We present a formula for computing proper pushforwards of classes in the Chow ring of a projective bundle under the projection $\pi : \mathbb{P}(\mathcal{E})\rightarrow B$, for $B$ a non-singular compact complex algebraic variety of any dimension. Our formula readily produces generalizations of formulas derived by Sethi,Vafa, and Witten to compute the Euler characteristic of elliptically fibered Calabi-Yau fourfolds used for F-theory compactifications of string vacua. As an application, we show that their is a an orientifold limit of a $D_5$ model of F-theory admitting a unique configuration of smooth branes satisfying the tadpole matching condition between F-theory and type IIB.