Three-Hinged Gable Frame — Vertical and Horizontal Load

1

Geometry and loads

Span, eave/crown height, vertical UDL and horizontal point load

Span ℓ

m

Eave height h

m

Crown rise r

m

Vertical UDL w (horizontal projection)

kN/m

Horizontal load P (at B)

kN

Define different section inertia / elastic modulus for column and rafter

Model: Bases A and D are pinned, and a THIRD hinge sits at the crown (E) — the system is STATICALLY DETERMINATE (not indeterminate). Reactions follow from equilibrium alone plus the crown condition (M_E=0); no section inertia or elastic modulus (E, I) is needed at all. w (per unit horizontal length, the usual snow/roof-load convention) and P (horizontal, at eave B, representing wind) are combined by superposition — either one may be left at zero.

2

Reactions

Direct from equilibrium + crown condition (closed form)

Contributionshorizontal/vertical component produced separately by each load type

H_w=wℓ²/[8(h+r)] · H_A,P=P(h+2r)/[2(h+r)] · H_D,P=Ph/[2(h+r)]

— — —

R_x,A= H_w − H_A,P (superposition) = —kN

R_x,D= −(H_w + H_D,P) = —kN

R_y,A= wℓ/2 − hP/ℓ = —kN

R_y,D= wℓ/2 + hP/ℓ = —kN

Because this frame is STATICALLY DETERMINATE, H_w and H_A,P/H_D,P depend on GEOMETRY only — the column/rafter stiffness ratio (k_r) never appears. For w>0, H acts INWARD at both supports (arch-like thrust); for P>0, both supports pick up a −x contribution (resisting the wind); the net sign depends on which load governs.

3

Bending moments

Eave corners (B, C) and the crown hinge (E)

M_corner,B= |R_x,A|·h = —kN·m

M_corner,C= |R_x,D|·h = —kN·m

M_crown (E)— by definition, a hinge = 0 (always, exactly)

In each column, the moment is zero at the base (hinge) and grows linearly to M_corner (depending only on that column's own horizontal reaction — the same general "V_column×h" rule as in Types 1/2/6/7). Thanks to the third hinge at the crown, M_crown is exactly zero under EVERY load combination — a built-in check you can use to verify the result.

Bending Moment Diagram (M) — on the frame

4

Shear and axial force

V, N in columns and rafters

Column AB · Column CD

V_AB=— kN N_AB=— kN

V_CD=— kN N_CD=— kN

Rafter B-E(eave B → crown E)

N_B=— kN V_B=— kN

N_E,left=— kN V_E,left=— kN

Rafter E-C(crown E → eave C)

N_E,right=— kN V_E,right=— kN

N_C=— kN V_C=— kN

N and V on a rafter are found by projecting the forces to the left of the cut onto that rafter's local axis (tangential/normal). At the crown (E), the LEFT and RIGHT values are generally DIFFERENT — a natural consequence of the roof slope changing direction at E (even with no point load applied there). Positive N = compression.

Shear Force Diagram (V) — on the frame

Axial Force Diagram (N) — on the frame