Column Analogy in Multi-Span Hinged Frames

: A procedure is presented in which the method of column analogy, normally applicable to the analysis of single-span and closed frames, is extended for the analysis of multi-span frames with columns hinged to the ground. The extension involves consideration of conditions of rotations at the hinges. For illus-tration, a two-span hinged frame is solved in details. Results are in excellent agreement with values obtained using classical structural methods.


INTRODUCTION
The objective of this paper is to extend the use of the method of column analogy (Cross, 1930, Cross & Morgan 1945, Sözen 2002 to multi-span hinged frames. In a recent paper (Badir & Badir 2012) a column analogy procedure was presented for the analysis of multi-cell structures with fixed columns. The analysis presented herein involves consideration of conditions of rotations at the hinges to replace conditions of moments at the fixed ends. Together with the previous published work (Badir & Badir 2012), it constitutes a generalization of the column analogy method for the analysis of multicell structures.

METHOD OF ANALYSIS
For a description of the suggested method of analysis, consider the two-span hinged frame of Fig. 1(a) with moments of inertia of the columns: I in exterior columns and 2I in interior column. This is a simplifying but not a necessary assumption.

Division of Multi-Span Frame: Case "0"
In Fig. 1(b) the multi-span hinged frame is divided into two isolated spans a and b with their inertia and loads by slicing column 1-2 into two halves; Case "0". The moments Mio at the various sections i of each frame including Mk0 and Mk0' at column sections k = 1 and 1' are computed by column analogy as usual. At hinges k = 2 and 2' the rotations rko and r'ko are also computed. These are simply the reactions at hinges 2 and 2' of the elastic load Mi0/EI, where E is Young's Modulus.

Correction Forces and Couples
Each isolated frame will now deform independently under the action of its external forces. In order to restore continuity of the multi-span frame, it is necessary to add corrections. These corrections are taken as the moments resulting from: two equal and opposite forces X1, and two equal and opposite couples X2, acting at the top of columns 1-2 and 1'-2' as shown in Fig. 1(c).

Continuity Restoration
In the multi-span frame of Fig. 1(a), the two halves of the sliced column 1-2 must undergo identical deformations in order that they fit in together forming the original column. This situation will be satisfied only when the rotations at the two hinges and the bending moments in the two halves of the sliced column are identical. These two conditions of continuity may be stated as follows: (1) , and m * kn = mkn + m'kn is the continuity moment-coefficient of case Xn = 1, and (2) another condition dealing with rotations at the bottom hinge k of the column, namely: in which r * k = rk + r'k, r * k0 = rk0 + r ' k0, and r * kn = rkn + r'kn is the continuity rotation-coefficient of case Xn = 1. The subscripts of the rotation r have the same meaning as those in moments (Badir & Badir 2012). In general, for every column there are two unknowns and two conditions of continuity, giving as many equations as the number of unknown forces and couples Xn. In the frame of Fig. 1(a) there are two equations and two unknowns X1 and X2. Solving Eqs. (1) and (2) simultaneously, the corrections Xn are obtained.

Bending Moment in Multi-Span Frame
In general, the moment at any section of the multispan frame is determined by superposition as the sum of moment due to external loads in Case "0" and the moment due to correction forces and couples in Cases "n". The final bending moment in the multi-span frame is given by Eqs. (8) and (9)

SOLVED EXAMPLE
The dimensions, inertia and loads of a two-span hinged frame are given in Fig. 2(a). The frame is divided into two identical frames: a and b of Fig.  2(b). The analogous column section and its properties are given in Fig. 2(c). In Figs. 2(d) and (e) are sketched the statical moments, on the tension side, in frames a and b, respectively. The straining actions and equations of indeterminate moments are given beside each frame. The computed moments M i0 , M 10 , and M' k0 are registered in column (3) of Table 1. The resulting rotations r20 and r'20 are recorded in column (4) of the table, but the calculations are not shown.
In this frame, one unknown correction force X1 and one couple X2 as shown in Fig. 1(c) are needed for continuity restoration. In Figs. 2(f) and (g) are given the statical moments in frames a and b in Case "1", i.e. X1 = 1. The straining actions and the indeterminate moments are given in the figure. The resulting moment-coefficients and rotation-coefficients are shown in columns (5), (6) and (7) Table 1. Case "0", and Moment and Rotation Coefficients; Cases "1 and 2".

Frame
(1) Case "0" Case "1" Case "2"  given in column (20) of Table 3. Eq. (4) gives the final bending moment at section 1 of column 1-2 with the aid of columns (17), (18) and (19) of Table  3. The moment 1 M is given in column (21). The final bending moment diagram is drawn in Fig. 3 on the tension side. Results are in excellent agreement with values obtained using classical structural methods.

CONCLUSION
The forgotten method of column analogy used to analyze statically indeterminate single span and closed frames is extended to the analysis of multispan frames with columns hinged to the ground. The procedure presented here, together with the previously published paper (Badir & Badir 2012), constitute a generalization to Professor Hardy Cross's method of column analogy (Cross, 1930, Cross & Morgan 1945) commonly applied to "one cell" frames, arches, and curved beams.