# Three Member Frame - Fixed/Fixed Side Uniformly Distributed Load

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## Three Member Frame - Fixed/Fixed Side Uniformly Distributed Load formulas

### Needed Values

$$\large{ e = \frac{h}{L} }$$
$$\large{ \beta = \frac{I_h}{I_v} }$$

### Support Reactions

$$\large{ R_A = R_D = \frac{ w\;h\; \beta\; e^2 }{ 6\; \beta \;e\;+\;1 } }$$
$$\large{ H_A = \frac{w\;h}{4} \; \left( \frac{8\; \beta\;e\;+\;17 }{2\; \left( \beta\;e\;+\;2 \right) } - \frac{4\; \beta\;e\;+\;3 }{6\; \beta\;e\;+\;1 } \right) }$$
$$\large{ H_D = \frac{w\;h}{4} \; \left( \frac{4\; \beta\;e\;+\;3 }{6\; \beta\;e\;+\;1 } - \frac{1 }{2\; \left( \beta\;e\;+\;2 \right) } \right) }$$

### Bending Moments

$$\large{ M_A = \frac{w\;h^2}{4} \; \left( \frac{4\; \beta\;e\;+\;1 }{6\; \beta\;e\;+\;1 } + \frac{\beta \;e \;+\; 3 }{6\; \left( \beta\;e\;+\;2 \right) } \right) }$$
$$\large{ M_B = \frac{w\;h^2 \; \beta \;e}{4} \; \left( \frac{ 6 }{6\; \beta\;e\;+\;1 } - \frac{ 1 }{6\; \left( \beta\;e\;+\;2 \right) } \right) }$$
$$\large{ M_C = \frac{w\;h^2 \; \beta \;e}{4} \; \left( \frac{ 2 }{6\; \beta\;e\;+\;1 } - \frac{ 1 }{6\; \left( \beta\;e\;+\;2 \right) } \right) }$$
$$\large{ M_D = \frac{w\;h^2}{4} \; \left( \frac{4\; \beta\;e\;+\;1 }{6\; \beta\;e\;+\;1 } - \frac{\beta \;e \;+\; 3 }{6\; \left( \beta\;e\;+\;2 \right) } \right) }$$

### Where:

 Units English Metric $$\large{ h }$$ = height of frame $$\large{in}$$ $$\large{mm}$$ $$\large{ H }$$ = horizontal reaction load at bearing point $$\large{lbf}$$ $$\large{N}$$ $$\large{ I_h }$$ = horizontal member second moment of area (moment of inertia) $$\large{in^4}$$ $$\large{mm^4}$$ $$\large{ I_v }$$ = vertical member second moment of area (moment of inertia) $$\large{in^4}$$ $$\large{mm^4}$$ $$\large{ w }$$ = load per unit length $$\large{\frac{lbf}{in}}$$ $$\large{\frac{N}{m}}$$ $$\large{ M }$$ = maximum bending moment $$\large{lbf-in}$$ $$\large{N-mm}$$ $$\large{ A, B, C, D, E }$$ = point of intrest on frame - - $$\large{ L }$$ = span length under consideration $$\large{in}$$ $$\large{mm}$$ $$\large{ R }$$ = vertical reaction load at bearing point $$\large{lbf}$$ $$\large{N}$$

## 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.

Tags: Frame Support