Beam Fixed at Both Ends - 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.

Beam Fixed at Both Ends - Concentrated Load at Any Point formulas

$$R_1 = V_1 \; (max.\; when \; a < b ) \;=\; (P\;b^2\;/\;L^3) \; ( 3\;a + b )$$

$$R_2 = V_2 \; (max.\; when \; a > b ) \;=\; (P\;a^2\;/\;L^3) \; ( a + 3\;b )$$

$$M_1 \; (max.\; when \; a < b ) \;=\; P\;a\;b^2\;/\;L^2$$

$$M_2 \; (max. \;when \; a > b ) \;=\; P\;a^2\;b\;/\;L^2$$

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

$$M_x \; ( x < a ) \;=\; (R_1 \; x) - (P\;a\;b^2\;/\;L^2)$$

$$\Delta_{max} \; (at \; x = \frac{2\;a\;L}{3\;a \;+\; b} \;when \;a > b ) \;=\; 4\;P\;a^3\;b^2\;/\; [\;3 \;\lambda\; I \; ( 3\;a \;+ \;b )^2 \;]$$

$$M_a \; (at \;point \;of \;load ) \;=\; 2 \;P\;a^3\;b^3\;/\;6\; \lambda\; I \;L^3$$

$$\Delta_x \; ( x < a ) \;=\; (2 \;P\;b^2\;x^2\;/\;12\; \lambda \;I \;L^3) \; ( 3\;a\;L - 3\;a\;x - b\;x )$$

$$x \; ( point\; of\; contraflexure\;between\;supports ) \;=\; a\;b^2\;P\;/\;L^2\;R_1$$

Symbol English Metric
$$\Delta$$ = Deflection or Deformation $$in$$ $$mm$$
$$x$$ = 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 $$lbf\;/\;in^2$$ $$Pa$$
$$I$$ = Moment of Inertia $$in^4$$ $$mm^4$$
$$R$$ = Reaction Load at Bearing Point $$lbf$$ $$N$$
$$L$$ = Span Length Under Consideration $$in$$ $$mm$$
$$a, b$$ = Span Length Under Consideration $$in$$ $$mm$$
$$P$$ = Total Concentrated Load $$lbf$$ $$N$$

Tags: Beam Support