A study on local stresses of corrugated steel webs in PC bridges under prestressing

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1 A study on loal stresses of orrugated steel webs in PC bridges under prestressing Shogo MORI *, Takao MIYOSHI **, Hisato KATOH *** Nobuo NISHIMURA ****, Satoshi NARA ***** Abstrat Generally, seldom orrugated steel webs in PC bridges do resist horizontal normal fores by prestressing and so on. However, bending deformation of the plates ours at a point in the web plates far from onnetion between steel flanges and PC slabs. Normal stress aused by plate bending of the webs has influene on the durability of welded joint in steel girders. In this paper, at first, plate bending stress alulated by a pratial equation is proposed. Proposed normal stress is verified by omparison with finite element analysis. Moreover, loal stress distribution of the plates obtained by numerial analysis is disussed. As a result, a proposed pratial equation is able to be applied in about middle 6% depth of the orrugated steel webs, and the plate bending stress in the vertial diretion is even larger than in the horizontal one at the viinity of onnetion between steel flanges and webs. Introdution PC bridge with orrugated steel webs, whih replaes onrete webs of traditional PC bo girder bridges by orrugated steel plates, is one of urrent strutures. The bridge dereases total weight of the superstruture and ost of onstrution in omparison with traditional PC bridges. Moreover, advantage of using orrugated webs is to make shear bukling strength of the webs higher, prestressing works for PC slabs more effiient beause of little horizontal stiffness, and to omit stiffeners in steel webs ),). In reent years, these advantages promote onstrution of many PC bridges with orrugated steel webs in Japan. However, loal stresses ourrimg in the orrugated web plates are not larified. Generally, bending moment auses uniform shear and vertial normal stresses in the orrugated steel webs near middle supports or loading point, due to negligible horizontal stiffness of the webs ),). However, the deformation of PC slabs by aial prestressing work or bend introdues plate bending deformation to orrugated steel webs. Therefore, it is important to evaluate plate bending stress of the webs, whih is negleted in general design, beause the stress has influene on the durability of welded joint in steel girders. It is not always pratial to use eperiments or finite element analysis (FEA) as a means of evaluating the loal stress 3). This study aims at proposal of a simple equation whih is effetive for pratial design work, and evaluation of the horizontal plate bending stress of the orrugated webs under prestressing. In this paper, a simple equation to alulate the horizontal plate bending stress in orrugated webs is derived, and its verifiation and appliable range in vertial diretion are shown using FEA. Also, the loal stress distribution obtained by FEA is disussed. * Graduate Student, Dept. of Civil Eng., Osaka University ** Dr. Eng., Designated Instrutor, Dept. of Civil Eng., Osaka University *** Dr. Eng., Manager, Bridge Engineering and Design Dept., JFE Engineering Corporation **** Dr. Eng., Professor Emeritus, Osaka University ***** Dr. Eng., Professor, Dept. of Civil Eng., Osaka University

2 Proposed bending normal stress derived from a simple equation The orrugated steel webs have symmetrially regular shape with a onstant wave length in horizontal diretion, as shown in Fig... To derive a simple equation, it is assumed that the plate with bending deformation is fiedly supported by PC slabs and flanges, and that aial deformation of the flanges is represented as the bending deformation of the webs on whih a plane rahmen is modeled as shown in Fig... Horizontal strain ε, whih ourred in PC slabs due to prestressing, is given as follows, σ o ε = (.) E where, σ : Magnitude of prestress introdued to PC slabs, and E : Elasti modulus of the onrete If the aial deformation in eah plate element of the orrugated web is negligible, the horizontal displaement L ourring in a wavelength L of the orrugated web is aused by only rotation R of slant plate elements as shown in Fig... Then, L is represented by the following equation, L = { os( + R) os} (.) As R is very small, it is possible to substitute relationships of osr and sinr R, for eq.(.), and, eq.(.3) is obtained as follows, L = Rsin (.3) L a b d a Fig.. A wavelength of orrugated steel web θ θ 4 y a os(+r) y L R y a y L os(+r) Fig.. Deformation of orrugated steel web in aial diretion R 3 Fig..3 Plate bending deformation in a wavelength -M - -M M M Fig..4 Distributions of plate bending moment M + 3 M -M 4 a M M

3 Using eq.(.3), the apparent horizontal normal strain under prestressing is represented as follows. L Rsin Rsin = = (.4) L ( a + os) a + os From eq.(.) and (.4), equation (.5) is obtained by assumption that the plate bending deformation is not onstrained by PC slabs and flanges. R σ E a os sin = (.5) + As shown in Fig..3, ompatibility ondition of rotation at point- is represented as follows, θ = R (.6) θ Also, as shown in Fig..4, defletion angles θ and θ at point- and - are given, respetively, as follows, = Ma EI θ (.7) = M 6EI θ (.8) where, E: Elasti modulus of a orrugated web, I: Moment of inertia of area, and M : Magnitude of the plate bending moment at point on folded line of a orrugated web. Substituting eq.(.7) and (.8) to (.6), R is represented as follows. M a = + EI 6 R (.9) By substitution of eq (.9) to (.5), the plate bending moment M is epressed by eq.(.). σ EI( a + os) 6 M = (.) E 3a + sin Maimum absolute value of the plate bending stress ours in both sides of a orrugated plate, and the simple equation is represented as follows using plate thikness t. σ M I + os t σ = E a E + 3a t sin p = (.) Moreover, the stress alulated from eq.(.) is different from atual one as a thin walled struture, beause eq.(.) is derived using assumption that a orrugated web is regarded as a plane rahmen. The stress-strain relation for plane stress field is given as follows, E σ ν = νe σ y ν τ y νe ν E ν E ( + ν ) ε ε y γ y (.) where, ν: Poisson s ratio of the orrugated web, and subsripts and y of σ, τ, ε and γ are orresponded with the loal oordinate system, as shown in Fig.., respetively. From eq.(.), when ε y is muh smaller than ε under prestressing, σ is evaluated approimately using the following equation. 3

4 E ν σ ε = (.3) Also,from eq.(.), the simple equation to alulate the horizontal plate bending strain ε p is represented as follows. ε + os σ a E + 3a t sin p = (.4) By regarding ε in eq. (.3) as ε p in eq.(.4), the simple equation to alulate the horizontal plate bending stress σ b is represented as follows. σ M I + os t σ E t = a E ν + sin 3a b = (.5) 3 Analytial models In order to evaluate the bending normal stress σ b of the orrugated web represented by eq.(.5), FEA is eamined. Moreover, in order to larify the appliable range of eq.(.5), FEA models based on ahievements of PC bridge with orrugated steel webs in Japan are set up. In this hapter, set up of analytial ases and models are shown. 3. Setion parameters for models Stress distributions in the orrugated web under prestressing are affeted by the following fators; () height H w, () thikness t, (3) folded angle (see Fig..), (4) dimensions a, b,, and d (see Fig..) of the web plate, (5) width and (6) thikness of the steel flange, (7) thikness and (8) (effetive) length of the PC slab, and (9) a number of reinforing bar in PC slabs. Stiffness of the orrugated web is affeted by the parameters from () to (4), and onstraint for plate bending deformation is dominated by the parameters from (5) to (9). It assumed that upper and lower flange are fied to the PC slabs, and rigid against the out-of-plane deformation. Nine above parameters are determined in adequate ranges, based on ahievements 3) with regard to PC bridges with orrugated webs in Japan and the eisting test speimen 6) whih is modeled on atual PC bridges with orrugated steel webs (9m in span length), as follows. At first, H w is equal to.m,.5m, and.m based on Fig.3. whih shows relationship between maimum span length L ma and the averaged height of the web H wa. Seondly, frequeny of is high at 3deg. and 37deg. as shown in Fig.3.. Therefore, of the analytial model is set up 3, 36.87, and 53.3deg. Finally, no-dimensional parameter t/h w and H w /a are defined using H w whih is determined from a speimen 6), and a and t of this analytial model are set up from results of the linear regression analysis between t/h w and H w /a as shown in Fig.3.3. Dimensions of the orrugated web, no. of wave in a span, name of analytial ase and series are shown in Table 3. respetively. 4 analytial series, as shown in Table 3., have different parameters, a for Series-P, for Series-P, t for Series-P3, and H w for Series-P4, respetively. 4

5 (m) wa H L ma (m) Fig.3. Relationship between H wa and L ma Frequeny ~3.9 Hw=.5m Ref. 3.4) Regression 3.~ ~35.9 Hw=.m Ahivements orrelation oeffiient = ~ ~ ~4.9 4.~ ~45.9 t /Hw (deg) Fig.3. Histogram of bending angle in orrugated web plate orrelation oeffiient =.83 Ahievements Model-P Model-P Model-P3 Model-P6 Model-P7 Model-P8 Model-P9 Regression H w/a Fig.3.3 Relationship between t/h w and H w /a Effetive width Effetive width Flange Reinforing bar PC slab PC tendon Corrugated steel web Reinforing bar PC slab PC tendon Fig.3.4 Cross-setion of analytial model 3. Models for numerial analysis As shown in Fig.3.4, in many eisting researh works 4)~7) with regard to mainly behavior in vertial plane of PC bridges with orrugated steel webs, analytial models and speimens are modeled as a orrugated web and upper and lower PC slabs plaed in one side. The simple equation (.5) is derived from the model, shown by Fig.3.4. Analytial model for FEA is shown in Fig.3.5. Rigid onnetion between upper and lower flanges and PC slabs are postulated. 4 Results and disussion of numerial analysis The plate bending stresses of the web obtained by FEA are disussed in omparison 5

6 Table 3. Dimensions of analytial models and analytial series for parametri analysis under prestressing H w (mm) H w /a t/h w a (mm) t (mm) (deg) Wave length L (mm) Span length L s (mm) No. of wave in a span No. of analytial model Model-P Model-P Model-P Model-P Model-P Model-P Model-P Model-P Model-P Model-P Model-P No. of analytial series Series-P Series-P Series-P3 Series-P4 Reinforing bar and PC tendons (Rebar modeling) (Prestress is introdued by temperature drop) Z (W) Setion- A B A A B A A B A Setion- A A' z y W= at this point Y(V) V= on this line U= on this plane X (U) Setion- PC slab (-node solid element) Setion- Setion A-A' Fig.3.5 Analytial model under prestressing (Model-P) Flange (8-node shell element) Vertial stiffner (8-node shell element) Corrugated web (8-node shell element) z Top side Middle line Bottom side with those by the simple equation. Also, harateristi of the stress and strain distribution in the orrugated web is disussed, where loation of ross-setions is foused at viinity of /4-point in span length, beause of negligible influene of load or boundary onditions on the behavior. 4. Charateristi of stress distribution Figs.3.6 and 3.7 represent the stress distribution for the basi analytial model-p along vertial diretion ourring at setion- and as shown in Fig.3.5 respetively. Loal oordinate system (-y-z) to desribe stress omponent and two sides of the orrugated 6

7 FEA-P.B.-bot. FEA-P.B.-top T-bot. T-top Stress (N/mm ) (a) Horizontal stress σ (b) Vertial stress σ y () Shear stress τ y Fig.3.6 Stress distributions at setion- of Model-P along vertial diretion FEA-P.B.-bot. FEA-P.B.-top T-bot. T-top Stress (N/mm ) FEA-P.B.-bot FEA-P.B.-top T-bot. T-top Stress (N/mm ) FEA-P.B.-bot. FEA-P.B.-top T-bot. T-top Stress(N/mm ) FEA-bot. FEA-top - 3 Stress (N/mm ) (a) Horizontal stress σ (b) Vertial stress σ y () Shear stress τ y Fig.3.7 Stress distributions at setion- of model-p along vertial diretion FEA-bot. FEA-top - 3 Stress (N/mm ) plate are defined as shown in Fig.3.5. The stress omponent due to plate bending and membrane transformed from vertial and horizontal stress σ and σ y are shown in Figs.3.6 and 3.7. FEA and T in their eplanatory note represent the results of FEA and eq.(.5), respetively. P.B. represents the plate bending stress, and bot., top, and mid. in their eplanatory note represent normal stresses ourring at bottom and top side of a orrugated plate, and mid-surfae, respetively. Additionally, the vertial ais in Figs.3.6 and 3.7 represents the normalized oordinate divided Z-oordinate in analytial models (see Fig.3.5) by height of the orrugated web H w. As shown in Figs.3.6(a) and 3.7(a), omparison between theoretial and FEA value of the horizontal bending stress shows that theoretial value get good agreement with FEA value in a range from =-.3 to =+.3. Aording to distributions of vertial plate bending stress as shown in Figs.3.6(b) and 3.7(b), plate elements parallel to bridge ais are subjeted to onstant bending stress around bridge ais in a range from =-.3 to =+.3. Figs.3.6() and 3.7() show that the in-plane shear stress τ y due to prestressing are almost negligible in a range from =-.38 to =+.38. Figs.3.6 and 3.7 also show that the orrugated web is in bi-aial stress state onstituted of the horizontal and vertial plate bending stress in the range from =-.3 to =+.3. However, the stress distribution lose to the flanges varies immediately in the distane to the flanges. Careful investigation will be needed, beause the stress omponent of vertial plate bending ourring at viinity of onnetion portion between flanges and a 7

8 ρ FEA-bot. FEA-top Strain (µ ) (a) Horizontal strain ε FEA-bot. FEA-top - 3 Strain (µ ) (b) Vertial strain ε y Fig.3.8 Strain distributions at setion- of model-p along vertial diretion ρ 5 5 H w /a (a) Series-P ρ ρ FEA-bot. FEA-top - 3 Strain (µ ) () Shear strain γ y (deg) t /H w (b) Series-P () Series-P3 Fig.3.9 Changes of ρ for eah analytial ase under prestressing H w /L s (d) Series-P4 orrugated web is about 6 times larger than that of vertial membrane, and the vertial plate bending stress is larger than horizontal one, in order to evaluate the durability of welded joints between a orrugated web and flanges. 4. Charateristi of strain distribution As shown in Fig.3.8, eah strain omponent represents onstant distribution from =-.35 to =+.35, and eah strain is onentrated in the area far from this range and loated at viinity of onnetion portion between flanges and a orrugated web. This trend is similar in the ase of stress distribution. The vertial strain is almost zero in a range from =-.36 to =+.36. An assumption that ε y is muh smaller than ε under prestressing is appropriate shown by the figure. The vertial plate bending stress σ yb is represented the following equation from eq.(.) and (.4), using ε y =. σ yb + os σ νe a E ν + 3a t sin = (4.) Distributions of σ yb obtained by eq.(4.) are shown in Figs.3.6(b) and 3.7(b) using T in their eplanatory note. Theoretial and FEA value of σ yb orrespond well eah other in a range from =-.3 to =+.3. Loal deformation of the orrugated web in onve diretion brings about a large σ yb, ourring at viinity of onnetion portion between flanges and a orrugated web. 8

9 4.3 Appliable range of the equation Changes of ρ for eah analytial series are shown in Fig.3.9. ρ denotes maimum value of no-dimensional oordinate whih represent the range held 5% in error is defined as an inde in this researh work. ρ is alulated as an averaged value at setion- and shown in Fig.3.5. As shown in Fig.3.9(a), an appliable range of the simple equation is sensitive to the length of the plate element. Figs.3.9(b) and 3.9() show that folded angle and plate thikness of the orrugated web are independent on an appliable range whih is onstant from =.3 to =.35. Moreover, as shown in Fig.3.9(d), an appliable range of the simple equation is slightly affeted by differene of the height of the orrugated web. Therefore, the equation is able to be applied in the range from =.3 to = Conlusions In this researh work, the simple equation to alulate the horizontal plate bending stress ourring in the orrugated web under prestressing is derived, and validity and the appliable range of the equation are investigated using FEA. Moreover, harateristi of loal stress distribution ourring in a orrugated web is disussed. Ahievements gained from this researh work are as follows. () Although the appliable range of the simple equation to alulate the horizontal plate bending stress varies slightly by hanges of plate length parallel to bridge ais and height of the orrugated web, the range is from 3% to 34% up and down from entral line of a orrugated web in vertial diretion. () Eah in-plane stress ourring in a parallel plate element of the orrugated web to bridge ais under prestressing is distributed onstantly at entral portion of the orrugated web in vertial diretion, and stress is in bi-aial stress state whih is onstituted of horizontal and vertial plate bending stresses. (3) Eah in-plane stress ourring at viinity of onnetion portion between flanges and the orrugated web represents the onentration, beause the orrugated web is loally deformed in onve diretion and the bending deformation of plate elements is onstrained elastially by upper and lower steel flanges. (4) Careful investigations will be needed in order to evaluate the durability of welded joint at the onnetion portion, beause the plate bending omponent of vertial stress ourring at viinity of onnetion portion between flanges and a orrugated web represents about 6 times larger value than membrane omponent of that. Referenes ) J. Combault (translated by T. Ohura) : The Maupre Viadut near Charolees, Frane, Journal of Japan Conrete Engineering Assoiation, Vol.34, No., pp.63-7, 99. (in Japanese) ) J. Combault, J.-D. Lebon and G. Pei : Bo-Girders Using Corrugated Steel Webs and 9

10 Balaned Cantilever Constrution, Proeedings of FIP Symposium 93, Kyoto, Japan, pp.47-44, ) Kinki branh of orporation of onstrution onsultants : A Report to Construt the Performane-based Design Method for Steel and Composite Strutures, 4. (in Japanese) 4) K. Yamaguhi, T. Yamaguhi and S. Ikeda : Mehanial Behavior of Composite Prestressed Conrete Girders with Corrugated Steel Webs, Conrete Researh and Tehnology, Vol.8, No., pp.7-4, 997. (in Japanese) 5) K. Uehira, F. Yanagishita, T. Ebina and K. Sonoda : Eperimental Study for Joint Methods of the Corrugated Steel Web of PC Bo Girder, Conrete Researh and Tehnology, Vol.9, No., pp.9-7, 998. (in Japanese) 6) Y. Ata, M. Ohiai, Y. Mizoe and F. Mahida : A Stati and Fatigue Test with Full Sale Models for PC Bo Girder Bridge Using Corrugated Steel Webs, Journal of Prestressed Conrete, Japan, Vol.43, No.4, pp.7-8,. (in Japanese) 7) H. Shiratani, H. Ikeda, Y. Imai and K. Kano : Fleural and Shear Behavior of Composite Bridge Girder with Corrugated Steel Webs around Middle Support, Journal of Strutural and Earthquake Engineering, JSCE, No.74(I-6), pp.49-67, 3. (in Japanese)

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