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Final lining in mechanized excavation includes the precast concrete segments. These segments are designed for applied loads during construction, moving, depot, assembling in the tunnel and service loads that are affected by earth conditions. One of these loads that are applied to the segments after assembling in the ring, are TBM jack loads especially when the TBM should excavate squeeze zones with single mode. As the jack pad section is smaller than the segment section, it causes splitting loads in the segment. Symmetric prism method is an approximate solution to evaluate these forces. In this paper, calculated results by this method are compared to that of numerical solution by ANSYS software. It shows 10 - 20 percent difference between numerical and analytical results.

Final lining in mechanized excavation includes the precast concrete segments. After the excavation, these linings will be installed within shield by Tunnel boring machine (TBM). TBM auxiliary jacks aimed to function after each excavation cycle to install precast segmental linings, moving back part and pulling TBM back up [

Since the plane sections are not remained plane exactly beyond the concentrated load, the Euler-Bernoulli beam theory cannot be applicable. Contours of compressive stress on beam due to concentrated loads are shown at

centerline of the member and perpendicular to that are shown at

Totally, the applied load is distributed along the depth and width of the restrained area where transverse reinforcements are required at each perpendicular ends (vertically and horizontally at sections of restrained areas). The required reinforcements at each direction are obtained by two dimensional analysis where vertical and horizontal transverse tensile loads are calculated and obtained by vertical and horizontal load distribution, respectively. The truss model is one of the simple models to analyze such sections. The truss action is considered at this simple model which has been shown in

The resulting vertical tensile load, T_{b}, is located at a distance from load application plane at curve centroid of transverse tensile stress, according to _{b} and T_{b} is approximately equal to D/2. It seems that this estimation can be used for cracked concrete area. Where,

The approximate method of symmetric prism can be used for multiple concentrated loads. The depth of symmetric prism (D_{e}) for a concentrated load can be considered the smallest distance along the transverse tension from restrained center to the closest adjacent restrained center and two times of the distance along transverse tension from restrained center to the closest edge of the restrained area. It can be observed that tensile load due to over-break moment (M_{s}) at edges of the beam can be developed as well as tensile loads due to splitting moments (M_{b}) while concentrated load and also concentrated load out of the axis exist. Where, maximum overbreak moment and internal tensile load due to moments can be obtained from following equations.

According to this method, the reinforcements are applied vertically and horizontally and the transverse reinforcements which are calculated thereof shall be placed through all parts of the restrained area that is vulnerable to cracking. Therefore, the steel cross section A_{sb} shall be uniformly distributed at the distance of 0.2 D_{e} to D_{e} from end area of the loading.

Where, the concentrated load is applied at a small surface of the beam section, the tensile loads along horizontal and vertical directions are developed. Closed stirrups are used to meet the horizontal and vertical reinforcements which are extended to a distance of the smaller dimension of the beam.

Based on analog truss of the segment, the jack loads along longitudinal direction and also segment thickness are distributed as compressive stress. The tensile loads along longitudinal direction of the segment and segment thickness are carried by longitudinal reinforcements and reinforcements along segment thickness, respectively.

The transverse tensile load will be considered and

calculated along longitudinal and radial direction for the segment. Using symmetric prism method, the transverse tensile load for multiple concentrated loads is obtained as follows.

where, F_{p}: the maximum load of each jack, L_{pi}: the width of the pad under the jack, L_{e}: effective length of the segment or the depth of the symmetric prism. The depth of the symmetric prism for calculating the transverse tensile load along longitudinal direction of the segment is equal to the smallest distance of two concentrated loads of the jacks (106 cm) and two times of the distance of influence point of concentrated load of the jack to the closest edge of the concentrated load application area (114 cm). Thus, the value of L_{e} is considered 106 cm.

The depth of the symmetric prism is considered equal to segment thickness (30 cm) to calculate the transverse tensile load along radial direction of the segment.

On the other hand, as it was described for stress distribution due to two concentrated loads, the tensile load due to over-break moment is developed at edge of the segment and between two concentrated compressive loads.

where,

The full scale segment is modeled by ANSYS software to control the symmetric prism equations to design reinforcements of segment under jack loads during TBM advance (

Concrete having compressive strength of 40 MPa, tensile strength of 4 MPa and modulus of elasticity equal to 30.2 GPa is used. The reinforcements having yield tensile strength of 400 MPa is modeled. Three dimensional elements, SOLID65 and SOLID45 have been used to mesh and model concrete and pad, respectively [

According to

The loading gradual trend within 31 steps has been written by APDL commands to approach to each jack thrust having 155 tons and apply to the model.

The Von Mises stress at the end of the loading at Step 31 is shown in

symmetric prism pattern to consider the transverse tensile loads (splitting loads) along longitudinal direction (T_{b1}), radial direction of the segment (T_{b2}) and also tensile loads due to over-break moment between two jacks (T_{s}), (

According to analytical calculations, this load is equal to 23 tons in accordance with symmetric prism pattern that longitudinal reinforcements are controlled within transverse range 0.2L_{e}-L_{e} (i.e. 20 - 96 cm) to carry the load. Stress distribution (S_{X})_{ }at_{ }section A-A has been used to calculate this load within this surface (

where,

According to analytical calculations, this load is equal to

9.5 tons in accordance with symmetric prism pattern that longitudinal reinforcements are controlled within transverse range 0 - 0.2L_{e} (i.e. 0 - 20 cm) to carry the load. Stress distribution (S_{X}) at section B-B has been used to calculate this load within this surface (

where,

According to analytical calculations, this load is equal to 18 tons in accordance with symmetric prism pattern that ladder reinforcements are controlled within transverse range 0.2L_{e}-L_{e} (i.e. 6 - 30 cm against jack pad having the width of 45 cm) to carry the load. Stress distribution (S_{Y}) at_{ }section C-C has been used to calculate this load within this surface (

where,

One of these loads that are applied to the segments after assembling in the ring, are TBM jack loads especially

when the TBM should excavate squeeze zones with single mode. As the jack pad section is smaller than the segment section, it causes splitting loads in the segment. An approximate solution to evaluate these forces is used of the symmetric prism method. In this paper, calculated results by this method are compared to that of numerical solution by ANSYS software. It shows 10 - 20 percent difference between numerical and analytical results.

The authors would like to thank the Dr. Mohsen Gerami for his kind help.