Three Member Frame - Pin/Roller Central Bending Moment

Three Member Frame - Pin/Roller Central Bending Moment formulas

\(\large{ R_A = R_B = \frac{M_C}{L} }\)
\(\large{ H_A = 0 }\)
\(\large{ M_{max} \;(at \; C) = \frac{M_C}{2} }\)
\(\large{ \theta \;(at \; C) = \frac{M_C\;L}{12 \; \lambda \; I} }\)

Units	English	Metric
\(\large{ h }\) = height of frame	\(\large{in}\)	\(\large{mm}\)
\(\large{ H }\) = horizontal reaction load at bearing point	\(\large{lbf}\)	\(\large{N}\)
\(\large{ I_h }\) = horizontal member second moment of area (moment of inertia)	\(\large{in^4}\)	\(\large{mm^4}\)
\(\large{ I_v }\) = vertical member second moment of area (moment of inertia)	\(\large{in^4}\)	\(\large{mm^4}\)
\(\large{ M }\) = maximum bending moment	\(\large{lbf-in}\)	\(\large{N-mm}\)
\(\large{ \lambda }\) (Greek symbol lambda) = modulus of elasticity	\(\large{\frac{lbf}{in^2}}\)	\(\large{PA}\)
\(\large{ A, B, C, D, E }\) = point of intrest on frame	-	-
\(\large{ \theta }\) = slope of member	\(\large{deg}\)	\(\large{rad}\)
\(\large{ L }\) = span length under consideration	\(\large{in}\)	\(\large{mm}\)
\(\large{ R }\) = vertical reaction load at bearing point	\(\large{lbf}\)	\(\large{N}\)

Bending moment diagram (BMD) - Used to determine the bending moment at a given point of a structural element. The diagram can help determine the type, size, and material of a member in a structure so that a given set of loads can be supported without structural failure.
Free body diagram (FBD) - Used to visualize the applied forces, moments, and resulting reactions on a structure in a given condition.
Shear force diagram (SFD) - Used to determine the shear force at a given point of a structural element. The diagram can help determine the type, size, and material of a member in a structure so that a given set of loads can be supported without structural failure.
Uniformly distributed load (UDL) - A load that is distributed evenly across the entire length of the support area.