According to the basic theory of mechanics of materials, two problems in plane bending of the beam are analyzed: the influences of different sections on the flexural normal strength of the beam and on the maximum shear stress of the cross section of I-beam. The influence of various cross-section shapes on the flexural normal strength and the effect of various I-beam section sizes on the maximum shear stress were obtained quantitatively. The main results are: the flexural normal strength of rectangular section is lower than that of circular section when the rectangular section is transverse and the aspect ratio is greater than 1.369 26, when the vertical degree of I-section is smaller than that of rectangular section and the web of I-section is very short or very wide, the flexural normal strength of I-section is lower than that of rectangular section, for the general section size the maximum shear stress on the I-section is not equal to the nominal average shear stress on the I-section web, only under the specific section size the maximum shear stress on the I-section is equal to the nominal average shear stress on the I-section web. The research results have corrected some vague concepts in the teaching of mechanics of materials and the research process and the research method provide a good case for improving students' scientific literacy and cultivating students'scientific research ability in the teaching of material mechanics which of great significance.