ca ca + ILcb1cb

n=1 Lgn where

-ca = modulus of elasticity of column above top lateral support point -cb = modulus of elasticity of column below top lateral support point Ica = moment of inertia of column above top lateral support point Icb = moment of inertia of column below top lateral support point Lca = unsupported length of column in direction under consideration above top lateral support point

Lcb = unsupported length of column in direction under consideration below top lateral support point

Egn = modulus of elasticity of beam, n, at top lateral support point Lgn = major moment of inertia of beam, n, at top lateral support point nb = number of beams that connect to the column at lateral support level d = angle between the column direction under consideration and the beam, n

For the K-factor calculation, the unsupported lengths are generally based on full member lengths and do not consider any rigid end offsets.

The calculation for Gbot is similar, as it corresponds to the bottom lateral support point. The column K-factor for the corresponding direction is then calculated by solving the following relationship for a:

This relationship is the mathematical formulation for K-factor evaluation assuming the sides way is uninhibited. The following are some important aspects associated with the column K-factor.

Cantilever beams and beams and columns having pin ends at the joint under consideration are excluded in the calculation of the stiffness EI/L summations because they do not contribute to the rotational stiffness of the joint. A column or beam that has a pin at the far end from the joint under consideration will contribute only 50% of the calculated EI/L value. If a pin release exists at a particular end of a column, the corresponding G-value is 10.0 in both directions. If there are no beams framing into a particular direction of a column, the associated G-value will be infinity. If rotational releases exist at both ends of a column, the corresponding K-factors are equal to unity.

Observe that the foregoing procedure for the calculation of K-factor can generate artificially high K-factors under certain circumstances. For example, in Fig. 7.5a, column line C2 has no beams framing in a direction parallel to the column minor direction. Similarly, column B3, shown in Figs. 7.5b, has no beams framing into the columns major direction. The Gtop and Gbot values for these columns are infinity. Such columns are considered to be laterally supported by the floor diaphragms with column K-factor of unity. Now consider the conditions shown in Fig. 7.6a and b when the beams framing into a column are slightly inclined with the column major axis, as shown in Fig. 7.6b for column line C2. The small components of the beam stiffness in the column minor direction will generate small Gtop and Gbot values for the column minor direction, resulting in a large minor direction K-factor. In general, such columns are laterally supported by the floor diaphragms in minor directions and should be assigned a K-factor of unity. For braced frames, the K-factors for the beam and brace elements are generally assumed to be unity.

7.1.3.4. Secondary Bending: PA Effects

Frame columns in buildings are in effect "beam-columns," i.e., they are subject to simultaneous bending caused by lateral loads, and axial compression due to gravity loads. Consider the column shown in Fig. 7.7a subjected to simultaneous action of axial load and moments at the ends. At any point, the total moment M can be considered as a combination of the moment M0 due to end moments plus the addition of the moment caused by P acting at an eccentricity y (Fig. 7.7b-d). Thus, M = M0 + Py. Since the deflection is maximum at midheight, the secondary moment also reaches its maximum

value at that height. A similar effect is caused when bending is produced by a lateral load as shown in Fig. 7.8. Since the deflection y and hence the magnitude of the secondary moment are functions of the end moments, a differential equation formulation is required for determining the stresses in beam-columns. Simple cases of beam-columns subjected to end moments and concentrated loads, uniformly distributed loads, etc., have been solved by differential equation techniques. In a practical structure, such a closed-form solution is extremely complicated if not impossible. Therefore, various design standards such as the ACI code and AISC specifications give provisions for approximate evaluation of the slenderness effect. The method in essence requires that the moments obtained by a so-called first-order analysis be magnified by a moment magnification factor.

The direct addition of the maximum PA moment to the maximum primary moment is valid only when the beam-column is subjected to equal moments at the ends subjecting the column to bend in a single curvature. For all other cases, it represents an upper bound, giving a moment magnification factor much larger than that in a real structure. If the two end moments are unequal but of the same sign, producing single curvature, the primary

Deflected shape of column

Deflected shape of column

Figure 7.8. Behavior of building column: (a) building frame showing deflected shape of column; (b) column subjected to the simultaneous action of axial loads and moments; (c) moment diagram due to end moment and PA effect.

Figure 7.8. Behavior of building column: (a) building frame showing deflected shape of column; (b) column subjected to the simultaneous action of axial loads and moments; (c) moment diagram due to end moment and PA effect.

movement M0 is certainly magnified but not to the same extent as when the moments are equal. If the end moments are of opposite sign, producing a reverse curvature in the column, the moment magnification effect will be very small. A moment magnification coefficient Cm is therefore used to take into account the relative magnitude and sense of the two end moments. It is given by the expression

In this equation, M1 and M2 represent the smaller and larger end moments, respectively. The ratio M1/M2 is positive when the column bends in a reverse curvature and negative when the moments produce a single curvature. As can be expected, when M1 = M2, as in a column subjected to equal end moments, the value of Cm becomes equal to 1.0. The foregoing expression applies only to members braced against side sway. For columns which are part of the lateral resisting system, the maximum moment magnification occurs, i.e., Cm = 1, as illustrated in the following discussion.

Consider Fig. 7.9a, which shows the deflected shape of an unbraced portal frame subjected to the simultaneous action of gravity and lateral loads. Considering only the lateral

Figure 7.9. PA effect in laterally unbraced frames: (a) deflected shapes due to horizontal load H and vertical load P; (b) moment at column ends due to horizontal load H; (c) moment at column ends due to axial loads P; (d) combined moment diagram due to H and P. Maximum moment due to H and P occurs at the ends of columns resulting in Cm = 1.0.

Figure 7.9. PA effect in laterally unbraced frames: (a) deflected shapes due to horizontal load H and vertical load P; (b) moment at column ends due to horizontal load H; (c) moment at column ends due to axial loads P; (d) combined moment diagram due to H and P. Maximum moment due to H and P occurs at the ends of columns resulting in Cm = 1.0.

loads, the deflection of the portal frame may be represented by solid lines as shown in Fig. 7.9a. The corresponding moments at the ends of a typical column are as shown in Fig. 7.9b.

When axial load is imposed on the deflected shape of the frame, additional sway occurs in the frame, as shown by dashed lines in Fig. 7.9a. This additional deflection imposes secondary moments in the column, as shown in Fig. 7.9c. It is seen that both the primary and secondary moments are of the same sign and have maximum values at the same locations, namely, at the two ends of the columns. They are, therefore, fully additive, as shown in Fig. 7.9d, meaning that the value of Cm = 1 for unbraced frames.

In American practice, for both steel and concrete buildings, the approach to the stability problem is to modify individual member design in a manner that approximately accounts for frame buckling effects. This is done by isolating a compression member together with its adjoining members at both ends and determining its critical load in terms of effective length factor K. The member is then analyzed as a beam-column by a simplified interaction equation which accounts for the moment magnification caused by the PA effect. Instead of frame analysis for the PA method, a member analysis is substituted.

We have seen earlier that using a total moment obtained by the direct addition of secondary and primary moments results in an overdesign if both these moments do not occur at the same location. The coefficient Cm in the interaction equation prevents overdesign by reducing the design moment by taking into account the relative magnitude and sense of the moments occurring at the ends of columns.

Values of Cm less than 1.0 increase Fb, offsetting the effects of axial load when the shape of the elastic curve increases stability. When there is no joint translation and where the shape of the curve is not affected by transverse loading, reverse curvature bending may reduce Cm to as little as 0.4.

To prevent a dramatic increase in Fb, which can result in unsafe designs, an interaction equation that does not contain the term Cm is also required to be satisfied.

The calculation of stress ratios in frame columns is essentially an exercise in the evaluation of stresses due to simultaneous axial and bending action.

7.1.3.5. Interaction Equations

Prior to 1963, structural engineers could have made peace with the entire design process of beam-columns by using the formula f + f < 1.0

Fa Fb

Since then, engineers have had to deal with many seemingly formidable factors that have been added onto the above interaction equation. For example, the allowable bending stress Fb now has a factor (1 - faIF') to account for the reduction in the bending capacity because of axial loads. The more the axial load in the column, the greater the reduction of Fb. Reducing the allowable stress is mathematically equivalent to increasing the design moment for the PD effects. F' is the familiar Euler's stress divided by the same factor of safety, 23I12, that governs the allowable stress of long columns.

Consideration of only uniaxial bending reduces the AISC equations to the less intimidating format as follows:

fa fb

0.6Fy Fb

where fa = axial stress in the column due to vertical loads Fa = allowable axial stress fb = bending stress in the column Fb = allowable bending stress

Cm = coefficient for modifying the actual bending moment to an equivalent moment diagram for purposes of evaluating secondary bending F'e = Euler's stress divided by safety factor, 23/12 Fy = yield stress of column steel

As mentioned previously, a stress ratio greater than 1.0 indicates overstress, requiring the redesign of the column.

For the general case of axial load plus biaxial bending, the interaction equations for calculating the stress ratios are as follows: If fa is compressive and fa/Fa > 0.15, the compressive stress ratio CR is given by the larger of CRla and CRlb, where f C f

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