Appendix A – Evaluation of VHM at Load Reference Point Appendix A Evaluation of VHM at Load Reference Point The model legs were instrumented with levels of axial gauges (A1 & A2) and levels of bending gauges (B1 to B3), as shown in Fig. A1. Calibration of the gauges and transformation of the measured raw readings (micro-strain) into axial force and bending moment are elaborated in Section 3.3.1.5. Rigid connection x l l1 Rigid connection Load reference point (L.R.P.) l2 V Mo Ho l3 Notes: A denotes axial gauges B denotes bending gauges Fig. A1 Instrumented spudcan leg The axial force measured is the force required to penetrate the spudcan into soil under undrained condition. Either one of the axial forces measured at 284 Appendix A – Evaluation of VHM at Load Reference Point A1 & A2 can be used to find V as both give similar values. On the other hand, the rotational moment, Mo and horizontal force, Ho at the load reference point (L.R.P.) cannot be measured directly from the experiments but they can be evaluated through extrapolation of the bending moment profiles on the leg as measured by MB1 to MB3 at levels B1 to B3. Simple Beam Theory The model leg has a uniform geometry and constant flexural rigidity, EI. With no consideration of the axial force, simple beam theory can be used to calculate the Ho and Mo at L.R.P as follows: Take moment about B3 M B M o H o l3 ……………………………… (A.1) Take moment about B2 M B2 M o H ol2 ……………………………… (A.2) => (A.2) – (A.1) M B M B H o (l3 l ) Hence , Ho M B2 M B3 l3 l ……………………………… (A.3) Substitute (A.3) to (A.1) or (A.2) to calculate M o M o M B H o l3 or M o M B2 H ol2 The other bending moment, MB1, can be used to crosscheck the Mo and Ho obtained above. The above equations give sufficiently accurate approximation of Mo and Ho provided the structure deformation is sufficiently small where 285 Appendix A – Evaluation of VHM at Load Reference Point the P-∆ effect is negligible. When this is not the case, beam-column theory has to be employed to calculate Mo and Ho. Beam Column Theory: A beam-column is a member where both bending moment and compressive (axial) force are acting at the same time. On some occasions, the deflections may be large enough to add a significant additional moment on the structure. The analysis of a beam-column therefore requires consideration of the effect on the equations of equilibrium of the change of geometry of the bar due to deformation. The governing equation: EIw Pw q (x) …………………………… (A.4) ………… (A.5) w A cos kx B sin kx Cx D f ( x ) in which P , P axial load EI & f ( x ) is a particular solution for the lateral load q ( x ) k The model leg has a uniform geometry and material property (constant EI). The general solutions have the following forms: Deflection w A cos kx B sin kx Cx D f ( x ) Slope , w k ( A sin kx B cos kx ) C f ( x ) Moment , M EI w P ( A cos kx B sin kx ) EI f ( x ) Shear force , Q EI w kP ( A sin kx B cos kx ) EI f ( x ) ( A.5a ) ( A.5b ) ( A.5c ) ( A.5d ) Horizontal force , H EI w Pw EI f ( x ) Pf ( x ) PC ( A.5e ) The integration constants A, B, C and D are to be determined from the boundary conditions of each beam-column. 286 Appendix A – Evaluation of VHM at Load Reference Point For all cases studied in this thesis, q(x) = and the axial load, P is V that were measured by the axial gauges (see Fig. A1). Boundary conditions : 1. No deflection at rigid connection : w( x 0) 0; A D …………… (A.6) …………… (A.7) 2. No rotational displacement at rigid connection : w( x 0) Bk C Consider the above two boundary conditions, (A.5c) and (A.5e) become: M (x l) M o V ( A cos kl B sin kl ) M o at x l ; ……………. (A.9) (P V ) Ho V hence H B o kV & A (A.8) H Ho PC H o C …………… H sin kl Mo o V cos kl k ………… (A.10) …………… (A.11) D A H o sin kl Mo V cos kl k Hence, Deflection , w Moment , M … (A.12) H sin kl H H Mo o cos kx 1 o sin kx o V cos kl k kV Vx H sin kl H o cos kx sin kx Mo o cos kl k k (A.13) (A.14) 287 Appendix A – Evaluation of VHM at Load Reference Point Rearrange (A.14) General forms: k Ho tan kl a tan kl b Mo Ma Mb cos kl a cos kl b H cos kl M a o sin kl tan kl a cos kl cos kl a k ……………… (A.15) ……………… (A.16) Substitute any pair of MB1, MB2 and MB3 with the corresponding distance from the rigid connection, l into equations (A.15) and (A.16) to compute Ho and Mo (which are horizontal force, H and moment, M acting at L.R.P.). To calculate the spudcan displacement with respect to L.R.P., the following equation is derived: Tip displacement , wo H sin kl 1 H o sin kl l Mo o 1 V k cos kl V k ………… (A.17) As illustrated above, the three major load components (VHM) at L.R.P can be evaluated from the axial and bending strain measurements of the model leg. Derivation of load inclination angle, and normalized eccentricity, e/D H Load inclination angle, tan 1 V Normalized eccentricity , e M D VD …………………… (A.18) ……………………. (A.19) 288 . time. On some occasions, the deflections may be large enough to add a significant additional moment on the structure. The analysis of a beam-column therefore requires consideration of the effect on. (A.12) (A.13) (A.14) 0 )0( :.2 0 ;0)0( :.1 : CBk xw connectionrigidatntdisplacemerotationalNo DA xw connectionrigidatdeflectionNo conditionsBoundary k klH M klV A kV H B hence V H C VPHPC HHlxat MklBklAV MlxM o o o o o o o o sin cos 1 & )( ; )sincos( )( . and material property (constant EI). The general solutions have the following forms: The integration constants A, B, C and D are to be determined from the boundary conditions of each beam-column. )()(& , )(sincos xqloadlateraltheforsolutionparticularaisxf loadaxialP EI P k whichin xfDCxkxBkxAw )5.()()(, )5.()()cossin(, )5.()()sincos(, )5.(