Elastic Buckling Loads of Hinged Frames by the Newmark Method

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2 Centre for Promoting Ideas, USA The column AB is subjected to three forces at B, namely: vertical force P, end couple X 1, and horizontal force X. The end rotation φ at B or C can be expressed in terms of the dimensions and properties of the horizontal beam BC: φ = (L b /EI b ) X 1, neglecting the effect of axial force X. Our goal is to determine P, X 1 and X (X is directly obtained from X 1 ) which maintain the column in its assumed buckling shape (y a ) and satisfy the end conditions at B. These end conditions are: (1) rotation φ at B of both column BA and beam BC is equal to (L b /EI b ) X 1, () horizontal displacement at B equal zero. Figure 3(a) shows the column AB and the end forces at B. The deformation of the column can be regarded as the superposition of Fig. 3(b) and Fig. 3(c) multiplied by X 1. On the basis of the assumed values of the chosen deflection shape y a, the pin ended column AB under only the axial force P is considered, Fig. 3(b), and the resulting deflection shape y p and rotation at B, φ p, are calculated. Again, the pin ended column AB is subjected to a unit couple (X 1 =1) at end B, Fig. 3(c) and the resulting deflection, y x 1 1, and rotation at B, x 1 1, are also determined. Then, the value of the couple X 1 can be obtained from the relation L b p X1 x 1 1 X1 (1) EIb And the new deflection of the column (y) is calculated directly from the following equation y yp X1yx 1 1 () Finally the ratios of assumed deflections y a to the resulting values y are determined and the critical load suggested by this first cycle is obtained. The results can be obtained by repeating the cycle of calculations. An illustration of the method is given by the following example. The critical load and the corresponding buckling mode are to be determined for the symmetrical two hinged frame of Fig. 1. The solution is found in Fig. 4 where the pin ended column AB is divided into seven sections (6 segments), each segment of length equal to λ, where L c /6. Line 1 assumed set of deflections y a representing a sine curve of amplitude 1000, which is the assumed deflection at the middle of the column Line to 6 corresponding values of moments M p, angle changes α, equivalent concentrated elastic load, average slopes av, and deflections y p are recorded, respectively. Line 7 the slope at B, due to the axial load P is thus found to be p P / EIc 1914 P / EIc. The pin ended column AB is then subjected to a unit couple, X1 1, at its end B. Normal calculations are shown from line 8 to 10. In line 11 are given the values of average slope φ av in the different segments after assuming a value of 6 in the first segment. In line 1 trial deflections are obtained starting with zero value at A. The resulting trial deflection at B has a value of 1 / 6EI c instead of zero. Line 13 correction deflections y c are applied with values varying linearly over the length of the column from 1 / 6EI c at B to zero at the left end. Line 14 true deflection, y x 1 are obtained, line 14 = line 1 + line13 1 Line 15 true value of the average slope between section 6 and 7 is equal to / EIc Line 16 slope at B due to a unit couple, x 1 1 is equal to ( ) / 6EI c or 1 / 6EI c From Eq. (1), 1914 / 1 / 6 / P EIc X1 EIc Lb EIb X1, where L c /6. By rearranging we get X 1 = (1914 P)/( + 3 K c /K b ) in which K EI / L and K EI / L. Hence, for K / K 1, c c c b b b X1 38.8P. Line 17 new deflections y calculated from Eq. (). Thus, line 17 = line 6 + (38.) line 14, for example y 3650 P / EI / 789 c P EIc P / EIc. (at the middle section) = line 18 gives the ratio ya / y at every division point. This ratio when equal to unity gives an estimate of the critical load, which appears to be not yet the same at all the division points ranging from to EI / c P giving a critical load between 1.39 and 14.5 EI / c L c. c b 15

3 International Journal of Applied Science and Technology Vol. 1 No. 3; June 011 The critical load calculated from the better ratio ya / y is EI / c L c, which is about one and half percent greater than the exact value of 1.88 EI / c L c (Horne and Merchant 1965). Figure 5 shows a complete second cycle. The assumed buckling configuration y a is obtained by reducing the deflections of line 17 of the previous cycle in the ratio of 1000/789. It is noticed that the ratio ya / y now ranges from to EI / c P giving a critical load between 1.69 and 13.9 EI / c L c. The critical load calculated from the better ratio ya / y is 1.94 EI / c L c with a difference of 0.47%. This result demonstrates the rapid convergence of the procedure. Antisymmetrical mode of buckling of hinged frames Consider the simple two hinged frame shown in Fig. 6 with antisymmetrical mode of buckling. The end forces and rotation for each member are separately shown in Fig. 7. The column AB is subjected to only two forces at B, namely: vertical force P and a couple. The end rotation φ at B or C can be easily obtained from the horizontal beam: φ = (L b /EI b ) X 1. Our goal is to determine P and X 1 which can maintain the column in its assumed buckling shape and at the same time produce an angle of rotation at B equal to (L b /6EI b ). Fig. 8 shows the column AB with an arbitrary sidesway Δ of 1000 units at B, hence X 1 = 1000 P. A complete numerical solution of the considered frame is illustrated in Fig. 9. Details of different steps are given as follows. First Cycle Line 1 assumed set of deflection representing a sine curve of amplitude 1000 at the right end. Line equivalent concentrated elastic loads kc / kb 1000 P / EIc and for kc / kb 1, 1000 P / EIc. This value is entered at the right end of line 3 between brackets and hence, line 3 is calculated from right to left. Line 4 new deflection y Line 5 line 1/line 4 Two subsequent cycles are also completely recorded, and the final one gives a critical load with a value of Line 3 the desired value of slope at B is equal to (L b /6EI b ) X 1. Therefore EI / c L c which is equal the exact value of EI / c L c (Timoshenko and Gere 1961). The suggested starting buckling mode of the fourth cycle (last line) is almost identical to the third one, thus the corresponding shape of the column at the critical condition is also determined with a high degree of accuracy. Conclusions The Newmark s double integration procedure is extended for use in computing critical loads and buckling modes of hinged frames. Results obtained show very good agreement with well-known methods. The elastic line of the mode of buckling is determined as a major part of the solution, which gives a clear insight of the behavior of the structure. The method presented here can be used to study buckling of frames with varying cross sections. References Horne, M. R., & Merchant, W., (1965). The stability of frames, Pergamon, Oxford. Newmark, N. M., (1943). Numerical procedure for computing deflections, moments, and buckling loads, Transactions of the. American Society of Civil Engineers, vol. 108, Timoshenko, S. P., & Gere, J. M., (1961). Theory of elastic stability, McGraw-Hill, New York, nd ed. 16

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