Paper: SurteedYeap

Paper

Load-strength charts for pitched roof, haunched steel portal frames with partial base restraint J. 0. Surtees, BSc, CEng, FIStructE, MICE University of Leeds

S . H. Yeap, BEng, MSc(Eng1 Aoki Construction, Taiwan

Synopsis Load - strength charts have proved useful for preliminary sizing of members in pitched roof portal frames. They also provide an independent means of checking simple plastic collapse strength derived from computer analysis. Unlike previous charts, those presented here allow for the efSect of partial column base restraint and enable the minimum length of eaves haunch to be determined directly. The new charts are applicable to a wide range of frame geometry and rajier/column strength ratios. Nominal base restraint is often used in the UK to prevent inward collapse of frames and consequent damage to external walls, in the event of afire. More substantial restraint can also be used to improve sway stability and eaves deflection. The basis of the charts is defined and may be incorporated into a more direct computer-aided design scheme if desired, but the real intention of the charts is to provide a computer-independent means of determining member sizes. Three examples of their use are given. The first sets out with preconceived rafter/column strength ratio, assuming pinned bases. The second imposes a restricted set of preferred sections, also using pinned bases, and seeks to improve on the previous total weight of steel. In the final example, use of base restraint equivalent to 20% of the column plastic moment leads to a further 7.5% saving in total weight of steel.

Notation a is the depth of haunch below rafter centreline b is the horizontal distance from apex to rafter hinge D is the depth of rafter section h is the vertical height to eaves k, kl are working constants L is the span of frame Lh is the haunch length (horizontal) Mpcolurnn is the minimum plastic moment of column Mpcolurnn' is the actual moment resistance of column Mprafrer is the minimum plastic moment of rafter Mprafterc is the actual moment resistance of rafter Mx is the bending moment in rafter at distance x p y is the design strength in bending Sx is the plastic modulus about principal axis of UB section X is the horizontal distance from column centreline a is the ratio of base restraining moment to M p c a l u m n

J? is the ratio of Mprafter to Mpcolumn

v is the shape factor for rafter section o is therafter slope (in deg.) o is the uniform vertical loadhnit length of rafter

Introduction Charts for assisting the design of haunched, pitched roof portal frames have been available for several These are derived from simple plastic collapse equilibrium and provide non-dimensional relationships between frame geometry, member moment resistances, and collapse load. Portal frame construction has been standardised to a large extent in the UK, and design procedures follow a common pattern. In essence, rafter and column sections are chosen on the basis of simple plastic collapse equilibrium and increased if necessary to prevent member buckling or overall frame instability. The charts provide a simple basis for initial sizing and indicate consequential properties such as the apex hinge position and horizontal

thrust at base level. They are also useful for providing an independent check on computer output.

One limitation of the charts is that they are applicable only to pinned-base frames. Not infrequently in portal frame construction, it is necessary to provide nominal base restaint to resist inward collapse of the frames when weakened by fire4. As the base restraint is normally reversible, advantage may be taken of its presence when sizing rafter and column sections. Base restraint has a secondary benefit in improving sway stability and also reduces eaves deflection under service load.

The charts also make fixed assumptions with regard to haunch length and magnitude of bending moment at the shallow end of the haunch which, together with frame geometry, determine the ratio of column moment to rafter moment for a given case. If the actual resistance moments of the column and rafter material do not conform to this ratio, it is necessary to check that the assumed haunch length is sufficient to suppress a plastic hinge at the rafter/haunch junction.

In contrast, the charts presented below provide for base restaint up to 30% of column moment capacity. They also allow any ratio of rafter and column bending capacity to be used (subject to MP column > Mpdter). Furthermore, haunch length and rafter hinge position are determinable from the charts, once this ratio is established.

Basis of charts Initial assumptions The primary form of collapse addressed in the charts is shown in Fig l . This corresponds to a single-bay frame loaded vertically by a uniformly distributed load, m, per unit length on plan. Formation of a plastic hinge at the shallow end of the haunch is suppressed by ensuring that the haunch extends to a point where the bending moment is below the first-yield capacity of the rafter section, i.e. Mprafter/v.

It is further assumed that the depth of the column hinge occurs at 1.5 D below the rafter centreline, where D is the depth of the rafter section'. This provides for the haunch to be cut from the same universal beam material as the rafter. The rafter depth D is approximated by L/60, which is deemed to represent current practice more typically than U 5 5 used in the charts.

Thus, in Fig. l , a = l .5 x L / 60 = L/40 and the following general equation may be derived for rafter bending moment:

\BM5 Mp rafterIuhere I l

Fig I. Primary form of collapse

The Structural EngineerNolume 74/No 1/2 January 1996 9

Paper: SurteedYeap

M , = OSm( L - .U) + MP - - + a) (h + x tan 8)Mp column ....( l ) h - a

Using appropriate boundary conditions, this equation may be used to establish values for the key design parameters, i.e. Mpco~umn, M p r a f t e r , position of rafter hinge b and length of rafter haunch Lh. Expression of these relationships in chart form enables the effect on b and Lh of practical choices for MPcolumn and M p r a f t e r to be assessed very quickly.

Position of rafter hinge Differentiating eqn. ( l) , maximum moment occurs when

L (1 + a)Mp column tan6 2 ( h - a)w

x=--

or putting x = - - b, - - L b - (l + a) tane column

2 L ( h a ) wL2 ...

h - a

r 1 ....( 3 )

Combining eqns. (2) and (3 ) ,

L l h = 2.0 kr L I h = 3.0

0.041 1

Charts for rapid determination of blL for design purposes are presented in Fig 5 . It can be shown that the influence of a is approximately linear for constant Llh, and correction factors are given on the charts to provide for non-zero values of a .

Minimum bending capacity for columns MPcolurnnlOL' may be obtained by substituting for blL in eqn (2). This result is embodied in the charts shown in Fig 2.

The main curves provide values of MPcolurnnlOL' at a = 0 for ranges of l3,8 and Llh. Ml,co~umn may be modified by a factor kl, shown on the same charts, for non-zero values of a. In the latter curves, both a and 13 affect the magnitude of kl. Three pairs of lines are shown corresponding to a = 0.1, 0.2 and 0.3, respectively. Each solid line refers to l3 = 1 .O, and each broken line refers to 13 = 0.2. Values of kl for intermediate values of B may be interpolated linearly between these extremes.

Minimum haunch length to prevent rafter end hinges The minimum length of haunch necessary to prevent formation of plastic hinges at the junction of the rafter and haunch may be found by putting

x = L, and M , = MP rafter / v (i.e. p M,, / v ) in eqn. (1).

Thus : P Mp column v = 0.5wLh ( L - ) + Mp column

-- (l + a) ( h + L, tan e) MP column ....( 4) h - a

The charts for minimum haunch length Lh in Fig 4 have been derived using eqn. (4) together with the previously found solution for M p c o l u m n .

Practical use of the charts The charts in Figs 2 to 5 represent a precise balance between external load and internal resistance. Practical frames, on the other hand, possess surplus strength by virtue of the limited choice of steel section sizes. As external load is known at the outset, the collapse mechanism will be incomplete if surplus strength exists, and it will be necessary to decide which of the potential hinges develop to full member capacity. The safest procedure is to assume that the column hinges form before the apex hinge. This ensures

1 .m

0.95

0.90

0.85

kl L / h = 4.0 kr 1 0 0

0.95

0.90

0.85

0.C30 i 5 10 15 20 25 30

Rafter slope ( O )

L I h = 5.0 kr 1 .m 0.95

0.90

0.85

0.03' J 0 5 10 15 20 25 30

Rafter slope ( O )

L l h = 6.0 kr 1 .m

0.95

0.90

0.85

0.04-- - .

0.030 i 5 10 15 20 25 30

Rafter slope ( O )

L I h = 7.0 kl 0.1 I , 1.02

0.03; 5 10 15 20 25 30

Rafter slope ( O ) Rafter slope ( O ) Rafter slope ( 0 )

Fig 2. Required column moment capacity MP for various rafter slopes, rafter-column M P ratios P and base restraining moments a

10 The Structural Engineer/Volume 74/No 1 /2 January 1996

Paper: SurteedYeap

L 1 h = 2.0 kl L l h = 3.0 kl L l h = 4.0 kr

0.01 " - ~ .iL 5 10 15 20 25 30

Rafter slope ( O )

L I h = 5.0 kt 1 .m 0.95

0.90

0.85

0 '0 5 10 15 20 25 30

Rafter slope ( 0 )

i.00

0.55

6.90

G. 85

5 '0 5 10 15 20 25 30

Rafter slope ( O )

1 .m

0.95

C.90

0.85

L I h = 6.0 kl 1 .m 0.95

0.90

0.85

"0 5 10 15 20 25 30

Rafter slope ( O )

"0 5 10 15 20 25 30

Rafter slope ( O )

L l h = 7.0

0 5 10 15 20 25 30

Rafter Slope ("1

Fig 3. Required rafter moment capacity MP for various rafter slopes, rafter-column MP ratios p and base restraining moments a

L I h = 2.0

0.35

0.3! - - l U = 0 0 a = 0 3 _ _ _ _ _

0.251 - . .

0.2 B 0.2

0.15 0.3

0.1

0.05

0.4

0.6 , . . . . - 0.8

1 .o '1, 5 10 15 20 25 3b Rafter slope ( 0 )

L l h = 5.0

0.35 a = 0.0 a = 0.3

~ - - . . . - .

0.2

0.1 5

0.1

0.05

O:, 5 10 15 20 25 40

Rafter slope ( O )

L I h = 3.0

0.3 a = 0 0 D = 0.3

0.251 . . . l 0.2

0.15

0.7

0.05

'0' 5 10 15 20 25 3'0

Rafter slope ( 0 )

L l h = 6.0

B

0.2

0.3 0.4

0.6

0.8 1 .o

5 10 15 20 25 30

Rafter slope ( O )

Fig 4. Haunch lengths Lh for various rafter slopes, rafter-column MP ratios p and base restraining moments a

The Structural EngineedVolume 74/No 1/2 January 1996

1.00

0.95

0.90

C. 85

1 B

0.2

0.3 0.4

0.6 0.8 l .o

O1, 5 10 15 20 25 40

Rafter slope ( 0 )

L I h = 7.0

. _ .......

U = 0.3

5 10 15 20 25 ;Cl Rafter slope ( 0 )

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Paper: SurteedYeap

L I h1 = 2.0 L I h1 = 3.0 0.16

0.14

0.12 - . b/L = ( l + 0.750) b/L ._-

0.1 - B . .

$ 0.08- h

'0 5 10 15 20 25 30

Rafter slope ( O ) Rafter slope ( O )

-1 . h

L / h1 = 5.0 P

.. - .

'0 5 10 15 20 25 30

L I h i = 6.0 P

0.2

Rafter slope ( O ) Rafter slope ( O )

Fig 5, Apex hinge locations b/L for varying rafter slope, rafter-column MP ratio p and base restraining moment a

that the haunch length L+, is always sufficient to suppress an end hinge in the rafter, whatever the real distribution of bending moment.

The procedure for using the charts may therefore be summarised as follows:

(1) Calculate the spanleaves height ratio uh. (2) Decide rafter slope (deg.). (3) Decide the M p r a ~ m J M p c o ~ u m n ratio B. (4) Decide the degree of base restraint. (5 ) Calculate mL2. (6) Determine the required value of MP column from Fig 2.

(8) Choose rafter and column sections from the range of available material. (9) Obtain the minimum necessary length of haunch Lh from Fig 4, (10) Determine the distance b to the apex hinge using Fig 5.

If a significantly larger than necessary section is chosen for the column, a reduced rafter section may be determined as follows:

(la) Use the selected column capacity Mpcolumn', with Fig 2 to determine B. (2a) Multiply Mpcolumn' by 13, to obtain Mprafter.

(3a) Obtain L h and b as before but using Mpcolumn*.

If the rafter section chosen in step (l) is significantly stronger than the MP rafter requirement, a reduced minimum value for Mpcolumn may be obtained as follows:

(l b) Use the selected rafter capacity MP raner' with Fig 3 to determine B (2b) Divide M P rafter', by 0, to obtain Mpcolumn.

(3b) Choose a column section to suit Mpcolumn.

(4b) Obtain Lh and b as before, using B based on the chosen column capacity MP column';

It must be stressed that the above procedure is concerned only with simple plastic collapse strength. Checks for column, rafter and haunch stability must be carried out after section properties and member geometry have been established. A comprehensive account of wider issues in portal frame design, with particular reference to Eurocode 3, has been published recently5.

(7) Multiply Mpcalomn by B, to obtain M P rafter.

12

L I h1 = 4.0 0.16 P

0.12- - - - -

0.1 . -

. . . . . . - .

Rafter slope ( O )

Rafter slope ( O )

The following example, taken from ref. 2, illustrates the design procedure.

Design example l

3.750

7.600

25.000

Fig 6.

L& = 2517.6 = 3.29 Rafter slope = tan-l (2hdL) = tan-' (2 x 3.75 /25) = 16.7" B = 0.3 say a = 0.0 for a pinned base uL2 = 9.48 x 252 = 5925 kNm From Fig 2, MpcolumrJL* = 0.068 M p m l u m n = 0.068 X 5925 = 402.9kNm and MPraftw = 402.9 x 0.3 = Assuming grade 43 steel with py = 275 Nlmm2

120.9 k Nm

Minimum plastic modulus Sx for column = 402.9 10.275 = 1465.1cm3 Minimum plastic modulus Sx for rafter = 120.9 l 0.275 = 439.64cm3 :. use 457 X 191 x 67kg/m UB for the column (actual Sx = 1472cm3)

and 356 x 127 x 33kglm UB for the rafter (actual Sx = 539cm3) These are not significantly different from the required minimum values and the present B value will still be valid. Using this value of 13 in conjunction with Fig 4, LdL = 0.16 :. minimum haunch length = 0.16 x 25 = 4.0m Similarly from Fig 5 , b/L = 0.065 and b = 1.6m

The Structural EngineedVolume 74/No 1/2January 1996

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In the above example it was possible to select a column capacity close to the minimum value given in the chart. If the range of sections is restricted, for example because of material rationalisation, this will not always be possible. The next example illustrates how to apply the charts to selection from a limited range of preferred sections.

Design example 2 Using the same geometry and loading as design example 1, this example will consider the effect of restricting the choice of section sizes to the following:

305 x 102 x 25kg/m (92.4) 457 x 152 x 52kg/m (303.0) 356 X 127 X 33kg/m (148.0) 457 x 152 x 60kg/m (352.0) 356 x 127 x 39kg/m (180.0) 457 x 191 x 67kg/m (404.0) 406 x 140 x 46kg/m (245.0) 457 x 191 x 74kg/m (457.0)

(Figures in parentheses are the plastic moment capacity Mcx in kNm - for grade 43 steel)

If the lightest section is chosen for the rafter, i.e. 305 x 102 x 25 kg/m UB.

:. minimum value for B =0.22 (from Fig. 3) and MPca~urnn = 92.4/0.22 = 420 kNm :. use 457 x 191 x 74 kg/m UB for the column (Mcx = 457 mm). For the purpose of deriving minimum haunch length, MpcolurndOL’= 457/5925 = 0.077 1 B = 92.4 / 457 = 0.202 and, from Fig 4, LJL = 0.24 :. minimum haunch length = 0.24 x 25 = 6m .

Thus, rationalisation of rafter and column sections is, in this instance, accompanied by a 5% saving of steel. Some of this saving wquld, however, be cancelled by increased fabrication cost as a result of the longer haunch.

Design example 3 The final example considers the possibility of reducing the column size of design example 2 by introducing a small restraining moment at the base. Assuming a restraining moment of 0.2 Mpcolurnn, the minimum value for B = 0.24 (from Fig 3)

:. use 457 x 191 x 67 kg/m UB for the column (Mcx = 404 m m ) .

B = 92.4 / 404 = 0.23 (assuming the same rafter size as design example 2) and from Fig 4, Lh /L = 0.18 :. minimum haunch length = 0.18 x 25 = 4.5m

Therefore, a base restraint of 0.2 Mpcolurnn has effected a further reduction of 7.5% in the weight of steel. If base restraint was not necessitated by fire considerations, increased fabrication cost would, of course, have to be set against the saving in steel.

Conclusion A set of charts has been presented which provides greater flexibility in sizing primary material for pitched roof portal frames than was available with previous charts. In particular, the effect of partial base restraint and a facility to determine minimum haunch length have been incorporated.

The basis of the charts is also defined and may readily be expressed in the form of an interactive computer program. However, the intention behind formulation of the charts is that they should provide a computer-independent basis for designing or checking portal frames, as well as lending insight into the influence of design parameters such as base restraint and rafter-column strength ratio on rafter and column section size.

Examples are given to illustrate use of the charts. Significant savings in weight of steel have been quoted but it should be borne in mind that reductions in cost of raw steel may be partly cancelled by increased fabrication cost.

References 1. Weller, A. D.: Lecture 14: ‘Portal frame design’, Zntroduction to Steelwork

Design to BS59.50 .- Part I, Steel Construction Institute, 1988 ’ 2. Manual for the design of steelwork building structures, Institution of

Structural Engineers, 1 989 3. Owens, G. W., and Knowles, P. R. (eds): Steel designer’s manual, 5th

ed., Blackwell Scientific Publications, Oxford, 1992 4. Newman, G. M.: Fire and steel construction: the behaviour of steel

portalfiames in boundary conditions, 2nd ed., Steel Construction Institute, 1990

MprndOL’ = 92.4 / 5925 = 0.0156

Mpcolurnn = 92.4 / 0.24 = 38 1 kNm

Mpcolurnn/OL2 = 404 / 5925 = 0.0682

The Structural Engineer/Volume 74/No 1/2 January 1996

5. King, C. M.: ‘Plastic design of single-storey pitched-roof portal frames to Eurocode 3’, Technical Report, SCI Publication 147, Steel Construction Institute, 1995

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