Maximum Stress Estimation Model for Multi-Span Waler Beams with Deflections at the Supports Using Average Strains

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1 Sensors 05, 5, ; do:0.3390/s Artcle OPEN ACCESS sensors ISSN Maxmum Stress Estmaton Model for Mult-Span Waler Beams wth Deflectons at the Supports Usng Average Strans Sung Woo Park, Byung Kwan Oh, and Hyo Seon Park,, * Center for Structural Health Care Technology n Buldngs, Yonse Unversty, Seoul 0-749, Korea; E-Mal: paksungwoo@naver.com Department of Archtectural Engneerng, Yonse Unversty, Seoul 0-749, Korea; E-Mal: aeoobk@yonse.ac.kr * Author to whom correspondence should be addressed; E-Mal: hspark@yonse.ac.kr; Tel.: ; Fax: Academc Edtor: Vttoro M.N. Passaro Receved: 8 January 05 / Accepted: 0 March 05 / Publshed: 30 March 05 Abstract: The safety of a mult-span waler beam subjected smultaneously to a dstrbuted load and deflectons at ts supports can be secured by lmtng the maxmum stress of the beam to a specfc value to prevent the beam from reachng a lmt state for falure or collapse. Despte the fact that the vast majorty of accdents on constructon stes occur at waler beams n retanng wall systems, no safety montorng model that can consder deflectons at the supports of the beam s avalable. In ths paper, a maxmum stress estmaton model for a waler beam based on average strans measured from vbratng wre stran gauges (VWSGs), the most frequently used sensors n constructon feld, s presented. The model s derved by defnng the relatonshp between the maxmum stress and the average strans measured from VWSGs. In addton to the maxmum stress, support reactons, deflectons at supports, and the magntudes of dstrbuted loads for the beam structure can be dentfed by the estmaton model usng the average strans. Usng smulaton tests on two mult-span beams, the performance of the model s evaluated by estmatng maxmum stress, deflectons at supports, support reactons, and the magntudes of dstrbuted loads. Keywords: average stran; vbratng wre stran gauge; maxmum stress; structural health montorng

2 Sensors 05, Introducton Safety montorng of mult-span beam structures under dstrbuted loads has been an ssue of nterest snce structural health montorng (SHM) has been ntroduced [ 8]. If the maxmum stress n a beam structure exceeds the specfed strength or allowable stress due to unexpected loads, then t can be consdered that the safety of the beam structure has reached a lmt state. Therefore, structural responses, ncludng the maxmum stress n beam structures must be montored n the long term to assess the safety of a structure [9 3]. To estmate the maxmum stran or stress of mult-span beams, mathematcal models based on measured average strans have been developed and appled to montorng the safety of beam structures [4]. In the mathematcal models, the maxmum stran of a mult-span beam s obtaned by defnng the relatonshp between the average stran measured from sensors and the estmated maxmum stran. It s shown that the maxmum stran of a mult-span beam can be accurately estmated by usng average strans obtaned from long gauge fber optc sensors (LGFOS) or vbratng wre stran gauges (VWSG) [5 8]. However, for the case of beam structures subjected to deflectons at the supports, the mathematcal models of mult-span beam wthout consderaton of the effect of deflectons at supports on the magntude of the maxmum stress cannot be used n the assessment of the safety of beam structures. A typcal example of a beam structure subjected to deflectons at ts supports s the waler beam n an anchored retanng wall system shown n Fgure. The waler beam s used to transfer retaned loads due to lateral pressures from the retaned sol evenly between the ground anchors [9]. (a) (b) Fgure. A mult-span waler beam supported by ground anchors n a retanng wall system: (a) Front vew; (b) Sde vew. As shown n Fgure, the ground anchors n a retanng wall can be modelled as supports for the waler beam. Snce the ground anchors are subjected to the axal tenson due to lateral sol pressure, axal dsplacement along the axs of the ground anchor occurs and then the waler beam supported by the ground anchors s subjected to dfferental support deflectons. Excessve structural deformaton n waler beams due to lateral pressure and dfferental support deflectons s a man cause of damage or collapse of retanng wall systems [0]. The safety of the retanng wall system can be assessed by montorng the safety of the waler beam subjected to deflectons at the supports. Thus, t s necessary

3 Sensors 05, to develop a generalzed mathematcal model for estmaton of the maxmum stress n a mult-span beam subjected to support deflectons. Fgure. Secton A-A of a waler beam supported by ground anchors n a retanng wall system. Therefore, n ths paper, an analytcal model to estmate the maxmum stress for a mult-span beam wth deflectons of supports under dstrbuted loads s presented. For the estmaton, the relatonshp between the maxmum stress and the average stran of each span s derved and the average stran s measured at the mddle of each span. In addton to estmaton of the maxmum stress, usng average strans from VWSGs, estmaton models for deflectons at supports, support reactons, and the magntudes of dstrbuted loads are presented. For the safety montorng of mult-span beam structures n retanng wall systems, the model s appled to estmaton of the maxmum stress and dstrbuton of moments along the length of a beam.. Average Stran of Waler Beams Measured by VWSGs In Fgure 3, a typcal waler beam supported by n + ground anchors n a retanng wall system s modeled a lnear elastc n-span contnuous beam wth n + supports. l and w are the length and the magntude of dstrbuted lateral load of the th span, respectvely. k and δ are the stffness of the anchor and the deflecton at the th supports, respectvely. It s assumed that the deflecton at each support has a dfferent value. For a lnear elastc mult-span beam as shown n Fgure 3, the relatonshp between the bendng stress, σ( x ), and stran, ε( x ), as a functon of the poston of x n the th span along the axs of the beam can be wrtten as: M( x ) σ( x) = = ε( x) E () Z where M( x ) s the bendng moment, Z s the elastc secton modulus and E s the modulus of elastcty of the beam []. Usng the predetermned value for E, the stress n Equaton () can be estmated by measurng the stran ε( x ). For the measurement of strans, VWSGs are the most frequently used sensors n the constructon feld of buldng structures snce operatng prncple s

4 Sensors 05, smple and nstallaton cost s relatvely low compared to other sensors [6,7]. In addton, VWSGs have excellent endurance and are free from electromagnetc nterference (EMI) so that VWSGs sutable for long-term montorng n the feld [ 4]. Fgure 3. n-span contnuous beam supported by n + ground anchors. Fgure 4. Typcal vbratng wre stran gauge attached to the surface of a steel secton for measurement of average strans. As shown n Fgure 4, a VWSG conssts of three components of two mountng blocks whch are attached to the surface of a structural member to be measured, a vbratng wre whose frequency changes due to tenson or compresson between the two mountng blocks, and pluckng and pckup col, whch excted the vbratng wre and measure ts resonance frequences. The change of length of a vbratng wre whch s fxed at the two mountng blocks causes the varaton of the natural frequency of the wre. For ths reason, a VWSG measures the average stran over the gauge length between the two mountng blocks. As shown n Fgure 3, f the gauge length s the dstance between the mountng blocks and the locatons of the mountng blocks for the VWSG n the th span of the waler beam are set to x and x, then the average stran for the th span of the waler beam, ε avg,, can be gven as: ε x ε( x ) dx x avg, = x x ()

5 Sensors 05, Then, the relatonshp between the bendng moment at the th span, M ( x ) n Equaton () and the measured average stran ε avg, n Equaton () can be expressed by: ε x M ( x ) dx (3) ( ) x avg, = EZ x x 3. Estmaton Model for Maxmum Stress Based on the three moment theorem, the relatonshp among the bendng moments, M, M and M +, at three consecutve supports of the mult-span beam n Fgure 3 s gven by Equaton (A) n Appendx A. The beam s assumed to have constant moment of nerta and elastc modulus. For the contnuous beam wth n + supports, n three moment equatons can be formulated usng Equaton (A) n Appendx A and expressed n matrx form as: [ A]{ M} = [ B]{ w} + [ C]{δ} (4) where [ A ] s a square matrx of dmensons ( n ) ( n ), [ B ] s a matrx of dmensons ( n ) n, and [ C ] s a matrx of dmensons ( n ) ( n+ ), gven by Equatons (B) (B3) n Appendx B. Also, {M}, {w} and {δ} are the bendng moment vector of dmenson ( n ), the magntude of dstrbuted lateral load vector of dmenson n, and the deflecton at support vector of dmenson ( n+ ), respectvely. Then, the equatons n Equaton (4) can be solved smultaneously for the moments at supports. Havng found the moments at n + supports, the vector of dmenson ( n+ ) for the reactons at the supports of the contnuous beam, { R }, can be found by: { R} = [ G]{ w} + [ C ]{ M} (5) where [G] s a matrx of dmensons ( n ) n and [C ] s a matrx of dmensons ( n+ ) ( n ), gven by Equatons (B4) and (B5) n Appendx B. As shown n Fgure 3, the reacton at the th support can also be found by multplyng the stffness k by the deflecton δ of the support. Then, the reacton vector n Equaton (5) can be expressed by substtutng the moment vector {M} n Equaton (4) nto Equaton (5). = = + + (6) ' { R} [ K]{δ} [ G]{ w} [ C ] [ A B]{ w} [ A C]{δ} Where [K] s the stffness matrx for the n + supports n Fgure 3. From Equaton (6), the relatonshp between the vectors {w} and {δ} s found by: + = (7) - - [ G CA B]{ w} [ K CA C]{δ} Usng the support reactons {R} and dstrbuted lateral loads {w}, the bendng moment of each span of the beam { M ( x )} can be gven by: { M ( x) } = [ P]{ R} [ Q]{ w} (8) where [ P ] s a matrx of dmensons ( n+ ) ( n+ ) and [ Q ] s a matrx of dmensons ( n+ ), gven by Equatons (B6) and (B7) n Appendx B. By consderng the gauge length of a VWSG n Equaton (3), the relatonshp n Equaton (8) for the th span can be evaluated by numercal ntegraton over the gauge length of x x as below:

6 Sensors 05, { ( ) } x M xdx = [ P]{ R} [ Q]{ w} x (9) where [ P ] and [ Q ] are ntegrals of [ P ] and [ Q ] n Equatons (B6) and (B7) n Appendx B over the gauge length, respectvely. The ntegral of the moment over the nterval of the gauge length n Equaton (9) can be expressed n term of the average stran n Equaton (3): x { ( ) } { ε } x M xdx = avg, EZx ( x ) = [ P]{ R} [ Q]{ w} (0) Then, by substtutng { R } n Equaton (6) and { w } n Equaton (7) nto Equaton (0), Equaton (0) can be gven by: {, } {} ' - T ' - ' - T ' - avg = EZ( x x) ε [ P][ K] δ [ Q] [ G CA B] [ G CA B] [ G CA B] [ K CA C]{δ} () From Equaton (), deflectons at supports of the waler beam { δ } can be found from the measurement of average strans from VWSGs { ε avg, } as below: T T {δ} = EZ( x x ) [ P][ K ] + [ Q] [ Y ] [ Y ] [ Y ] [ X ] {ε avg, } () ' - ' - where [ X] = [ K C A C] and [ Y] = [ G+ C A B]. Thus, by substtutng the deflectons determned from the R and the average strans n Equaton () nto Equatons (6) and (7), reacton at each support { } magntude of the dstrbuted lateral load at each span { w } can be dentfed. Consequently, the bendng moment of each span can be determned by substtutng { R } and { δ } nto Equaton (9). Fnally, based on the relatonshp n Equaton (), the maxmum stress of an n-span waler beam can be determned from the average strans measured by VWSGs. 4. Test of the Model on Mult-Span Beams 4.. Four-Span Beam The setup for smulaton of the four-span test beam s shown n Fgure 5. The secton of beam s H , wth a depth of 300 mm, a flange wdth of 300 mm, a web thckness of 0 mm, and a flange thckness of 5 mm. The materal for the beam secton s SS400 grade steel wth a modulus of elastcty of 05 GPa and yeld strength of 35.3 MPa. The test model s desgned to smulate the collapse of the waler beam n a retanng wall at a constructon ste n 009. The waler beam n the retanng wall was desgned to carry a unformly dstrbuted load of 06.5 kn/m. The waler beam was supported by ground anchors of prestressed concrete (PC) strands of Φ.7 mm 4 at a horzontal spacng of.6 m. The axal stffness of the anchor was gven by 0,7.6 kn/m. Wth these propertes, the four-span beam n the constructon ste can be modeled as a conceptual model n Fgure 5c.

7 Sensors 05, VWSG 6400 VWSG (a) Unt: mm (b) (c) Fgure 5. Expermental setup for the test of 4-span beam: (a) Sde vew; (b) Bottom vew; (c) Conceptual model. To test the conceptual model, t s assumed that the vbratng wre stran gauges wth the gauge length of 5 cm are nstalled to measure the average strans n Fgure 5b. As an nput, the measured average strans are assumed to be ε = ε4 = 344 με and ε = ε3 = 457 με. For the gven values for average strans obtaned from the numercal analyss wth the propertes of the model n Fgure 5c, the magntudes of deflectons of supports, reactons, and dstrbuted loads n Table are estmated from the analytcal model n Equatons (6), (7) and (), respectvely.

8 Sensors 05, Table. Estmated values for deflectons, reactons, and dstrbuted loads for the 4-span beam. Deflecton (m) Reacton (kn) Dstrbuted load (kn/m) δ = 0.00 R = 0.9 w = 06.5 δ = 0.09 R = 94.3 w = 06.5 δ = R 3 = 37. w 3 = 06.5 δ = R 4 = 94.3 w 4 = 06.5 δ = R 5 = 0.9 The deformed shape based on estmated support deflectons, support reactons, and appled loads of the test beam s shown n Fgure 6. As can be seen n Fgure 6, a maxmum deflecton of 3.3 cm and a maxmum reacton of 37. kn were calculated at the thrd support. Fgure 6. Deformed shape of 4-span beam subjected to the dstrbuted lateral loads. In Fgure 7, the bendng moment dagram wth the consderaton of support deflectons are compared wth the bendng moment dstrbuton wthout the consderaton of deflectons at the supports. As can be seen n Fgure 7, maxmum bendng moments of 7.63 knm were estmated at the ponts of.4 m and 3.93 m measured from the left end of the beam. Usng the modulus of elastcty of 05 GPa n Equaton (), maxmum stresses of MPa were estmated at the same ponts of the four-span beam where the maxmum moment occurs. Thus, for a gven value of the average strans measured by VWSGs, t s confrmed that the reactons, dstrbuted loads, bendng moments, and maxmum stress of the 4-span test beam can be estmated by the proposed analytcal model. As a practcal applcaton of the analytcal model presented n the paper, the measured values of the average stran can be lmted to a specfc value to prevent the waler beam from reachng one of lmt states for falure or collapse of waler beams. For the safety montorng of the four-span waler beam, the average strans at the second and thrd spans, ε and ε3, may be lmted to the value of 780 με to prevent the waler beam from reachng the allowable stress of 60 MPa for SS400 grade steel beam used n the constructon ste.

9 Sensors 05, Fgure 7. Bendng moment dagram for a 4-span beam for estmaton of maxmum stress. 4.. Eght-Span Beam A schematc dagram for the eght-span test beam s shown n Fgure 8. The materal for the beam secton of H s SS400 grade steel wth a modulus of elastcty of 05 GPa and yeld strength of 35.3 MPa. To test the applcablty of the estmaton method, the test beam n Fgure 8 s desgned to have rregulartes n geometry, mechancal propertes of anchors, and the magntudes of dstrbuted lateral loads. w w w 3 w 4 w 5 w 6 w 7 w 8 Average stran ε 85.7 με ε.5 με ε με ε με ε με ε με ε με ε με Stffness of supports (kn/m) k =0,000 k =0,000 k 3 =0,000 k 4 =0,000 k 5 =3,000 k 6 =3,000 k 7 =,000 k 8 =,000 k 9 =0,000 Length of spans l =m l =m l 3 =3m l 4 =3m l 5 =4m l 6 =4m l 7 =5m l 8 =5m Fgure 8. Schematc dagram for the 8-span beam wth rregulartes n geometry, mechancal propertes, and magntudes of lateral loads. The waler beam was desgned to carry three dfferent magntudes of dstrbuted desgn loads of w = w = w7 = w8 = 00 kn/m, w3 = w4 = 00 kn/m, and w5 = w6 = 300 kn/m. As nput for the estmaton model n Equaton (), the average strans from VWSGs are lsted n Table. For the gven values for average strans obtaned from the numercal analyss wth the propertes of the model

10 Sensors 05, n Fgure 8, the magntudes of deflectons of supports, reactons, and dstrbuted loads n Table 3 are estmated from the analytcal model n Equatons (6), (7) and (), respectvely. Table. Span length, stffness of anchor, and average stran for an 8-span beam. Span Length (m) Stffness of Anchor (kn/m) Average Stran (με ) l =.0 k = 0,000.0 ε = 85.7 l =.0 k = 0,000.0 ε =.5 l = k = 0, ε = l = k = 0, ε = l = k = 3, ε =, l = k = 3, ε =, l = k =, ε = l = k =, ε = k = 0, Table 3. Estmated deflectons, reactons, and magntudes of dstrbuted loads for an eght-span beam. Deflecton (cm) Reacton (kn) Dstrbuted Load (kn/m) δ = 3.0 R = 75.3 w = 00 δ = 8.88 R =.0 w = 00 3 δ = 4.83 R 3 = w 3 = 00 δ = 3.3 R 4 = w 4 = 00 4 δ = 3.85 R 5 = w 5 = δ = 4.9 R 6 =87.7 w 6 = δ = R 7 = 7.4 w 7 = 00 7 δ = R 8 = w 8 = 00 8 δ =.00 9 R 9 = 0.0 The deformed shape based on estmated support deflectons, support reactons, and appled loads of the test beam are shown n Fgure 9. As can be seen n Fgure 9, maxmum deflecton of 47.5 cm and maxmum reacton of 87.7 kn were calculated at the 7th and 6th supports, respectvely. In Fgure 0, the bendng moment dagram wth the consderaton of support deflectons are compared wth the bendng moment dstrbuton wthout the consderaton of deflectons at the supports. As can be seen n Fgure 0, a maxmum negatve bendng moment of knm was estmated at the 5th support. Usng the modulus of elastcty of 05 GPa, a maxmum stress of 3.67 MPa was estmated at the same ponts where the maxmum moment occurs, therefore, t s found that the magntude of the estmated maxmum stress exceeded the allowable stress of 60 MPa for a SS400 grade steel beam.

11 Sensors 05, Loads 00 kn/m 00 kn/m 300 kn/m 00 kn/m Deflectons m m m 0.33 m m 0.49 m m m 0.00 m l =m l =m l 3 =3m l 4 =3m l 5 =4m l 6 =4m l 7 =5m l 8 =5m Reactons 75.3 kn.0 kn kn kn kn 87.7 kn 7.4 kn kn 0.0 kn Fgure 9. Deformed shape of an 8-span beam subjected to dstrbuted lateral loads. Fgure 0. Bendng moment dagram for an 8-span beam for estmaton of maxmum stress. 5. Conclusons In ths paper, an analytcal model for estmaton of the maxmum stress of a mult-span beam wth deflectons of ts supports s presented. The analytcal model s derved by defnng the relatonshp between the maxmum stress and the average stran of each span of the mult-span beam. Based on the use of the average strans obtaned from VWSGs as the nput to the analytcal model, the analytcal model allows estmaton of deflectons at supports, support reactons, and the magntudes of dstrbuted loads. Usng tests on two mult-span beams, the performance of the model s evaluated by estmatng maxmum stresses, support deflectons, support reactons, and the magntudes of dstrbuted loads. From the smulaton test, t can be concluded that safety montorng of mult-span waler beams s made possble by comparng the gven allowable stress of the beam wth the estmated maxmum stress wth consderaton of any deflectons at the supports. Through perodc estmaton of the maxmum stress by the proposed method, the safety of retanng walls at constructon stes wll be mproved. Acknowledgments Ths work was supported by the Natonal Research Foundaton of Korea (NRF) Grant funded by the Korea government (MSIP) (No ).

12 Sensors 05, Author Contrbutons S.W. Park and H.S. Park developed the proposed method. S.W. Park, H.S. Park and B.K. Oh wrote the paper. Appendx A. Three Moment Equaton where Ml + M ( l + l ) + M l = wl + w l + 6EI δ + + δ δ 3 3 ( ) ( ) ( ) ( ) l l l+ l+ M, l, w and δ are the bendng moment, length, dstrbuted lateral load and deflecton at the th support of the beam, respectvely. E and I are elastc modulus and moment of nerta, respectvely. B. Matrces n Secton 3 (A) 0 ( l+ l) l 0 l ( l + l3) l3 0 0 l3 ( l3+ l4) l4 0 [ A] = 0 0 l n 0 ln ( ln + ln ) 0 0 ln ( ln + ln) 3 3 l l l l3 [ B] = ln l n l l l l 0 + l [ ] 6 l l3 l C = EI ln ln ln l n [ G] l 0 0 l l 0 0 l l ln l n 0 0 l n 3 = (B) (B) (B3) (B4)

13 Sensors 05, [ P] [ C ] = [ C ] 6 EI x x x l 0 x l l 0 3 = 0 xn xn l xn l l xn l ln x 0 T (B5) (B6) x 0 l l( x ) x 0 0 l l l( x3 ) l( x3 l ) x3 0 [ Q] = 0 l l ln l( xn ) l( xn l ) ln ( xn l ln ) x 0 In Equatons (B6) and (B7), x ( l + + l n x < l + + l n ) s the dstance from the left end of the beam, as shown n Fgure 3, and x 0 ( 0 x0 < l ) s the dstance from the rght sde of the beam to the n poston n the last span. (B7) Conflcts of Interest The authors declare no conflct of nterest. References. Jang, X.; Adel, H. Pseudospectra, musc, and dynamc wavelet neural network for damage detecton of hghrse buldngs. Int. J. Numer. Meth. Eng. 007, 7, L, H.; Huang, Y.; Ou, J.; Bao, Y. Fractal dmenson-based damage detecton method for beams wth a unform cross-secton. Comput. Aded Cv. Inf. 0, 6, Rezae, D.; Taher, F. Damage dentfcaton n beams usng emprcal mode decomposton. Struct. Health Mont. 0, 0, Park, H.S.; Lee, H.M.; Adel, H.; Lee, I. A new approach for health montorng of structures: Terrestral laser scannng. Comput. Aded Cv. Inf. 007,, Mtchell, R.; Km, Y.; El-Korch, T. System dentfcaton of smart structures usng a wavelet neuro-fuzzy model. Smart Mater. Struct. 0,, Cho, S.W.; Lee, J.; Oh, B.H.; Park, H.S. Measurement Model for the Maxmum Stran n Beam Structures Usng Multplexed Fber Bragg Gratng Sensors. Int. J. Dstrb. Sens. N 03, Hong, K.; Lee, J.; Cho, S.W.; Km, Y.; Park, H.S. A Stran-Based Load Identfcaton Model for Beams n Buldng Structures. Sensors 03, 3, Ng, C.-T. Bayesan model updatng approach for expermental dentfcaton of damage n beams usng guded waves. Struct. Health Mont. 04, 3,

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