Collapse load of portal frame, P = 2ql

Assumptions

• Portal frame A-B-D-E with width $l$ and height $l$: A hinged, E fixed. Columns AB and DE have $M_p$, beam BD has $2M_p$; C is the mid-point of BD.
• Loads: UDL $q$ per unit length on AB acting horizontally (towards the right), and $P = 2ql$ vertical at C. Both increase together; find the collapse value of $q$ (and $P$) in terms of $M_p$ and $l$.
• Take $l = 1$ for the working (all lengths in units of $l$); restore $l$ at the end. Reactions: A hinged (2), E fixed (3), so degree of indeterminacy = 2; critical sections B, C, D, E (and the hinge A) give 4 - 2 = 2 independent mechanisms (beam, sway) and their combination.

Mechanisms

Mechanism 1: beam mechanism

No sway. Hinges at B (in the column, the weaker member), under P (at mid-span C) and at D (in the column DE). Joint rotations: $\theta/2$ at B and D, $\theta$ at C.

Internal work $= 3\,M_p\,\theta$.

Equation: $q\times \frac{1}{2}\,\theta = 3\,M_p\theta$, so $q = 6\,M_p$ per $l^2$.

Mechanism 2: sway mechanism

The beam translates by $l\theta$; both columns rotate $\theta$ (A is hinged). Hinges at B, D (in the columns) and E.

Internal work $= 3\,M_p\,\theta$.

Equation: $q\times \frac{1}{2}\,\theta = 3\,M_p\theta$, so $q = 6\,M_p$ per $l^2$.

Mechanism 3: combined mechanism

The beam and sway mechanisms are combined so that the hinge at B cancels. Hinges at C, D and E.

Internal work $= \frac{7}{2}\,M_p\,\theta$.

Equation: $q\times \frac{3}{4}\,\theta = \frac{7}{2}\,M_p\theta$, so $q = \frac{14}{3}\,M_p = 4.6667\,M_p$ per $l^2$.

A mechanism with the sway hinge inside the loaded column (at 0.618 of the height, the section of maximum moment) gives 5.236, which is higher.

Collapse load

The lowest value governs (combined mechanism):

$$q_c = \frac{14}{3}\frac{M_p}{l^2} = 4.667\,\frac{M_p}{l^2},\qquad P_c = 2q_cl = \frac{28}{3}\frac{M_p}{l} = 9.333\,\frac{M_p}{l}$$

Answer: $q_c = 14M_p/(3l^2)$ and $P_c = 28M_p/(3l)$.