What Is Rn In Structural Analysis

This equation represents the average value of stress in the cross-section which will induce the yield stress at mid-height of the column for any given value of n. Experimental evidence obtained by Perry and Robertson indicated that the hypothetical initial curvature of the column could be represented by;

which was combined with a load factor of 1.7 and used for many years in design codes to determine the critical value of average compressive stress below which overall buckling would not occur. The curve of stress/slenderness for this curve is indicated in Figure 6.15 for comparison with the Euler and Tangent Modulus solutions.

Perry Robertson Buckling Formula

Figure 6.15

6.5 European Column Curves

Whilst the Perry-Robertson formula does take into account many of the deficiencies of the Euler and Tangent Modulus approaches, it does not consider all of the factors which influence the failure of columns subjected to compressive stress. In the case of steel columns for example, the effects of residual stresses induced during fabrication, the type of section being considered (i.e. the cross-section shape), the material thickness, the axis of buckling, the method of fabrication (i.e. rolled or welded), etc. are not allowed for.

A more realistic formula of the critical load capacity of columns has been established following extensive full-scale testing both in the UK and in other European countries. The Perry-Robertson formula has in effect been modified and is referred to in design codes as the Perry strut formula and is given in the following form:

from which the value of pc may be obtained using:

PkPj bPy)

where:

py is the design strength X is the slenderness

The Perry factor n for flexural buckling under axial force should be taken as:

X0 is the limiting slenderness below which it can be assumed that buckling will not occur.

The European Column curves are indicated in graphical form in Figure 6.16.

The Robertson constant a should be taken as 2.0, 3.5, 5.5 or 8.0 as indicated in design codes depending on the cross-section, thickness of material, axis of buckling and method of fabrication.

Figure 6.16

Typically, the values of 'a' are allocated to various cross-sections as indicated in Table 6.1.

Type of section

Maximum thickness (see Note 1)

Robertson's constant 'a'

Axis of buckling x-x y-y

Hot finished structural hollow

2.0

2.0

section

Cold-formed structural section

5.5

5.5

Rolled I-section

<40mm

2.0

3.5

>40mm

3.5

5.5

Rolled H-section

<40mm

3.5

5.5

>40mm

5.5

8.0

Welded I or H-section (see Notes 2

<40mm

3.5

5.5

and 4)

>40mm

3.5

8.0

Rolled I-section with welded flange

<40mm

2.0

3.5

cover plates 0.25<U/B<0.8 (see Figure 6.17a)

>40mm

3.5

5.5

Rolled H-section with welded

<40 mm

3.5

5.5

flange cover plates 0.25<U/B<0.8 (see Figure 6.17a)

>40mm

5.5

8.0

Rolled I or H-section with welded

<40 mm

3.5

2.0

flange cover plates U/B>0.8 (see Figure 6.17b)

>40mm

5.5

3.5

Rolled I or H-section with welded

<40 mm

3.5

5.5

flange cover plates U/B<0.25 (see Figure 6.17c)

>40mm

3.5

8.0

Welded box section (see Notes 3

<40 mm

3.5

3.5

and 4)

>40 mm

5.5

5.5

Round, square or flat bar

<40 mm

3.5

3.5

>40mm

5.5

5.5

Rolled angle, channel or T-section

Any axis: a=5.5

Two rolled sections laced, battened

or back-to-back Compound rolled

sections

Note 1 : For thicknesses between 40 mm and 50 mm the value of pc may be taken as the average of the values for thicknesses up to 40 mm and over 40 mm for the relevant value of py.

Note 2 For welded I or H-sections with their flanges thermally cut by machine without subsequent edge grinding or machining, for buckling about the y-y axis, a=3.5 for flanges up to 40 mm thick and a=5.5 for flanges over 40 mm thick.

Note 3 The category 'welded box section' includes any box section fabricated from plates or rolled sections, provided that all of the longitudinal welds are near the corners of the cross-section. (This is to avoid areas in the cross-section which have locked in residual compressive stresses which induce premature failure at a reduced buckling strength). Box sections with longitudinal stiffeners are NOT included in this category.

Note 4 For welded I, H or box sections pc should be obtained from the Perry strut formula using a py value 20 N/mm2 below the normally assigned value. (This is a simplification to avoid the use of a different set of curves which are required for fabricated sections).

Table 6.1

Figure 6.17

The design of the majority of concrete and timber column members is usually based on square, rectangular or circular cross-sections, similarly with masonry columns square or rectangular sections are normally used. In the case of structural steelwork there is a wide variety of cross-sections which are adopted, the most common of which are shown in Figure 6.18.

Figure 6.18

In all cases, irrespective of the material or member cross-section, an assessment of end and intermediate restraint conditions must be made in order to estimate effective buckling lengths (Le) and hence slenderness X. It is important to recognise that the effective buckling length is not necessarily the same about all axes. Typically, it is required to determine two LE and X values (e.g. LEy, Xy and LEx, Xx), and subsequently determine the critical compressive stress relating to each one; the lower value being used to calculate the compressive resistance of a member. In the case of angle sections other axes are also considered. The application of the Perry strut formula to various steel columns is illustrated in Examples 6.1 to 6.4 and Problems 6.1 to 6.5.

6.6 Example 6.1 Slenderness

The square column section shown in Figure 6.19 is pinned about both the x-x, and y-y axes at the top and fixed about both axis at the bottom. An additional restraint is to be provided to both axes at a height of L1 above the base. Determine the required value of L1 to optimize the compression resistance of the section.

Tor optimum compression restdaoce tint maximum itendfirncis for lengths Al! and l)C must be ih.e ssnie. i.e. " ^tH1

Cow Etkr the rffcoUvo lengths ofAB anil uc = 0 K5/-1 and Lvjk = 1 -Oii 0.85/,i-l.OLi

"lite total height tiflh? column (J_i + /..}■ jit Li + 0.85/, i = 6.0 and hence Lt = 3.2-1 m ^ = 2.76 m The rcqulrvil mint otii m

Figure 6.19

6.7 Example 6.2 Rolled UC Section

A column, which is subjected to a concentric axial load 'P', is shown in Figure 6.20. Restraint against lateral movement, but not rotation, is provided about both axes at the top and the bottom of the column. Additional lateral restraint is also provided about the y-

y axis at mid-height as shown.

Using the data provided determine the compression resistance of the column using the Perry strut formula.

pinned base

Figure 6.20

Data:

Yield Stress py = 355 N/mm2 E = 205 kN/mm2 Robertson Constants: y—y axis a = 5,5

Section Property

Section

203 x 203 x 60 UC Universal Coin inn

Cross-sectional Area (A)

76 4 crrT

Radius of Gyration (rw)

5.2 cm

Radius of Gyration {rX!()

Solution:

Perry strut formula:

Buckling Length L 2.25)= 2.25 m

The liiTettivi: huckliug [L'ii^th .", ■ 2.25 m

52.0

Eulcr slresijPi

43,27s

- LOKO.fi NAiinr

Limiting ikncteOBSS k = O.ltfe/pyf* = 10 2 * (n; x 205000055^"]

t) = CI{A - ^yiOOO = 5,5(43.27 - 15.1 y] 005 = 0.155

ftspv f+ ^ - ftfy J* so 1.5 + (801.5:3 -10SflL 6 * .3 5 Sf'

Buckling length a (1.0 n 4.5) = J.5 n l ilt ellective buckling length = 4.5 nt

S9.G

UnntipgslenddniB Z,, = 0.H,Tii/pffs= 15.1 If = a{A - jfeyi 000 = 3.5[50.22 - 15.1 yt 000 = 0.123 , IK +{l?4t)/>E _ 3554 (0.123+ 1)502.:

Piift

627.0

= iSVmm1

4> + - pvPj]P Gl 7.9 + {627.9= - S02.2 x 3 55J" Crilieal value ofp, — 192 .Ä Nl/iniir

Co mpress Eon resistance P1 = (pe*At) = (292 8 * 76,4 x 1 0J = 1231 kN

6.8 Example 6.3 Laced Section

A column comprising two Universal Beam sections laced together to act compositely is shown in Figure 6.21. The restraints to lateral movement about both the A-A and B-B axes are as indicated. Using the data given determine the compressive resistance of the section using the Perry strut formula.

Data:

Section Property

Section

533x210x82 UB Universal Beam

Yeild Stress Py=

=275 N/mm2

Cross-sectional Area (A)

105 cm2

Young's Modulus E=205 kN/mm2

Radius of Gyration (ryy)

4.38 cm

Robertson

Constants:

Radius of Gyration (rxx)

21.3 cm

x-x axis

a=5.5

2nd Moment of Area (Ixx)

47500 cm4

y-y axis

a=5.5

2nd Moment of Area (Iyy)

2010 cm4

Laorra I

3C.MI.I 11

Ptmcd

I accrn rentrainxs

lltHniin! iibc/ut

liiiij^iujj -I- 3.9 m ccilrcs

Composite Section Properties:

= 2 x [201 Ox 10" + (] 0500 x 1)] = 512.7 x ltf5 mm4

The possibility of buckling of the individual UB sections and the composite section must be considered in this problem as follows:

(Extract from to BS 5950-1:2000 Structural Use of Steelwork in Building)

"The slenderness X of the main components (based on their minimum radius of gyration) between consecutive points where the lacing is attached should not exceed 50. If the overall slenderness of the member is less than 1.4Xc the design should be based on a slenderness of 1.4Xc."

Perry strut formula:

Note: Since the same curve is used for both the A-A and the B-B axes in this case (i.e. a=5.5), the compression resistance will correspond to the axis with the highest slenderness value, i.e. the one which produces the lowest pc value.

Bolt For Angle Bar Dimension

Consider an individual UB section:

Consider the composite section:

>(0.85 x 11.0) = 9.35 m The effective buckling length .. = 9.35 m

Slendemess

156.25

0 0

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