Question 1 of 9: Minimum number of structural degrees of freedom (8 marks)

Question 1: Minimum number of structural degrees of freedom (8 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Four structures, all with inextensible members: (a) a four-span continuous beam under a UDL over its whole length, pinned at the left end and on rollers elsewhere, with internal hinges $3L/4$ in from each end; (b) a two-storey two-bay frame joined to a taller single-bay frame, four fixed bases, UDL on the upper beam of each part; (c) a symmetric hipped frame on pinned bases under a symmetric UDL over the full projection; (d) an unsymmetric bent on two fixed bases under a UDL over the full projection.

Find. The minimum number of structural degrees of freedom $k$ for each structure, together with the arrows that represent them.

Approach. Count the independent joint rotations and the independent joint translations separately, then delete every rotation that a zero-moment release makes redundant and every pair that symmetry ties together.

1. State the counting rule. The slope-deflection unknowns are the joint displacements, so $$k=n_{\theta}+n_{\Delta},\qquad n_{\Delta}=2\,j_{\text{free}}-m_{\text{inext}}-n_{\text{support}}$$ where $j_{\text{free}}$ counts joints free to move, $m_{\text{inext}}$ the inextensible members and $n_{\text{support}}$ the translation components held by supports. A rotation is not counted at a joint where every member framing in is released, because the modified stiffness $3EI/L$ absorbs it.
2. (a) Strip the determinate end pieces. The left hinge sits $0.75L$ from the pin, so the piece between them carries $M=0$ at both ends and has no interior support: it is a simple span of length $0.75L$ that hands a known force $$V_{\text{hinge}}=\frac{w(0.75L)}{2}=0.375\,wL$$ to the hinge. The mirror-image piece at the right-hand end behaves identically, so neither contributes an unknown.
3. (a) Count what is left, then use symmetry. The residue is a three-support beam with a $0.25L$ overhang at each end, giving $\theta_{B}$, $\theta_{C}$, $\theta_{D}$. Structure and load are symmetric about the centre support, so $\theta_{C}=0$ and $\theta_{D}=-\theta_{B}$: $$k_{(a)}=3-2=\boxed{1}$$

Structure (b) offers none of those reductions, so it must be counted in full. Every beam is inextensible, which locks all the joints on one floor to a single horizontal movement, and every column is inextensible on a fixed base, which stops any joint moving vertically.

4. (b) Count joints and storeys. Three rigid joints at the lower floor, four at the intermediate floor and two at the roof of the tall bay give nine rotations; three floors give three sways: $$k_{(b)}=9+3=\boxed{12}$$
5. (c) Count the unreduced totals. Four free joints give four rotations (the pinned bases contribute none). Their eight displacement components are tied by five inextensible members, and the two pinned bases fix nothing else, so $$n_{\Delta}=8-5=3,\qquad k_{(c),\text{unreduced}}=4+3=7$$
6. (c) Apply symmetry. A mirror about the centre line maps the frame and the load onto themselves, so the two eave rotations pair off and so do the two ridge-beam rotations. The symmetric translation field requires $u_{3}=-u_{4}$ while the horizontal member joining those joints is inextensible, so $u_{3}=u_{4}=0$ and only the spread of the eaves survives: $$k_{(c)}=(4-2)+(3-2)=\boxed{3}$$
7. (d) No reductions are available. Three free joints give three rotations; their six components less four inextensible members leave two translations, and nothing is symmetric or released: $$k_{(d)}=3+2=\boxed{5}$$ A convenient independent pair is the horizontal movement of the left knee and the vertical movement of the top of the inclined member; every other component follows from the four member constraints.

The arrows the question asks for are drawn on the four sketches: a single curved arrow at the first interior support in (a); nine curved arrows and three straight storey arrows in (b); two curved arrows on one half of (c) plus one straight arrow showing the eaves spreading; and three curved arrows plus two straight arrows in (d).