08851.021.32491.521.672 72.12.16394.64.60090.54

1.09541.041337 O1.541.68842.22.255 O4.74.700 80.56

1.10241.061.34921.561.704 12.32.34674.84.800 70.58

1.10971.081.361 61.581.720 O2.42.43984.94.900 50.60

1.11721.101.37411.601.736 O2.52.53395.05.000 50.62

1.125 O1.121.38691.621.75212.62.62885.15.100 40.64

1.13291.141.39981.641.76822.72.72455.25.200 30.66

1.141 11.161.41281.661.78452.82.82035.35.300 30.68

1.14961.181.426 11.681.800 92.92.91765.45.400 20.70

1.15821.201.43941.701.817 43.03.01495.55.5002

0.72

1.167 11.221.453 O1.721.834 O3.13.11265.65.6002

0.74

1.17621.241.46671.741.85063.23.210 75.75.700 1

0.761.18551.261.48051.761.86743.33.30995.85.800 1

0.78

1.195 O1.281.49451.781.88423.43.40765.95.900 1

0.80

1.204 81.301.50861.801.901 13.53.50646.06.000 1

0.82

1.21471.321.52291.821.918 13.63.60546.16.100 1

0.84

1.22481.341.53731.841.93523.73.704 56.26.200 1

0.86

1.23521.361.551 81.861.95243.83.80386.36.300 O

0.88

1.24571.381.56651.881.96963.93.90326.46.4000

0.90

1.25651.401.581 31.901.987 O4.04.00276.56.500 O

0.92

1.26741.421.59621.922.004 34.14.10236.66.6000

0.94

1.27851.441.611 31.942.02184.24.20196.76.700 O

0.96

1.28981.461.62651.962.03944.34.30166.86.800 O

0.98

1.30131.481.641 81.982.057 O4.44.40136.96.900 O

Also, the function k cotanh k can be expressed, for low values of k, in seriesform as:

k2 3k4k cotanh k = 1 + - - - + ....

3 135(3.30)

For values of k < 1.5, the third and successive terms of the series may beneglected, Le.

100

k2k cotanh k ~ 1 + -.

3(3.31 )

(3.32)

Methods oj Ana/ysis

Whenj = 1, the value of k = 1.5 corresponds to a width/span ratio bjL ofapproximately 2/3; most practical girders will have ratios below this value, so thesimplification in Eq. (3.31) can be used.

For a steel girder without concrete layers, EjG = 2.6 and v = 0.3, so that thevalue of k from Eq. (3.28) becomes

jreb ~kj = 2L ~ 2.6 ; - 0.3 .

Then, making use of the approximation for k cotanh k from' Eq. (3.31) andsubstituting into Eq. (3.29), the expression for the amplitude in the steel girderis simplified as

(b) [j2L jreb ( t )]Nj - = -Qej - + - 0.87 - - 0.1 .2 'reb 2L t

(3.33)

For example, for a steel box girder having the cross-section shown in Fig. 3.6,

and with a span length of 18.29m, the accurate value of Nj obtained from Eq.(3.27) is -6.398 N/mm, whenj = 1. The corresponding value obtained fromthe simplified expression in Eq. (3.33) is - 6.406 N/mm, the difference being only0.1 %.

The flanges of plated girders with general arrangements of cross-sections(multi-cellular girders, overhanging flanges, flanges with thickness variable in thetransverse direction, flanges of girders with inclined webs, etc., Fig. 3.4a) areconsidered as systems of longitudinal strips (Fig. 3.4b). The general expression

(a)

Fig. 3.4. (a) Cross-section of a multi­cellular girder, (b) strip layout of thetop ftange, (c) shear ftow diagram.

(C)

~I[Ir101

(3.32)

Methods oj Ana/ysis

When j = 1, the value of k = 1.5 corresponds to a width/span ratio biL ofapproximately 2/3; most practical girders will have ratios below this value, so thesimplification in Eq. (3.31) can be used.

For a steel girder without concrete layers, EIG = 2.6 and v = 0.3, so that thevalue of k from Eq. (3.28) becomes

j7tb ~kj = 2L './ 2.6 ; - 0.3.

Then, making use of the approximation for k cotanh k from· Eq. (3.31) andsubstituting into Eq. (3.29), the expression for the amplitude in the steel girderis simplified as

(b) 82L j7tb ( t )]Nj - = - Qej - + - 0.87 - - 0.1 .2 '7tb 2L t(3.33)

For example, for a steel box girder having the cross-section shown in Fig. 3.6,

and with a span length of 18.29m, the accurate value of Nj obtained from Eq.(3.27) is - 6.398 N/mm, when j = 1. The corresponding value obtained fromthe simplified expression in Eq. (3.33) is - 6.406 N/mm, the difference being only0.1 %.

The flanges of plated girders with general arrangements of cross-sections

(multi-cellular girders, overhanging flanges, flanges with thickness variable in thetransverse direction, flanges of girders with inclined webs, etc., Fig. 3.4a) areconsidered as systems of longitudinal strips (Fig. 3.4b). The general expression

(a)

(Ó) ... :: ."" , ,~'.,

Li! IL.JULJL.JUULm-th strip

Fig. 3.4. (a) Cross-section of a multi­cellular girder, (b) strip layout of thetop flange, (c) shear flow diagram.

(C)

ttI[rr101

Shear Lag in Wide F/anges

of the amplitude of the normal longitudinal force in the m-th strip - charac­terized by coeffi.cient m(j - is given in accordance with Eq. (3.8) as

mNAY) = mClj cosh m(jY + mC2j sinh m(jY'

Tbe shear flow in the m-th strip is expressed as

(3.18a)

which represents a general form of Bq. (3.20).Tbe shear flows ~qe(x) and ~qe(x), which are transmitted to the left and right

!edges ofthe m-th strip (Fig. 3.4c), are again evaluated by the simple beam theoryand expressed in the form of Fourier series

(3.23a)

(3.23b)00 j1tX

~qe(x) = L ~Qej cos - .j=l L

If the origin of the local transverse coordinate y for the m-th strip is located inthe middle of the width b of the strip, then the following system of two equationsfor the constants mClj and mC2j can be formed:

bb- sinh m(j

-cos~(j-

rlj, 2

( .,t""E _ r) -1/2

2~+bbt*G

sin~(j

-coshm(j-mC2jIII ,22

= O,

'hose solution is

(3.25a)

(3.25b)

102

Methods oJ Ana/ysis

Having thus determined the two constants for the m-th strip, the amplitudeof the normal longitudinal force for any particular harmonic is obtained bysubstituting them into Eq. (3.18a) and the remaining calculation is .similar to thecase for the symmetrical arrangement.

When applying this theory to a particular girder, the fi.rst step in the calcula­

tion of the shear lag effect is to evaluate the coefficient Qej appropriate to thespecified loading and support conditions. Table 3.2 can be used for some loadingcases frequently encountered.

This is then substituted into Eq. (3.27) or, for a steel girder, into the approxim­ate altemative form, Eq. (3.33), to give the amplitude of the normal force for anyharmonic. This, in tum, is substituted into Eq. (3.16) to give the normal forceper unit width at any position on the ftange nxtx, y).

The corresponding value of longitudinal stress in the stee1 component (thesheet and stiffeners) is then obtained simply as:

(3.34a)

and the longitudinal stress in the concrete layer may be evaluated as

(3.34b)

Should the shear stress values also be required, Eq. (3.20) can be evaluated togive the shear ftow at any position, q(x, y). The corresponding shear stress in thesteel sheet is obtained as:

sr(x, y) = q(x, y)t* '

and the shear stress in the concrete layer is

Gccr(x, y) = sr(x, y) -.

G

(3.35a)

(3.35b)

It should again be noted that the modified ftange thickness t(see Eq. (3.1)),which includes the contribution of the stiffeners and that of the concrete layerto the axialload-carrying capacity of the ftange is used in Eq. (3.34). Since thestiffeners do not contribute to the shear-carrying capacity, the ftange thicknesst* is used in Eq. (3.35).

The numerical calculations are seen to be very simple. However, for addedconvenience, a program in BASIC for a personal computer is presented in Table

103

Shear Lag in Wide F/anges

3.1 together with its imput data requirements. The program given is intendedfor the analysis of a simply supported girder under a loading distributed overa prescribed length, thus covering a range of practical cases from theuniformly distributed to the concentrated loading case. Figure 3.5 illustrateshow different loading and support conditions can be taken into considera­tion.

The accuracy of the normal force per unit width obtained from Eq. (3.16)

a) 09 09/2~ ~ = lS.'lllllllllllllrlllllll~

~ -

09 9C) gmn mmn S

~

09/2-- glllllllllllllllllllll~

9/2- g"lllllllllllllllllllll ~

d)H -nLJ : 'L :

09

cjg d'"II1III1I1II1I1II1II1~

e) -~<~. R-2glf)

~ 11 11 <~ ~= ~ I fl/JI c?1/JIc?q'J.1R-909l/J ~

999fi-il1l 11 11 < E'" "'" ~g) = _ WS~~~~~~~, ~

ljfO

Fig. 3.5. Applieation of harmonie analysis to a variety of loading andsupport eonditions.

104

Methods oj Ana/ysis

depends upon the number of series terms taken. The rate of convergence of thesolution depends on three factors: .

(a) The type oj loading

The convergence is very much more rapid for a case of distributed loadingthan for a concentrated load. This is ilIustrated in Fig. 3.6, where the values ofthe longitudinal stresses at the edge of the flange, calculated using differentnumbers of series terms, are expressed as the ratio of the calculated value to thecorrect one and plotted for three loading conditions. The convergence is rapid,both for the case when the load is distributed over the full span length and whenthe load is distributed over one-quarter of the span. However, the rate isconsiderably slower when the load is distributed over only one-tenth ofthe span.Jt should be noted that, in each case, the proposed solution converges to thevalue given by Moffat and Dowling [3.2].

90

80

70

If-o.zl

c.!!Eom -.•.....-'-'.•....•...,.

........••\/' ../ "\..

.~.L1­ó-Li'o

,-

60

Jf50

5 7 9 ff ~ ~ nApplied number 01 ferms o( rOllrier series

Fig. 3.6. Convergence of harmonie solution for different loading cases.

(b) The widthlspan ratio (biL)

The argument of the cotanh term in Eq. (3.27) is a function of biL. Since thevalue of cotanh 'Jb12 converges rapidly to I, the values of Nj(bI2) for the highervaIuesof the width/span ratio cease to be dependent on the number of terms O)

105

fOO

Shear Lag in Wide Flanges

and are governed only by the coefficient Qej. In Fig. 3.7, the inftuence of thewidthjspan ratio upon the convergence of the solution is shown for the uniform­ly distributed loading case. For all values of widthjspan ratio ranging from 0.2to 0.8, convergence is seen to be rapid.

Unlformly distributed load 1 b-J658mmI

. t:;f'S."mm;DIC % l b/L -0.8 12.7 mjmct ./". ~IfO yb/L -0.# ~~ bjL= 0.3 25.1,mm

biL - 0.2

90

80

70

60

50f J 5

Applied numóer of termsof rour/er series

Fig. 3.7. Convergence of harmoniesolution for different width/span

ratios.

(c) The cross-sectional area oj the stiffeners (A )•This factor inftuences the ratio t/t* in Eq. (3.1) and thus the coefficient Ci in

Bq. (3.27); high values of Ci are obtained when the cross-sectional area of thestiffeners is large. Since the coefficient Ci appears in the argument of the cotanhterm in Eq. (3.27), the effect of a large stiffener area upon the convergence issimilar to that of a high widthjspan ratio, as discussed in point (b).

It is not possible to establish specific rules to define the rate of convergencefor allloading and support conditions, but the three factors discussed above havethe greatest inftuence. In any analysis, additional terms of the series should betaken until a satisfactory degree of convergence has been achieved.

Obviously, the number of terms used in any calculation will depend upon therequired degree of accuracy. Since the method is intended for preliminary designcalculations, only a few terms of the series will normally be required. Fordistributed loading, only the first and third terms will usually be needed toachieve an accuracy of around 3 %.

106

Methods oj Ana/ysis

One source of approximation in the method presented has already been notedand discussed; this related to the effect of the additional flexibility due to shearlag upon the overall bending moment and shear force diagrams for staticallyindeterminate structures, as in Fig. 3.2.

There is an additional approximation arising from the fact that the methodemploys an expression obtained from simple beam theory, Eq. (3.22), to deter­mine the shear flow qe(x) transmitted from the web to the flange. This simpleformula assumes a uniform distribution of longitudinal normal stress across theflange width, Le. that there is no shear lag. As a consequence, the calculatedvalues of edge shear flow are slightly overestimated and the longitudinal edgestresses resulting from shear lag are also overestimated.

This approximation could be removed by using an iterative approach. In thisapproach, the results of the first shear lag analysis by this method could be usedto provide a more accurate picture of the edge shear flows and a second shearlag analysis could then be carried out; the process could be repeated untilsatisfactory convergence was obtained. However, this added complexity is notconsidered to be justified because the method is intended for use as a design tool.

3.1.5 Harmonie Analysis of the Shear Lag Effeetin Stiffened Flanges

The previous method (Section 3.1.4) was developed primarily for unstiffenedflanges. However, it may also be applied to flanges with closely-spaced stiffenersas in Fig. 3.3b, where the properties of the stiffeners can be assumed to bedistributed continuously, or "smeared", over the flange width. Such an ap­proximation is not justified for the more typical, practical case of a flange platestiffened by a few, widely-spaced, large stiffeners, as in Fig. 3.8.

b

r -I

I · · · 1Fig. 3.8. Flange with widely-spaced stiffeners.

ln such a case, the markedly non-homogeneous character of the system ofplate panels and stiffeners has a significant influence upon the shear lag effect.ft is evident that different stiffener arrangements, such as those shown in Fig. 3.9,will cause different shear lag characteristics within the flange itself even thoughthey may provide the same total cross-sectional stiffener area and the samesecond moment of area of the complete box cross-section. These local shear lag

107

Shear Lag in Wide F/anges

effeets, which are dependent upon the aetual stiffener arrangement, eannot betaken into aeeount by any approaeh that idealizes the flange as a homogeneousplate.

The method presented in this Seetion has been developed to deal with the easeof a flange plate stiffened by a few, widely-spaeed, large stiffeners [3.7]. Themethod has the great advantage that, by employing harmonie analysis, it enablesthe shear lag effeets to be predieted direetly from hand ea1culations. It isassumed that the axialload-earrying eapacity of the stiffened flange is eon­centrated at a number of longitudinal bar elements, situated at the stiffenerpositions as shown in Fig. 3.10, and the eentroid of eaeh bar is assumed to beloeated at the mid-thiekness ofthe flange plate. The flange sheet itselfis assumedto be eapable of earrying shear stresses only. The shear flow within eaehindividual segment of the sheet, between longitudinal bars, must then be eon­stant and the variation of the shear flow aeross the flange width may be repre­sented by a step diagram, as in Fig. 3.10e.

Ó)

a.......nH7f®

OJÍ

... n"H'~®

C)

d)

108

a

M a"1 a 'I r1.1.,.2.········~®i (j) 0 ® ~ II

Fig. 3.9. Idealizations for com­mon stiffener arrangements.

Methods oj Ana/ysis

The equivalent area A of eaeh bar is taken as the eross-seetional area A. ofthe stiffener, together with the area of the adjaeent flange sheet. Thus, as shownin Fig. 3.10,

A = A. + ta, (3.36)

(3.37)

where t is the thiekness of the sheet and a is the stiffener spacing.Since the longitudinal stress at the edge of the flange must normally be

determined, the last bar (Le. the r-th) is situated at the edge of the flange, asshown in Figs. 3.9 and 3.10. Figure 3.9 also illustrates the idealizations fordifferent stiffener arrangements. Eaeh segment of the sheet, in between the bars,is in a state of plane stress. Thus, the shear strain y may be expressed in termsofthe displaeements u, v in the longitudinal (x) and transverse (y) directions as

au avy = - +-.

ay ax

Harmonie analysis ean be applied directly to the ease of simply supportedbeams. Also, as diseussed in Section 3.1.4, eantilevers and eontinuous beams

.may be analysed by fi.rstestablishing a system of substitute beams.ln harmonie analysis, all desired funetions may be expressed in the form of

Fourier series with unknown amplitudes. Since the applied extemalloads canalso be expressed by Fourier series, the entire analysis ean be eondueted for eaeh

(a) I~b

I" a I a"1 'I1

· ~ · I,1As

(h)1_~ [A,

j, -As+fa

(C)

Fig. 3.10. Idealization of a typical flange: (a)actual flange, (b) idealized flange, (c) shear

flow diagram.

109

Shear Lag in Wide F/anges

term of the series independently and the results simply added together. In thisway,it becomes possible to operate only with the amplitudes of the series termsinstead of having to deal with the fulI funetions.

Using the equations derived in detail in Seetion 3.1.4, the longitudinal straindeveloped at any position x, y in any portion of the tlange sheet may beexpressed in the following form for eaeh harmonie:

vj1tto ['

. J1t

L ~sInh _ ~ ] . j1tXv~+v LV~TVY SIn-oL

[j1t ~] j1tXtx = to eosh L v 2 + v y sin L;wherej is the number of a Fourier series term, andL is the length of simply supported span.The eorresponding transverse strain is

[j1t ~] j1tXty = - Vtx = - vto eosh L v 2 + v y sin Land, sinee ty = ovjoy, the transverse displaeement may be written as

v = _ VtoL ['

. J1t. ~~ .}1tft+; L ft+; Y] sin J:X

so that

o~

From Bq. (3.38)

Otx j1t ~ [j1t;;:;-;-:] j1tX- = - to V 2 + v sinh - v 2 + 1t Y sin-oy L L L

and by eomparing Eqs. (3.39) and (3.40)

o2V v otx

OX2 2 + v oy

Equation (3.37) ean be differentiated to give

oy o (Ou) o2v otx o2vOX = oy OX + ox2 = oy + OX2

110

(3.38)

(3.39)

(3.40)

(3.41)

Methods o! Ana/ysis

and, by substituting from Eq. (3.41),

a')' 2(1 + v) aex------ax 2 + v ay

By rearranging this equation, the rate of change of longitudinal strain ex in thesheet with respect to the transverse coordinate direction y may be expressed interms of the rate of change of the shear strain ')'with respect to the longitudinalaXlSx as

aex 2 + v a')'-= -ay 2(1 + v) ax

Thus, the total change in longitudinal strain Aex across any width Ay ofthe sheetmay by expressed as

aex 2 + v a')'Aex = -Ay = ----Ay.

ay 2(1 + v) ax(3.42)

The normal elastic stress-strain relationships enable the shear strain ')' in thesheet to be expressed in terms of the shear flow q as

_ q _ q 2(1 + v)')'-------tG t E

Equation (3.42) then gives

2 + vaqAex = ---Ay.

tE ax(3.43)

If a typicallongitudinal bar i has a cross-sectional area Aj (see Eq. 3.36) andcarries an axial10ad Fj, then the direct axial strain in the bar may be written as

(3.44)

where uj is the longitudinal displacement of the bar. Displacements of typicalelements of sheets and bars are shown in Fig. 3.11. For compatibility of lon­gitudinal displacements, the difference in the longitudinal strains in two adjacent

111

Shear Log in Wide F/anges

(3.45)

bars, as obtained from Eq. (3.44), must be equal to the total change in longitudi­nal strain in the area of sheet connecting the two bars, as given in Eq. (3.43).Thus, for bars i and (i + 1), separated by a sheet of width aj,

1 (Fj+l Fj) 2 + v oqex,i+l - ex,j = E Ai+l - Aj = ill ox aj'

It is not necessary for the spacing aj between the bars to be uniformo

v:+!ft. dx dv,.•., d, dxy;•.,+ dX x

(b)(a) n Il

-JdUj•.,

f d/515+-' dxui•.,'" -;jXdX

, dx

II{; dx i/ II!li+,dx IliidX Idx......••.

~.. I I~

~Uj,,'

I IJL.

J-LIF,.v:

Vi•.,

I',a·,II

Fig. 3.11. Typical sheet and bar elements: (a) typical sheet element, showing dispacements,(b) typical bar element, showing forces.

(3.46)

By considering the equilibrium of an elementallength dx of a typical har (i),as in Fig. 3.11h, another expression relating the axial force in the bar to the shearftows in the adjacent sheet segments can be obtained

dF._I + qj _ qj-l = O.dx

For simply supported end conditions, the axial forces in the bars and the shearftows in the sheet segments may be written in the form of the Fourier series:

bar force,

OCJ

F. = "S . j1CXI L. j.sm-. J 'J=l L

sheet shear ftow,

OCJ

q. = " Q j1Cx1 .L. jj cos - .

J=l L

112

Methods 01 Ana/ysis

Equation (3.45) may then by written for any term of the series as

(Si+lj Sij), l1tx 2 + v j1t . j1tX-- - - sm- = - --a.-Q ..sm-- - . 1 IJAj+l Aj L t L L

and, similarly, Eq. (3.46) may be written as

j1t j1tX j1tXS .. - cos - + (Q .. - Q. 1') cos - = O.

IJ L L IJ 1- J L

These two equations can be simplified and rearranged to give

_ (Sjj 2 + v )S'+I' = A'+1 - - aa· -- Q..

I J 1 A. 1 J t IJ1

and

(3.45a)

(3.46a)

(3.45b)

(3.46b)

where (X,j = j1t/ L.The solution of these equations depends partly upon the arrangement of thestiffeners. The most common stiffener arrangements are shown in Fig. 3.9.

ln the two cases illustrated in Figs. 3.9a and b, where a bar is situated at themid-width position of the flange, the shear flow in the sheet segments on either

side of the middle stiffener (Le. sheets 1 and 1') will be equal and oppositebecause of transverse symmetry. Thus

so that, from Eq. (3.46b),

(3.47)

Thus the unknown amplitude ofthe shear flow in the first sheet segment Qlj hasbeen expressed in terms of the unknown amplitude of the axial force in the

central bar Slj" .By substituting from Eq. (3.47) into Eq. (3.45b), the unknown amplitude of

the axial force in the second bar may be expressed as

Thus, the force in the second bar has again been expressed in terms of theunknown force in the central bar.

113

Shear Log in Wide F/anges

This procedure may be continued and the general equations (3.46b) and(3.45b) utilized to relate the shear ftows in all the sheet segments and the axialforces in all the bars to the first unknown bar force Slj'

The value of Slj is determined from the last equation established, Le.when theedge of the ftange is reached. At the edge, the shear ftow Qrj on the outside ofthe last (Le.the r-th) bar may be equated to the known amplitude of shear ftowQej transmitted from the web to the edge of the ftange, i.e. flOm Eq. (3.46b)

(3.49)

The calculation of the edge shear ftow Qej was described fully in Section 3.1.4and the values for certain loading cases that frequently occur in practice arepresented in the second column of Table 3.2.

Once the values of the forces in each of the bars are known, the axial stressesin the bars are simply ca1culated, e.g. for the edge bar

(3.50)

The shear ftow in each individual sheet segment may also be calculated ifrequired.

The two bar arrangements shown in Figs. 3.9c and 3.9d differ from those ofFigs. 3.9a and 3.9b in that there is no stiffener positioned at the mid-width ofthe ftange plate. In such a case, the shear stresses developed in the central sheetsegment are zelObecause of symmetry.Therefore QOj = O, and from Eq. (3.46b)

(3.47a)

This equation replaces Eq. (3.47)for a ftange that does not have a stiffener at themid-width position.

By substituting from Eq. (3.47a) into Eq. (3.45b), the unknown amplitude ofthe axial force in the second bar S2j is obtained as

The remainder of the ca1culation plOcedure is then as described earlier for theftange with a central bar. The required ca1culations can be carried out directlyon any pocket ca1culator without the need to solve any large systems of equa­tions.

The most-convenient procedure to adopt is to assume, in the first instance,that the amplitude of the axial force in the first bar is unity, Le. Síj = 1. Thenusing Eqs. (3.47), (3.48) (or (3.47a) and (3.48a) for the alternative bar arrange-

114

The Main Features oj Shear Lag and the lnfluence oj Various Parameters

ment), (3.46b) and (3.45b), the amplitudes ofsheet shear tlows Qíj' Q2j' ... , andbar forces S2j' S3j' ... , corresponding to the assumed unit value of Síj, can bedetermined.

Working systematically through the successive bars and sheets, the amplitude

ofthe last shear tlow Q'rj is eventually obtained, corresponding to the assumedunit value of Síj. Since the actual amplitude of the edge shear tlow Qej is known(see Table 3.2), the following equation can be written

Q;jSlj = Qej'

so that the actual amplitude of the force in the fi.rst bar is obtained as

(3.51 )

Having obtained Slj' the corresponding trne values of all the other bar and sheetforces can be calculated.

These numerical calculations must be carried out for each term of the Fourier

series taken and the accuracy of the solution depends upon the number of termsof the series considered. The rate of convergence of the solution depends uponthree factors, viz.

(a) the type of loading - the convergence is more rapid for a distributedloading case (where the fi.rst and third terms ofthe series are normally sufficient)than for a concentrated load,

(b) the tlange widthjspan ratio of the girder - an increase in this ratio leadsto a slower rate of convergence,

(c) the cross-sectional area of the stiffeners - an increase in stiffener areaagain reduces the convergence rate.

The convergence characteristics are very similar to those for the methodpresented in Section 3.1.4 (Figs. 3.6 and 3.7), where the results of parametricstudies of convergence rate were discussed. It should be appreciated that, evenif several terms of the series have to be considered, the calculations involved foreach term are very simple, so the complete solution can still be obtained veryconveniently.

3.2 The Main Features of Shear Lag and theInfluence of Various Parameters

3.2.1 Variation or WidthjSpan Ratio

It is well known that as the tlange width increases in relation to the span, theshear lag effect becomes more pronounced. This finding is of general validityand can be clearly illustrated, for example, by using the simple analytical method

115

The Mam Features oj Shear Lag anáihe lrifluence oj Various Parameters

presented in Section 3.1.4. The results of such a parametric study are shown inFig. 3.12, where the edge stress (Je for a box girder under distributed loading isplotted against the girder span. The calculated stress is compared to the stress(Jo predicted by simple beam theory.

%1.75

1.5

1.25

1.0

f5mfOm5mS 20m'Pan

Fig. 3.12. Influence of span length upon the shear lag effect.

3.2.2 The Type and Position of Loading

The non-uniformity of distribution of longitudinal stresses increases rapidlyin the region of a point load or a support (see also Section 3.2.4). Moving theload system away from the mid-span results in a reduction of the effectivebreadth ratios (Moffat and Dowling [3.2]). The effectivebreadth ratios are onlysensitive to the loaded length if this length is less than half of the span.

3.2.3 Effect of Stiffeners

Because stiffeners contribute to the axialload-carrying capacity of a flangewithout increasing its shear capacity, shear lag is more pronounced in a stiffenedflange than in a flange without stiffening. As an example, Fig. 3.13 shows theresults of a parametric study investigating the influence of the variations of theratio tJt in a steel girder (without any concrete layer) where, as in Eq. (3.1),t = t + AJa. The unstiffened steel girder is represented by the case whent = t and the ratio is then increased to a value of 2, representing a heavily

116

Shear Lag in Wide F/anges

stiffened girder. Within this range, the stress at the edge of the flange is seen toincrease almost linearly.

For practical purposes, it is desirable to simplify the numerical analysis bysmearing the stiffener properties over the flange width. This approach must beverified from the point of view of:

(i) the number of longitudinal ribs;(ii) the effect of th~ir own flexural rigidities and the eccentricity of their

connections to the flange sheet;(iii) the regularity of stiffener spacing;(iv) the shape of the cross-section of longitudinal ribs.

~8

0.26

0.25

0.24

I ([."'1- (."';t"t12.7 12.7]l~[~mm~. ~f

~f a .lIeaYl1y stlffeneáI \ glrder

on UfJs!tffened . \

girder ~ \~~'"'' ':

.......~.MOff~t I. lJow/i/}.!l[.ul

0.21

2 tjtFig. 3.13. Influence of flange stiffening upon the shear lag effect.

It is obvious that with an increase in the number of stiffeners, the moment ofinertia of the whole cross-section also increases and, consequently, the mag­nitude of the stress drops. It has been found, however that, in spite of theinfluence of the rate of flange stiffening discussed above, the general characterof the stress distribution remains similar. This is c1early seen from Fig. 3.14,which, as an example, gives the distribution oflongitudinal normal stresses overthe flange breadth for various numbers of longitudinal flat ribs. Hence, it seemsthat the concept of smearing the stiffener properties is acceptable not only forclosely-spaced stiffeners but also, with little loss of accuracy, for large, ratherwidely-spaced stiffeners. The conditions for the acceptability of such an ap­proach are the regularity ofthe stiffener arrangement (see Section 3.1.5) and theassumption that the stiffeners are concentrated in the flange plane (Fig. 3.l5a).

Longitudinal stiffeners are generalIy welded to the inner side of the flangesheet (Fig. 3.15b). Due to the eccentricity ofthe stiffener connection, individualportions of the flange with stiffeners, which are eccentricalIy affected by shearflowsacting in the plane of the flange sheet (Fig. 3.15c),tend to exhibit addition­al flexure, as shown in Fig. 3.16d.

117

The Main Features oj Shear Lag and the /njluence oj Various Parameters

p::1.

L

lY (HPa)0.04

0.06

o.ot,

0.02

Fig. 3.14. Distribution oflongitudina! normal stresses for various num­bers of longitudina1 fiat ribs.

Q) Ó)

11 · · ·

I•

It J'IIIII

'J,a. D. a Q 1.1111 I'~ .1

Fig. 3.15. Stiffeners concentrated at the Hangc sheet pla:neand eccentrica1ly-connected stiffeners.

118

Shear Lag in Wide Flages

b-Jó58mm

SWfening fOctor(Maffa! &. Dowling !;r'lf) (C -f.o

a)t-

IW/2

~r~

I f2.7, a ~

IJ)

tfOJ

C)

d)

l'flange dellection

Fig. 3.16. Influence of eccentrically-connected stiffeners uponthe stress distribution and deflections.

The influence of eccentricity of the stiffener connection is illustrated in Fig.3.16.A steel box girder without intermediate diaphragms, under uniform load­ing (w = 1N/mm), with span L = 9144 mm and the cross-section shown in Fig.3.16a is studied. The distribution oflongitudinal stresses across the flange widthis shown by the solid line in Fig. 3.16b; the dashed line corresponds to that ofthe stiffeners concentrated in the flange sheet. Different flexural actions ofindividual stiffeners with adjacent flange portions are shown in Fig. 3.16c,whichdepicts distributions of the longitudinal stresses along the stiffener depths.

It can be seen from Fig. 3.16c that the eccentrical1y connected stiffeners,particularly those near the mid-point of the flange, exhibit stress distributiontending to that of a beam stressed by bending. The eccentricity of the stiffener

119

The Main Features oj Shear Lag and the lnjluence oj Various Parameters

connections thus results in a loss of efficiency of the total stiffened flange. Thisis also the reason why the solid line in Fig. 3.16b, indicating the stress distribu­tion for the eccentrically connected stiffeners, falls completely (Le. along the totalwidth of the flange) above the dashed curve, which corresponds to a fully actingflange with stiffeners concentrated at the flange sheet.

The transverse flexure of the stiffened flange due to stiffener eccentricity isshown in Fig. 3.16d. This kind of additional deformation is only partiallyrestrained by a rather flexible flange.

It is seen that the stiffener eccentricity influences the distribution of thelongitudinal stresses adversely, unless closely-spaced sufficiently rigid transversediaphragms are used to ensure equal deflection of all stiffeners. Thus thediaphragms indirectly influence the shear lag effects. Their presence is essentialto allow use of the methods that do not regard stiffener eccentricity.

It has been found that the regularity of stiffener spacing (even in cases of thesame total cross-sectional area of stiffeners and thus the same total second

moment of area of the whole cross-section) influence the shear lag behaviour ofthe stiffened plate.

~e= -a 088 N/mm2

Fig. 3.17. Inftuence of stiffener positions.

Figure 3.17 shows the cross-section of two steel box girders having a· spanL = 9144 mm, with stiffeners atdifferent postitions, loaded by uniformly distri­buted loading of an intensity w = 1 N/mm. Although the longitudinal stress onthe edge of the flange at mid-span for the case shown in Fig. 3.17a is - 0.087 55N/mm2, the stress for the stiffener arrangement shown in Fig. 3.17b reaches a

magnitude of - 0.081 8 N/mm2 only. According to Moffat and Dowling [3.2],

where no distinction is made between the stiffener arrangements, for the !stress)effective breadth ratio 0.67 the corresponding stress is - 0.082 N/mm . Thisrepresents an excellent agreement with the results obtained for the case shown inFig. 3.17b. Here, the stiffeners are situated at mid-points of adjacent flangeportions in accordance with a regular stiffener system asassumed in [3.2].

120

t

\~

Shear Log in Wide Ffanges

However, the stiffener arrangement shown in Fig. 3.17a (with the samedistances between all stiffeners and between the first stiffener and the web),where the stiffeners are situated more at the middle region of the ftange, resultsin a 7 % increase in the values of the longitudinal stresses compared with thecase shown in Fig. 3.17b. The reason for this is that the shear lag effect dependson shear deformability of those ftange segments where the shear stress is ofhighest intensity, Le. in the regions close to the webs. The width of the ftangesegment between the web and the first stiffener (and thus its shear deformability)is considerably lower in the case shown in Fig. 3.17b than in that shown in Fig.3.17a.

The results clearly confirm the necessity of accounting for the actual stiffenerpositions for ftanges with large, widely spaced stiffeners, e.g. by using the methodpresented in Section 3.1.5. This fact cannot be accounted for by any method ofanalysis which assumes a regularly arranged structure, even if the finite elementmethod is used.

Most studies of the impact of the shear lag phenomenon characterize lon­gitudinal ribs solely by their area, and the effect of stiffener configuration is nottaken into account.

Investigations dealing with the stability problem of longitudinally stiffenedftanges, in Chapter 5, proved the great inftuence of stiffener cross-sectional shapeupon the buckling of ftange plates. Thus it is of interest to find out whether thestiffener cross-section configuration also shows a significant effect on the shearlag phenomenE>n.

For this reason, the effect of various stiffener shapes was investigated, whileother stiffener parameters (the number and location of stiffeners, and - at leastapproximately - the moment of inertia) were kept constant. Four stiffenerconfigurations were studied: (i) a ftat stiffener (Fig. 3.18a), (ii) an angle stiffener(Fig. 3.18b), (iii) a T-section stiffener (Fig. 3.18c) and (iv) a trapezoidal closed­section stiffener (as shown in Fig. 3.18d). The overall dimensions of the girdersanalysed are shown in Fig. 3.14.

Fig. 3.18. The stiffener configurations consideredin the study.

a)

STI"-II-f2.7mm

lO C)

~;FJ!mmltci

Ó)

12Hm!>==::;==~~.\)(.,J'--176 mm

7~mm

121

The Mam Features oj Shear Log and the lnftuence oj Various Parameters

The results show that, unlike the case of flange buckling, the effect of stiffenerconfiguration (when the stiffener area is kept constant and its moment of inertiadoes not vary much either) is not significant. This means that the torsionalrigidity of the stiffeners does not play a substantial role in the phenomenon ofshear lag.

3.2.4 Shear Lag in Support Regions of Continuous Beams

When analysing the shear lag in box girders, the differences in performanceof the top and bottom flanges are usually tacitly ignored in practice. It hasbeen found, however, that this concept gives satisfactory results only for boxgirders of ordinary dimensions (i.e. for girders which are not too short) wmchare subjected to uniformly distributed and similar loadings. In reality, theextemalloading is usually first transmitted by a system of transverse ribs tothe upper edges of webs, while on the other hand the lower edges of the websare subjected to support reactions representing very large point loads (Fig.3.19a,b)

The webs of box girders are usually reinforced by vertical stiffeners located atthe cross-section of application of the reaction, Fig. 3.19a,b. Substantial por­tions of the support reactions are transmitted (as shear flows) by these verticalstiffeners into the webs (Fig. 3.19c). In order to evaluate the influence ofvariousdistributions of these shear flows upon the shear lag effects, the influence linesof the peak values of the longitudinal normal stresses at the edges of the bottomand top flanges of the cross-section above the support can be constructed. Suchan influence line thus indicates by its ordinates the value of the peak stress dueto a unit vertical force acting at the support cross-section at the vertical positiontJ (Fig. 3.20a).

As an example, the influence lines of the peak stresses in the flanges of a girderwith cross-section having dimensions as indicated in Fig. 3.12 are shown in Figs.3.20b,c. The full curves indicate the stresses occurring directly at the supportcross-section. It is seen there that the application of a load very close to theflange investigated results in a considerable increase in the peak stresses. Thisfact is also documented in Fig. 3.20d, where the distribution of longitudinalnormal stresses across the flange breadth is plotted for several position tJ of theforce acting. On the other hand, it has been found that this stress increase hasonly a local character, disappearing very fast as one proceeds further away fromthe support cross-section. For instance, the influence lines for the peak stress atcross-sections close to the support are shown in Fig. 3.20b,c by dashed lines.

ln practice, cases are also encountered where the girder is supported under

122

Shear Lag in Wide Flanges

(a)[xtema/ /oad

(b)

(C)

~

Fig. 3.19. (a) Action of extemalload on box girder bridges, (b)action or reactioDSin tbe support cross-section witb verticalstiffeners, (c) shear tlow transmitted from the vertical stiffener

into tbe web.

123

The Mam Features oj Shear Lag and the Injluence oj Various Parameters

III,.L.­- f.006.,(O·4

(b) (C)

124

Fig. 3.20 a, h, c

Shear Lag in Wide Flanges

-f.006K(0-~

[N/mm~

"I-o

-0.297.fO -#

;1_0.2J4.l0-+tJ-o.2MO-+~:TI

yFig. 3.20. Influence lines of longitudinal normal stress values at

the flange edges: (a) construction of the influence lines - posi­tions of a unit loading force, (b) influence line of the value oflongitudina! norma! stress at the edge of the botton flange, (c)influence line of the va!ue of longitudina! normal stress at the

edge of the top flange, (d) distributions of longitudinal norma!stresses for various positions of the unit loading force.

diaphragms, which then transfer the reactions into webs (Fig. 3.21). The distri­bution of the shear flow acting between the diaphragm and the web depends onthe positions of the supports. When the support is situated just under the web(Fig. 3.22a), the shear flow is distributed very non-uniformly, and the distribu­tion of longitudinal stresses across the bottom flange is quite different

\

125

The Main Features oj Shear Lag and the lnjluence oj Various Parameters

(a)p

II' '" "'"'''''r''' "uu, '" "" "" i~~:"'""''''í"'" 111111

.1 LIL

(D)

(t)

Supportdiaphragm

(d)

Fig. 3.21. (a) A continuous beam with many spans, (b) action of asupport diaphragm, (c)a support diaphragm in a single-cellbox girder,

(d) a diaphragm supporting the central girder.

126

Shear Lag in Wide Flange.l

•l.(a)IIIIII.

I:I

tI I'92.6 t

0.5

O

-lli,

(Ó) x-ZffJZ.5mm I j6:+.91

+15.26~. 0.86

I

ITwmm~

(C) /IlIIII11II,.99.6

IIz !li

t" [Nim] ,

0.5O

I 355 ~Fig. 3.22 a, b, c 127

2 1.5

t"[N/mJ

The Main Features oj Shear Log and the Injluence oj Various ParametersI'

(d) x-2ffJ2.5mm

j5.25+

!5.2.3I!(e)

/1r

I'

~I

~II82.J

III

I,I'(II

z IJ~ I

)

II LfJ

f a5o

I.t:[N/mJ '829~I(f)

X -2f fJ2;5mm

d+

i5.22

Fig.3.22.Effect of a diaphragm supported at various positions(no connection to tlanges): (a), (c),(e) shears loading the web, (b), (d), (f) corresponding longitudinal normal stress distributions.

128

Shear Lag in Wide Flages

(Fig. 3.22b) from that in the top flange. An improvement is achieved when thediaphragm is supported at some distance from the web (Fig. 3.22c,d), and thebest state occurs with the support located in the middle of the diaphragm (Fig.3.22e,f).

These results (Fig. 3.22) are based on a study carried out for a continuousbeam with rather long spans. The effects discussed would be much more pro­nounced with girders having higher width-to-span ratios and/or flanges withlongitudinal stiffeners.

3.2.5 Influence of Loads Acting above Longitudinal Stiffeners

When calculating shear lag effects, it is usually assumed that the load acts directlyabove the webs. This requires a perfect transfer of allloads in the transverse direction

(usually through a system of transverse diaphragms). However, cases may be en­countered in design practice where short-span bridges provided with rather stifflongitudinal ribs (e.g. railway bridges) are loaded directly above these stiffeners.

/II =fN/mm

g'" ,/. ~ "A

L dl'ph"',,,, I

L -11## mm •

x

e

C) tF [N/mm~

0.061 a)0.0t,0.02 .

e

I

I

.-'-.-. -.-. -+-.-.-. -. -.-.I

Fig. 3.23. Effect of direct loading of longitudinal stiffeners.

129

The Main Features oj Shear Log and the lnfluence oj Various Parameters

An example of such an arrangement is shown in Fig. 3.23b. Here, a uniformlyloaded girder with central transverse diaphragm is considered. The distributionof longitudinal stresses at the quarter-span is compared in Fig. 3.23c for a loadacting above the stiffeners (Fig. 3.23b) and the more common case when the loadacts above the webs (Fig. 3.23a). An entirely different character of the stress­distribution pattern is apparent. The directly loaded stiffeners behave almost asindependent girders with typical stress distribution along their depths accompa­nied by their own shear lag effects.

3.2.6 Inftuence of Overhanging Flanges

It is common practice in the design:of steel box girder bridges that the websand flanges are interconnected as shown in Fig. 3.24c. The influence of thisarrangement upon the distribution of longitudinal stresses in comparison withthe distribution corresponding to the case usually considered (Fig. 3.24b) isshown in Figs. 3.24d and e. As can be expected, the interconnection shown inFig. 3.24c exhibits a favourable influence upon the distritution of longitudinalstress (in this case, a decrease in the peak value by 9 %).

C)

~ 9fltltmm

Q) W-fN/mm( ~~.T , •............

-I

J8f f.UB -fUDmm

j;;;1~.

c::il

...,I~I

~_ 'Ha ~

Fig. 3.24. Effect of the arrangement of interconnection of flanges to webs.

130