Fixed End Beam Calculator

End moments (hogging, therefore negative):

\[ M_A = M_B = -\frac{wL^2}{12} \]

Positive midspan moment (sagging):

\[ M_{\text{mid}} = +\frac{wL^2}{24} \]

Vertical support reactions (equal by symmetry):

\[ R_A = R_B = \frac{wL}{2} \]

Maximum deflection at midspan:

\[ \delta_{\max} = \frac{wL^4}{384\,EI} \]

ⓘ Units: w in kip/ft converted to kip/in; L in ft converted to inches; E in ksi; I in in⁴. Result in inches. Compare with simply supported: 5wL⁴/384EI — fixed beam deflects 5× less.

\[ \delta(x) = \frac{w\,x^2(L-x)^2}{24\,EI} \]

x and L in inches, w in kip/in. Returns deflection in inches at any point x along the beam.

Left support fixed-end moment:

\[ M_A = -\frac{P\,a\,b^2}{L^2} \]

Right support fixed-end moment:

\[ M_B = -\frac{P\,a^2\,b}{L^2} \quad \text{where } b = L - a \]

Left reaction:

\[ R_A = \frac{P\,b^2\,(L + 2a)}{L^3} \]

Right reaction:

\[ R_B = \frac{P\,a^2\,(L + 2b)}{L^3} \]

Note: When a = b = L/2 (central load), Mₐ = Mₛ = −PL/8 and Rₐ = Rₛ = P/2. Max deflection at center = PL³/192EI.

For x ≤ a:

\[ \delta(x) = \frac{P\,b^2\,x^2\bigl(3a\,L - (3a+b)\,x\bigr)}{6\,EI\,L^3} \]

For x > a:

\[ \delta(x) = \frac{P\,a^2\,(L-x)^2\bigl(3b\,L - (3b+a)(L-x)\bigr)}{6\,EI\,L^3} \]

All dimensions in inches, P in kips, E in ksi, I in in⁴. Deflection result in inches.

\[ M_A = -\frac{wL^2}{20}, \qquad M_B = -\frac{wL^2}{30} \]

\[ R_A = \frac{3\,wL}{20}, \qquad R_B = \frac{7\,wL}{20} \]

w = maximum intensity at right end (kip/ft). Reactions are unsymmetrical because the load is heavier toward the right.

The fixed-end moments for a UDL w acting from x = c₁ to x = c₂ are derived by integrating the moment influence function:

\[ M_A = -\frac{w}{L^2}\int_{c_1}^{c_2} x(L-x)^2\,dx \]

\[ M_B = -\frac{w}{L^2}\int_{c_1}^{c_2} x^2(L-x)\,dx \]

Closed-form expansion used in the calculator:

\[ M_A = -\frac{w}{L^2}\left[L^2\frac{c_2^2-c_1^2}{2} - 2L\frac{c_2^3-c_1^3}{3} + \frac{c_2^4-c_1^4}{4}\right] \]

This is the exact closed-form result of integrating the fixed-end moment influence function. When c₁ = 0 and c₂ = L, this reduces to the full-span UDL formula wL²/12.

\[ f_b = \frac{M_{\max}}{S_x} \]

Mₚ₊ₓ = maximum moment (in kip·in, converted from kip·ft × 12); Sₓ = elastic section modulus (in³). Result in ksi. This is the extreme fiber bending stress.

Plastic moment capacity:

\[ M_p = F_y \cdot Z_x \]

LRFD design strength (φₛ = 0.90):

\[ \phi_b M_n = \phi_b M_p = 0.90\,F_y Z_x \]

ASD allowable strength (Ωₛ = 1.67):

\[ \frac{M_n}{\Omega_b} = \frac{F_y Z_x}{1.67} \]

Valid for compact sections fully braced laterally. Fʏ in ksi, Zₓ in in³. Result in kip·in, converted to kip·ft for display. AISC 360-22, Section F2.

\[ \delta_{\text{allow}} = \frac{L}{n} \]

where n = 360 (live load floor), 240 (total load), 180 (roof), 480 (sensitive finishes), or user-defined.

\[ \text{Utilization Ratio} = \frac{\delta_{\max}}{\delta_{\text{allow}}} \leq 1.0 \implies \textbf{PASS} \]

δₚ₊ₓ in inches (or mm), L in same units. Per IBC / AISC Design Guide 3 recommendations.

For n load cases, total reactions and moments at each point x are:

\[ M_A^{\text{total}} = \sum_{i=1}^{n} M_{A,i}, \qquad V(x)^{\text{total}} = \sum_{i=1}^{n} V_i(x) \]

\[ M(x)^{\text{total}} = \sum_{i=1}^{n} M_i(x), \qquad \delta(x)^{\text{total}} = \sum_{i=1}^{n} \delta_i(x) \]

This is the principle of superposition, valid for linear elastic materials and small deflections. The calculator solves each load case separately and then sums all contributions. This is the standard approach used in AISC tables.