Relationships between fixed end moments and deformation values in the simply-supported beam

Abstract

For the young statician who knows Clapeyron’s theorem of three moments for continuous beams, Fig. 1, where, for constant moment of inertia, it takes the form

$$M_{1}l+ 2M_{2}(l+{l}') + M_{3}{l}' + 6(\alpha_{0}^{'}+\beta_{0})=0$$

it often happens that using the equation for this case presents difficulties in connection with imagination, due to the disappearance of the angles at the support when the ends are fully builtin. This can be overcome by the following artifice: If the middle span $\overline{BC}$ of length l₂ of the three-span beam shown in Fig. 2 is examined, it will immediately be realised that the angles of rotation α and ß are smaller than would be the case with a single beam BC with no lateral spans. If we assume for the time being that only the span BC is loaded, which would be the case anyway if there were no lateral spans (Fig. 2 b) for the fixed span l₂, it is immediately obvious that the lateral spans l₁ and l₃ act as elastic fixings on the middle span l₂. The degree of elasticity of these fixings is determined by the beam properties of the lateral spans.