Gable Frame — Pinned Bases, Horizontal Load at Eave

1

Geometry, load and stiffness

Span, eave/crown height, horizontal load at B, and section moments of inertia

Span ℓ

m

Eave height h

m

Crown rise r

m

Horizontal load (wind) P

kN

Define separate moment of inertia / modulus of elasticity for column and rafter

Model: Bases A and D are pinned; joints B, E, C are rigid. P is applied horizontally exactly at B (the eave facing the wind). Unlike Type-2 (a flat portal frame), the SLOPED roof means the horizontal load is NOT split equally between the two supports — the B side (loaded side) and the C side carry different H values. The system is indeterminate to the first degree; solved with the unit-load method.

2

Reactions and horizontal thrust

H_A, H_D, R_v

δ_D0 · δ_D1unit-load flexibility coefficients (absolute value)

δ_D0=Ph³/(3E_cI_c)+Phℓ(2h+r)/(4E_bI_bcosθ) · δ_D1=2h³/(3E_cI_c)+ℓ[(h+r)³−h³]/(3rE_bI_bcosθ)

— —

H_Dat support D (redundant) = δ_D0/δ_D1 = —kN

H_Aat support A = P − H_D = —kN

R_vat each base, vertical (force couple) = Phℓ = —kN

In Type-2 (flat roof), the axial stiffness of the columns and beam made H equal to P/2 at both supports. On a sloped roof the rafters' axial stiffness no longer equalizes the horizontal displacements so simply (rafter elongation has both a horizontal and a vertical component) — hence H_A≠H_D. Uplift (tension) on the A side, compression on the D side.

3

Bending moments

Eave corner moments (B, C) and crown moment (E)

M_k,B= H_A·h = —kN·m

M_k,C= H_D·h = —kN·m

M_mcrown (E) = H_D(h+r) − Ph/2 = —kN·m

M_k,B and M_k,C are GENERALLY DIFFERENT (unlike Type-2's equal values on a flat roof) — the loaded-side corner moment depends on the column shear (H_A). Column moment is zero at the base and increases linearly to the corner (pinned base, no base moment).

Bending Moment Diagram (M) — on the frame

4

Shear and axial force

V, N in the columns and rafters

Column AB (B side)

V=— kN N=— kN (tension)

Column CD (C side)

V=— kN N=— kN (compression)

Rafter B-E · Rafter E-C(compression, both of constant magnitude)

N_BE=— kN N_EC=— kN

V_BE=— kN V_EC=— kN

V and N are CONSTANT along each rafter (no distributed load, only point effects at the ends) — they jump between the two rafters at the crown.

Shear Force Diagram (V) — on the frame

Axial Force Diagram (N) — on the frame