# Simple Beam - Concentrated Load at Any Point

on . Posted in Structural Engineering

### diagram Symbols

• 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.

### Simple Beam - Concentrated Load at Any Point formulas

$$R_1 \;=\; V_1 \; ( max.\; when \;\; a < b ) \;=\; P\;b\;/\;L$$

$$R_2 \;=\; V_2 \; ( max.\; when \;\; a > b ) \;=\; P\;a\;/\;L$$

$$M_{max} \; (at \;point \;of \;load ) \;=\; P\;a\;b \;/\; L$$

$$M_x \; ( x < a ) \;=\; P\;b\;x \;/\; L$$

$$\Delta_a \; (at \;point \;of \;load ) \;=\; P\;a^2\;b^2 \;/\; 3\; \lambda \;I \;L$$

$$\Delta_x \; ( x < a ) \;=\; ( P\;b\;x \;/\; 6\; \lambda \;I \;L ) \; ( L^2 - b^2 - x^2 )$$

$$\Delta_{max} \; (at \; x = \sqrt{ \frac{ a\; ( a \;+\; 2\;b ) }{3} } \; when \; a > b ) \;=\; P\;a\;b \; ( a + 2\;b) \; \sqrt{ 3\;a \; ( a + 2\;b ) } \;/\; 27\; \lambda \;I \;L$$

Symbol English Metric
$$\Delta$$ = deflection or deformation $$in$$ $$mm$$
$$a, b$$ = distance to point load $$in$$ $$mm$$
$$x$$ = horizontal distance from reaction to point on beam $$in$$ $$mm$$
$$M$$ = maximum bending moment $$lbf-in$$ $$N-mm$$
$$V$$ = maximum shear force $$lbf$$ $$N$$
$$\lambda$$   (Greek symbol lambda) = modulus of elasticity $$lbf\;/\;in^2$$ $$Pa$$
$$R$$ = reaction load at bearing point $$lbf$$ $$N$$
$$I$$ = second moment of area (moment of inertia) $$in^4$$ $$mm^4$$
$$L$$ = span length of the bending member $$in$$ $$mm$$
$$P$$ = total concentrated load $$lbf$$ $$N$$

Tags: Beam Support