Continuous Beam Analysis

Direct-stiffness (matrix) method · Euler–Bernoulli beam elements · 1–5 spans

Units
Sign convention: loads P and w positive downward · concentrated moments positive counter-clockwise · bending moment positive sagging · shear positive when the net force left of the cut acts upward · deflection reported positive downward · reactions positive upward.
Elevation view — live model
w = 15.00 kN/m50.0 kN40.0 kNS1 · PinnedR1 = 42.51 kNS2 · PinnedR2 = 110.60 kNS3 · FixedR3 = 103.17 kNM3 = 30.72 kN·mS4 · PinnedR4 = 13.72 kN0.004.009.0013.00L1 = 4.00 mL2 = 5.00 mL3 = 4.00 m
Material & section properties
N.A.h = 900 mmb = 400 mm
Shape
Rectangular
Area A
3.600e+5 mm²
Inertia I
2.430e+10 mm⁴
Modulus S
5.400e+7 mm³
Geometry & supports
Span lengths (m)
L1
L2
L3
L4
L5
Support type at each span-end node (4 nodes)
S1 @ x = 0.00 m
S2 @ x = 4.00 m
S3 @ x = 9.00 m
S4 @ x = 13.00 m
Point loads & concentrated moments
x (m)
P (kN) ↓+
M (kN·m) CCW+
1
2
3
4
5
6
7
8

A row with P = 0 and M = 0 is disabled. Loads may sit anywhere along the beam, including part-way through a span.

Uniformly distributed loads
x start (m)
x end (m)
w (kN/m) ↓+
1
2
3
4
5

Set w = 0 to disable a segment. Segments may be partial-span and may overlap.

Assumptions & limits

Linear-elastic, small-displacement, prismatic (constant EI within each span), in-plane bending only, supports at the same level (no settlement).

Euler–Bernoulli elements: shear deformation is neglected. No internal hinges. Up to 5 spans, 8 point loads/moments, 5 distributed-load segments.

Sign convention: loads positive downward, concentrated moments positive counter-clockwise, moment positive sagging, shear positive when the net force left of the cut acts upward.

Adequate restraint is required (two vertical restraints, or one vertical plus one rotational); otherwise the model is a mechanism and analysis is refused.