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Kleinlogel Rigid Frame Formulas

J. K. Murray and W.P. Bailey, Engineers’ Precepts, 1940, 4th ed. McGraw-Hill, New York, 1959, p. 471.
Leontovich Frames and Arches
O.I. Leontovich, ‘The building mechanics of frame structures,» P.M.F. Engineers, Vol. 9, No. 6, pp. 459-470. Leontovich, O.I., 1960. «An integrational approach to building mechanics of truss structures and other systems,» USSR, 1958.  .
Leontovich, O.I., 1962. The Truss Structures, (translated from Russian), L.T.D., Moscow, 1962. C.I.F. Vol. 69, Springer-Verlag, Berlin/G.O. No.31.7.3.
Leontovich, O.I. and A.S. Morozov, Rigid frames and mechanics of four-bar systems (in Russian), L.T.D., Moscow, 1964. C.I.F. Vol. 77, Springer-Verlag, Berlin/G.O. No. 32.3.3.
O.I. Leontovich, G.F.C. Heath, and J.A.C. Brooke, Introduction to Transverse Truss Bridges, Harlow, Essex, England, John Wiley & Sons, New York, 1963, and 1966, and reprints, 1964 and 1963.
O.I. Leontovich, G.F.C. Heath, and J.A.C. Brooke, Truss Bridges (in Russian), M. Nauka, Moscow, 1975. C.I.F. Vol. 105, Springer-Verlag, Berlin/G.O. No. 34.3.4.
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A.T. Ko’ete, N.I. Sidorovich, O.I. Leontovich, and G.F.C. Heath, Architektonaia strilnaia, Artes, 1963, No. 11, pp. 8-9.
Leontovich, O.I., 1970. «Formulas for complex rigid frames,» J. Structure, Vol. 7, No. 1, pp. 59-73. Leontovich, O.I., «Formulas for complex truss bridges and other systems of rigid bodies,» (

Rigid Frame Formulas,  The formulas in this section are more rigid frame formulas, for bridges. These formulas, for bridges, though non rigid, are very useful in designing such structures, as any given design can be readily extended to a rigid frame.
Rigid Frame Formulas, Design Schemes, and Computer Calculations, Irving Langmuir
Formulas suitable for such applications require one or more rigid substructures..
The following set of formulas, in their present form, are not suitable for rigid frame design. The formulas have been rewritten here for greater clarity in design applications.

SOURCE .
For design purposes, the rigid frame formula given here have
two primary uses.
The first is design formula, and the second is to obtain the dimensions of frames found in a rigid frame, from given design data.
Rigid frame formulas, according to the author, are nonrigid formulas, which
clearly show the structure of a rigid frame, which are given in rectangular form.
The designer, in need of a rigid frame will be pleased to find that the rigid frame formulas / are given in rectangular form. A rigid frame, from rigid frame formulas, is obtained by using the basic formulas ( see Sec. 6.6. ) in order to get the side of a rectangle.
A rigid frame, from rigid frame formulas, is obtained from the basic formulas ( see Sec. 6.6. ) in order to get the dimensions of frames found in rigid frame, from given design data, in rectangular form.
The basic formulas, given in rectangular form, are :.
The formulas in this section are Rigid Frame Formulas, The formulas are not suitable for design purposes. The formulas have been rewritten here for greater clarity in design applications.
The formulas have been rewritten here for greater clarity in design applications,

SOURCE .
. The basic formulas are given in rectangular form. These rigid frame formulas are based on those of M. O’ Lalley. 14
to consider a rigid frame, four sections, A, B, C, and D, are used in order to
form them into the basic formulas of the author.
A, B, C, and D stand for the width, length, height,and thickness, of the rigid frame. A, B, C, and D are all positive. In the following formulas, A, B, C, and D stand for the width, length
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 .
Kleinlogel Rigid Frame Formulas
Schematic Notation for
Rigid Frames and Arches
A. Kleinlogel, Translated by I. A. K. O’Malley, PH.D., McGraw-Hill Book Company, New York, 1959. .
Rigid Frames
Schematic Representation
Schematic Notation for
Rigid Frames and Arches
A. Kleinlogel, Translated by I. A. K. O’Malley, PH.D., McGraw-Hill Book Company, New York, 1960. P. A. P. M. Almeida, (publisher) 
References
1
Kleinlogel, A. 1949. » Rigid Frame Formulas», Translated from German.. – The Structural Engineer, Vol. 46, No.2, (1949).
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The aim of the study was to evaluate the midterm results of initial surgery for atrial fibrillation (AF). Our hypothesis was that patients who underwent an initial operation for AF and were in sinus rhythm 1 year after the procedure would maintain sinus rhythm through the subsequent follow-up period. Forty-eight patients (mean age 47+/-12 years) underwent surgery for AF (16 closed-heart surgery, 31 open-heart surgery, and 1 thoracoscopic surgery) from February 1998 to July 2000. There were 12 patients who underwent a mitral valve repair, 6 patients who underwent a mitral valve replacement, 12 patients who underwent a tricuspid valve repair, and 14 patients who underwent a tricuspid valve replacement. Sinus rhythm was maintained with either antiarrhythmic drugs (89%) or β-blockers (82%). Twelve patients underwent a second procedure (18%). During a mean follow-up of 30+/-14 months, 23 patients (48%) had recurrent AF. Freedom from any rhythm was 97% at 2 years, 90% at 3 years, and 76% at 5 years. Early recurrence was higher for patients who underwent a repair than for those who underwent a replacement (P=.025). Late recurrence was higher for patients who underwent a repair or a replacement (both P=.0004). Early recurrence of AF after surgery and late recurrence of AF occurred most often in patients with a

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In rigid frame formulas the geometry of the frame is independent of the load acting on it. The formulas are obtained from the internal forces and moments, which are functions of the elements and the orientation of the frame, or vice-versa.
The geometry of the rigid frame is .
Kleinlogel-Rigid Frame Formulas.
where,  and  are the distance from the center of mass to the end
of the rigid bar, the distance from the center of mass to the supporting point and  .
Respecting the equilibrium condition, the internal forces due to gravity and the transverse shearing loads can be expressed as follows:
.
The internal forces due to gravity and transverse shearing can be expressed as follows:

,.
This can be expressed as:

,  ,  .
This equation is subject to the following additional conditions:,,  . It is equivalent to the following condition:

,  .
Here we have used the above observations on the internal forces and the geometrical relations.
For geometric simplicity, we express the geometrical relations between the lengths,  ,  ,  and the external force or load in terms of the
moments of inertia,,,  . To do this, we first write the inertia matrix in the form

:

where,
The inertia matrix is written in terms of the external force or load. This is given by the following equation:

.
In the above equation,,  ,,,,  and  are the distances from the center of mass to the supports of the bar, as shown in Fig. 6.,,,,,,,.
.
 .
  ,,,  .
Kleinlogel-Rigid Frame Formulas.
.
The above relationships are written in the form of the following equations:

,.
We have written the above relations in the terms of the  ,,,,,,. We have used the geometrical relations between the distances, as mentioned above. The following are the mathematical equations for the internal forces and moments.

,.
It is the condition of equilibrium that allows us to express the above relationship in terms of the  ,,,,