PERFORATED METAL DECK DESIGN

Transcription

1 PERFORATED METAL DECK DESIGN with Commentary Prepared By: L.D. Luttrell, Technical Advisor Steel Deck Institute November 18, 2011 Copyright 2011 All rights reserved P.O. Box 25 Fox River Grove, IL Phone: (847) Fax: (847)

2 Section Properties. Cold-Formed steel deck is produced from thin flat material commonly supplied in steel coils. The sheet material is passed through a forming process that leads to corrugated profiles that will possess strength and stiffness as dictated by geometry introduced in the forming process. Such panels can have a range of thicknesses, various steel properties, and may be either open corrugated or have a bottom unit added to form a cellular profile. The image below represents one unit from a corrugated panel. Such panels are described in a series of Steel Deck Institute publications, Design Manual for Composite Decks, Form Decks and Roof Decks (). The structural properties for these coldformed steel panels are developed in accordance with American Iron and Steel Institute Standards (). Elements of a panel cross section usually are not compact in that they can have large width-tothickness ratios, w/t. When compressed, they may exhibit plate buckling and out-ofplane movement as indicated by the wavy line along the element length as in Figure. When such a corrugated panel is loaded in bending, the top compression element often will not be fully effective. Material nearer the webs may remain essentially straight and thus can resist compression efficiently. Farther away from the web, longitudinal strips will try to buckle out of the way as may be visualized following the Figure model where the corners; marked a, b, c, and d; correspond to the similarly marked points on Figure. Nearer the center, strips are less efficient in compression with similar overall shortening though they can t freely buckle because they are in plates, not columns. The buckled pattern becomes a square half-wave type as shown in the sketch.. See numbered references, last page.

3 Viewed from the top, a-b-c-d points of Figure form a square halfwave with the transverse lines a-b and c-d remaining straight. These lines are at an inflection and represent crossstrips that might be thought of as little columns holding the webs apart. The intermediate curved line at the middle of a-b-c-d is a transverse tension band limiting the bulge or outof-plane buckling. The square-wave buckling indicates that the central strip along the top flange is not nearly so efficient in resisting compression as is the strip nearest the web. The result of this is a non-compact and non-linear condition that makes a closed mathematical solution most difficult. The AISI Standard () contains the numerical method for dealing with this non-linear condition which is illustrated below. The curved line in Figure describes the stress distribution across the width dimension, w. With the maximum edge stress established from design conditions, the product of the area under the curve and the material thickness, t, describes the total compression force, C, developed in the element. The AISI Standard (), Section B., substitutes an effective width, b, such that f max (b)(t) is equal to the compression force C. Most steel deck sections will have flat compression elements which may be described following Figure above. Deck bending properties then are developed by representing the cross-section as a series of line elements and then multiplying these line results by a thickness, t, to develop completed properties.

4 The introduction of perforations into a compression element may reduce the effective area beyond those reductions associated with Figure. Each element with perforations is first evaluated as an un-perforated element and the effective width, b, determined. A secondary evaluation then is used that will further modify the effective width when perforations are present within the b/ widths. Deck Section Bending Properties For a deck profile, the bending property analysis often will require an iterative process that starts from a presumed compression stress. At this compression stress, an element s effective width, b, can be established. With that value in hand, the section s moment of inertia and the section modulus can be determined and, finally, the tension stress calculated. If the tension stress is within allowable limits, the determined section properties are suitable for use. If not, the presumed compression stress is lowered and the process repeated until both tension and compression stresses are within specified limits. With the top in compression and stresses within the design limits, the positive section properties are set. This same operation is repeated with the bottom elements in compression for setting the negative section properties. Based on the section modulus, S p, the nominal positive moment capacity, M np, is calculated using Equation a. The nominal negative moment capacity, M nn, is similarly calculated using Equation b. M S F (a) np nn p n y y M S F (b) Deck Webs. The deck web height, h, is described as that flat portion of material between the upper bend into the top flat and the similar bend into the lower flat. The element, marked W w in subsequent illustrations, is oriented at an angle measured from the horizontal. The performance of a web may be limited by its direct shear capacity or its stability over a support bearing. Web Crippling Web crippling limitations for solid deck webs are set in Section C.. of the AISI Standard () as reproduced here in Equation. Three slenderness terms are contained in this equation: R/t, N/t, and h/t. R/t is the radius-to-thickness ratio where the horizontal flat element transitions into the deck web. With increasing R/t ratios, web stability decreases due to the tendency for deck bends to roll out under bearing loads. The N/t ratio measures the influence of support bearing length against the element thickness and h/t indicates the influence of web slenderness on web crippling loads.

5 R N h P n =Ct Fy Sinθ 1-CR 1+CN 1-Ch t t t () R: The inside bend radius at the web bottom N: Support bearing length h: The web height (also =W w elsewhere) t: Base metal thickness of web Angle between plane of web and plane of the bearing surface C, C R, C N, and C h are coefficients listed in AISI Standard (), Section C.. Equation has three parenthesized parts and each of these has additive parts. The second part, containing N, tends to be much larger than either the first, containing R, or the last containing the web slenderness, h/t. If web perforations were to be introduced, they would be located within the flat part of the web, h. Note the t term in the leading part of the equation which indicates that web crippling is very sensitive to web thickness. Some view of the components in Equation may be developed by considering specific cases with a 2 deep deck section having: R = 3/16, N = 4, and h = 2 with the web slope, θ = degrees. Case with t = 0.0 inches and using an interior reaction for example where: P n R N h = t -CR +CN -Ch CFy Sinθ t t t (a) From AISI S00 Table C..-, C =, C R = 0., C N = 0., C h = 0.00 for interior one flange loading. There are parameter limits in the Table and for design these can define the useable value of N. = (.) The second right side expression is./0.0 =. times larger than the first. For Case, a moderate decrease in thickness to 0% of that above or 0.0 inches: P n CFy Sinθ (b) The second right side expression is./0. =. times larger than the first. In these comparisons, the last three sets of terms in Equation a are not particularly sensitive to changes in panel thickness but the complete expression is as evidenced by the leading t term. Web crippling clearly is influenced by bearing conditions at the web s lower edge. With bearing, the lower bend radii tend to roll out, to increase the lower contact width at deck valleys,

6 and to move adjacent webs a bit further apart. Such roll out would be increased by the presence of perforations if they were in the corner bend. Perforations in the flat portion of the web will lower web transverse stiffness that helps resist roll. This effect is involved with the h/t term of Equation a. Perforated Sheet Properties The perforation process is one of long history in the steel industry and is described in detail by The Industrial Perforators Association, the IPA. The IPA group standardizes punching patterns for products used in agricultural, industrial, and commercial screening for sorting and screening materials. They publish the Designers, Specifiers and Buyers Handbook for Perforated Metals () which lists standard perforation patterns with round holes up to ¾ diameter and several patterns with smaller holes. A commonly used deck pattern, as listed in the IPA manual is shown below in Figure and gives definitions for two dimensions, Length and Width. These are pattern terms and do not necessarily correspond to deck panel length and width definitions. Figure 4. The standard 60 degree staggered pattern showing length and width orientations. The IPA describes perforated sheet tensile properties as a fraction of the behavior that would have developed if the plate had not been perforated. For the case when the holes in Figure are 1/8 diameter and spaced at 3/16 centers, the IPA designated this pattern as No. and it has a 40% open area. The IPA efficiency term is determined using Equation 3 and simply means that this plate responds to loads as if it were a solid plate but at % of the original unperforated capacity. k e= 0.+p o -.p o 0. po 0. () p o perforated area percentage in a perforated band width, and expressed as a decimal k e efficiency of the perforated area relative to a solid area of the same material See reference list No. and Appendix A.

7 Figure illustrates a deck component with five bands of perforations running along the panel length. Perforations are made in strips along flat sheet coils prior to the formation of the deck profile. The strips are located so they are within the flat width of the element of the finished deck unit. The perforating process removes some material and reduces the structural properties of the strip to some new level. For example, a strip that is W tp wide with a thickness, t, and a yield strength, F y, can have tensile strength T = F y (W tp) (t). With perforations, the effective tensile area is reduced and the new strength will be (k e )( F y W tp t) where k e is established from Equation. Figure 5. Deck unit illustrating perforation strips in the top, web, and bottom flats. Deck Panel Elements and Sub Elements. It is convenient to describe a panel element using the Symbols in Figure where the double subscripted dimensions represent perforated bands within the element. Figure 6. Cross-section dimensions and definitions.

8 W t, W w, and W b : The flat dimensions of the top, webs, and bottom elements. W tp, W wp, and W bp : The widths of perforation bands when perforations are present. D: Deck depth. P: Panel corrugation width. AISI Effective Widths Deck section property analysis is first done as if the section had no perforations and this would lead to a compression element effective width as indicated by the b-dimension in Figure depending on which side of the deck is in compression. Figure 7. Section showing possible overlap of W tp on to the b/2 strip. Application of Perforation Factors. Top Flange Tension From Figure, the T o width is outside the perforated strip. However, W tp has perforations. The effective top flange width is twice T o plus k e (W t ) with k e found from Equation. This can be expressed as W t W tp + (W tp )( k e ). The perforated strip in tension shows a reduction directly related to its efficiency as related to its perforation pattern. Then the effective top width, W tef, is: Top Flange in Compression tef o e tp t tp e W T k W W W k () The effective width, b, is first established for the un-perforated case. With W t being the top flat width and referring to dimensions in Figures and, T o = (W t W tp )/. If b/ is less than T o, the perforated band does not intrude on the effective width and b is unchanged.

9 If b/ is greater than T o, that part of b extending into the W tp strip is reduced leading to: b W tef = T O+ - TO ke () Bottom Flange Tension The bottom flange effective width is B o + k e W bp. W bef =Wb -W bp(- k e) () Bottom Flange in Compression The effective width, b, is first established for the un-perforated case. With W b being the bottom flat width and referring to dimensions in Figures and, B o = (W b W bp )/. If b/ is less than B o, the perforated band does not overlay the effective width. Use the effective width, b, with no further reductions. If b/ is greater than Bo, that part of b extending into the W bp strip is reduced and the effective bottom width, W bef, is: b W bef = B O+ - BO ke () Deck Webs in Shear The effective web area for webs with acoustic perforations is: ew w wp wp e w wp e A =(W -W )t + W k t = W - W - k t () It can be more convenient to define the effective area in terms of a reduction factor, q S, as follows: Wwp A ew = - - k e ( ) Ww t = qs WW t () Ww AISI () S00, C.. for webs without holes, is used to define a nominal web shear strength, F v. The web slenderness ratio, W w /t, dictates which of the three ranges that is applicable for shear strength. When perforations are present, a reduction factor, q s, is used in defining the shear strength.

10 AISI () Section C.. addresses webs with holes typical of those present in wall studs. For these, the shear strength is established as if there are no holes and an overlay or multiplying term, q s, is applied to define the perforated section strength. While this does not apply to acoustic deck perforations, the overlay approach is the same. Web Shear Example A three inch deep deck has a 2.75 deep web with a base metal thickness of 0.0 in. and a yield strength, F y = 0 k/in. Web W w /t =./0.0 =. 0 The perforated mid-depth web band has W wp = 2.25 and 1/8 diameter perforations on 3/16 centers. From Appendix formulas: Equation leads to: p o = 0.0 (D/C) = 0.0 and k e = 0. q =- - k s e wp w (W /W )=.0-0.(./.)=0. Nominal web shear stresses, F v, are established by AISI () S00 Eq. C..- through Eq. C..-b for solid webs in terms of the height-to-thickness ratio, h/t. Here h/t is equivalent to W W /t and equal to. The transition points within the F v range all involve multiples of Ek /F = E(.)/0. which is the first transition. The second transition is at v y.(.) =.. The present example problem has W w /t = which places it in the third range, AISI () S00 Eq. C..-b leading to: F v=0.0ek v/(w w/t) =. kips/in The nominal web shear in the presence of the perforations is: V n = qsww t F v=(0.)(.)(0.0)(.)=0.0 kips Web Crippling in Perforated Decks The earlier examination of Equation, indicated the relative values of right-side terms. The h/t term is one measuring a vertical column-like slenderness where the addition of web perforations would impact the h-term. The associated increase in slenderness is measured through an equivalent thickness, t a, described in Equation.

11 R N h P =Ct F Sinθ 1-C 1+C 1-C n y R N h t t t a where t q t a s (c) Tests and Comparisons. Reference () lists the results from a series of interior-two-flange web crippling tests for three inch deep steel deck. Interior two flange loading is defined in the AISI Standard (). Half the panels were perforated and half were not allowing the ideal direct comparisons between solid and perforated deck manufactured in the same profile. Table. x 8 Perforated Floor Deck Profiles t Th. P n Test P n Shear * * Theoretical V n based on equivalent t a from Equation () and AISI () Section C. Table contains results from tests on six assemblies having perforated webs. The perforation open area as a decimal, p o, is 0.0. The interior bearing lengths were inches. Column lists the Eq. c theoretical bearing strength, lbs/ft. of width, using the t a thickness from Equation. No resistance () factors are included. In all six cases test results were above the theoretical values indicating the conservative nature of Equation. Table 2. 3 x 8 Floor Deck Profile Without Perforations Perf P n -to- Perf-to-solid t Th. P n Solid P n Shear * Shear * Theoretical V n based on t and AISI () Section C. 10

12 Table lists comparisons between solid and perforated web values. Note particularly that the P n values were little influenced by the presence of web perforations. However, such perforations did have a marked influence on shear capacity as indicated in the last-column ratios of Table. Though the perforations have virtually no impact on bearing capacity they have a significant impact on the web shear strength. The un-factored web shear strength per foot of panel width is given in the Shear columns of both Tables and. The perforations led to a % reduction in shear strength for this particular perforation pattern. Conclusion Perforated decks in bending are evaluated in two phases. They are first evaluated as un-perforated units to find preliminary effective width for each element in the cross section. The perforation pattern is then imposed on the preliminary findings to further modify the effective widths. The reduced properties are then used in design in the same manner as with un-perforated decks. 11

13 References Design Manual for Composite Decks, Form Decks and Roof Decks, Steel Deck Institute, Fox River Grove, IL North American Specification for the Design of Cold-Formed Steel Structures, AISI Standard, 2007 Edition with Supplement No. 2, 2010, and with Commentary. Designers, Specifiers and Buyers Handbook for Perforated Metals, Industrial Perforators Association, 1993, Milwaukee, WI, (web avail: Web Crippling of Cold Formed Steel Multi-Web Deck Sections, S. Routledge, M. Pope, and S. Fox, University of Waterloo, Ontario, Canada, Apr Perforated Deck Testing, VicWest Division of Jannock Steel Co., B. Mandelzys and M. Sommerstein, Ontario, Can., October Flexural Strength and Stiffness of Acoustic Roof Decks, Canadian Sheet Steel Building Institute, Ontario, Fox, Schuster, Kafi, and Gillis, Univ. of Waterloo, Ontario, Can. 1½ Deep Wide Rib Acoustical B-Deck 20 Gage Wheeling BW-36, Farabaugh Engineering and Testing for Wheeling Corrugating Co., T237-09, September, Perforated Metal Deck Diaphragm Design, L.D. Luttrell, Luttrell Engineering Co., Morgantown, WV, August 15,

14 Appendix A. Perforation Properties Deck panels may be formed from coils that have been pre-punched or perforated with bands of holes along the coil length. These bands are spaced such that the finished deck panel will have the perforation patterns in preselected locations. They usually are in top flats, web flats, and bottom surfaces or in closure panels in cellular decks and used for an acoustic effect. The perforation process is one of long history in the steel industry. In steel roof deck, holes reduce the solid area of the plate, introduce circuitous stress paths through the plate and increase its flexibility over a similar un-punched plate. The Industrial Perforators Association, the IPA, publishes the Designers, Specifiers, and Buyers Handbook for Perforated Metals (). This publication lists standard punch patterns with round perforations up to ¾ diameter in several patterns and three patterns with 1/8 diameter perforations. A commonly used deck pattern, as listed in the IPA manual is shown in Figure A-. Figure A-. Typical perforation pattern. IPA describes the structural properties for a perforated sheet as a fraction of those found for a similar un-perforated plate. The fractional measure is related to the hole pattern and the percentage of open area of the element. The most common punch pattern is one involving three adjacent holes centered on the tips of equilateral triangles as in Figure A-. With this standard pattern, holes are spaced apart at a C- distance both left-to-right and on the diagonals. The inclined lines of holes are sloped 0 degrees to the horizontal. The distance between lateral lines along the hole centers then is (C sin 0 o ) = 0. C. Thus the tributary area to a single hole is 0.C. Given the perforation hole diameter, D, the fraction of open area can be expressed as: πd 1 o = 0.0 D p = 0.C C (A-) When the exhibited pattern is punched in a No. 113 IPA pattern with 1/8 diameter holes on 3/16 centers, the open area is found as 40% or p o = 0.0. From their development projects, the IPA related the behavior of a perforated plate to a fraction of that for the unperforated plate. The 13

15 relating fraction, k e, is directly related to p o and reduced stiffness. It may be found from IPA tables or from the following expression. k e = 0.+p o -.p o 0. p 0. o (A-) The IPA pattern mentioned above, for example, had p o = 0.0. Using Equation A- leads to k e = 0.0. This may be thought of as an effective width factor where the perforated plate is about % as stiff as a similar non-perforated plate. The better view is to use the original plate dimensions for setting a particular property such as T max = F y A. In the perforated plate, the maximum usable value is T = k e F y A. 14