when we make a case , say e.g.-
operating ( T3, D3)+ dynamic hogging (D1)+ static hogging (D4) + Normal ship accleration/ deflection in X direction (U1,D6)+Normal ship accleration/ deflection in Z direction (U3,D8)which is -
W+D1+D3+D4+D6+D8+T3+P1+H+U1+U3 -
My question is- U1 is acceleration in +ve X and U3 is acceleration in +ve Z. Corresponding structural deflections for tall structures supported from main deck are D6 and D8. Normal practice is D6 and D8 used in these cases are also positive as received from structural design group corresponding to +ve value of U1 and U3 respectively.But as per dynamic behavior when D6 is positive the corresponding inertial acceleration on pipe has to be in opposite direction and hence both U1 and D6 can not have same sign in this case. The signs should be opposite . Same logic for U3 and D8. PLEASE ADVICE .

Let's see if I made sense of all your inputs.
T3 = operating temperature

D1 = static hogging, which would mean up/down if at the midline of the vessel, but everywhere else would mean up/down and rotate in/out, but happens once per trip.
D3 = dynamic hogging, which would mean up/down if at the midline of the vessel, but everywhere else would mean up/down and rotate in/out, and this happens all day, every day.

U1 = acceleration forward (increasing velocity forward with propellers)
U3 = deceleration forward (decreasing velocity forward with propellers)

D6 & D8 = pipe in tall structures waving around above as an effect of hogging. Are the structures capable of waving in a chaotic fashion, or strictly as a function of D3, U1, and U3, which is modified by D1?

Ultimately, your goal is to get CAESAR to simulate the shape the pipe is going to be in at any phase of its journey, calculate stresses, and as applicable, fatigue life.

I would caution you that you need to scrutinize the addition of like vectors (e.g. displacement vectors). I believe CAESAR averages them by default, and thus e.g. D1+D2 might not give you what you want.