Distortional analysis of simply supported box girders with inner 1 diaphragms considering shear deformation of diaphragms using 2 initial parameter method 3

In this paper, the distortion of simply supported girders with inner diaphragms subjected to concentrated eccentric loads is investigated using initial parameter method (IPM), in which the in-plane shear deformation of diaphragms is fully considered. A statically indeterminate structure was modeled with inner redundant forces, where the interactions between the girder and diaphragms were indicated by a distortional moment. Considering the compatibility condition between the girder and diaphragms, solutions for the distortional angle, warping displacements and stresses were derived and further simplified by establishing a matrix equation system. The validity of IPM was intensively verified by a finite element analysis and distortional experiments. Parametric studies were then performed to examine the effect of the diaphragm number on the distortional angle, warping displacements and stresses under various ratios of height to span of the girder and the diaphragm thicknesses. Besides, stabilities of the local web plate and mid-span diaphragm were analyzed based on IPM for box girders with symmetrical three inner diaphragms. Results show that the local web plate will buckle before overall yielding with the increment of the eccentric loads Pj, and the mid-span diaphragm is constantly stable in the whole deformation process. It shows that more attentions should be paid on the stability of the local web plate than overall yielding for girders subjected to eccentric loads.


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During the past several decades, box girders have been widely applied in buildings and 34 bridges due to their large bending and torsional stiffness. However, they are generally susceptible 35 to the cross-sectional distortion [1] under eccentric loads due to their quadrilateral instability. 36 Therefore, excessive distortional warping and transversal bending stresses will be produced 37 besides the torsional and bending ones in box girders. In a special case, the distortional warping 38 stresses may be significant to the torsional and bending ones. In order to control the distortion, 39 diaphragms are installed at the interior of the girder, which can increase not only the stability of 40 diaphragm was generally made in most studies [16,20,27], where the in-plane deformation of 85 diaphragms was totally restrained and warping was free. Similar assumptions can be found in 86 distortion of curved box beams [28], where the distortional angle at the location of diaphragms is 87 set as zero. However, the infinite-rigidity assumption is just an approximation, which is not 88 applicable to thin flexible diaphragms. The main objective of this work is to investigate the 89 distortion of simply supported girders with inner flexible diaphragms under concentrated eccentric 90 loads, where the in-plane shear deformation of diaphragms is fully considered. Interactions 91 between the girder and diaphragms are indicated by a distortional moment. Based on the 92 compatibility condition between the girder and diaphragms, solutions for both the distortional 93 angle and warping function are obtained from the IPM. Taking a simply supported girder with 2, 5 94 and 9 diaphragms, respectively, as an example, the distortional solutions from IPM were obtained, 95 then verified by a FE analysis and experiments. This was followed by a parametric study, in which 96 distortional deformations and stresses were investigated in terms of the diaphragm number and 97 thickness and the height to span ratio of the girder. Based on the proposed IPM, stabilities of both 98 the local web plate and mid-span diaphragm were examined for girders with three symmetrical 99 inner diaphragms. A series of curves were obtained for the relations between the critical buckling 100 load and the position of diaphragms under various height to width ratios of the cross section. 101 and vertical displacements at node M, respectively. The variation of the right angle at node N is 118 defined as the distortional angle χ, given by χ=χ1+χ2. Moreover, the warping displacement wd and 119 the stress σd, produced by the distortional moment Bd, are shown in Fig.2d. There also exists the 120 shear stress τd along the cross-section profile, developed by the distortional moment Md, as shown 121

Structural model
in Fig.2e. 122 This article will focus on the distortional deformation and stresses of a simply supported box 123 girder with inner diaphragms subjected to concentrated eccentric loads. 124 Flexural loads x y x y b P j n 1 b Eccentric load P j x y P j n 1 2b P j n 1 2b P j n 1 2h P j n 1 2h x y P j n 1 2b P j n 1 2b P j n 1 2h The boundary conditions for a simply supported girder are 150 Correspondingly, the state vectors on both ends are 153 The jth distortional load in IPM is indicated by a vector Zj, given by  Mpi (i=1,2,…,n). Only Mpi,, opposite to Mj, will resist the distortional deformation and stresses. In 191 IPM, Mpi is indicated by a vector Zpi, given by 192 (2) Compatibility condition between the girder and diaphragms 194 In-plane shear strains of diaphragms are considered, given by γpi = Mpi /(Gbhtpi). The compatibility condition is that the distortional angle at the mid line of diaphragms is opposite to 196 the in-plane shear strain of diaphragms. That is χ(zpi)= -γpi (0≦i≦n). This is a key aspect for the 197 distortion of girders with inner diaphragms. 198 Combining Eq.(8) and Eq.(12), the state vector Z(z) can be expressed as 199 where R and S are the numbers of diaphragms and moments Mj before point z, respectively. The 201 transfer matrices P(z-zi) and P(z-zj) are obtained from P(z) by substituting the variable z by 'z-zi' 202 and 'z-zj'. 203 For z=l, Eq.(13) changes into 204 where the vectors Z(l) and Z(0) are referred to Eq.(6); W(0) and Md(0) in vector Z(0) can be 206 calculated from the first and third simultaneous equations in Eq.(14). Then, substituting W(0) and 207 Md(0) into Eq. (13), the distortional angle and the warping function can be obtained as 208 ( ) where φm (i) (x) and φn (i) (y) are the ith differentiation of functions φm(x) and φn(y). φn (-1) (y) is the 219 integral of function φn(y), given by 220 When the calculated point z is located in the thickness of (R+1)th diaphragm (zp(R+1)-tp(R+1)/2 222 ≦z≦zp(R+1)+tp(R+1)/2), an additional angle χadd and function Wadd should be involved, 223 Since Mj has been given in Eq.(10), solutions rest in Mpi. 229

Derivation of Mpi 230
The compatibility condition gives the equation 231 kT is the number of distortional loads before the Tth diaphragm. 236 The and the diagonal elements in matrix η is 242 ( , ) 248 Substituting Eq.(21) into Eq.(15), and the angle χ(z) changes into 249

Simplification of χ(z) and W(z)
where n and m are the total numbers of diaphragms and distortional loads, respectively. 251 The number of calculation steps is m×n in Eq. (22) In this approach, the distortional angle χ(z) can be simplified as 259 ( ) Therefore, the function W(z) can be simplified as 264 Taking the node N (see Fig.1b  IPM-t p /t=0.5 IPM-t p /t=1 IPM-t p /t=2 FEA-t p /t=0.5 FEA-t p /t=1 FEA-t p /t=2 FEA-t p /t=1 306 Fig.8 The distortional angle, warping displacements and stresses between IPM and FEA for a simply supported 307 girder with two diaphragms of different diaphragm thicknesses 308 Good agreements are observed between IPM and FEA in Fig.8 to Fig.10 for the distortional 309 angle, warping displacements and stresses for simply supported girders with inner diaphragms. 310 Compared the girders braced by 2 diaphragms with those by 5 and 9 diaphragms, it's worth noting 311 that the mid-span diaphragm effectively restrains the transversal deformation. 312 For the distortional angle, the largest error between IPM and FEA occurs at the loading 313 sections, where the FEA result is 23.68% higher than the IPM one for girders with two diaphragms, 314 and reduces to 13.86% for those with five diaphragms and 10.18% for those with nine diaphragms. 315 Since there is no diaphragms or stiffeners at the loading sections, the error between IPM and FEA 316 can be attributed to the local stress concentration. So the distortional angle obtained from IPM is 317 susceptible to the influence of stress concentration.   In addition, the influence of shear strains of the cross section on distortional deformations 328 and stresses are examined in Fig.11 for simply supported girders with 2 and 5 diaphragms, where 329 the compatibility condition between the girder and diaphragms is considered. It is seen that the 330 shear strain of the cross section makes little effect on warping displacements and stresses, but a 331 large influence on the distortional angle. The largest difference occurs at the loading sections 332 z=0.45l and z=0.55l, which is 14.9% for girders with 2 diaphragms and 17.8% with 5 diaphragms. 333 Also, the error at the mid span for girders with 2 diaphragms is 13.3%. Thus shear strains of the 334 cross section cannot be ignored when the transversal deformation of girders is considered. 335

Verifications with experiments 336
For further verification of the IPM, a series of experiments were performed using four groups 337 of samples -girders with no diaphragms, one, two and three diaphragms, subjected to distortional 338 loads. Diaphragms are equally distanced along the span. Both girders and diaphragms were 339 fabricated from carbon structural steel plates (yield strength 235MPa) of 8mm thickness. All 340 girders are 3 meters long, with height h=0.6m and width b=0.346m, giving a 30° angle between 341 the diagonal and the web. All girders are sealed by a steel plate of 6mm thickness on both ends. 342 To simultaneously produce two concentrated distortional loads, as in Fig.4d, two steps were 343 taken as follows 344 (1) For distortional loads, as shown in Fig.12, units were designed with two groups of wheels 345 anti-symmetrically about the shear center O of the cross section. This is to decompose the 346 horizontal power force Psource produced by FCS hydraulic servo system into two orthogonal 347 loading components Ph and Pv.

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(2) For concentrated loads, as shown in Fig.13, four stiff units of 1.5m in length and 0.15m 349 and 0.350m in width, respectively, were welded onto flanges and webs symmetrically over the 350 mid span, and the loading sections were hence located at z1=0.75m and z2=2.25m, respectively. 351 During the experiments, compressive loadings were applied along the diagonal of the cross 354 section by two anti-symmetrical wheel units, as shown in Fig.12a. The entire experimental setup is 355 shown in Fig.14a, including two FCS hydraulic servo loading systems, two connection beams, two 356 wheel units, four stiff units, the tested girder and two supports. The maximum loading was set to 357 5t with a loading speed of 2mm/s, controlled by a loading equipment in Fig.14b.   Fig.15 The scheme of tested points 370 Compared with those from IPM, the transversal displacements and distortional angle are 371 depicted in Fig.16 for girders without diaphragms under the source loading of 5t. There exist some 372 errors between the IPM and experimental results at the singularity points A1 for x-axial 373 displacement and B10 and B11 for y-axial displacement, which are mainly caused by the residual 374 strains of welding and manufacturing. Eliminating the influences induced by singularity points, 375 fitting lines were obtained from experimental results by applying the quadratic fitting method 376 provided in the software MATLAB. Fig.16     π π π π π π π β π π π π σ π β π π σ η where the definite integrals on variables u and p are given in Tab indicating increased resistance to buckling. 517 Back to the girder with equally-distanced three inner diaphragms in Fig.22, the girder will 518 reach its yield strength limit of 235MPa when the eccentric load Pj is 650kN, which is much 519 smaller than the critical buckling load Pcr2 of 8743kN at zp1/l=0.25 for the ratio h/b=2 according to 520 the stability of the mid-span diaphragm. Simultaneously, the plastic yield load of 650kN is much 521 larger than the critical load Pcr1 of 12.2kN at zp1/l=0.25 according to the stability of the local web 522 plate. This implies that buckling of the local web plate will be the primary failure mode with the 523 increment of eccentric loads Pj. Hence attentions should be paid on the stability of the local web 524 plate for the design of girders subjected to eccentric loads. 525

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In this paper, the initial parameter method (IPM) is applied to investigate the distortion of 529 simply supported girders with inner diaphragms subjected to concentrated eccentric loads, where 530 in-plane shear deformation of diaphragms is considered. The main conclusions can be drawn as 531 follows 532 (1) Compared with results from FEA and experiments, accurate analyses can be obtained by 533 IPM for the distortional angle, warping displacements and stresses for simply supported girders 534 with inner diaphragms. Both the in-plane shear deformation of diaphragms and the compatibility 535 condition between the girder and diaphragms are taken into account in IPM. Besides, comparison 536 of results between considering the shear strain of the cross section and not shows that the shear 537 strain of the cross section cannot be ignored when calculating the distortional angle. 538 (2) Both the distortional angle and warping stresses decrease with the increment of the ratio 539 of height to span, the number of diaphragms and their thickness. The warping displacement 540 converges to a fixed value between 0.1 and 0.2 for large diaphragm numbers. And the mid-span 541 diaphragm plays a key role in reducing the distortional deformations and stresses for girders under 542 symmetrical loads. 543 (3) Stabilities of the local web plate and the mid-span diaphragm were both investigated for 544 box girders with symmetrical three inner diaphragms. Results show that both the local web plate 545 and the mid-span diaphragm increase their resistance to buckling when the diaphragm I is located 546 close to the loading sections. Moreover, the local web plate will buckle first as the primary failure 547 mode. Therefore, attentions are needed on the stabilities of the local web plate for simply 548 supported girders under eccentric loads. 549 Based on the IPM, it is possible to improve the warping displacements and stresses of simply 550 supported girders through optimizing the positions and wall thickness of diaphragms. Future work 551 are needed for (1) optimization of the distortion of girders with diaphragms; (2) mechanical 552 properties of girders with perforated diaphragms. 553