Previous studies have shown that when using carbon nanotubes (CNTs) as tapping-mode AFM probes, their tilt angle with respect to vertical (denoted 0) must be close to 0 degrees to obtain high-quality images and that very poor images are obtained for phi > 30 degrees. Here we present a quantitative theoretical investigation of the effect of 0 on tapping-mode AFM imaging for single-wall and multiwall nanotube (SWNT and MWNT, respectively) probes of diameters 3.4-5.5 nm and aspect ratio 7.5, which have been found ideal for imaging via TEM. Using molecular and classical dynamics, we investigate the effect of phi on CNT probe stiffness (quantified through the maximum gradient of the tip-sample interaction force) and show that it decreases linearly with increasing phi, becoming negligible at around phi approximate to 40 degrees, thus confirming the conclusions of previous studies. We find that MWNT probe stiffness is proportional to the number of walls, but that the difference in stiffness between SWNTs and MWNTs also decreases linearly with increasing 0 and becomes negligible at around phi approximate to 40 degrees. The simulated cross-sectional scans of a sample SWNT using two different values of phi show that the image can be distorted and shifted laterally when phi is large, in some cases giving measured heights appreciably greater than the sample dimensions. We show analytically that the tip-sample forces that occur during imaging can be significantly lower when CNT probes are used instead of conventional probes, even in the absence of buckling, and that they can be further reduced by increasing phi. On the basis of this result, we propose the design of free-standing kinked probes for the characterization of sensitive samples, whereby the probe approaches the sample at a vertical orientation and possesses a tilted section that regulates the tip-sample interaction forces.