Elastic Buckling Loads of Fixed Frames By The Newmark Method

: A numerical method for the solution of the elastic stability of fixed frames is presented. The Newmark iterative procedure to perform elastic buckling analyses for isolated columns is extended for use in computing buckling loads and buckling modes in frames with fixed columns. The method is illustrated in details by means of different cases of single-storey portal frames commonly used in commercial buildings. Compari-sons of obtained results with other well known methods show very good agreement.


INTRODUCTION
The elastic critical load of a bar with uniform or non-uniform cross section can be calculated by a numerical method of double integration (Newmark 1943). Instead of assuming the deflection y as some function of x, the beam is divided into segments and a numerical value of deflection is assumed at each division point along the beam. The subsequent calculations are made, determining ordinates to the M/EI diagram (M is the moment, E is the elastic modulus of the material and I is the area moment of inertia), and new values of deflections at each site. If these are equal to the assumed deflections at every division point, then the required critical load P and the buckling mode are determined. If they are not equal, the new set of deflections is assumed and the calculations are repeated. This procedure is successful because the results of each cycle yield better deflections and the procedure converges to the exact buckling mode after a few numbers of cycles of iteration.
The technique of Newmark's numerical method (Newmark 1943), applied to columns, has been extended for use in computing buckling loads and buckling modes of hinged frames (Badir 2011). In this paper, Newmark's method is further extended for the case of frames with fixed columns. The analysis and results are reported herein with detailed calculations in order to illustrate the method for two cases; namely, (1) rigidly jointed elastic portal frames with fixed columns without sway, as in symmetrical modes or when translation of the joints is prevented for, and (2) rigidly jointed elastic frames with fixed columns, with sway as in anti-symmetrical modes. The philosophy of the method described herein can be summarized as follows: the buckling load of the structure is the load just enough to maintain it in an assumed buckling configuration. The method involves cycles of iteration in which a new configuration better than the assumed one is obtained at the end of each cycle. The calculations can be repeated until the required degree of accuracy is obtained. In most cases, accurate results are obtained after only few cycles.

SYMMETRICAL MODE OF BUCKLING OF FIXED FRAMES
Consider the fixed frame shown in Fig. 1, the end forces and rotations of each member are separately shown in Fig. 2. The column AB is subjected to three forces at B, namely: vertical force P, end couple X1, and horizontal force X2. The end rotations φ at B or C can be easily determined from the horizontal beam BC: φ = (Lb/2EIb) X1, neglecting the effect of axial force X2.
Our goal is to determine P, X1, and X2 which are just enough to maintain the structure in its assumed buckling shape (ya) and satisfy the end conditions at B, namely:   (2) Figure 4 shows the calculations of the last cycle of the frame shown in Fig. 1. Line 1 in Fig. 4 represents the assumed set of deflections obtained from the previous cycle where the fixed column AB is divided into seven sections (6 segments), each segment of length equal to λ. The assumed set of deflection ya of the first cycle, not shown in Figure 4, represents a sine curve of amplitude 1000 at the middle of the column. The final result does not depend on the starting assumed function, the method converges to the correct buckling mode after a few number of cycles. Normal calculations (Newmark 1943) are recorded from line 2 to line 18 as follows Line 2 corresponding values of angle changes α (α = M/EIc) commonly known as the elastic load. A common factor is shown at the end of each line, for line 2 the common factor is P/EIc Line 3 equivalent concentrated elastic load acting on each section. The values of theses concentrations are computed with sufficient accuracy from the formulae given in the work of Newmark (1943).
Lines 4 and 5 The shearing forces and bending moments in the conjugate beam are calculated from the concentrated loads (in line 3). They represent the average slopes ϕ, and the deflections yp, respectively.
Line 6 the slope at B, due to the axial load P is thus found to be ϕp = (3368 + 104)Pλ/EIc = 3472 Pλ/EIc. The fixed ended column AB is then subjected to a unit couple, 1 1 X = , at its end B. Normal calculations are shown from line 7 to 9. In line 10 are given the values of average slope φav in the different segments. In line 11 the deflections are obtained starting with zero value at the fixed end. The resulting deflection at B has a value of -18λ 2 /EIc.
The following variables of Eqs. (1) and (2)  The critical load calculated from the better ratio Σya/Σy is 0.70186 EIc/Pλ 2 for the case of stiffness ratio / 1 c b K K = , giving a critical load with a value of 25.267 EIc / L 2 c, which is the same as the value of 25.266 EIc / L 2 c given by Horne and Merchant (1965). The assumed buckling mode of a next cycle, see last line of Fig. 4, is almost identical to the previous one. One may notice that line 7 to line 18 are unaltered, hence they are only calculated in the first cycle.  The column AB is subjected to only two forces at B, namely: vertical force P and a couple X1. The end rotation φ at B or C is equal to (Lb/6EIb) X1. Figure 7 shows the column AB with an arbitrary sidesway of Δ = 1000 units at B. The columns can be regarded to as the superposition of Fig. 7(b) and Fig. 7 Where p ϕ and p y are the rotation and deflection at B due to the axial force P, 1 1 x ϕ = and 1 1 x y = are the rotation and deflection at B due to unit couple acting at B. From Eq. (3) X1 is determined and then y is calculated from Eq. (4). Figure 8 shows complete calculations of the last cycle, in which column AB is assumed to buckle in the shape reached at this cycle. The assumed set of deflection ya of the first cycle, not shown in Fig. 8, represents a straight line varying from a value of zero at A to a value of 1000 at B. However, the final result does not depend on the starting deflection assumption. The better ratio Σya/Σy is