Gable Frame — Pinned Bases, UDL on Rafters

1

Geometry, load and stiffness

Span, column height, ridge height, UDL and section inertias

Span ℓ

m

Eave height h

m

Ridge rise r

m

UDL w (horizontal projection)

kN/m

Define different section inertia / elastic modulus for column and rafter

Model: Bases A and D are pinned; B-E-C (eave-ridge-eave) are rigid joints. w is defined as a uniform load per unit HORIZONTAL PROJECTION (the snow/roof-load convention — also used by the Kleinlogel tables). The system is indeterminate to the first degree (same as Type-1 with pinned bases) and is solved by the unit-load method. As r decreases (a flatter ridge), the results converge to Type-1's flat-beam portal frame.

2

Reactions and horizontal thrust

R_v, H

R_vat each base, vertical = wℓ2 = —kN

δ_D0 · δ_D1unit-load flexibility coefficients (absolute value)

δ_D0=w(8h+5r)ℓ³/(96·E_bI_b·cosθ) · δ_D1=2h³/(3E_cI_c)+ℓ[(h+r)³−h³]/(3r·E_bI_b·cosθ)

— —

Hhorizontal thrust, inward = δ_D0/δ_D1 = —kN

θ = roof slope angle, tanθ=r/(ℓ/2). As r→0 (a flat roof), the rafter term in δ_D1 converges to Type-1's h²ℓ/(E_bI_b) term, and the H formula converges to Type-1's H=wℓ²/[4h(3+2k_r)]. Axial forces: N=R_v (compression) in the columns; on the order of N=H·cosθ+R_v·sinθ (compression) in the rafters.

3

Bending moments

Eave-corner (M_k) and ridge (M_m) moments

M_keave corner (B, C) — outer-fiber tension = H·h = —kN·m

M_mridge (E) — bottom-fiber tension = wℓ²8 − H(h+r) = —kN·m

Along the rafter, the moment as a function of horizontal position x is M(x)=M₀(x)−H·y(x) — M₀(x)=(w/2)x(ℓ−x) is the simple-beam parabola (same as Type-1, thanks to the horizontal-projection convention), and y(x) is the height at that point. In the column, the moment is zero at the base and increases linearly to M_k at the corner (same behavior as Type-1).

Bending Moment Diagram (M) — on the frame

4

Shear and axial force

V, N in column and rafter — at the eave and ridge

At the eave (B, C)V, N — at the column-rafter joint

V_eave=— kN N_eave=— kN

At the ridge (E)V, N — on the axis of symmetry

V_ridge=— kN N_ridge=— kN

Column: V=H (constant), N=R_v (compression). Along the rafter, V and N vary linearly from eave to ridge (between the end values above) — both keep a consistent compression/shear sense.

Shear Force Diagram (V) — on the frame

Axial Force Diagram (N) — on the frame