Simple Beam - Uniform Load Partially Distributed at One End

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 - Uniform Load Partially Distributed at One End formulas

$$R_1 \;=\; V_1 \;=\; (w \;a\;/\;2\;L) \; ( 2\;L - a )$$

$$R_2 = V_2 \;=\; ( w \;a^2 \;/\;2\;L )$$

$$V_x \; ( x < a ) \;=\; R_1 - w\;x$$

$$M_{max} \; ( at \; x = R_1\;/\;w ) \;=\; R_{1}{^2} \;/\; 2\;w$$

$$M_x \; ( x < a ) \;=\; (R_1 \; x) - (w\;x^2\;/\;2)$$

$$M_x \; ( x > a ) \;=\; R_2 \; ( L - x )$$

$$\Delta_x \; ( x < a ) \;=\; ( w\; x \;/\; 24\; \lambda \;I \;L) \; [\; [\; a^2 \; ( 2\;L - a )^2\;] - [\; 2\;a\;x^2 \; ( 2\;L - a )\;] + L\;x^3 \;]$$

$$\Delta_x \; ( x > a ) \;=\; [\; w\; a^2 \; ( L - x ) \;/\; 24\; \lambda \;I \;L \;] \; ( 4\;x\;L - 2\;x^2 - a^2 )$$

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

Tags: Beam Support