Limited output voltage swing in lm358 opamp
Here is what is probably a better simulation using LTspice and a TI model for the LM358. I modified the circuit slightly, using 5.6K/3.9K for the bias divider to center it near the middle of the no-load output range (~3.7V). Input is 370mV. 

You can see it saturating badly on the first cycle, then it settles down towards the middle of the range.