COLUMN DESIGN

5.20 COLUMN DESIGN

As stated before, the column requirements are given in CHAP. 22, DIV. IV, under 6.1, ‘‘Column Strength’’:

6.1.a. Axial compression loads:

1.2PD_⫹0.5PL_⫹0.2PS_⫹0.4R_⫻PE_ⱕ␾cPn (6-1) where the term 0.4R is greater or equal to 1.0.

Exception: The load factor on PL in Load Combination 6-1 shall equal 1.0 for garages, areas occupied as places of public assembly, and all areas where the live load is greater than 100 psf.

6.1.b. Axial tension loads:

0.9PD_⫺0.4R_⫻PE_ⱕ␾tPn (6-2) where the term 0.4R is greater or equal to 1.0.

6.1.c. The axial Load Combinations 6-1 and 6-2 are not required to exceed either of the following:

1. The maximum loads transferred to the column, considering 1.25 times the design strengths of the connecting beam or brace elements of the structure.

2. The limit as determined by the foundation capacity to resist overturning up-lift.

Before applying (6-1) or (6-2) let us reflect on the following:

PE means the axial load induced by both lateral Eh and vertical E_v com-ponents of the earthquake.

The effect of Eh itself is magnified by_␳. In our case _␳equals unity.

The E_v is defined as E_{v ⫽} (0.5CaI)D. For our project:

Ca_⫽ 0.48 From our project earthquake analysis.

D Tributary dead load on structure

I _⫽1 Seismic importance factor (per Table 16-K) Thus the coefficient for E_v, in parentheses is

(0.5C I)_{a ⫽} (0.5)(0.48)(1.0)_⫽ 0.24

According to formulas (6-1) and (6-2), both horizontal and vertical earthquake components of Pe must be magnified by the 0.4 _⫻ R _⫽ 0.4 _⫻ 6.4 _⫽ 2.56

158 SEISMIC STEEL DESIGN: BRACED FRAMES

magnification factor. The earthquake force magnification has been taken care of in computer analysis load combinations [12] and [13].

Note that 1.2D _⫹ (2.56)(0.24)D _⫽ 1.8144D applied to load combination [12] and formula (6-1), CHAP. 22, DIV. IV, 6.1.a, can be written as

1.8144PD_⫹ 0.5PL_⫹2.56PE_{hⱕ ␾}cPn

where PEh indicates the axial component caused by the lateral (horizontal) earthquake load. Similarly, formula (6-2) of CHAP. 22, DIV. IV, can be re-written as

0.9PD_⫺ (0.4 _⫻6.4)0.24PD_⫺ 2.56PE_{h ⱕ␾}tPn

0.2856PD_⫺ 2.56 PE_{hⱕ ␾}tPn

Design of First-Story Column for Compression (Structural Component 4)

(a) Design of W12 _⫻ 96 for Combined Axial Force and Bending about Major Axis The maximum axial compression complying with CHAP. 22, DIV. IV, formula (6-1), is

Pu _⫽691.0 kips Computer load combination [12]

coupled with

Mu_⫽ 16.0 kip-ft The column is bent in single curvature so that

Cm_⫽ 1.0

For both major and minor axes, KL_⫽ 13.5 ft_⫽ 162 in., with Mnt _⫽16.0 kip-ft

The design parameters of the W12_⫻ 96 are

2 3

Ag _⫽28.2 in. rx_⫽5.44 in. Zx_⫽ 147 in.

Using the LRFD equation (C1-1) the magnified moment is Mu_⫽ B M1 nt

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5.20 COLUMN DESIGN 159

where B₁is given by the LRFD equation (C1-2),*

C_m B_{1 ⫽}

1 _⫺P /P_{u e1}

where Puis the maximum limit axial load and P_{e1 ⫽} ␲²EI / (KL)², the Euler buckling strength. Values of P_e1/Ag can be determined from Table 8 of the Specification, then multiplied by A_gross to obtain P_e1.

KLx 162

⫽ ⫽ 29.78

rx 5.44

(KLxmust be modified as the tables are built for the minor axis) P_e1

P^e1 _⫽冉 冊^Ag A^g_⫽ (323)(28.2)_⫽9108 kips

B_{1 ⫽} 1.0 _⫽ 1.082

1 _⫺691 / 9108

M_{u ⫽}1.082(16)_⫽ 17.3 kip-ft Since Lp ⫽ 12.9 ft _⬍ Lb⫽ 13.5 ft _⬍ Lr ⫽ 61.4 ft,

␾bMp _⫽ 397 kip-ft _␾bMr _⫽255 kip-ft

and BF_⫽ 2.91 (load factor design selection table 4-18, Part 4 of the Speci-fication). Applying the LRFD equations (F1-2) and (H1-1a) yields

␾bMnx_⫽C [b ␾bMp_⫺ BF(Lb _⫺ L )]p

⫽1.0 [397_⫺ 2.91(13.5_⫺12.9)]_⫽ 395 kip-ft

Having evaluated the moment term for the LRFD equation (H1-1a), the axial force component_␾cPnneeds to be determined from the column design tables of Part 3 of the Specification. To achieve this, the value of KLxobtained above must be modified as the tables are constructed for the minor axis. The mod-ification factor is

* The presence of axial force coupled with bending will give rise to secondary moments that augment the initially applied bending moment Mnt. The magnification factor to account for the overall effect is B1. The reader will recognize that the secondary moment is caused by axial force times the eccentricity (P⌬ effect) represented by the deflected elastic curve caused by the initial bending Mnt.

160 SEISMIC STEEL DESIGN: BRACED FRAMES

rx

⫽1.76 ry

Therefore we will enter the column table with an effective KL:

KLx 1.0(13.5)

KL_{⫽ ⫽ ⫽} 7.67 ft

r /rx y 1.76

␾cPn _⫽823 kips

Using the LRFD equation (H1-1a) from the Specification,

Pu 8 Mux 691 8 17.3

⫹ 冉 冊⫽ ⫹ 冉 冊⫽ 0.878_⬍ 1.0

␾cPn 9 ␾bMnx 823 9 395

(b) Design of Column about Minor Axis From the column design table 3-24, Part 3 of the Specification,

␾_cP_ny⫽747 kips _⬎691 kips applied

The column has adequate strength to counter the maximum compressive design loads about the principal axes.

Design of Third-Story Column for Compression (Structural Component 14)

(a) Design of W12_⫻ 40 Column for Combined Axial Force and Bending about Major Axis The maximum axial compression complying with CHAP.

22, DIV. IV of the 1997 UBC, formula (6-1), gives

P_u⫽ 128 kips Computer analysis load combination [12]

with end moments

M_{1 ⫽} 25 kip-ft M_2⫽ 32 kip-ft

The column is bent in double curvature. By the LRFD equation (C1-3),

M₁ 25

C^m_⫽ 0.6_⫺ 0.4冉 冊^M2 _⫽0.6 _⫺0.4冉 冊³² _⫽0.287

Note that the ratio of moments is positive if beam and column are bent in double curvature. The design parameters of the W12_⫻ 40 are

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5.20 COLUMN DESIGN 161

2 3

Ag_⫽ 11.8 in. rx_⫽ 5.13 in. Zx_⫽ 57.5 in.

From Table 8 of the Specification with KLx/ rx⫽ 162 / 5.13_⫽ 31.58, P_e1⫽ 287.2 (11.8)_⫽ 3389.0 kips

Cm 0.287

B_{1⫽ ⫽ ⫽} 0.298_⬍ 1.0

1_⫺ P /Pu e1 1_⫺ 128 / 3389

Use B_{1 ⫽} 1, the magnified moment by the LRFD equation (C1-1), Mux_⫽ B M1 nt_⫽ (1.0)(32.0)_⫽ 32.0 kip-ft

Since Lp⫽ 8.0 ft _⬍ Lb⫽ 13.5 ft _⬍ Lr⫽ 26.5 ft,

␾_bM_{p ⫽} 155 kip-ft _␾bM_{r ⫽}101 kip-ft

and BF_⫽ 2.92 (load factor design selection table 4-19, Part 4 of the Speci-fication). Thus

␾bMnx_⫽C [b _␾bMp _⫺BF(Lb _⫺L )]p

Because the moments cause a reversed deformation curvature, we will use the LRFD equation (F1-3) to determine Cb:

12.5 M_max Cb_⫽

2.5M_{max ⫹}3MA_⫹ 4MB _⫹ 3MC

12.5(32)

⫽ ⫽ 1.09

2.5(32)_⫹ 3(30)_⫹4(29)_⫹ 3(27)

where MA, MB, MCare moment values at the– –¹₄, ,¹₂ and–³₄ points of the segment.

Thus

␾_bM_nx⫽ 1.09[155_⫺2.92(13.5_⫺ 8.0)]_⫽151 kip-ft

For the LRFD interaction equations we need the value of _␾cPn. To obtain this value from the column design tables of Part 3 of the Specification, we will enter a modified KL value into the table:

KLx 1.0(13.5)

KL_{⫽ ⫽ ⫽} 5.08 ft

r /rx y 2.66 Thus

162 SEISMIC STEEL DESIGN: BRACED FRAMES

Figure 5.11 Control points of the idealized stress–strain diagram.

␾cPn_⫽ 340 kips By the LRFD interaction equation (H1-1a), we obtain

Pu 8 Mux 128 8 32

⫹ 冉 冊⫽ ⫹ 冉 冊⫽ 0.565_⬍ 1.0

␾cPn 9 ␾bMnx 340 9 151

The column has adequate strength for the combined axial compression and bending about the major axis.

(b) Design of Column About Minor Axis In this step no moment is cou-pled with the axial compression. From column design table 3-25, Part 3 of the Specification,

␾ePny_⫽ 249 kips_⬎ 128 kips applied

The column has adequate strength to resist the maximum design loads about the principal axes.

Design of Columns for Tension

At this point the reader is encouraged to review the concepts and steps out-lined in Section 4.26, ‘‘Design of Columns.’’ Our next step is to design the columns for tension; this will involve structural elements 1 and 21 of the mathematical model of the computer analysis. The same LRFD equations (H1-1a) and (H1-1b) presented for compression design will apply for tension.

No P_⌬ effect, and hence no magnification factor such as B₁, will be used because, due to cyclic reversals caused by earthquakes and unlike in the case of axial compression, any initial postbuckling or moment-induced crooked-ness will be reduced, if not eliminated, by the tensile force, which tends to straighten out the deflected chord under tension (See Figures 5.11, 5.12, and 5.13.)

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5.20 COLUMN DESIGN 163

Figure 5.12 Deformed beam shapes determining unsupported length design param-eters.

Figure 5.13 Postbuckled deformed shapes: (a) under compression; (b) under ten-sion.

Design of First-Story Column for Tension (Structural Component 1)

(a) Design of W12_⫻ 96 for Combined Axial Force and Bending about Its Major Axis (Ag⫽28.2 in.²) The maximum axial tensile force, mandated by formula (6-2) of CHAP. 22, DIV. IV, 2211.4, UBC 1997, is Pu ⫽ 636 kips, load combination [13] of the computer analysis, which includes vertical uplift component Ev⫽0.5CaID, formula (30-1), CHAP. 16, DIV. IV, of UBC 1997, associated with 16.0 kip-ft bending moment:

1. Bending component term of the interaction equation:

L_p⫽ 12.9 ft_⬍ L_b⫽ 13.5 ft_⬍ L_r⫽ 61.4 ft

␾bMp_⫽ 397 kip-ft From LRFD selection table 4-18, Part 4

␾bMr_⫽ 255 kip-ft From LRFD selection table 4-18, Part 4 BF_⫽ 2.91 From LRFD selection table 4-18, Part 4

164 SEISMIC STEEL DESIGN: BRACED FRAMES

Moment connection corresponding to Lp ⬍ Lb ⬍ Lr:

␾bMnx_⫽ C [b _␾bMp _⫺ BF(Lb _⫺L )]p _{ⱕ ␾}bF Zy

␾bMnx_⫽ 1.0[397_⫺ 2.91(13.5_⫺12.9)]_⫽ 395 kip-ft

2. Axial force term of the LRFD interaction equation:

␾tPn_{⫽ ␾}tF Ay g _⫽(0.9)(36)(28.2) _⫽ 914 kips Pu 636

⫽ ⫽ 0.70_⬎ 0.2 Use Equation (H1-1a)

␾tPn 914

Using the Specification interaction equation (H1-1a) yields

Pu 8 Mux 8 16.0

⫹ 冉 冊⫽ 0.70_⫹ 冉 冊_⫽ 0.74_⬍ 1.0

␾tPn 9 _␾bMnx 9 395

(b) Design for Tension about Minor Axis (No Moments Present) To eval-uate_␾tPn, two conditions must be checked:

1. _␾tPn ⫽ ␾tFyAg with_␾t⫽ 0.9

␾tPn ⫽ (0.9)(36)(28.2)_⫽ 914 kips _⬎ 636 kips _← GOVERNS 2. _␾tPn ⫽ ␾tFuUAg with_␾t⫽ 0.75

␾tPn ⫽ (0.75)(58)(0.9)(28.2)_⫽ 1104 kips _⬎ 636 kips

The column is safe against uplift-caused tension.

Design of Third-Story Column for Tension (Structural Component 21) The maximum design axial tensile force, mandated by CHAP. 22, DIV. IV, 2211.4, formula (6-2), is represented by computer analysis load combination [13]. Two issues will be addressed:

(a) Combined Axial Force and Bending about Major Axis The maximum axial tensile force is

P_{u ⫽}106 kips associated with

M_ux,max⫽ 18.0 kip-ft M_ux,min⫽15.0 kip-ft

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5.20 COLUMN DESIGN 165

12.5 M_max Cb_⫽

2.5M_{max ⫹}3MA_⫹ 4MB _⫹ 3MC

12.5(18.0)

⫽ ⫽ 1.07

2.5(18.0)_⫹ 3(17.3)_⫹4(16.5)_⫹ 3(15.8) The properties of the W12 _⫻ 40 column are

Ag_⫽ 11.8 in.2 rx_⫽ 5.13 in. ry_⫽1.93 in.

L_p⫽ 8.0 ft_⬍ L_b⫽ 13.5 ft_⬍ L_r⫽ 26.5 ft

␾bMp_⫽ 155 kip-ft From LRFD selection table 4-19, Part 4

␾bMr_⫽ 101 kip-ft From LRFD selection table 4-19, Part 4 BF_⫽ 2.92 From LRFD selection table 4-19, Part 4 The moment correction corresponding to

␾bMn_⫽ C [b _␾bMp_⫺ BF(Lb _⫺ L )]p

␾bMn_⫽ 1.07[155_⫺ 2.92(13.5_⫺ 8.0)]_⫽ 149 kip-ft Using the Specification interaction equation (H1-1a) yields

Pu 106

⫽ ⫽ 0.278_⬎ 0.2

␾tPn 382

P_u 8 M_ux 8 18.0

⫹ 冉 冊 ⫽ 0.278_⫹ 冉 冊_⫽ 0.385_⬍ 1.0

␾_tP_n 9 _␾bM_nx 9 149

(b) Design of Column for Tension about Minor Axis In the absence of bending moment,

Pu_⫽ 106 kips The following capacity equations apply:

␾_tP_{n ⫽ ␾t}F A_{y g} with_{␾ ⫽t} 0.9

␾tPn _{⫽ ␾}tF UAu g with_{␾ ⫽}t 0.75

␾_tF A_{y g ⫽} 0.9(36)(11.8)_⫽ 382 kips_← GOVERNS

␾tF Au g _⫽ (0.75)(58)(0.9)(11.8)_⫽462 kips

166 SEISMIC STEEL DESIGN: BRACED FRAMES

The balance of this calculation yields

␾tF Ay g_⫽ 382 kips_⬎ 106 kips applied

The column has adequate tensile resistance about the principal axes.

Note: Detailed design presentation of the second- and fourth-story columns (structural elements 9 and 19) is not included since it can be verified that these components are less critically stressed than their counterparts beneath at the corresponding lower story. For instance, while size of the second-story column was kept unchanged—as was the first-story column—the reader will note a dramatic drop in the member force as compared to the column below.

Reducing column size at each floor was avoided because frequent splicing would increase cost and slow down construction.

In document Earthquake Engineering - Application to Design Charles k. Erdey (Page 172-181)