Concept:
In a plate girder, the web primarily resists shear, while the flanges resist bending moments.
The fillet weld joining the web to the flanges must transfer the horizontal shear developed due to bending between these components.
The horizontal shear flow is given by
\[
q=\frac{VQ}{I},
\]
where
\[
V=\text{Shear force},
\]
\[
Q=\text{First moment of area},
\]
and
\[
I=\text{Moment of inertia of the section}.
\]
Step 1: Identify the function of the weld.
The weld transfers the longitudinal (horizontal) shear between the web and the flange.
Without adequate weld strength,
\[
\text{Flange and web would act independently.}
\]
Step 2: Determine the governing force.
The fillet weld is designed so that it safely carries
\[
\boxed{\text{Horizontal shear force between the flange and web}.}
\]
It is not designed primarily for bending stress, vertical shear, or buckling forces.
Therefore, the correct option is
\[
\boxed{(C)\;\text{The horizontal shear force between the flanges and the web plate}.}
\]