Alignment charts for columns- easy approach.
Our subject as of today is the use of Alignment charts for columns whether braced or unbraced columns.
Columns when attached to frames, exist in two conditions braced columns or unbraced columns, for each case.
If you like to search for this subject by searching on the internet, you will write the Alignment chart or Nomograph or Andersons and Woodward, but this method is based on certain assumptions, such that the column is elastic and the beam is elastic if there is a violation of such conditions and an adjusted factor is used if the column is inelastic. What is the alternative to this method? the alternative method is the direct analysis method. First, in the shown figures, at the left, there is a braced column that is part of a frame.
The side-sway is zero due to bracing or sometimes a small value that can be neglected This expression is difficult and is called side-sway inhibited and for the next figure at the right.
The unbraced frame is acted upon by a horizontal force, there is no side-sway when a horizontal force P acting on the unbraced frame side-sway of large value due to non-bracing is called side-sway uninhibited, which matches the unbraced expression, and inhibited means braced.
From the book of Prof. Salmon For the first figure on the left side, the braced column, due to bracing, the k value is always <1, the joint is rigid, the angle between the girder and the column is 90 degrees, and this angle is kept unchanged and the K is <1, KL=0.7 L and < L.
In all the cases, K<1, This is the case e of a frame with pinned support.
Let us investigate the braced column in a frame with fixed support, which is case C, what will be the k factor for the column? K>0.50<0.70, unlike the case where 0.7, the inflection point distance becomes smaller.
While for the unbraced frame, case b, with hinged supports, while there is the last case, with the unbraced frame with fixed base supports.
For the unbraced frame, there will be a movement to the right and the deflected shape, the beam curve will be coming out of the frame right edge, there will be a deflection curve moving upwards and curvature returns to the hinge at the support, the k value of the column will be > 1.
While for the column at the right side from the inflection point to the next inflection point The k value >1, the last case of unbraced frame kL > L, and but <2L. Finally, for the braced frame, the k is always < 1 and for the braced frames, K will be >1.
is > 2L. In the next slide, the author explains the reasons for the k values. To understand why the minimum value of k in an unbraced is theoretically 1.0 by referring to Figure no. d, and back to shape.
The inflection point will be at the mid-height and the buckled shape will be in fig 6-9-3, while for case b, there is a side-sway, k=2, but in case of no c, if the base is hinged with a partial restraint, the k*L
Based on these conclusions, two alignment charts for columns were derived, The first chart Side-sway inhibited, or braced column in a frame, our k is from 0.5 to 1, Our column is composed of two joints, joining together, and the upper one is GA and the lower one is GB, the G value calculation is the summation of EI/L for columns, for instance for a column, the upper part, there is no other column above then, then EI/L is only for one column.
The alignment charts for columns, the first chart is for the braced columns.
But for two columns above each other, for the common joint, G value the summation of EI/L for both will be estimated, this is the nominator / the summation EI/L for beams at the same joint. At the right and left of the joint, for one side beam, EI/Lbeam Will be for one beam based on the same material used E for column and beam, the value is the same.
E will cancel each other, There is a comment if the column framing on a beam which has higher inertia, stiff beam, G will be increased approaching to zero theoretically.
The alignment charts for columns, the second chart for the unbraced columns.
Let us have a look at the other chart for the alignment chart for columns side-sway uninhibited.
The figures, for alignment char for columns, have been changed starting from 0 to infinity for both GA and Gb k is from 1 to 20, unlike the values in the other chart, since k is > 1. The deformed shape is shown.
The p delta will be later considered, the equation for the chart, parameters are herewith shown, for the side-sway inhibited, for different values of GA and Gb, and the corresponding k value. If the girders at joints are very stiff they have a huge value of EI/L, the G will approach zero, and the K factor will be small The column moment cannot rotate the joint very much If G is very small.
This means the girder is rigid the column moments cannot rotate the joints this joint is close to a fixed-end situation, usually, G is > zero.
The conditions of development of charts.
The conditions for which the alignment charts for braced and unbraced columns were developed were based on a certain set of assumptions.
First, the members are elastic having constant cross-sections and no variable sections exist. and connected by rigid joints.
In elastic column and elastic beam, all columns buckle simultaneously, for braced frames, the rotations at opposite ends of each beam are equal in magnitude, and each beam bends in a single curvature, single curvature from one end to another in a single wave, but for the unbraced frame. Double curvature is as shown in the sketch.
For unbraced frames, the rotations at opposite ends of each beam are equal and opposite in direction. Axial compression forces in the girders are negligible.
If these conditions are not fulfilled, a modification is done for the side-sway inhibited frames, and a braced frame for a girder BC attached to a column.
The video I used for illustration.
In the next post, we will discuss solved problem-7-1.
This is the pdf file used in the illustration of this post.