Why do airplane wings have limit load of both positive and negative g's (e.g., the Weedhopper ultralight has a load factor of +5 and -2gs). Why not just have a positive limit load since most airplanes (like the Weedhopper ultralight) don't fly upside down? In particular, why the negative load factors?
You can get negative load factors (g forces) in different ways than just flying upside down:
Change in pitch: When you push on the control column, the pitch will start to decrease. Depending on how fast you do this, the load factor can even become negative from this. Some aircraft do this intentionally to reduce the g force to exactly zero:
(image source: Wikimedia)
By pushing the control column further forwards, you would get negative gs.
Wind gusts: The limiting factor for airliners1 (which rarely do the maneuver described above) is mostly due to wind gusts and up-/downdrafts. This can (for a short amount of time) result in large changes of the load factor, even into negative gs for a strong gust in the vertical direction. When experiencing strong turbulence, the airspeed should be reduced to turbulent air penetration speed (VB), which reduces the impact of the gusts.
Turbulent Air Penetration
Severe turbulence should be avoided if at all possible. However, if severe turbulence is encountered, use the Severe Turbulence procedure listed in the Supplementary Procedures chapter of the FCOM. Turbulent air penetration speeds provide high/low speed margins in severe turbulent air.
(Boeing 737 NG FCTM 1.50 General Information)
1 Limits for large aircraft from EASA CS-25:
(1) The positive limiting load factor must not be less than:
(i) 2.5 g with the EFCS functioning in its normal mode and with the high-lift devices retracted up to VMO/MMO. The positive limiting load factor may be gradually reduced down to 2.25 g above VMO/MMO.;
(2) The negative limiting load factor must be equal to or more negative than:
(i) -1.0 g with the EFCS functioning in its normal mode and with the high-lift devices retracted;
Five short, generic reasons (i.e., not specific to the Weedhopper):
Fatigue reduction: some highly stressed parts of the airframe (particularly mainspars and engine mounts) are susceptible to fatigue failure from cyclic loading. The key word here is 'cyclic'; a spar which is subjected to loads between (say) -5g/+5g will fail faster than one which is subjected to loads of only -1g/+5g. This can be demonstrated easily by bending a paperclip until it snaps: bending it fully one way and then fully the other will cause it to snap faster than simply bending it fully one way, then straightening it and bending again, etc.
Fuel systems: fully aerobatic aircraft typically have some form of pressurised fuel tank, fuel injection, or similar system to prevent fuel starvation under negative g. Case in point; Merlin engines in the early part of WW2 (Spitfires, Hurricanes, etc.) used a simple updraft carburettor. This would rich-cut under negative g as the fuel in the float bowl moved to the top of the carb, forcing the float to move downward and open the fuel valve all the way. This was fixed later in the war with the addition of pressure carbs, however early in the war this gave the German pilots a potential combat advantage: their fuel injected engines didn't have this flaw, meaning they could simply pull negative g to cause a pursuing allied aircraft to lose power.
Oil and coolant systems: related to point 2, if an engine has an oil pickup at the bottom of a sump (as most car oil systems do) then this can potentially be starved of oil under certain manoeuvres if the sump is improperly baffled. Aircraft intended for aerobatics may be fitted with a different oil system (such as a dry-sump system, which uses a separate oil tank and pressure pumps) to limit damage to the engines due to oil starvation.
Cargo storage: items (overnight bags, life jackets, fire extinguishers, etc.) stored loose under/behind seats may come free and move around the cockpit under negative g if they're not properly secured (admittedly this point involves some degree of common sense, although I've seen pilots at the local airfield who simply put their overnight bag behind the pilot's seat).
Airfoil profile: This point is very dependent on specific aircraft, but a negative g is often accompanied by a negative angle of attack of the wing. Depending on the aircraft in particular, this may lead to flow instabilities over the wing itself, and also the elevator section. This can result in buffeting of the stabilizer/elevator, or a loss of control authority of those surfaces, resulting in (worst case) a departure from controlled flight (stall/spin, etc.) or damage to/structural failure of the tail.
First, I would like to point out the question was about limit load on airplane wings, which is different that load limits for the entire aircraft. Therefore, this sounds like an airframe question, not propulsion.
Let's look from the wing frame of reference: it does not care if aircraft is going level, up, down, sideways, or upside down. All it cares is whether the vertical component of the force is pointing up (positive) from its frame of reference or down (negative).
Non aerobatic wings tend to have non symmetrical profiles since they are optimized for flying right side up. This structural non-symmetry also leads to it being able to handle different loads in the positive vs negative directions.
Flying an aircraft is all about knowing what are the safe limits and staying inside them, be it the minimum takeoff speed for a given load, how much fuel you need to reach a given destination, how much weight you can have in the aircraft while landing, or how far you can push the wings before they snap or at least become rather fatigued.
Incidentally, a continuous maneuver at a given wing load is much less stressful to the wings/airframe than vibrating the same wing to the same wing load and back. Also, modern wings can handle wing load better because they distribute the load instead of concentrating it close to the root, which is why when an old style wing break off it is close to the fuselage.