FRAME ANALYSIS BY ANALYTICAL METHODS

For continuous beams and one-way slabs not meeting the limitations of ACI 8.3.3 for analysis by coefficients, an elastic frame analysis must be used. Approximate methods of frame analysis are permitted by ACI 8.3.2 for

“usual” types of buildings. Simplifying assumptions on member stiffnesses, span lengths, and arrangement of live load are given in ACI 8.7 through 8.11.

2.4.1 Stiffness

The relative stiffnesses of frame members must be established regardless of the analytical method used. Any reasonable consistent procedure for determining stiffnesses of columns, walls, beams, and slabs is permitted by ACI 8.7.

The selection of stiffness factors will be considerably simplified by the use of Tables 2-8 and 2-9. The stiffness factors are based on gross section properties (neglecting any reinforcement) and should yield satisfactory results for buildings within the size and height range addressed in this book. In most cases where an analytical procedure is required, stiffness of T-beam sections will be needed. The relative stiffness values K given in Table 2-8 allow for the effect of the flange by doubling the moment of inertia of the web section (b_wh). For values of h_f/h between 0.2 and 0.4, the multiplier of 2 corresponds closely to a flange width equal to six times the web width. This is considered a reasonable allowance for most T-beams.^2.2For rectangular beam sections; the tabulated values should be divided by 2. Table 2-9 gives relative stiffness values K for column sections. It should be noted that column stiffness is quite sensitive to changes in column size. The initial judicious selection of column size and uniformity from floor to floor is, therefore, critical in minimizing the need for successive analyses.

As is customary for ordinary building frames, torsional stiffness of transverse beams is not considered in the analysis. For those unusual cases where equilibrium torsion is involved, a more exact procedure may be necessary.

Chapter 2 • A Simplified Design Approach

2.4.3 Design Moments

When determining moments in frames or continuous construction, the span length shall be taken as the distance center-to-center of supports (ACI 8.9.2). Moments at faces of supports may be used for member design purposes (ACI 8.9.3). Reference 2.2 provides a simple procedure for reducing the centerline moments to face of support moments, which includes a correction for the increased end moments in the beam due to the restraining effect of the column between face and centerline of support. Figure 2-10 illustrates this correction.

For beams and slabs subjected to uniform loads, negative moments from the frame analysis can be reduced by w_u˜²_a/6. A companion reduction in the positive moment of w_u˜²_a/12 can also be made.

A B C D

A B C D

A B C D

(2) Loading pattern for negative moment at support B

(3) Loading pattern for positive moment in span BC w_d + w_˜

w_d

w_d w_d + w_˜

(1) Loading pattern for negative moment at support A and positive moment in span AB

w_d + w_˜

w_d w_d

Figure 2-9 Partial Frame Analysis for Gravity Loading

2.4.2 Arrangement of Live Load

According to ACI 8.11.1, it is permissible to assume that for gravity load analysis, the live load is applied only to the floor or roof under consideration, with the far ends of the columns assumed fixed. In the usual case where the exact loading pattern is not known, the most demanding sets of design forces must be investigated. Figure 2-9 illustrates the loading patterns that should be considered for a three-span frame.

Table 2-8 Beam Stiffness Factors

Moment of inertia of T-section =~ 2I

=b_wh³

12 = 2

I 10 I

K* ˜

Values of K for T-beams

Span of beam, l ^(ft)** Span of beam, l ^(ft)** 19 6485 162 130 108 93 81 65 54 43 21 81648 2040 1630 1360 1170 1020 815 680 545 6 2916 73 58 49 42 36 29 24 19 8 49392 1230 990 825 705 615 495 410 330 8 3888 97 78 65 56 49 39 32 26 10 61740 1540 1230 1030 880 770 615 515 410 10 4860 122 97 81 69 61 49 41 32 11.5 71001 1780 1420 1180 1010 890 710 590 475 18 11.5 5589 140 112 93 80 70 56 47 37 42 13 80262 2010 1610 1340 1150 1000 805 670 535 13 6318 158 126 105 90 79 63 53 42 15 92610 2320 1850 1540 1320 1160 925 770 615 15 7290 182 146 122 104 91 73 61 49 17 104958 2620 2100 1750 1500 1310 1050 875 700 17 8262 207 165 138 118 103 83 69 55 19 117306 2930 2350 1950 1680 1470 1170 975 780 19 9234 231 185 154 132 115 92 77 62 21 129654 3240 2590 2160 1850 1620 1300 1080 865 6 4000 100 80 67 57 50 40 33 27 8 73728 1840 1470 1230 1050 920 735 615 490 8 5333 133 107 89 76 67 53 44 36 10 92160 2300 1840 1540 1320 1150 920 770 615 10 6667 167 133 111 95 83 67 56 44 11.5 105984 2650 2120 1770 1510 1320 1060 885 705 20 11.5 7667 192 153 128 110 96 77 64 51 48 13 119808 3000 2400 2000 1710 1500 1200 1000 800 13 8667 217 173 144 124 108 87 72 58 15 138240 3460 2760 2300 1970 1730 1380 1150 920 15 10000 250 200 167 143 125 100 83 67 17 156672 3920 3130 2610 2240 1960 1570 1310 1040 17 11333 283 227 189 162 142 113 94 76 19 175104 4380 3500 2920 2500 2190 1750 1460 1170 19 12667 317 253 211 181 158 127 106 84 21 193536 4840 3870 3230 2760 2420 1940 1610 1290 6 5324 133 106 89 76 67 53 44 36 8 104976 2620 2100 1750 1500 1310 1050 875 700 8 7099 177 142 118 101 89 71 59 47 10 131220 3280 2620 2190 1880 1640 1310 1090 875 10 8873 222 177 148 127 111 89 74 59 11.5 150903 3770 3020 2510 2160 1890 1510 1260 1010 22 11.5 10204 255 204 170 146 128 102 85 68 54 13 170586 4260 3410 2840 2440 2130 1710 1420 1140 13 11535 288 231 192 165 144 115 96 77 15 196830 4920 3490 3280 2810 2460 1970 1640 1310 15 13310 333 266 222 190 166 133 111 89 17 223074 5580 4460 3720 3190 2790 2230 1860 1490 17 15085 377 302 251 215 189 151 126 101 19 249318 6230 4990 4160 3560 3120 2490 2080 1660 19 16859 421 337 281 241 211 169 141 112 21 275562 6890 5510 4590 3940 3440 2760 2300 1840

*Coefficient 10 introduced to reduce magnitude of relative stiffness values

Chapter 2 • Simplified Frame Analysis

Table 2-9 Beam Stiffness Factors

h

b

Values of K for T-beams

Span of beam, (ft)** Span of beam, (ft)**

Figure 2-10 Correction Factors for Span Moments ^2.2

2.4.4 Two-Cycle Moment Distribution Analysis for Gravity Loading

Reference 2.2 presents a simplified two-cycle method of moment distribution for ordinary building frames.

The method meets the requirements for an elastic analysis called for in ACI 8.3 with the simplifying assump-tions of ACI 8.6 through 8.9.

The speed and accuracy of the two-cycle method will be of great assistance to designers. For an in-depth discussion of the principles involved, the reader is directed to Reference 2.2.

2.5 COLUMNS

In general, columns must be designed to resist the axial loads and maximum moments from the combination of gravity and lateral loading.

For interior columns supporting two-way construction, the maximum column moments due to gravity loading can be obtained by using ACI Eq. (13-7) (unless a general analysis is made to evaluate gravity load moments from alternate span loading). With the same dead load on adjacent spans, this equation can be written in the following form:

M = 0.07 q_⎡⎣

(

)

(B) = Computed moment ignoring stiffening effect of column support (C) = Modified moment at face of column

= uniformly distributed factored load (plf)

= span length center-to-center of supports

= width of column support

= c/˜

Chapter 2 • Simplified Frame Analysis

where:

q_Du = uniformly distributed factored dead load, psf

q_Lu = uniformly distributed factored live load (including any live load reduction; see Section 2.2.2), psf

˜_n = clear span length of longer adjacent span, ft

= clear span length of shorter adjacent span, ft

˜₂ = length of span transverse to ˜_nand , measured center-to-center of supports, ft For equal adjacent spans, this equation further reduces to:

The factored moment M_ucan then be distributed to the columns above and below the floor in proportion to their stiffnesses. Since the columns will usually have the same cross-sectional area above and below the floor under consideration, the moment will be distributed according to the inverse of the column lengths.