# Simple Beam - Concentrated Load at Center

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 Center formulas

$$\large{ R = V = \frac{P}{2} }$$

$$\large{ M_{max} \; \left(at\; point \;of\; load \right) = \frac{P\;L}{4} }$$

$$\large{ M_x \; \left( x < \frac{L}{2} \right) = \frac{ P\;x}{2} }$$

$$\large{ \Delta_{max} \; \left(at \;point\; of\; load \right) = \frac{ P\;L^3}{48\; \lambda\; I} }$$

$$\large{ \Delta_x \; \left( x < \frac{L}{2} \right) = \frac{P\;x}{48 \;\lambda\; I} \; \left( 3\;L^2 - 4\;x^2 \right) }$$

### S B Concentrated Load at Center - Solve for R

$$\large{ R = V = \frac{P}{2} }$$

### S B Concentrated Load at Center - Solve for Mmax

$$\large{ M_{max} \; \left(at\; point \;of\; load \right) = \frac{P\;L}{4} }$$

 total concentrated load, P span length, L

### S B Concentrated Load at Center - Solve for Mx

$$\large{ M_x \; \left( x < \frac{L}{2} \right) = \frac{ P\;x}{2} }$$

 total concentrated load, P distance from reaction, x

### S B Concentrated Load at Center - Solve for Δmax

$$\large{ \Delta_{max} \; \left(at \;point\; of\; load \right) = \frac{ P\;L^3}{48\; \lambda\; I} }$$

 total concentrated load, P span length, L modulus of elasticity, λ second moment of area, I

### S B Concentrated Load at Center - Solve for Δx

$$\large{ \Delta_x \; \left( x < \frac{L}{2} \right) = \frac{P\;x}{48 \;\lambda\; I} \; \left( 3\;L^2 - 4\;x^2 \right) }$$

 total concentrated load, P distance from reaction, x modulus of elasticity, λ second moment of area, I span length, L

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