Local stresses at the intersection of cross-beam flange with box-girder web

Thin-Walled Structures 17 (1993) 113-131

Local Stresses at the Intersection of Cross-Beam Flange with Box-Girder Web

Ichiro Okura & Yuhshi Fukumoto Department of Civil Engineering, Osaka University, 2-1 Yamadaoka, Suita, Osaka 565, Japan (Received 17 May 1992; accepted 3 November 1992)

ABSTRACT Local stresses at the intersection of a top flange of a cross beam with a boxgirder web are investigated analytically and experimentally. Finite element analyses of a T-shaped weldedjoint and a loading test of a cantilever beam of acrylic material reveal that a local stress which is different from the one caused by the stress concentration at a weld toe is developed at the intersection. Finite element analyses of an 1-section beam show that the restraint of the vertical deformation of the cross beam at the box-girder web induces the local stress. It is pointed out that the local stress must be considered in fatigue design. A method of determining the local stress is proposed.

1 INTRODUCTION To investigate the fatigue cracking at the cross-beam connections in boxgirder bridges for a straddle-type monorail, a fatigue test was carried out for a specimen shown in Fig. l(a). ] A cross beam of I-section is connected to a box girder. As shown in Fig. 1(b), the specimen was set upside-down, and the top of the box girder was fixed to the test floor. The vertical load was applied near the end of the cross beam. Contrary to the investigators' expectations, fatigue cracks were initiated very early at the intersection of the top flange of the cross beam with the box-girder web. 113 Thin-Walled Structures 0263-8231/93/$06.00 © 1993 Elsevier Publishers Ltd, England. Printed in Great Britain.

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(b) T e s t setup Fig. 1. Fatigue test of cross-beam connection with box girder.

Figure 2 shows the distribution of the strain measured on the top flange of the cross beam near the box-girder web. The distance from the weld toe is taken on the abscissa. The strain distribution estimated by the beam theory is also given in the figure. The measured strains near the weld toe are higher than those estimated by the beam theory, because of the stress concentration at the weld toe. As will be mentioned in the subsequent section, the stress concentration at the weld toe becomes almost negligible at x = 6.6 m m away from the weld toe. The magnitudes of the strains for

Stresses at the intersection of cross-beam flange with a box-girder web

115

j , B o x - g i r d e r web /Top flanse of ~ f c r o s s beam

4

r

--X

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x(m) Fig. 2. Strainsmeasured on the top flange of the cross beam near the box-girderweb.

x > 6.6mm are, however, much higher than those estimated by the beam theory. The objective of this paper is to provide the reason why such a stress is much higher than that estimated by the beam theory which is developed at the intersection of the top flange of the cross beam with the box-girder web.

2 R A N G E OF STRESS C O N C E N T R A T I O N AT W E L D TOE To examine the range of stress concentration at a weld toe, finite element analyses were carried out for a T-shaped welded joint shown in Fig. 3. The vertical and horizontal components of the joint correspond to the boxgirder web and the top flange of the cross beam of the fatigue test specimen shown in Fig. l(a), respectively. Two angles of 45 ° and 90 ° were taken into account for the weld toe of the joint. The displacements on the left side of

116

lchiro Okura, Yuhshi Fukumoto

% .m---

45*

S2

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Cross-beam flange

l

-(

110

Unit(!.) Box-girder web

I'

Fig. 3. T-shaped welded joint.

the vertical component were all fixed. Axial stress or bending stress was applied to the right-hand end of the horizontal component. As shown in Table 1, finite element analyses were done for 24 cases. Case 2 is under plane stress condition. The others are all under plane strain condition. Cases 1-17, 23 and 24 are for butt welds, and Cases 18-22 are for fillet welds. For the fillet welds, the size was specified by the following equation 2 tl > S _> X/~2

(1)

where S is the size of a fillet weld, tl is the thickness of a thinner plate, and t2 is the thickness of a thicker plate. In the finite element analyses for the fillet welds, double nodes were given on the thick short line, as shown in Fig. 4(a). In Cases 5 and 17, there is no welding reinforcement on the top and bottom surfaces of the horizontal component, as shown in Fig. 4(b). In Cases 23 and 24, an unsymmetrical welding shape was dealt with. In these cases there is no welding reinforcement on the bottom surface of the horizontal component, as shown in Fig. 4(c). In Case 12, the vertical component does not exist. In this case the displacements on the left side of the horizontal component were all fixed, as shown in Fig. 4(d). The elements which were used in the finite element analyses are isoparametric plane elements with 8 nodes. 3 Figure 5 shows the mesh division for Case 1. The size of the finest elements near the weld toe is 0.5mm x 0.5mm, which was determined from the relation between the element size and the accuracy of numerical results. Figure 6 shows the computed results for Case 1. Here tr is the surface

Stresses at the intersection of cross-beam flange with a box-girder web

117

TABLE 1 T-Shaped Welded Joints to Examine the Stress Concentration at a Weld Toe

e/eo Case Type of Weld-toe Si x $2 tw tf Type of x/tf = welding angle(°) (mm) (ram) (ram) loading 0.25 (1) (2) (3) (4) (5) (6) (7) (8) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23T~ Bb 24T B

Butt Butt Butt Butt Butt Butt Butt Butt Butt Butt Butt Butt Butt Butt Butt Butt Butt Fillet Fillet Fillet Fillet Fillet Butt

45 45 45 90

45 90 90 90 90 45

Butt

45

90 90 90 90 90 90 90 90 90 90 90

6x 6x 6x 6x 0x 3x 9x 3x 6x 9× 8x 6x 6x 6x 6x 6x 0x 6x 6x 7x 8x 9x 6x 0x 6x 0x

6 6 6 6 0 3 9 6 3 3 2 6 6 6 6 6 0 6 6 7 8 9 6 0 6 0

10 10 10 l0 I0 10 10 10 10 10 10 0 20 20 20 20 38 10 10 10 10 10 10

10 10 10 10 10 10 10 10 10 10 10 10 10 15 20 25 38 10 10 10 10 10 10

Axial Axial Bending Axial Axial Axial Axial Axial Axial Axial Axial Axial Axial Axial Axial Axial Axial Axial Axial Axial Axial Axial Axial

10

l0

Bending

1.033 1.035 1.007 1-031 1-026 1.032 1.028 1.025 1.044 1.050 1,050 1.015 1.026 1-030 1.030 1.028 1,024 1.098 1-097 1.059 1.036 1.023 1.011 1.003 1-024 0.982

°On the top surface. ~On the bottom surface. %

110 Uni t (,,-)

r

t

x/tf = 0.30 (9) 1-009 1.011 1.001 1-006 1.003 1.008 1.003 1.002 1.020 1.027

1.029 0.998 1.001 1.005 1.006 1-006 1.001 1-040 1.038 1.011 0.996 0-988 1.001 0-999 1.011 0.989

x/tf = 0.35 (10) 0.994 0.995 0.999 0.990 0.989 0.992 0.987 0.987 1-004 1.010 1.014 0.988 0.985 0.990 0.992 0.992 0.988 1.001 0.999 0-980 0.971 0.966 0.994 0.996 1.002 0.996

118

Ichiro Okura, Yuhshi Fukumoto

//Double

nodes

I

f

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(a) Cases

l

(b) Cases 5 and 17

18 to 22

(c) Cases 23 and 24

(d) Case 12

Fig. 4. Sketches of T-shaped welded joints.

stress of the horizontal component, ao is the axial stress applied to the right-hand end of the horizontal component, x is the distance from the weld toe, and tf is the thickness of the horizontal component. The ratio a/ao takes the largest value a t x/lf = 0, then rapidly decreases, and becomes

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Mesh

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(Case 1).

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Stresses at the intersection of cross-beam flange with a box-girder web

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Fig. 6. Distribution of a/ao (Case 1).

equal to 1.0 a t x/tf = 0.34. For 0.34 < x/tf < 1.56, altro is below 1.0 with a minimum value of 0-98 a t x/tf - 0.53. For x/tf > 1.56, the ratio a/ao is almost 1-0. Figure 7 shows the computed results for Case 24. The bending stress applied to the right-hand end of the horizontal component is used for ao on

0

1I oo o/o o 3.0--

~ I

, iOn

On the 2. bottom surface

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the top surface

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x/tf Fig. 7. Distribution of ~r/Cro(Case 24).

Ichiro Okura, Yuhshi Fukumoto

120

the ordinate. The distance from the weld toe on the top surface of the horizontal component is used for x on the abscissa. The distribution of ~/ao on the top surface is similar to that shown in Fig. 6. The ratio a/ao on the bottom surface decreases to 0.75, and then increases gradually. For x/tf >_ 0.39, ~r/ao is nearly 1-0. The a/go values at the locations of x / t f - - 0 . 2 5 , 0.30 and 0.35 are listed in columns (8), (9) and (10) in Table 1, respectively. The values are between 0.988 and 1.040 a t x / t f = 0.3. Hence, the stress concentration at the weld toe almost disappears at the location 0.3 times the thickness of the horizontal component away from the weld toe. As shown in Fig. l(a), the top flange of the cross beam of the fatigue test specimen is 22 mm thick. In Fig. 2, thus, the stress concentration at the weld toe would almost vanish at the location x -- 6.6 m m (= 22 m m x 0.3) ahead of the weld toe. The magnitudes of the strains for x > 6.6mm, however, are still much higher than those estimated by the beam theory.

3 L O A D I N G TEST OF C A N T I L E V E R BEAM OF A C R Y L I C MATERIAL 3.1 Outline of test As can be seen in Fig. 1(a), a number of welds gather at the intersection of the top flange of the cross beam with the box-girder web. Besides, some copes for welding exist there. To confirm that a stress which is higher than that estimated by the beam theory can be produced even without such welds and copes, a loading test was carried out for a T-section cantilever beam of acrylic material shown in Fig. 8. The cantilever beam is connected to the vertical plate of 280 m m x 360 m m x 20 m m on the left side by an adhesive. Hence, neither welds nor copes exist at the connection of the cantilever beam with the vertical plate. The vertical plate was fixed to a steel column with six bolts. A loading weight was hung with a string of nylon near the right-hand end of the cantilever beam. The Young's modulus and Poisson's ratio of acrylic material are very temperature-dependent. Hence, a tension test specimen of the same material as the cantilever beam was prepared. Loading tests of the cantilever beam and the tension test specimen were carried out at the same time, and Young's modulus and Poisson's ratio were obtained from the tension test specimen. Acrylic material is deformed greatly due to creep. A strain measurement was carried out after the deformation caused by creep stopped under load, and it was again done immediately after the string of nylon to hang the

Stresses at the intersection of cross-beam flange with a box-girder web

121

4,

if=6

Ng. 8. Cantilever beam of acrylic material. weight was cut. The difference between the two strains is the elastic strain caused by the load. 3.2 Test results

Figure 9 shows the strains on the top surface of the flange just above the web. The distance from the vertical plate is taken on the abscissa. The strain distribution given by the beam theory is also drawn in the figure. The measured strains are almost identical with those estimated by the beam

c(p) 2030~

1

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0 0

P-9.81 N E=3.62x10s HPa ~-0. 343

~ 1O0

, 200 x

(,,,.,,)

300

Fig. 9. Strains on the top surface of the flange just above the web.

122

Ichiro Okura, Yuhshi Fukumoto

theory away from the vertical plate. N e a r the vertical plate, the measured strains are higher than the estimated ones. The location o f the strain gauge attached nearest the vertical plate is at x = 4 mm. As the flange is 6 m m thick, the stress concentration caused by the c o m e r formed by the flange and the vertical plate almost vanishes at x = 1.8mm (= 6 m m × 0-3). Hence, the strain gauge at x = 4 m m does not contain the stress concentration at x = 0. Figures 10 and 11 show the strains on the top surface o f the flange and on the web at x -- 100 mm, respectively. In each figure, the measured strains are very close to those estimated by the beam theory. As can be seen from Fig. 10, the phenomenon of shear lag is not observed in the flange. Thus, the increase in stress near x = 0 is not due to shear lag. Figure 12 shows the strains on the top and b o t t o m surfaces of the flange at x = 4 m m . The strains on each surface do not show a uniform distribution over the width o f the flange. The strain on the top surface around y = 0 is much higher than that estimated by the beam theory. The measured strains in Fig. 12 can be decomposed into membrane and plate-bending strains by em -

el + e2 2

(2)

eb

E1 --E2 2

(3)

-

where referring to Fig. 13, ~;mis the m e m b r a n e strain, eb is the plate-bending strain, el is the strain on the top surface o f the flange, and e2 is the strain on the b o t t o m surface of the flange. y

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Fig. 10. Strains on the top surface of the flange at x = lOOmm.

Stresses at the intersection of cross-beam flange with a box-girder web

123

z(,.=)

P=9.81 1¢ E=3.62x 103 xJ=O. 343

/

~Beam theory

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Fig. 11. Strains on the web at x = 100ram.

• 0

Hx=4 I~.

On t h e t o p s u r f a c e On t h e b o t t o u s u r f a c e = g s t l a m t e d by beam t h e o r y on.the top surface

....

E s t i m a t e d by beam t h e o r y on t h e b o t t o m s u r f a c e

II

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Fig. 12. Strains on the top and bottom surfaces of the flange at x = 4 mm.

Figure 14 shows the membrane and plate-bending strains calculated by eqns (2) and (3) for the measured strains in Fig. 12. In the figure the membrane and plate-bending strains estimated by the beam theory are also given. These strains are provided by

Ichiro Okura, Yuhshi Fukumoto

124

Plate

thickness

c2

-

~b

Fig. 13. M e m b r a n e and plate-bending strains.

•

Membrane s t r a i n

0

Plate-bendlng s t r a i n Membrane s t r a l n estimated by beam theory

....

strain estimated by beam theory

Plate-bending

v-0.343

8

•

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-80

I , ,

-50

@@

m

o

08

, ; I-1 0

@ i t t

@o l-T-I--

50

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S0

y (ram)

Fig. 14. Membrane and plate-bending strains in the flange at x = 4 mm.

M z] ~m

--

E1

M Z1 ~b -- E 1

+zz 2

(4)

-- Z 2

2

(5)

where M is the bending m o m e n t at x = 4 mm, E is Young's modulus of acrylic material, I is the m o m e n t of inertia of the cantilever beam, zl is the distance from the neutral axis to the top surface of the flange of the cantilever beam, and z2 is the distance from the neutral axis to the bottom surface of the flange of the cantilever beam. As can be seen from Fig. 14, the measured membrane strains are higher than those estimated by the beam theory around y = 0, and the measured plate-bending strains are also higher than those estimated by the beam theory over the almost whole width of the flange. The results of the loading test of a cantilever beam of acrylic material have confirmed that a local stress beyond the magnitude estimated by the beam theory can be produced at the intersection of the top flange of the

Stresses at the intersection of cross-beam flange with a box-girder web

125

cross beam with the box-girder web for some other reasons which are not related to welds and copes.

4 REASON FOR I N I T I A T I O N OF LOCAL STRESS In order to give a reason why such a local stress is much higher than the magnitude estimated by the beam theory which is developed at the intersection of the top flange of the cross beam with the box-girder web, finite element analyses were conducted for an I-section beam shown in Fig. 15. The beam is fixed at the left end. A load was applied to the righthand end of the beam. The elements which were used in the finite element analyses are isoparametric shell elements with eight nodes. 4 Figure 16 shows the mesh division for the beam. From symmetry, half of the beam was divided into finite elements. The size of the finest elements of the top and bottom flanges near the left end is 3.125 m m x 3.125 mm, which was determined from the relation between the element size and the accuracy of numerical results. Referring to Fig. 17, the following three different boundary conditions were considered at the left end: (I) (II)

The displacements in the x, y and z directions are all fixed on the top and bottom flanges and the web. On the top and bottom flanges, the displacement in the y direction is free, but the displacements in the x and z directions are both fixed. On the web, the displacements in the x, y and z directions are all fixed.

Load Fig. 15. Beam o f I-section.

126

Ichiro Okura, Yuhshi Fukumoto

Fig. 16. Mesh division.

(III) On the top and b o t t o m flanges, the displacement in the x direction is fixed, but the displacements in the y and z directions are both free. At one node in the middle of the web, the displacements in the x, y and z directions are all fixed. At the nodes on the web other than this node, the displacement in the z direction is free, but the displacements in the x and y directions are both fixed. These are all regarded as the fixed condition in the beam theory, because each case gives the boundary condition of zero vertical displacement and zero rotation at the left end of the neutral axis of z = 0. Figure 18 shows a comparison of the stress distributions on the top surface of the flange just above the web. In the figure the stress distribution given by the beam theory is also drawn. The distributions for Cases I and II are almost identical. Remote from the left end, they are close to the straight line given by the beam theory. In the vicinity of the left end, they start to depart from this straight line. The distribution for Case III is almost the same as given by the beam theory over the entire region of the flange. Figure 19 shows a comparison of the stress distributions on the top surface of the flange at x = 0. The stress distribution given by the beam theory is also provided in the figure. Case I takes a large value at both edges of the flange. This is because the displacement in the y direction of the flange is fixed. Cases I and II take a m a x i m u m value at y = 0, which is much higher than the magnitude given by the beam theory. Case III shows

Stresses at the intersection of cross-beam flange with a box-girder web

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~x

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0

x

o

x

0

x

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(a) C a s e (I)

o

x

(b)

C a s e (~r)

I1~ ¢ ~---Fixed

0

x

----------=

(c) Case(]]I) Fig. 17. Various boundary conditions.

50Eo(HPa)

t0

x

40

(I)

bCases 30

and

(1T)

C.ase ( I n )

P=9.81 kN s MPa I E-2.06x10 u-0.3

P

"20 10['-

|

0

. . . .

Beam t h e o r y

I

50

I

1O0

I

I

200

x(==)

Fig. 18. Comparison of distributions of stress on the top surface of the flange just above the web.

Ichiro Okura, Yuhshi Fukumoto

128

I

+

E=2.06x10$ HPa v=O.3 o (HPa) 54~

~Case

20 ~

0

I I i I I I I I I - I00

-50

i

(]~)

Case (11I) ....

Beam theory

I I I I I I I I I I 0

50

100

y (ram)

Fig.

19. C o m p a r i s o n of distributions of stress on the top surface o f the flange at x = 0.

an almost uniform distribution with the magnitude being nearly the same as given by the beam theory. In Cases I and II, a local stress beyond the magnitude estimated by the beam theory is produced on the flange in the vicinity of the left end. In Case III, such a stress is not developed anywhere on the flange. In Cases I and II, the vertical displacement of the top and bottom flanges and the web is fixed at the left end. In Case III, it is free except at one point in the middle of the web. Therefore, it can be concluded that the restraint of the vertical deformation of the beam at the left end induces a local stress on the flange. In the fatigue test specimen shown in Fig. l(a), the vertical deformation of the cross beam is restrained at the box-girder web by the box-girder web itself and the diaphragm inside the box girder. Hence, since the condition for the vertical deformation of the cross beam at the box-girder web was close to Case I, a local stress must have been induced at the intersection of the top flange of the cross beam with the box-girder web.

5 CLASSIFICATION OF LOCAL STRESSES Local stresses at the intersection of a top flange of a cross beam with a boxgirder web are classified as shown in Fig. 20. Remote from the weld toe, the

Stresses at the intersection of cross-beam flange with a box-girder web

129

! ! I !

all

,

°12

I

0.3tf

tf

,,.

I

Fig. 20. Local stresses at the intersection of the top flange of the cross beam with the box-girder web.

stress on the top flange is estimated by the beam theory, and it shows a straight line. Near the weld toe, the stress begins to depart from this straight line, as a local stress due to the restraint of the vertical deformation of the cross beam at the box-girder web is developed. Much nearer the weld toe or within 0.3tf (tf is the thickness of the top flange) from the weld toe, the stress concentration at the weld toe is initiated. The following three stresses occur at the location of the weld toe: - - The nominal stress trn estimated by the beam theory. - - The local stress tr12due to the restraint of the vertical deformation of the cross beam at the box-girder web. - - The local stress thl due to the stress concentration at the weld toe. The fatigue strength for the connection detail shown in Fig. 20 is usually provided by the fatigue test of a specimen shown in Fig. 21. In this specimen, the nominal stress tr, and the local stress all are produced, and the nominal stress tT, is used for S in the S - N relation. Hence, in fatigue design for the connection detail shown in Fig. 20, the sum of the nominal stress trn and the local stress th2 must be compared with the fatigue strength given by the fatigue test of the specimen shown in Fig. 21. In practice it is not easy to distinguish between the local stresses ~u and

lchiro Okura, Yuhshi Fukumoto

130

I I

oi I )

II 0.3tf C*--I

'K J

"V

J

l

tf

r

I

P

Fig. 21. Fatigue test specimen.

0-12, since they are initiated in a very limited region near a weld toe. As a method of determining the local stress thE, the following may be proposed. As shown in Fig. 20, at least three strain gauges should be mounted on the top flange as near the box-girder web as possible, but they must not be within 0.3tf from the weld toe. The value at the location of the weld toe, given by parabolic extrapolation of these three measured strains, is used for the local stress o'12. 6 CONCLUSIONS Local stresses at the intersection of a top flange of a cross beam with a boxgirder web were investigated analytically and experimentally. Finite element analyses of a T-shaped welded joint and a loading test of a cantilever beam of acrylic material revealed that a local stress which is different from the one caused by the stress concentration at a weld toe is developed at the intersection. Finite element analyses of an I-section beam showed that the restraint of the vertical deformation of the cross beam at the box-girder web induces the local stress. It was pointed out that the local stress must be considered in fatigue design. A m e t h o d of determining the local stress was proposed. REFERENCES .

Macda, Y., Fukuoka, T., Okura, I. & Isozaki, K., Fatigue test of cross beam connections in steel track girders for straddle-type monorail. Proc. JSCE, No. 404 (1989) 425-34 (in Japanese).

Stresses at the intersection of cross-beamflange with a box-girder web

131

2. Japan Road Association, Specification for Highway Bridges. 1990 (in

Japanese).

3. Zienkiewicz, O. C., The Finite Element Method. McGraw-Hill, New York, 1977, pp. 178-210. 4. Zienkiewicz, O. C., The Finite Element Method. McGraw-Hill, New York, 1977, pp. 398-422.