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E-1 Appendix E. Slab-Base Interface Shear Bonding Model The interface shear bonding, ï¤ , in Eq. E.1 is expressed by the ratio of in-situ shear stress, 2( ) fzx ï±ï´ , in the base course on the slab-base interface, and the shear stress, 2v , in the base layer on the interface when full shear is transferred. 2 2 2 ( ) , ( ) f f zx zx fsv ï± ï± ï´ ï¤ ï´ï½ ï£ (E.1) where fs is the shear strength of the base course. Figure E.1. Illustration of In-situ Shear Stress in the Base Course on the PCC-Base Interface Using a Mohr-Coulomb Failure Envelope. Figure E.1 shows a schematic figure of the developed shear stress in the Portland Cement Concrete (PCC)-base interface. The parameters 1v and 2v are shear stress on the interface in the slab and base layer, respectively, when full shear is transferred. The parameter 2( ) fzx ï±ï´ is the shear stress in the base course on the interface for in situ conditions, which is limited by the shear strength, fs , of the base course on the failure plane. When 2( ) fzx ï±ï´ is greater than 2v , the interface is considered as fully bonded. Depending on the ratio of the 2( ) fzx ï±ï´ and 2v , the partial bonding condition is defined in the PCC-base interface. The following section derives the expressions of 2v and 2( ) fzx ï±ï´ . FORMULATION OF V2 The shear stress acting on the two faces of PCC-base interface can be expressed using the theorem of elastic beam shear stress on the transformed section. It is assumed that full shear is transferred through the interface of the transformed section: 1 1 VQv Ib ï½ ; (E.2) 2 2 ( )b s VQv EIb E ï½ (E.3)
E-2 where V is the shear force acting on the cross section. The parameter Q is the first moment of area from the neutral axis of the transformed section; herein 1 1 1Q A dï½ and 2 2 2Q A dï½ . Substituting Eq. E.2 into Eq. E.3 yields: 2 2 1 1 * ( / )b s Qv v Q E E ï½ (E.4) where 1v is determined using the Boussinesq point load solution (97) Figure E.2 illustrates the shear stress acting on the PCC-base interface. Figure E.2. Stresses in Slab-Base Interface Caused by a Point Load. 2 2 2 5/2 2 2 1/ 2 2 2 1/ 2 3 (1 2 ) 2 ( ) ( ) [( ) ] s x s s s s h aP h a h a h a h ï®ï³ ï° ï© ï¹ï ï½ ïïª ïºï« ï« ï« ï«ï« ï» (E.5) 3 2 2 5/2 3 2 ( ) s z s hP h a ï³ ï° ï½ ï« (E.6) 2 1 2 2 5/2 3 2 ( ) s zx s ahPv h a ï´ ï° ï© ï¹ ï½ ï½ ïª ïºï«ï« ï» (E.7) where P is the surface point load. hs is the thickness of the slab.
E-3 a is the horizontal distance of target point from P. ï® is the Poissonâs ratio. From Eqs. E.4 and E.7, it is obtained that: 2 2 2 2 5/2 ( )3 2 2 ( ) ( ) 2 b b s s ss s hh h zahPv hh a h zï° ï« ïï© ï¹ ï½ ïª ïºï«ï« ï» ï (E.8) FORMULATION OF (Î¤ZX)2ÎF The expression of in-situ shear stress in the base course on PCC-base interface is derived using the Mohr-Coulomb failure envelope, as shown in Figure E.3. The failure envelope is defined by the shear strength parameters (i.e., cohesion, c, and friction angle, ï¦ ). Figure E.3. Maximum Shear Strength of Base Course in Mohr Coulomb Failure Envelope. 2( ) sin2fzx rï±ï´ ï±ï½ (E.9) where r is the radius of the Mohrâs circle. Herein, the state of plane stress on the slab-base interface is defined by xï³ , zï³ , and zxï´ , which is rotated by an angle of 2ï± from principal plane of stress. The angle of rotation, 2ï± , is expressed in Eq. E.10.
E-4 ï¨ ï© 2 2 2 2 3 2 2 2 2 1/22 2 3 ( ) tan2 3( )1 (1 2 )[ ]2 2 ( ) s szx z x s s s s s ah h a h h a h a h a h ï´ï± ï³ ï³ ï® ï© ï¹ ïª ïºï«ï« ï»ï½ ï½ ï ï ï ï« ï« ï© ï¹ï« ï«ïª ïºï« ï» ï¨ ï© 2 2 2 2 1 3 2 2 2 2 1/22 2 3 ( ) 2 tan ( ) 3( )1 (1 2 )[ ] 2 ( ) s s s s s s s ah h a h h a h a h a h ï± ï® ï ï© ï¹ ïª ïºï«ï« ï»ï ï½ ï ï ï« ï« ï© ï¹ï« ï«ïª ïºï« ï» (E.10) As illustrated by Figure E.3, the shear strength of the base course is calculated by Eqs. E.11 and E.12: c o sfs r ï¦ï½ (E.11) ta n [ s i n ] t a nf fs c c rï³ ï¦ ï³ ï¦ ï¦ï½ ï« ï½ ï« ï (E.12) where fï³ is the normal stress on the failure plane. Therefore: ( tan )cosr c ï³ ï¦ ï¦ï½ ï« (E.13) Herein, 3 2 2 2 5/2 2 2 1/2 2 2 1/2 3( ) (1 2 )[ ] 2 4 ( ) ( ) [( ) ] z x s s s s s s h h aP h a h a h a h ï³ ï³ ï®ï³ ï° ï« ï« ï ï½ ï½ ï ï« ï« ï« ï« (E.14) According to Eqs. E.9, E.10, and E.13, it is obtained that: ï¨ ï© 2 2 2 2 1 2 3 2 2 2 2 1/22 2 3 ( ) ( ) ( tan )cos sin[tan ( )] 3( )1 (1 2 )[ ] 2 ( ) f s s zx s s s s s ah h a c h h a h a h a h ï±ï´ ï³ ï¦ ï¦ ï® ï ï© ï¹ ïª ïºï«ï« ï»ï½ ï« ï ï ï« ï« ï© ï¹ï« ï«ïª ïºï« ï» (E.15) The expression of interface shear bonding, ï¤ , is derived from Eqs. E.1, E.8, and E.15:
E-5 ï¨ ï© 2 2 2 2 1 3 2 2 2 2 1/22 2 2 2 2 5/2 3 ( ) ( tan )cos sin[tan ( )] 3( )1 (1 2 )[ ] 2 ( ) ( )3 2 2 ( ) ( ) 2 s s s s s s s b b s s ss s ah h a c h h a h a h a h hh h zahP hh a h z ï³ ï¦ ï¦ ïµ ï¤ ï° ï ï© ï¹ ïª ïºï«ï« ï»ï« ï ïï« ï« ï© ï¹ï« ï«ïª ïºï« ï»ï½ ï« ïï© ï¹ ïª ïºï«ï« ï» ï (E.16) The ranges of the parameters in ANN model are given in Table E.1. Table E.1. Selected Range of Input Parameters in ANN Training Dataset Input parameters Level Input values PCC thickness (mm) 3 178, 254 and 348 Base thickness (mm) 3 101.6, 203.2 and 254 PCC modulus (MPa) 3 14420, 41400, and 82737 Base modulus (MPa) 4 69, 690, 6894 and 25000 Subgrade modulus (MPa) 3 34.5, 282 and 551 PCC-base interface bonding 4 0, 0.3, 0.6, and 1