Expected norm of matrix product with random unit vector

Question

Let $S_d = \{x \in \mathbb{R}^d : \|x\| = 1\}$ be the unit sphere in $\mathbb{R}^d$. Let $A \in \mathbb{R}^{n \times d}$. Note that \begin{align} \max_{x \in S_d} \|A x\| &= \max \sigma(A) & \text{(spectral norm)} \\ \min_{x \in S_d} \|A x\| &= \min \sigma(A) \\ \operatorname*{\mathbb{E}}_{x \sim \text{uniform}(S_d)} \|A x\|^2 &= \frac{\|\sigma(A)\|^2}{d} \end{align} where $\sigma$ yields the singular values of a matrix (note that $\|\sigma(\cdot)\| = \|\cdot\|_\text{F}$ is the Frobenius norm). Is there a similar expression for $\operatorname*{\mathbb{E}}_{x \sim \text{uniform}(S_d)} \|A x\|$?