Pinned-Base Portal Frame

1

Geometry, load and stiffness

Span, column height, UDL and section moments of inertia

Span ℓ

m

Column height h

m

UDL w

kN/m

Set different moment of inertia / modulus of elasticity for column and beam

Model: Bases A and D are pinned (no moment transfer); corners B and C are rigid (the beam-column angle stays fixed). A uniform load w runs the full span of the beam. The system is indeterminate to the first degree; the horizontal support reaction is taken as the redundant and solved by the unit-load method.

2

Reactions and horizontal thrust

R_v, H

k_rbeam/column relative stiffness ratio (E·I) = E_bI_b·hE_cI_c·ℓ = — = —

R_vat each support, vertical = wℓ2 = — = —kN

Hat each support, horizontal thrust (inward) = wℓ²4h(3 + 2k_r) = — = —kN

Axial forces: in the columns N = R_v (compression); in the beam N = H (compression) — the fixed-end moment from the rigid corners rotates the column tops slightly inward, putting the beam in longitudinal compression, much like a two-hinged arch.

3

Bending moments

Corner (M_k) and mid-span (M_a) moments

M_kcorner (B, C) — tension on the outer fiber = H·h = — = —kN·m

M_amid-span — tension on the bottom fiber = wℓ²8 − M_k = — = —kN·m

The simple-beam moment wℓ²/8 is reduced by the fixed-end moment M_k from the rigid corners — just as in a fixed-fixed beam. In the columns, moment is zero at the base (pin) and grows linearly to M_k at the corner.

Bending Moment Diagram (M) — on the frame

4

Shear and axial force

V, N in column and beam — on the frame

Vcolumn (constant) · beam: R_v − wx, + at x=0, − at x=ℓ = H · R_v − wx

Ncolumn: R_v (compression) · beam: H (compression), both constant = R_v · H

Shear Force Diagram (V) — on the frame

Axial Force Diagram (N) — on the frame