Continuous Steel Beam Calculator - Direct Stiffness Method Analysis
This professional Continuous Steel Beam Calculator uses the Direct Stiffness Method (matrix analysis) to deliver engineering-grade results for indeterminate multi-span beams (up to 8 spans) with cantilevers, support settlements, and complex loading.
Quickly compute support reactions, shear force diagrams (SFD), bending moment diagrams (BMD), and deflection curves. Includes full AISC 360 section checks for flexure, shear, and lateral-torsional buckling (LRFD/ASD), plus live load patterning.
Perfect for steel designers needing fast, reliable analysis of continuous beams without expensive FEA software. Free, instant, and mobile-friendly.
Continuous Steel Beam Calculator
Analyze multi-span indeterminate beams — reactions, shear, moment & deflection — using the Direct Stiffness Method. Free, instant, no signup.
● No Signup | ● AISC / Eurocode | ● Up to 8 Spans| Span # | Length (ft) | Left Support | Right Support | Settlement (in) |
|---|
Support settlement causes moment redistribution in continuous beams — enter 0 if no settlement.
Diagram updates automatically as you change inputs.
When checked, the calculator generates all skip-loaded cases and reports the governing envelope of moments, shears, and reactions.
For a prismatic beam element of length $L$, flexural rigidity $EI$, the element stiffness matrix (4×4, degrees of freedom: $v_1, \theta_1, v_2, \theta_2$) is:
$$[k_e] = \frac{EI}{L^3}\begin{bmatrix} 12 & 6L & -12 & 6L \\ 6L & 4L^2 & -6L & 2L^2 \\ -12 & -6L & 12 & -6L \\ 6L & 2L^2 & -6L & 4L^2 \end{bmatrix}$$This is assembled into the global stiffness matrix $[K]$ by superposition over all beam elements.
After applying boundary conditions (pinned: $v=0$; fixed: $v=0, \theta=0$; roller: $v=0$), the reduced system is:
$$[K_r]\{d\} = \{F_r\}$$where $\{d\}$ is the vector of unknown displacements/rotations, $\{F_r\}$ is the reduced load vector. Solved by Gaussian elimination.
For a full-span uniformly distributed load $w$ (force/length) on a span of length $L$:
$$R_{A,fixed} = R_{B,fixed} = \frac{wL}{2}$$ $$M_{A,fixed} = +\frac{wL^2}{12}, \quad M_{B,fixed} = -\frac{wL^2}{12}$$For a partial UDL from $a$ to $b$, fixed-end reactions are integrated accordingly.
For a point load $P$ at distance $a$ from the left end of a span $L$ (where $b = L - a$):
$$R_A = \frac{Pb^2(3a+b)}{L^3}, \quad R_B = \frac{Pa^2(a+3b)}{L^3}$$ $$M_A = +\frac{Pab^2}{L^2}, \quad M_B = -\frac{Pa^2 b}{L^2}$$The demand/capacity ratio (DCR) for flexure:
$$\text{DCR}_{\text{flex}} = \frac{M_u}{\phi_b M_n}, \quad \phi_b = 0.90 \text{ (LRFD)}$$For compact sections (LTB not governing), $M_n = M_p = F_y Z_x$
Plastic (no LTB) when $L_b \le L_p$:
$$L_p = 1.76 r_y \sqrt{\frac{E}{F_y}}$$Inelastic LTB when $L_p < L_b \le L_r$:
$$M_n = C_b \left[M_p - (M_p - 0.7F_y S_x)\frac{L_b - L_p}{L_r - L_p}\right] \le M_p$$Elastic LTB when $L_b > L_r$:
$$M_n = F_{cr} S_x \le M_p, \quad F_{cr} = \frac{C_b \pi^2 E}{(L_b/r_{ts})^2}\sqrt{1 + 0.078 \frac{Jc}{S_x h_o}\left(\frac{L_b}{r_{ts}}\right)^2}$$where $A_w = d \times t_w$ and $C_{v1} = 1.0$ for $h/t_w \le 2.24\sqrt{E/F_y}$.
$$\text{DCR}_{\text{shear}} = \frac{V_u}{\phi_v V_n}$$Deflection is computed by double integration of the moment diagram:
$$EI \frac{d^2 y}{dx^2} = M(x)$$ $$EI \frac{dy}{dx} = \int M(x)\, dx + C_1$$ $$EI\, y = \int\!\!\int M(x)\, dx + C_1 x + C_2$$Constants $C_1, C_2$ are determined by boundary conditions. For a simply supported reference span:
$$\delta_{max} = \frac{5wL^4}{384EI} \quad \text{(UDL, simply supported)}$$The serviceability check requires: $\delta_{max} \le \dfrac{L}{360}$ (or selected limit)
Support settlement $\Delta_s$ at an interior support introduces additional fixed-end moments. For a propped cantilever or interior support settling by $\Delta_s$:
$$M_{settlement} = \frac{6EI \Delta_s}{L^2}$$These are incorporated directly into the fixed-end force vector $\{F_0\}$ before solving $[K]\{d\} = \{F\}$.
| Application | Deflection Limit | Code Reference |
|---|---|---|
| Office/residential floor — live load | L / 360 | AISC 360, IBC |
| Floor — total load | L / 240 | AISC 360 |
| Roof — live load | L / 180 | AISC 360 |
| Sensitive equipment | L / 480 to L / 600 | Project specific |
| Steel (A36, A992, A572) | E = 29,000 ksi (200,000 MPa) | AISC Steel Manual |
| Aluminum 6061-T6 | E = 10,000 ksi (69,000 MPa) | ADM |
| Concrete (fc'=4000 psi) | E ≅ 3,605 ksi (24,855 MPa) | ACI 318 Eq. 19.2.2.1 |
| Douglas Fir-Larch #2 | E = 1,600 ksi (11,030 MPa) | NDS Supplement |
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