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

on . Posted in Structural Engineering

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

$$\large{ R_1 = V_1 \; \left(max.\; when \; a < b \right) \;\;=\;\; \frac {P\;b^2} {L^3} \; \left( 3\;a + b \right) }$$

$$\large{ R_2 = V_2 \; \left(max.\; when \; a > b \right) \;\;=\;\; \frac {P\;a^2} {L^3} \; \left( a + 3\;b \right) }$$

$$\large{ M_1 \; \left(max.\; when \; a < b \right) \;\;=\;\; \frac {P\;a\;b^2} {L^2} }$$

$$\large{ M_2 \; \left(max. \;when \; a > b \right) \;\;=\;\; \frac {P\;a^2\;b} {L^2} }$$

$$\large{ M_a \; \left(at \;point \;of \;load \right) \;\;=\;\; \frac {2\;P\;a^2\;b^2} {L^3} }$$

$$\large{ M_x \; \left( x < a \right) \;\;=\;\; R_1\; x - \frac {P\;a\;b^2} {L^2} }$$

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

$$\large{ M_a \; \left(at \;point \;of \;load \right) \;\;=\;\; \frac {2 \;P\;a^3\;b^3} {6\; \lambda\; I \;L^3} }$$

$$\large{ \Delta_x \; \left( x < a \right) \;\;=\;\; \frac {2 \;P\;b^2\;x^2} {12\; \lambda \;I \;L^3} \; \left( 3\;a\;L - 3\;a\;x - b\;x \right) }$$

$$\large{ x \; \left( point\; of\; contraflexure\;between\;supports \right) \;\;=\;\; \frac{a\;b^2\;P} {L^2\;R_1} }$$

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

## diagrams

• 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. 