In this section, the behaviour and design of beams acting primarily in bending will be considered. In most beams axial forces are small enough to be neglected, but where large axial forces may occur column design procedures should be employed.

Moment-curvature relationships for steel beams under monotonic loading

For the adequate seismic design of the steel beams, and the associated connections and columns, the moment-curvature or moment-rotation relationship should be known. A long stable plastic plateau is required which is not terminated too abruptly by lateral or local buckling effects, such as indicated by terminating at points A, B and C in Figure 10.6. The curves terminating at D and E are typical of the desired behaviour achieved by well-designed beams under moment gradient and uniform moment, respectively. The

Local buckling

M Mp

Local buckling

Ideal elasto-plastic

Available Oh for curve D

Theoretical Oh for curve D

Ideal elasto-plastic

Rotation, 6

Figure 10.6 Behaviour of steel beams in bending moments in Figure 10.6 have been normalized in terms of the plastic moment capacity, mp = Sfy

where S is the plastic modulus of the section, and fy the characteristic yield stress of the steel.

Moment gradient is the usual loading condition to be considered with plastic hinges forming at the ends of beams in laterally loaded frames. The localization of high stresses produced by the moment gradient causes strain hardening to occur during plastic rotation, resulting in an increase in moment capacity above the ideal plastic moment M (curve D in Figure 10.6). Strain hardening may increase the plastic moment by as much as 40% (Erasmus, 1984). Local buckling and lateral buckling arising from plastic deformation of the compression flanges generally produce a reduction of moment capacity in the later stages of rotation, as illustrated by curve D in Figure 10.6.

To predict the rotation capacity of a plastic hinge the following expression presented by Lay and Galambos (1967) for the monotonic inelastic hinge rotation 9h (Figure 10.7) of a beam under moment gradient may be used:

btf_ dtw

where b is flange width, d overall depth of section, tf flange thickness, tw web thickness, Af flange area, Aw web area, V1 and V2 are absolute values of shears acting either

Figure 10.7 Beam under moment gradient with plastic hinge deformations and the hinge rotation Oh of equation (10.5) as defined by Lay and Galambos (1967). (Reproduced by permission of the American Society of Civil Engineers)

Bending moment

Figure 10.7 Beam under moment gradient with plastic hinge deformations and the hinge rotation Oh of equation (10.5) as defined by Lay and Galambos (1967). (Reproduced by permission of the American Society of Civil Engineers)

side of the hinge, arranged so that Vi < V2, P is the ratio of strain at onset of strain hardening to strain at first yield, and ey is strain at first yield. Oh represents a substantial proportion of the total rotation capacity of the beam (Figure 10.6). For the American section 10 WF25 (A36 steel), equation (10.5) predicts that Oh = 0.07 radians. It should be noted that equation (10.4) incorporates simplifications which lead to underestimations of Oh of 20% or more.

The degree to which the plasticity of a section is utilized in rotation may be expressed by the rotation capacity R, which is a ratio of the plastic hinge rotation to the rotation at or near first yield. Under monotonic loading, the rotation capacity is a function only of the beam section properties and its lateral supports, and decreases as some inverse function of the slenderness ratio 1/ry. Using the definition

Up where Op = Mpl/EI, Takanashi et al. (1973) have shown for typical Japanese beam sections that R1 exceeds 10 for l/ry less than about 40, and R1 = 2 for l/ry of about 100.

A similar alternative to equation (10.6) calculates the rotation capacity from

Uy where Oy is the elastic rotation between the far ends of the beam segment up to the formation of the hinge.

The rotation capacity of plastic hinges may be subject to reduction under cyclic loading, as discussed below.

In steel frames designed to make good use of inelastic resistance in earthquakes, several reversals of strain of 1.5% or more may have to be withstood. As discussed in Section 10.2.2, stable repetition of the monotonic ductile capacity of beams, as measured by 9h or r in equations (10.5) and (10.7), may not be possible under cyclic loading to higher strains. The hysteretic degradation of strength observed by Vann et al. (1973) was mainly due to web buckling, but flange buckling and lateral torsional buckling, plus low-cycle fatigue, can also have similar effects.

The rate at which strength degradation occurs is, of course, significant to design. It is of interest that Vann et al. (1973) found that the strength of an American W8 x 13 I-section (Figure 10.8) had degraded to 72% of its plastic moment after 11 load cycles. This behaviour would be acceptable for full ductile design as strength degradation of up to 20% after four load cycles represents robust cyclic behaviour.

As may be concluded from the above discussion, the key factor in maintaining beam strength under seismic loading is the provision of stiffness or restraints to control local and lateral buckling. The requirements of the New Zealand code set out in Tables 10.3

Load (kips)

Monotonic

Cyclic

Monotonic

Load (kips)

Slenderness ratio - ~ 30 min

Figure 10.8 Hysteresis loops for a steel beam under moment gradient (after Vann et al., 1973)

Cyclic

Slenderness ratio - ~ 30 min

Figure 10.8 Hysteresis loops for a steel beam under moment gradient (after Vann et al., 1973)

Table 10.3 Values of plate element slenderness limits for hot rolled steel, simplified from NZS | |||||

3404 (1997). (Reproduced by permission of Standards New Zealand) | |||||

Case |
Plate Longitudinal |
Category 1 |
Category 2 |
Category 2 |
Category 4 |

number |
element edges |
members1 |
members2 |
members3 |
members4 |

type supported |
(Ki) |
&e2 ) |
(Ks) |
(^e4) | |

Flat One | |||||

1 |
(Uniform compression) |
9 |
9 |
10 |
25 |

Flat One | |||||

2 |
(Maximum compression at |
9 |
9 |
10 |
25 |

unsupported edge, zero stress | |||||

or tension at support edge) | |||||

Flat Both | |||||

3 |
(Uniform compression) |
25 |
30 |
40 |
603 |

Flat Both | |||||

4 |
(Either non-uniform compression |
30 |
40 |
55 |
75 |

or compression at one edge, | |||||

tension at the other) | |||||

5 |
Circular hollow section |
35 |
50 |
65 |
170 |

1 Fully ductile structure /> 3. 2Limited ductility, 3.0 > /> 1.25. V = 1.25. = 1.0.

1 Fully ductile structure /> 3. 2Limited ductility, 3.0 > /> 1.25. V = 1.25. = 1.0.

Table 10.4 General limit on N*/$Ns as a function of member category (ductility demand) from NZS 3404 (1997). (Reproduced by permission of Standards New Zealand)

Category member N*/$Ns not to exceed:

in a column member in a brace in an eccentrically braced frame

1 |
0.5 |
_1 |

2 |
0.7 |
_1 |

3 |
0.8 |
0.8 |

4 |
1.0 |
1.0 |

1This category of member is not appropriate for this application.

1This category of member is not appropriate for this application.

and 10.4, show the increasing stability needs with increasing ductility demands. In Table 10.3, b fy

e tV 250

where b is the clear width of the element, and t is plate thickness.

10.2.5 Steel columns

Columns are often required to resist appreciable bending moments as well as axial forces. The moment-curvature relationships for the so-called 'beam column' are similar to those for beams under uniform moment, except that the capacity is reduced below the beam plastic moment mp by the presence of axial load, as shown in Figure 10.9.

As indicated in Figure 10.9, the full moment mpc of columns may not be developed because of local buckling or lateral torsional buckling as for beams. Although columns should generally be protected against inelastic cyclic deformation by prior hinging of the beams, some column hysteretic behaviour is likely in strong earthquakes in most structures, and even with beam hinging mechanisms column hinges (or pins) are required at the lowest point in columns, as shown in Figure 8.9(b).

The behaviour of steel columns under cyclic bending is similar to that of beams without axial load, except that the axial force added to the bending moment concentrates the yielding in the regions of larger compressive stress. This leads to a more rapid decay of load capacity owing to more extensive buckling, as may apparently be inferred by comparing Figure 10.10 with Figure 10.8. Second-order bending (P x A effect) may also be important in the inelastic range.

Design recommendations which allow for the effects to axial load and restraint are discussed below.

Load (kips)

Load (kips)

Deflection (ins)

Slenderness ratio

Figure 10.10 Hysteresis loops for a steel member under cyclic bending with a constant axial force (after Vann et al., 1973)

Deflection (ins)

Slenderness ratio

Figure 10.10 Hysteresis loops for a steel member under cyclic bending with a constant axial force (after Vann et al., 1973)

According to NZS 3404 (1997), the degree of ductility which a column should be designed to supply is a function of the level of axial compression N * expressed as a fraction of its yield compression capacity $Ns, where $ is strength reduction factor (typically 0.8 or 0.9), and Ns = Afy is nominal section capacity, as given in Table 10.4.

In addition, for category 1, 2 and 3 column members, excluding brace members of concentrically and eccentrically braced frames, the following limitation on design axial compression applies. When capacity design (see Section 8.3.8) is not undertaken,

0.88

where N* is the design axial compression force, is 0 for columns forming part of a seismic resisting system and 0.5 for columns forming part of an associated structural system, and

I Ns Nol in which

JT2EI

with i being the second moment of area for the axis about which the design moment acts, i.e. the axis perpendicular to the plane of the structural system, and l the actual length of the member.

When capacity design is undertaken and the column is a secondary element,

0.88

^EYC

where NO C is the capacity design derived design axial compression force on the column when the column is a secondary element.

In earthquake resistant design, special considerations regarding effective length of columns arise through the effects of inelasticity and drift limitations. Alternatively, if the effective lengths of columns are to be calculated it should be noted that effective length factors for inelastic columns will be the same only when the column acts as an independent member. In other cases (i.e. when an inelastic column is part of a continuous frame) its effective length should be calculated appropriately. Estimating the effective length of a steel member can be quite complex, and the reader is recommended to read modern earthquake code requirements, such as in NZS 3404 (1997).

Columns designed to respond elastically in earthquakes may be designed to the normal non-seismic rules for lateral restraint against buckling, but extra precautions are required for the development of limited or full ductility as set out in Table 10.5. As shown in the

Table 10.5 Spacing of lateral restraints for steelwork (adapted from Walpole and Butcher, 1985)

Parts of members requiring Parts of members requiring full ductility R = 24, a = 0.75 limited ductility R = 10, a = 1.0

Flange length Ly where Ly > 480a Ly > 480a Ly > 640a Ly > 640a the compression flange is fully yielded

Flange length Ly where Ly > 480a Ly > 480a Ly > 640a Ly > 640a the compression flange is fully yielded

Spacing of braces with |
<480a |
one brace required |
<640a |
one brace required |

length Ly | ||||

Spacing to brace |
<720a |
<720a |
<960a |
<960a |

adjacent to length Ly |

1Parts of members responding elastically should be braced according to allowable stress. 2Symbols are defined in Section 10.2.5.

1Parts of members responding elastically should be braced according to allowable stress. 2Symbols are defined in Section 10.2.5.

table, appropriate levels of rotation capacity R, as defined by equation (10.7), are R = 10 for structures of limited ductility (R = 10 is normally used for non-seismic plastic design), and R = 24 for fully ductility structures. The spacings in Table 10.5 are based upon the length 640aa, where

Jy and Ly is the length of column (or beam) over which the compression flange is fully yielded, which may be taken as occurring where

where M*es is the design bending moment at the point under consideration for calculation of the length of yielding region, C1 = 0.85 for N*/$Ns < 0.15, C1 = 0.75 for N*/$Ns < 0.15, $ is a strength reduction factor, typically 0.8 or 0.9, Ms = Ms or Mr, and Mr is the section moment capacity reduced by axial load.

Local buckling

The section geometry limits for controlling local buckling of steel columns are the same as for beams as given in Table 10.3.

Forces in struts

The maximum compressive load capacity of struts not subjected to bending may be taken as

where Fac is the maximum compressive stress as a function of the slenderness ratio, calculated on a permissible stress basis, and As is the sectional area of the member.

Combined axial load and moment

Where uniaxial bending occurs about the major principal axis, the design bending moment M* should satisfy

where $ is the strength reduction factor and

is the nominal section moment capacity, reduced by axial force (tension or compression), with N* being the design axial force, Ns the nominal section capacity for axial force, and Msx the nominal section capacity for bending moment.

The equivalent y-axis expression is used for uniaxial being about the minor axis, and where biaxial bending occurs the following condition should be satisfied:

$ns $msx $mSy Shear in columns

In the unusual circumstances where the shear stress is high in a column (i.e. v/vp > 2/3), the interaction formula for moment, axial load and shear given by Neal (1961) is appropriate:

Was this article helpful?

## Post a comment