CalcHandbook

Portal Frame — Fixed Base, UDL on Beam

Frames · Fixed-Base Rectangular Portal · UDL on Beam · Slope-deflection closed-form (symmetry)
1

Geometry, load and stiffness

Span, column height, UDL and section moments of inertia
m
m
kN/m
Define separate moment of inertia / modulus of elasticity for column and beam
Model: Bases A and D are FIXED (transmit both rotation and translation restraint — see base moment), corners B and C are rigid. A uniform load w acts over the full span of the beam. The system is stiffer than the pinned-base version (3rd-degree statically indeterminate); symmetry (both frame and load are symmetric) means rotation and shear are zero at mid-span, which simplifies the solution via slope-deflection (see Report).
2

Reactions and horizontal thrust

Rv, H
krbeam/column relative stiffness ratio (E·I) = EbIb·hEcIc·ℓ = — = —
Rvat each base, vertical = wℓ2 = — = —kN
Hat each base, horizontal thrust (inward) = wℓ²4h(2 + kr) = — = —kN
Axial forces: N = Rv (compression) in the columns, N = H (compression) in the beam — the rigid corners and base moments rotate the column ends slightly inward, compressing the beam along its length.
3

Bending moments

Base (Mt), corner (Mk) and mid-span (Ma) moments
Mtbase (A, D) — moment developed at the fixed base = Mk / 2 = — = —kN·m
Mkcorner (B, C) — outer fiber in tension = 23·H·h = — = —kN·m
Mamid-span — bottom fiber in tension = wℓ²8 − Mk = — = —kN·m
The base moment is HALF the corner moment (Mt=Mk/2) — this is an EXACT algebraic result from symmetry and the zero base-rotation condition (θA=0), independent of kr. In the column, the moment varies LINEARLY from base to corner (same sign at both ends — outer fiber in tension at both corner and base). In the beam, the moment varies from Mk at the corner to Ma at mid-span.
Bending Moment Diagram (M) — on the frame
4

Shear and axial force

V, N in column and beam — on the frame
Vcolumn (constant) · beam: Rv − wx, + at x=0, − at x=ℓ = H · Rv − wx
Ncolumn: Rv (compression) · beam: H (compression), both constant = Rv · H
Shear Force Diagram (V) — on the frame
Axial Force Diagram (N) — on the frame