An external periodic load is considered to act on a fluid-conveying pipe clamped at both ends, and the nonlinear forced vibration for such a system is explored by the multidimensional Lindstedt-Poincaré (MDLP) method. According to the analysis, when the second natural frequency of the system is nearly thrice the first one, and the excitation frequency is near the middle of first two natural frequencies, a combination resonance with internal resonance may occur. The characteristics of this response are discussed, where the motions of first two modes are investigated in detail. The influence of excitation amplitude on the internal resonance is analyzed. Numerical examples reveal rich and complex dynamic behaviors caused by internal resonance and show that the occurrence tendency of internal resonance will die down and the response forms will vary with the excitation amplitude increasing. The convenience and efficiency of the MDLP method in predicting nonlinear dynamics are as well demonstrated by the results of the study.