VERTICAL VIBRATION ANALYSIS OF MANHOLE UNDER TRAFFIC LOADING

st International Conference on Microstructure Related Durability of Cementitious Composites 3-5 October 008, Nanjing, China VERTICAL VIBRATION ANALYSIS OF MANHOLE UNDER TRAFFIC LOADING Nanguo Jin (), Xianyu Jin (), Zongjin Li () and Shenhua Liu () () Department of Civil Engineering, Zhejiang University, Hangzhou 3007, China () The Hong Kong University of Science and Technology, Hong Kong, China Abstract The vibration problem of manhole under traffic force was investigated using elasticity. The analysis considered the influence of the soil at the bottom of the manhole, as well as its self weight. Using Laplace transform, the vibration equations of manhole under traffic load and its self weight were established. By using convolution theorem, inverse Fourier transform and superposition principle, the semi-analytical solution was obtained for the vibration equations. An effective structural configuration for minimizing the settlement of manhole was proposed by vibration analyzing of 0 types manholes.. INTRODUCTION The vertical vibration theory of manhole is the foundation of manhole settlement calculation. It is very important to investigate the vertical vibration of a manhole to correctly calculate the settlement of the manhole under traffic load. Currently the vibration problem of a manhole is only analyzed by engineers for consideration of construction. There is no systematic study on this issue. Considering the similarity between the vibration of manhole and that of concrete piles, referring to the vertical vibration theory of piles [-6], it has been applied here for analysis of manhole. As a result, the mechanical model of manhole is established. The displacement of the top surface of a manhole is calculated by integral transform. The parameters have been assigned so as to get the manhole s top displacement curve of its vertical vibration. At last, we analyzed the other inds of manholes.. ESTABLISHING VIBRATION FUNCTIONS The manhole consists of four main parts: manhole canister, manhole room, basement and underlay. 97

st International Conference on Microstructure Related Durability of Cementitious Composites 3-5 October 008, Nanjing, China Figure : Model of manhole For the simplification and according to the actual variation of the cross sections of manhole, following assumptions were adopted in the calculation. The interfaces among the manhole s sidethe manhole s top soil and the manhole were assumed as Voigt phase.. The action of the manhole s top soil on the manhole was considered as the viscous elastic combination of elastic coefficient K b and viscosity coefficient C s. 3. The action of the soil around element of the manhole on the element was considered as the viscous elastic combination of elastic coefficient K s, and viscosity coefficient C s,. The traffic load was simulated as Ft (). The setch map of the manholethe simplified model of the manhole and the mechanical model of element are shown in Fig.. The self weight of a manhole was not considered in previous vibration studies, however, it was included in the manhole s vertical vibration equation in this study. Consider an element of the manhole, the vibration equation can be written as: u u u () E A C s, P u Ks P m + m 0 g= z t t Equation () can be simplified as: u u u D B u C + C g = z t t 0 where C = m /( E A), D = ( Cs, P)/( E A) B = ( Ks P)/( E A), E is the elastic modulus of the element of the manhole, A is the area of the element of the manhole, m is its unit length mass, and K the elastic constant of the soil around the manhole, C s, is the viscosity coefficient of the element of the manhole, K b is the elastic coefficient of the manhole basement soil, P is the perimeter of the cross section of the element of the manholeand the traffic load is simulated as Ft ( ). () 98

st International Conference on Microstructure Related Durability of Cementitious Composites 3-5 October 008, Nanjing, China 3 THE SOLUTION OF THE VIBRATION FUNCTION 3. The solution of the first part manhole As showed in Fig., the bottom surface of the first section of manhole was loaded with and the top surface with f. By applying Laplace transform to f, F () s can be obtained. By applying Laplace transform to boundary conditions and vibration equations respectively, one can obtain: U C g (3) ( D s+ B+ C s ) U+ = 0 z s U F() s = z z= h E A U K b + Cs, s + U = 0 z= L z z= L E A Because f and self weight are different forces, the displacement can be divided into two parts. Let: U = V + W (4) Where : V is the displacement of the first part of the manhole under the action of f, and W is the displacement of the first part of the manhole under the action of self weight. By substituting Eq.(4) into Eq.(3), we can get the displacement impedance of the first section of the manhole under the action of f and self weight. Z() s = F ()/ s V (,) zs = E A γ tan( β γ H ) (5) zh = E A C g sin( β) Z() s = = E A /( ) swzs (,) sin( β γ H) γ zh = where tan( β) = ( Kb + Cs s) /( E A γ), H = L h γ = ( D s+ B + C s ). 3. The solution of the second section of manhole As showed in Fig., the bottom surface of the second section of the manhole was loaded with f, and the top surface with f. Applying Laplace transform to f and F () s were obtained. By applying Laplace transform to boundary conditions and vibration equations respectively, we get: (6) U C g ( D s+ B+ C s ) U+ = 0 z s U F() s = z z= h E A U Z U + = 0 z z= h E A z= h K b, 99

st International Conference on Microstructure Related Durability of Cementitious Composites 3-5 October 008, Nanjing, China Because f and self weight are different forces, the displacement can be divided into two parts. Let: U = V + W (7) Where V is the displacement of the second section of manhole under the action of f, and W is the displacement of the second part manhole under the action of self weight. By substituting Eq.(7) into Eq.(6), we can obtain the displacement impedance of the second part manhole under the action of f and that of self weight: Fs () (8) Z() s = = E A γ tan( β γ H ) Vzs (,) zh = E A C g/ s sin( β) Z() s = = E A /( ) Wzs (,) sin( β γ H) γ zh = where tan( β) = Z/( E A γ), H = h h, γ = ( D s+ B + C s ). In turn we can obtain the displacement impedance of the section of the manhole under the action of f + f and corresponding self weight: Z () s = F()/ s V (,) z s = E A γ tan( β γ H) (9) zh = E A C g sin( β) Z() s = E A/( ) sw = (,) zs sin( β γ H) γ zh = where tan( β) = Z /( E A γ), H = h h, γ = ( D s+ B + C s ). 3.3 The solution of the top section of manhole Concluding all the above we can obtain the displacement impedance function of the top section of the manhole under the action of Ft ( ), its self weight and support force: Zn () s = Fn()/ s Vn (,) zs = E tan( ) 0 n An γn β z n γn Hn (0) = En An Cn g sin( βn) Zn () s = = En An/( ) sw n(,) zs sin( β ) z 0 n γn Hn γ = n where tan( βn) = Zn /( En An γn), Hn = hn, γ n = ( Dn s+ Bn + Cn s ). Therefore top vertical vibration displacement response function of the manhole is: ctg( γn Hn βn) sin( βn) Wn () s = + = ( )/ E / n An Z Z E A γ sin( β γ H ) γ n n n n n n n n n Let: s= i w ϑ ( n = Dn i w+ Bn Cn w ), we can obtain top vertical vibration displacement frequency response function of the manhole: ctg( ϑn H n βn) sin( βn) () Hu ( iw) = ( )/ E / n An E A ϑ sin( β ϑ H ) ϑ n n n n n n n () 90

st International Conference on Microstructure Related Durability of Cementitious Composites 3-5 October 008, Nanjing, China By applying inverse Fourier transform on top displacement frequency response function of the manhole, we can obtain the displacement response function under unit load: ctg( ϑn H n β ) n iwt sin( βn) iwt (3) ut () = IFTH [ u( iw )] = e dw ( )/ E/ ) n An e dw π E A ϑ π sin( β ϑ H ) ϑ n n n n n n n According to convolution theorem, the manhole s top vibration displacement response function under traffic load Ft () is written as : ut () = Ft () ut () = IFTFi [ ( w) Hu ( i w)] (4) ctg( ϑn H n βn) iwt sin( βn) iwt = Fi ( w) e dw Fi ( w) ( )/ E/ ) n An e dw π + E A ϑ π sin( β ϑ H ) ϑ n n n n n n n Where, Fiw ( ) is the result of Ft ( ) s Fourier transform. 4 THE MANHOLE S TOP VIBRATION DISPLACEMENT CURVE 4. Parameters value assignments As an example, a manhole s response is analyzed within the assigned scope of typical parameters. The parameters used are shown in Table. Table : Parameters of manhole sections inner radius/m outer radius/m length/m masonry elastic modulus/gpa 0.0 3.5 0.45 0.7 3.5 3 0.5 0.5 3.5 3 The elastic coefficient of the soil surrounding the manhole K s = 7.5 0 N/m 3 viscosity 6 coefficient C s = 3.0 0 Ns/m 3 4 the underneath soil s spring constant K b = 5.0 0 N/m. The chosen traffic loading is cyclic loading: Ft= F+ F t= t (5) () 0 cos(9.09 ) 0 0cos(9.09 ) By applying Fourier transform to cyclic loading, we obtain: Fw ( ) = 0 πδ( w) + 0 π[ δ ( w+ 9.09) + δ( w 9.09)] (6) The period of vehicles passing manhole is.3s. 4. Manhole top displacement curve By substituting the values in Table and Eqs.(7)(8) into Eq.(5), we can obtain the manhole s top displacement curve, as shown in Fig.: As we can see in Fig., the displacement curve of the top of manhole can be divided into three stages: rapid-growing stage (with a slope larger than ), slow-growing stage (with a slope between 0.0~) and steady-state stage (with a slope smaller than 0.0). 9

st International Conference on Microstructure Related Durability of Cementitious Composites 3-5 October 008, Nanjing, China Figure : Displacement curve of manhole 4.3 The influencing factors on top displacement curve of the manhole Manholes can be varied in cross-section s area and construction materials. According to the survey of the currently used manholes, 0 types of manholes, made by combination of six inds of cross sections and two inds of materials are chosen for the analysis of the displacement. The height of each manhole is set as m, and the thicness of the concrete underlays 0.5m, the height of the concrete big-belly is 0.5m. The elastic modulus of the masonry materials is 3.5GPa, and the elastic modulus of concrete is 5.5GPa. The shapes of the manholes are shown in Fig.3, and the dimensions of the six inds of cross sections are showed in Table. Figure 3: 0 types of manhole Table : Size of manhole s cross section Cross sections 3 4 5 6 Inner radius/m 0.5 0.45 0 0.5 0.45 0 Outer radius/m 0.5 0.70.0.0.0 0.5 Using the same soil s parameters and the loading values in 3., the manhole s top vertical vibration displacement for 0 different manholes in Fig.3 can be obtained. Top vibration displacement (or settlement) of the manholes are showed in Fig.4 - Fig6. 9

st International Conference on Microstructure Related Durability of Cementitious Composites 3-5 October 008, Nanjing, China Figure 4: Displacement curve of masonry manholes Figure 5: Displacement curve of concrete manholes Figure 6: Displacement curve of other manholes As we can see in Fig.4, the vibration displacement of big-top-big-bottom manhole is the smallest, and that of small-top-small-bottom manhole is the biggest. However, the displacement of big-top-small-bottom manhole is more or less the same as that of big-top-bigbottom. Taing building cost into account, big-top-small-bottom manhole seems the optimized one. In Fig.5, concrete canister manhole has a smaller top vibration displacement than the manhole with a concrete foundation only. Moreover the building cost of concrete canister manhole is less than that of concrete foundation manhole, therefore the concrete canister manhole has better comprehensive advantages. As in Fig.6, concrete big-belly manhole has a smallest top vibration displacement, and its building cost is the same with that of concrete underlay manhole. Therefore, concrete bigbelly manhole has better comprehensive advantages considering both building cost and vertical vibration displacement. 93

st International Conference on Microstructure Related Durability of Cementitious Composites 3-5 October 008, Nanjing, China 5. CONCLUSION The following conclusions are obtained by analyzing the traffic loaded manhole s vertical vibration: The manhole s top vertical vibration displacement frequency domain response function and time domain response function have been derived; By utilizing typical manhole s parameters, a manhole s top displacement curve has been obtained. It can be divided into 3 stages: rapid-growing stageslow-growing stage and steady stage. By the analyse of 0 inds of manholes, the favorable structure configurations to resist manhole s settlement are selected: big-top-small-bottomconcrete canister and concrete big-belly manholes. By the analyse of factors affecting top displacement of the manhole, it is found that the manhole s top cross section area and the material s elastic modulus have larger impact on the manhole s top vibration displacement. ACKNOWLEDGEMENT Financial supports for this research project is from the Research Fund for the Doctoral Program of Higher Education (Grant No. is 0070335087) and the Hi-Tech Research and Development Program of China (863 Program. Grant No. is 006AA04Z45) are greatly appreciated by the authors. REFERENCES: [] Wang, T., Wang, K.H. and Xie, K.H. A quasi-analytical solution to longitudinal vibration of pile with variable sections and its application (in Chinese). Chinese Journal of Geotechnical Engineering, 000, (6): 654-658. [] Randolph, M.F. and Worth, C.P. Analysis of deformation of vertically loaded piles. Journal of Geotechnical Engineering Division, ASCE, 978, 04(), 465-488. [3] Randolph, M.F. and Worth, C.P. An analysis of the vertical deformation of pile groups. Geotechnique, 979, 9(4): 43-439. [4] Nova, M., Nogamt, T. and About-Ella, F. Dynamic soil reactions for plane strain case. J Eng. Mech, 978,04 (EM4) :953-959. [5] Lee, K.M. and Xiao, Z.R. A simplified nonlinear approach for pile group settlement analysis in multilayered soils. Can. Geotech, 00, 38(5): 063-080. [6] Nova, M. and Aboul-Ella, F. Impedance functions of piles in layered media. Geotech. Eng. Div. ASCE, 978. 04(6): 643-66. 94