I have a single angle connected to a double angle via a 3/16" weld 
I know there is a factored, applied moment at that connection. I would like to ask how I can go about checking the capacity of this connection for moment? There is a lot of stuff in AISC for welds in Chapter J, partially and fully restrained moment connections but I am not sure where to start. What chapter/section of AISC codifies this kind of problem?
What chapter/section of AISC codifies this kind of problem?
If you're looking for a specific code provision for this exact type of weld, you're out of luck.
However, it is possible to solve this problem using basic mechanics of materials principles. For this case, I'm going to use elastic design. You could also use the instantaneous center of rotation (ICR) method, but that's pretty complicated when you have loading in more than one direction.
The first thing to recognize is whether or not this joint is considered pinned or fixed. If fixed, you have moments to contend with that introduce loading out of the plane of the weld. First, let's introduce the equations that describe the weld section properties, as found in Blodgett's Design of Welded Structures:
If we call $x$ the axial direction of the member, $y$ the vertical, and $z$ transverse, you solve for the different weld force components like so:
$$ f_{xw} = \frac{F_x}{L_w} + \frac{M_y}{S_{yw}} + \frac{M_z}{S_{zw}}, $$
$$ f_{yw} = \frac{F_y}{L_w} + \frac{M_x(0.5b)}{J_w}, $$
$$ f_{zw} = \frac{F_z}{L_w} + \frac{M_x(0.5d)}{J_w}, $$
where $L_w$ is the total weld length and $ f_{xw}, f_{yx}, $ and $ f_{zw} $ represent the component forces of the weld about the noted axis.
To find the resultant weld force, you take
$$ f_w = \sqrt{f_{xw}^2 + f_{yx}^2 + f_{zw}^2}, $$
where $f_w$ is the resultant weld force. This you compare to the allowable weld strength per the AISC code.
Note: this is a simplified approach that does not take flexibility of the joint into account.
