Portal Frame — Point Load on Beam (Any Position)

1

Base type, geometry, load and stiffness

Span, column height, point-load position and section inertias

Span ℓ

m

Column height h

m

Point load P

kN

Load position a (from B)

m

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

Model: Corners B and C are rigid. P is applied vertically on the beam at distance a from B (b=ℓ−a from C). Depending on the base type, A and D are either pinned (no moment transfer) or fixed (moment is also transferred — base moment shown).

2

Reactions and horizontal thrust

R_v,A, R_v,D, H

k_rbeam/column relative stiffness ratio = E_bI_b·hE_cI_c·ℓ = — = —

R_v,A · R_v,Dvertical reactions = —

—kN —kN

Hhorizontal thrust, inward — pinned base = 3Pab2hℓ(2k_r+3) = —kN

Hhorizontal thrust, inward — fixed base = 3D_m/h = —kN

D_m=Pab/[2ℓ(2+k_r)], ψ_m=Pab(b−a)/[4ℓ²(1+6k_r)] — auxiliary quantities (see report)

Columns are not assumed identical here — H, R_v,A, R_v,D all depend on k_r AND on the load position (a, b). Axial forces: N=R_v (compression) in the columns, N=H (compression) in the beam.

3

Bending moments

Corner, base (fixed base) and under-load moments

M_kcorner (B, C) — equal = H·h = —kN·m

M_k,B · M_k,Ccorner moments — generally different = 2(D_m±ψ_m)

—kN·m —kN·m

M_t,A · M_t,Dbase moments = D_m∓2ψ_m

—kN·m —kN·m

M_Punder the load (x=a) — usually the maximum span moment = —kN·m

With a fixed base, the corner/base moments generally DIFFER between the B/C (and hence A/D) sides — the side closer to the load carries more moment. The column moment varies linearly from base to corner and usually changes sign (sway effect).

With a pinned base, both corners carry the SAME moment (M_k=Hh) — there is no base moment (hinge).

Bending Moment Diagram (M) — on the frame

4

Shear and axial force

V, N in the column and beam — on the frame

Vcolumn: H (constant) · beam: R_v,A (x<a), −R_v,D (x>a) — stepped

Ncolumn AB: R_v,A · column CD: R_v,D (both compression) · beam: H (compression)

Shear Force Diagram (V) — on the frame

Axial Force Diagram (N) — on the frame