Based on the Timoshenko beam theory, the finite element equation for bending and vibration of fluid-conveying pipes is derived using the principle of virtual work. The transverse acceleration of fluid is derived using the theorem of acceleration composition. The deflection and slope of the pipe subjected to the combined actions of gravity and fluid under two boundary conditions are obtained, and the influence of fluid velocity on the deflection and slope is analyzed. Under simply supported boundary constraint at both ends the pre-stress effect is transformed to integrate into the strain energy of the pipe, and the relationship between axial pre-stress and bending deflection is studied. The relationships between the first three natural frequencies and the flow velocity of the pipe under the simply supported and cantilever boundary conditions are obtained, and the influence of the axial pre-stress on the natural frequency under the simply supported condition is presented. The results show that under the condition of simply supported boundary, the deflection and slop increase with the increase of fluid velocity, and the deflection and slop decrease with the increase of pre-stress. With the increase of velocity, the first three natural frequencies decrease, while an increase in pre-stress results in higher natural frequencies. For the cantilever boundary condition, the deflection and slope decrease with the increase of fluid velocity, and the first three natural frequencies decrease with increasing flow velocity.