Parameters
Number of spans
Load pattern
Section & Load
Elastic modulus E200 GPa
Second moment I8000 cm⁴
Load intensity q20.0 kN/m
Span lengths (m)
—
M_max+ [kN·m]
—
M_max− [kN·m]
—
Max Reaction [kN]
—
Max Deflection [mm]
Bending Moment Diagram (BMD)
Deflection Diagram
Three-Moment Equation (Clapeyron)
For adjacent spans $i$ and $i+1$:
$$M_{i-1}L_i + 2M_i(L_i+L_{i+1}) + M_{i+1}L_{i+1} = -\frac{q_i L_i^3}{4} - \frac{q_{i+1} L_{i+1}^3}{4}$$(equal EI, uniform distributed loads $q$)
Support reactions (span i):
$$R_{i,L} = \frac{q_i L_i}{2} - \frac{M_{i+1}-M_i}{L_i}, \quad R_{i,R} = \frac{q_i L_i}{2} + \frac{M_{i+1}-M_i}{L_i}$$Location of max positive moment: $x^* = R_{i,L} / q_i$
CAE Note: The three-moment solution provides an ideal benchmark for validating FEM beam element models. It is also the conceptual foundation for the direct stiffness method used in structural FEM software.