Purlin Design-VP Design Manual

STANDARDS SECONDARY and STRUCTURAL PURLINS PRACTICES DESIGN

VP DESIGN MANUAL Section: 4.1 BUILDINGS, 11/15/04 Rev. 8 of 19 INC.

I. GENERAL

A. Purlin Material and Geometry: VP purlins are normally cold formed; "Z" shaped; 6½", 8½", and 11 ½" depth secondary structural members. Section properties may be found in Design Manual Section 20.1.1. Materials Specifications may be found in Design Manual Sections 16.ii and 16.5:

Yield Strength: Fy = 55 ksi Tensile Strength: Fu = 70 ksi

90o

50o

I.R. = 5/16”

d

b

h

Zee Purlin Geometry

h = 6 1/2" b = 2 1/8"

h = 8 1/2" b = 2 1/2"

h = 11 1/2" b = 3 1/2"

Design Thickness

Blank (in.)

Lip, d (in.)

Blank (in.)

Lip, d (in.)

Blank (in.)

Lip, d (in.)

0.059" 11.625 0.706 14.375 0.706 0.065" 11.750 0.780 14.500 0.780 19.500 0.780 0.073" 12.000 0.921 14.750 0.921 19.750 0.921 0.082" 12.000 0.938 14.750 0.938 19.750 0.930 0.092" 12.000 0.958 14.750 0.958 19.750 0.958 0.105" 12.000 0.983 14.750 0.983 19.750 0.983 0.120" 12.000 1.012 14.750 1.012 19.750 1.012

Note: Sections shaded are not available in VP Command.



II. PURLIN ANALYSIS

A. Simple Purlins: Simple purlins are analyzed as simple beams supported at each end by the main framing. The boundary conditions used for simple purlin analysis assume pinned connections at frame supports and span lengths for analysis equal to the distance between centerline of frame supports.

Simple Purlins are provided with 2-3/4" laps extending into each bay for 6½" & 8½" Zee purlins and 5-3/4" laps for 11½" Zee purlins. Additional stiffness in lap areas is not considered in analysis.

B. Continuous Purlins: Continuous purlins are analyzed as continuous beams utilizing the Direct Stiffness Method. Purlin continuity over frame supports is developed by lapping or extending purlins into adjacent bays and providing shear connections at purlin lap ends. The lapped purlin conditions over frame supports provide a double section in the region of high moment and provides additional stiffness over the supports which tend to reduce mid-span moments and increase support moments.

6½" & 8½" Continuous Zee Purlins are provided with 1’-0”, 1’-6", 2'-0", 2'-6", 3'-0", 3'-6" and 4'-0" laps extending into each bay. 11½" Continuous Zee Purlins are provided with 2'-0", 2'-6", 3'-0", 3'-6" and 4'-0" laps extending into each bay.



C. Boundary Conditions used for continuous roof purlin analysis assume that the purlins are fully continuous from building rake-to-rake with pinned connections to each frame support. Span length for analysis is equal to the distance between centerline of frame supports.

Purlins are normally extended past end frame locations to accommodate standard inset or outset endwall girt dimensions. These roof extensions are included in continuous and simple purlin analysis as cantilever elements. Span length for analysis is equal to the distance between centerline of end frame supports and building line (6" for inset girts, 1'-0" for outset girts), less 1/4".

Purlin extensions at end frame locations beyond that required to accommodate standard inset or outset endwall girt dimensions is accomplished by either (1) extending the endbay purlin with a One Piece Extension or (2) by adding a Bolt On Extension with purlin lap connections as shown below. These roof extensions are included in continuous and simple purlin analysis as cantilever elements. Span length for analysis is equal to the sum of the distances between end frame support and building line plus the roof extension length plus 1-1/4".

Outset Girts Inset Girts

Section Interior Lap Length

Exterior Lap Length

Interior Lap Length

Exterior Lap Length

6½" Zee 4'-7½" 11¾" 5'-1½" 5¾"

8½" Zee 4'-7½" 11¾" 5'-1½" 5¾"

11½" Zee 4'-1½" 11¾" 4'-7½" 5¾"



III. PURLIN DESIGN

A. General: Continuous and Simple roof purlins are designed for dead, live, snow, wind or other loads as required. Member design is in accordance with the NASPEC 2001 AISI STANDARD and VP Standard Practices. In general, purlin members are designed for bending, shear, and web crippling, combined bending and web crippling, combined bending and shear, axial and combined bending and axial loads as required. The following is a summary of AISI provisions used in the design of VP purlin members. A detailed explanation of design equations and assumptions follow.

B. Design Assumptions: Purlin analysis and design assume the following:

1. Continuous purlins are connected within the lapped portions in a manner that achieves full continuity between the individual purlin members.

2. The continuous beam analysis to establish the shear and moment diagrams assumes a continuous prismatic member in which the Ix values within lapped portions is the sum of the individual members.

3. The strength within lapped portions is assumed to be the sum of the strengths of the individual sections. This includes bending, shear, and web crippling capacities.

4. The attachment of through-fastened roof covering to the purlin provides continuous lateral support to the purlin top flange. (e.g. Panel Rib)

5. For gravity loads, the compression flange (bottom) between the frame supports and moment inflection point is treated as follows:

Segment A (lap area) is assumed to be fully braced Segment B is treated as an unsupported cantilever (Cb = 1.0) Segment C is fully braced for Panel Rib, B-deck, etc. Gravity load R-factors

apply for SSR, SLR or other standing seam type roof panels.

: : .Seg. A Seg. B

Moment diag.

Moment inflection point

Seg. C



6. For uplift loads, the compression flange (top) at or near the frame supports is assumed to be fully braced between frame supports and moment inflection points. Uplift R-factors apply to negative bending region between inflection points.

7. The last step in the design process of continuous purlins is lap optimization. After initial purlin member selection is made based on the proceeding design criteria in each bay, purlin laps are reduced until no further reduction is possible without producing overstress. Moment redistribution due to lap length reductions is considered.

Designer is cautioned to insure that other panel types that may be used with VP purlin systems provide adequate lateral support. If lateral support is not provided, purlins shall be designed considering unbraced length effects per AISI C3.1.1-3 . See Section III.D for procedures.



C. Allowable Bending Moment, Ma: Cold formed purlin members are designed for bending in accordance with AISI section C3.1

Ma = Mn / Ωb

Ma = Allowable Bending Moment (inch-kips) Ωb = Factor of Safety for Bending = 1.67 Mn = Nominal Section Strength per AISI C3.1 (inch-kip)

Mn based on initiation of yield (AISI 3.1.1)

This section strength assumes that the section is adequately braced such that failure occurs by yielding. This is typical in purlin designs in areas with through-fastened roof panels (e.g. Panel Rib) attached to the compression flange. It may also apply to purlins with discreet bracing spaced at sufficiently close intervals to preclude overall inelastic instability. For these cases:

Mn = Se Fy

Se = Elastic Section Modulus of Effective Section About Geometric axis calculated at f = Fy (in3)

Fy = Material Yield Stress (ksi)

Bending Moment Capacity, Ma = Se Fy / 1.67 AISI C3.1-1

Design Thickness (in.)

6½" Zee Purlin (in. kips)



0.059" 40.34 58.56 --- 0.065" 48.41 68.77 97.45 0.073" 56.28 84.34 122.78 0.082" 63.10 99.69 148.17 0.092" 70.54 111.84 176.20 0.105" 80.13 127.26 203.40 0.120" 91.10 144.91 246.31




Mn based on lateral elastic buckling (AISI 3.1.2)

This section applies to the bending strength of laterally unbraced segments of roof purlins. Unbraced conditions arise when roof panels that are not adequate for lateral support and for unbraced bottom compression flanges between support and inflection point of continuous purlins subjected to gravity loads. Unbraced compression flanges of purlin roof extensions fall within this design criteria.

Mn = nominal section strength per AISI C3.1.2 (inch-kip) Sf = elastic section modulus of full section (in3) Sc = elastic section modulus of effective section (in3) evaluated at f = Mc / Sf , ( VPC evaluates at f = Fy resulting in Sc = Se )

Mn = Sc Mc / Sf

Mc = critical moment strength (inch-kip) per AISI C3.1.2 In the event that roof panels are not adequate to brace VP purlins, purlins are assumed to be un-sheeted with neither purlin flange attached to sheathing. (See Design Manual Section 10.6 for panel requirements) Purlin brace and design requirements for unsheathed “Z’s” are driven by three sections of the AISI Specification: • AISI Section D3.2.2: Lateral Bracing Requirements when Neither Flange is connected to Sheathing. • AISI Section C3.1.2: Lateral Buckling Strength • AISI Section C4, C4.1, C5: Axial Load Strength Bracing Requirements AISI Section D3.2.2 requires top and bottom purlin flange bracing and gives provisions for brace member design forces. (See Design Manual Section 4.7 for bracing type and design) The quantity of discreet braces will be controlled by the purlin strength requirements of AISI Section C3.1.2. In order to achieve full bending capacity of VP Z purlins, the following maximum un-braced lengths Lu max. may be used as a guide.

Lu (max.) Section Cb = 1.0

6½” Z 3’-6” 8½” Z 4’-0”

11½” Z 5’-0”



As indicated by the above, it may not be possible to achieve full bending capacity for all VP section and span conditions with any economical scheme of bracing. As an aim at criteria, a bracing scheme has been developed that will achieve an average of 90% of the capacity of the purlin. This bracing scheme consists of three brace points with a spacing of L / 4 and centered in the bay as shown below. For bay spaces > 30 feet, L / 5 brace spacing will be required.

xx xL/4

L = Bay Space

L/4 L/4L/4

C.L. Frame C.L. Frame

Use L/4 Brace Space for 0' to 30' Bays Use L/5 Brace Space for > 30' Bays

Note: The purlin bending capacity for both uplift and gravity load cases must be evaluated for unbraced lengths. In some cases, a more optimum solution is to set the two interior brace spaces to L / 5 and the spaces adjacent to the frame to 0.3 L. This allows for smaller un-braced lengths near the maximum span moments where Cb = 1.

xx xL/

L = Bay Space

0.3 L 0.3 LL/

C.L. Frame C.L. Frame



Mn Purlins having one flange through-fastened to deck or sheathing (AISI C3.1.3)

This section applies to the bending strength of "Z" and "C" sections with Purlin tension flanges attached to the roof panels and with the compression flanges laterally un-braced. (i.e. uplift loading). This section does not apply to a continuous beam for the region between inflection points adjacent to a support, or to a cantilever beam.

Mn = nominal section strength per AISI C3.1.3 (inch-kip) Se = elastic section modulus of effective section @ f = Fy (in3) Fy = material yield strength (ksi)

Mn = R Se Fy

R = reduction factor per AISI C3.1.3 or full scale base method testing (1)

(1) Note: For standing seam roof systems the R-factor must be determined by test. Current VP testing indicates that the values given in AISI C3.1.3 are adequate for use with SSR and SLR roof systems without insulation. Mn Purlins with one flange fastened to a standing seam roof system (AISI C3.1.4) Standing seam roof systems are specially designed to allow thermal expansion of the panels without producing significant stress or displacement in the panels, panel clips or supporting structure. This is normally achieved through use of a sliding panel connection clip, which allows relative movement between the panels and the roof purlins. Due to the nature and diversity of the various standing seam roof systems, the nominal flexural strength for purlins supporting them must be determined by testing. A standard test method called "The Base Test Method for Purlins Supporting a Standing Seam Roof System" has been developed and is specified in the AISI Manual. The product of the Base Test Method is the reduction factor R required by AISI C3.1.4. Mn = RSeFy The 1996 AISI specification requires that insulation be included in the test assembly if insulation will be used in the as-built system. This applies to standing seam roof systems only. It is conservative to use R-factors derived from a test with thicker insulation for a system with insulation of lesser thickness. The converse is not true. It has been shown that the presence or absence of thermal blocks will not significantly affect the gravity load test results for our standing seam systems. The Base Test Method may also be used to determine the R-factor required for uplift loading per AISI C3.1.3. Thermal blocks will affect the test results for uplift loading. Table 3-1 provides R-factors for gravity and uplift for roofs without insulation.



TABLE 3-1 R-Factors for Roof Systems Without Insulation Between Panels and Purlins (1)

R Values for Simple Span Purlins R Values for Continuous Span Purlins Section /

Thickness Panel Rib SSR SLR Panel Rib SSR SLR

Gravity Uplift 1 Gravity 2 Uplift 2 Gravity 2 Uplift 2 Gravity 2 Uplift 2 Gravity 2 Uplift 2 Gravity 2 Uplift 2

6.5 Z(4) 0.059 1.00 0.70 0.93 0.50 0.81 0.50 1.00 0.70 0.93 0.70 0.81 0.70 0.065 1.00 0.70 0.93 0.50 0.82 0.50 1.00 0.70 0.93 0.70 0.82 0.70 0.073 1.00 0.70 0.94 0.50 0.83 0.50 1.00 0.70 0.94 0.70 0.83 0.70 0.082 1.00 0.70 0.95 0.50 0.83 0.50 1.00 0.70 0.95 0.70 0.83 0.70 0.092 1.00 0.70 0.96 0.50 0.84 0.50 1.00 0.70 0.96 0.70 0.84 0.70 0.105 1.00 0.70 0.97 0.50 0.85 0.50 1.00 0.70 0.97 0.70 0.85 0.70 0.120 1.00 0.70 0.98 0.50 0.85 0.50 1.00 0.70 0.98 0.70 0.85 0.70 8.5 Z 0.059 1.00 0.65 0.93 0.50 0.81 0.50 1.00 0.70 0.93 0.70 0.81 0.70 0.065 1.00 0.65 0.93 0.50 0.82 0.50 1.00 0.70 0.93 0.70 0.82 0.70 0.073 1.00 0.65 0.94 0.50 0.83 0.50 1.00 0.70 0.94 0.70 0.83 0.70 0.082 1.00 0.65 0.95 0.50 0.83 0.50 1.00 0.70 0.95 0.70 0.83 0.70 0.092 1.00 0.65 0.96 0.50 0.84 0.50 1.00 0.70 0.96 0.70 0.84 0.70 0.105 1.00 0.65 0.97 0.50 0.85 0.50 1.00 0.70 0.97 0.70 0.85 0.70 0.120 1.00 0.65 0.98 0.50 0.85 0.50 1.00 0.70 0.98 0.70 0.85 0.70 11.5 Z(5) 0.065 1.00 0.50 0.97 0.50 0.82 0.50 1.00 0.70 0.97 0.70 0.82 0.70 0.073 1.00 0.50 0.97 0.50 0.83 0.50 1.00 0.70 0.97 0.70 0.83 0.70 0.082 1.00 0.50 0.97 0.50 0.83 0.50 1.00 0.70 0.97 0.70 0.83 0.70 0.092 1.00 0.50 0.97 0.50 0.84 0.50 1.00 0.70 0.97 0.70 0.84 0.70 0.105 1.00 0.50 0.97 0.50 0.85 0.50 1.00 0.70 0.97 0.70 0.85 0.70 0.120 1.00 0.50 0.97 0.50 0.85 0.50 1.00 0.70 0.97 0.70 0.85 0.70

1. Panel Rib uplift R-factors based on Supplement NO. 1 to the 1996 AISI Specification. For simply supported purlins with Panel Rib with blanket insulation, multiply tabulated R-values by the reduction factor r = 1 – 0.01 ti. Where: ti = blanket insulation thickness (in.)

2. R-values for standing seam panels are based on full scale “Base Test Method for Purlins Supporting a Standing Seam Roof System” per AISI C3.1.4. Continuous span uplift values are those prescribed in the AISI specification for through-fastened panels. This is considered rational based on correlation of the simple span purlin test results with the values prescribed in the specification.

3. Values for 6-1/2" purlins assumed greater than or equal to 8-1/2" test values. 4. All testing was performed with purlins opposed. 5. All 11-1/2” purlins require sag angles because VP base tests were performed with sag angles only. 6. Anchorage analysis is required when any of the gravity values tabulated above are used including the Panels Rib

values. For uplift loading no anchorage analysis is required. Note: Sections shaded are not available in VP Command. Gravity load Base Tests with insulation For VP standing seam roofs with insulation, gravity load Base Tests were performed for three different systems with SSR and SLR. The R-factors for each system are given in Table 3-2. These values apply to roofs with 6" or less of high-density fiberglass insulation between the panel and purlin. It is only the thickness of the insulation between the panel and purlin that is of importance. Additional insulation between purlins will not affect the R-factors. Values are applicable both with and without thermal blocks.



The three systems tested are as follows: System I System 1 is the "plain" system. That is, it is just the panels and purlins without any additional materials to enhance stability. This is the simplest and easiest to erect. However, system 1 also produces the lowest R-factors. System II System 2 has two rows of purlin braces added to the bay. These braces consist of crossed sag angles as shown in the figure below. The two rows of sag angles are always located at six feet apart and centered on the bay no mater what the bay spacing. When braces are used they need only be placed in every-other purlin space. The braces serve to provide excellent torsional restraint to the purlins and significantly increase the R-factors. System III System 3 is the same as system 2 except anti-roll purlin clips are added to the adjacent frame lines. The number of clips required at each frame line depends on the anchorage force requirement. Refer to the anchorage section of this manual for more detail about this.

No braces this space

No braces this space



TABLE 3-2 Gravity Load R-Factors for Roof Systems With 6" or Less of HD Insulation Between Panels and Purlins R Values for SSR R Values for SLR

Section / Thickness

System I

System II

System III

System I

System II

System III

6.5 Z 0.059 .67 (5) (5) .71 (5) (5) 0.065 .67 - - .69 - -

0.073 .66 - - .68 - - (2) 0.082 .66 - - .67 - -

0.092 .66 - - .66 - - 0.105 .65 - - .64 - - 0.120 .65 - - .62 - - 8.5 Z 0.059 .67 .77 .77 .71 .72 .81

(1) 0.065 (.014)Tanθ + .67 ≤ 1 .77 .79 .69 .72 .8

0.073 (.073)Tanθ + .66 ≤ 1 (.115)Tanθ + .78 ≤ 1 .83 .68 .72 .79 0.082 (.149)Tanθ + .66 ≤ 1 (.425)Tanθ + .78 ≤ 1 (.123)Tanθ + .87 ≤ 1 .67 .72 .77 0.092 (.208)Tanθ + .66 ≤ 1 (.667)Tanθ + .79 ≤ 1 (.264)Tanθ + .90 ≤ 1 .66 .72 .76 0.105 (.282)Tanθ + .65 ≤ 1 (.972)Tanθ + .79 ≤ 1 (.442)Tanθ + .94 ≤ 1 .64 .72 .75 0.120 (.366)Tanθ + .65 ≤ 1 (1.32)Tanθ + .80 ≤ 1 (.643)Tanθ + .99 ≤ 1 .62 .72 .74

11.5 Z 0.065 .79 (6) N/A (4) .71 .75 N/A

0.073 .78 - N/A .67 .74 N/A 0.082 .77 - N/A .64 .73 N/A 0.092 .76 .82 N/A .60 .73 N/A 0.105 .76 .87 N/A .56 .72 N/A 0.120 .74 .94 N/A .50 .71 N/A 1. Equations are based on linear interpolation between zero roof pitch and 3.35:12 using values from test performed with purlins

oriented in the same direction at zero slope and purlins opposed at 3.35:12. (3.35:12 ≅ principal axis of 8-1/2" purlins) For angles above 3.35:12 use 3.35:12. R-values are based on full scale “Base Test Method for Purlins Supporting a Standing Seam Roof System” per AISI C3.1.4 with 6" HD insulation between panel and purlin compression flange. Values are applicable with or without thermal blocks.

2. Values for 6-1/2" purlins assumed greater than or equal to 8-1/2" test values. 3. All testing was performed with purlins oriented in the same direction and without any external restraints not found in the in-

place roof system. 4. System 3 is not applicable to 11-1/2" purlins since all 11-1/2" purlins are connected to rafters with PC-13 clips. The testing was

performed with PC-13 clips on both purlins. 5. System 2 and system 3 values not applied to 6-1/2" purlins since cross bridging is not practicable due to the shallow purlin

depth. Other effective means of providing stiff torsional braces would probably justify use of the system 2 values. 6. The system 2 values did not produce any benefit for the first three thicknesses of 11-1/2" purlins with SSR. 7. Anchorage analysis is required in all cases when the values tabulated above are used. Note: Sections shaded are not available in VP Command.



Uplift load Base Tests with insulation For VP standing seam roofs with insulation, uplift load Base Tests were performed for two different systems with SSR and SLR. These two systems are the same as system I and II above. The presence or absence of purlin anti-roll clips are not significant to uplift base tests since the uplift reaction tends to resist rollover at the frame lines. The R-factors for each system are given in Table 3-3 both with and without thermal blocks as applicable. The values apply to roofs with 6" or less of high-density fiberglass insulation between the panel and purlin. It is only the thickness of the insulation between the panel and purlin that is of importance. Additional insulation between purlins will not affect the R-factors.



TABLE 3-3 Uplift Load R-Factors for Roof Systems With 6" or Less of HD Insulation Between Panels and Purlins (2) R Values for SSR R Values for SLR

SYSTEM I SYSTEM II SYSTEM I SYSTEM II Section / Thickness W/O TB’s With TB’s W/O TB’s With TB’s W/O TB’s With TB’s W/O TB’s With TB’s

6.5 Z 0.059 0.55 0.58 (4) (4) N/A N/A (4) 0.065 0.54 0.55 N/A N/A

0.073 0.53 0.52 N/A N/A (3) 0.082 0.52 0.49 N/A N/A

0.092 0.52 0.46 N/A N/A 0.105 0.51 0.43 N/A N/A 0.120 0.50 0.39 N/A N/A 8.5 Z 0.059 0.55 0.58 0.68 0.65 N/A N/A 0.065 0.54 0.55 0.69 0.68 N/A N/A 0.073 0.53 0.52 0.71 0.71 N/A N/A 0.082 0.52 0.49 0.72 0.75 N/A N/A 0.092 0.52 0.46 0.73 0.78 N/A N/A 0.105 0.51 0.43 0.75 0.82 N/A N/A 0.120 0.50 0.39 0.77 0.87 N/A N/A 11.5 Z 0.065 0.74 0.77 N/A N/A 0.082 0.73 0.79 N/A N/A 0.073 0.72 0.81 N/A N/A 0.092 0.71 0.83 N/A N/A 0.105 0.70 0.85 N/A N/A 0.120 0.69 0.88 N/A N/A 1. Blanks in the table indicate testing not yet complete for these systems. Gravity load testing implies the following:

• R-factors for 11-1/2” Purlins with SSR are greater than or equal to those for 8-1/2” purlins • SLR R-factors may be somewhat less than the corresponding SSR values

2. Values for 6-1/2" purlins assumed greater than or equal to 8-1/2" test values. 3. Systems without thermal blocks are not applicable to SLR. 4. System 2 values not applied to 6-1/2" purlins since cross bridging is not practicable due to the shallow purlin depth. Other

effective means of providing stiff torsional braces would probably justify use of the system 2 values. 5. Anchorage analysis is not required for uplift loading. Note: Sections shaded are not available in VP Command. Each roof system has several "essential variables". These are system components and characteristics that, if modified will probably effect the value of the R-factor. Purlin geometry and roof system parts and characteristics are considered essential variables. As such, each different purlin profile requires a separate test. Also, if the panel clip or some other critical system characteristic is changed, the R-factors derived from the previous version of the product will no longer be valid. Any standing seam type roof system for which R-factors have not been provided by appropriate application of the Base Test Method must be considered completely ineffective for bracing purlins and the purlins will need to be designed per AISI C3.1.2 with discreet bracing. In some cases judgment will be required in deciding whether or not a particular roof system should be classified as a standing seam roof. Most insulated sandwich panel systems should be viewed as SSR's.



3. Axial Load Strength Purlin strut members shall be designed for axial load in accordance with AISI Section C4 considering the un-braced lengths as established above and assuming that the frame connection provides a point of support about both axes. In evaluating axial load strengths and considering quarter point bracing, a general rule can be made that strong axis buckling (KL/r)x will control. There are a few exceptions for the thinner 6½“ and 8½" Interior bay Z sections. However, the ratio of (KL/r)y to (KL/r)x never drops below .96.



E. Web Crippling: Roof purlin members are evaluated for web crippling at interior and exterior frame supports per the NASPEC 2001 AISI Standard. Web crippling checks are not made for purlin systems utilizing purlin clips for purlin-to-frame connections.

Table E – 1 Allowable Purlin Exterior Reaction Web Crippling Capacity, Pa


6½" Zee Purlin (kips)



0.059" 1.00 0.97 0.065" 1.22 1.17 1.11 0.073" 1.53 1.47 1.40 0.082" 1.91 1.85 1.77 0.092" 2.38 2.31 2.22 0.105" 3.06 2.98 2.87 0.120" 3.94 3.84 3.71

Table E – 2 Allowable Purlin Interior Reaction Web Crippling Capacity, Pa





0.059" 1.46 1.44 0.065" 1.83 1.80 1.76 0.073" 2.38 2.34 2.30 0.082" 3.08 3.03 2.98 0.092" 3.96 3.9 3.83 0.105" 5.25 5.18 5.09 0.120" 6.96 6.87 6.77

Notes: 1. Design Parameters: N = 5", Fy = 55 ksi, Ri = 0.3125", E = 29500 ksi, θ = 90° 2. Capacity within purlin lap regions is the sum of individual capacities. 3. 11½" purlins normally not checked for web crippling due to purlin clip attachments.




F. Combined Bending and Web Crippling: Nested Z-purlin members shall be designed for the interaction of bending and web crippling at supports in accordance with the NASPEC 2001 AISI Standard.

Ω

≤+65.185.0

0 nn PP

MM

Where: Ω = 1.75 M = Required flexural strength at section under consideration (in-k)

Mn0 = Nominal flexural strength of nested Z-purlins, i.e. sum of the two sections evaluated individually, per AISI C3.1.1 (in-k) P = Required strength for concentrated load or reaction in the presence of bending moment (kips) Pn = Nominal web crippling strength of nested Z-purlins, i.e. sum of the two sections evaluated individually. Pn is determined based on the values in table E – 2 as follows. Pn = PaΩw Where: Pa = See table E – 2 Ωw = 1.65

G. Shear: Purlin members are evaluated for shear in accordance with AISI section C 3.2 at purlin supports, point loads and end of purlin laps.

Purlin Shear Capacity, Va AISI C3.2





0.059" 3.05 2.26 0.065" 4.08 3.03 2.18 0.073" 5.80 4.30 3.10 0.082" 7.58 6.10 4.39 0.092" 9.55 8.64 6.22 0.105" 12.44 12.44 9.27 0.120" 14.88 16.24 13.87

Notes: 1. Design Parameters Fy = 55 ksi, E = 29500 ksi 2. Capacity within purlin lap regions is the sum of individual capacities.




H. Combined Bending and Shear: Combined bending and shear is evaluated per AISI section C3.3 and is checked at the end of the purlin laps and frame supports.

IV. PURLIN CONNECTIONS

A. Purlin-to-Purlin Lap Connections: Purlin-to-Purlin lap bolt connections are made with (2) A325 bolts for 6½" and 8½" depth Zee purlins and (3) A325 bolts for 11½" depth Zee purlins. The connection requirements needed to produce structural continuity at frame supports are based on bolt shear connections at the end of the purlin laps.

From the structural analysis results for the continuous purlin a bending moment (M) is assumed to exist directly over the center of the frame line. This moment is resisted by both of the lapped purlins in proportion to their individual strength. Thus, the portion of moment M in each purlin is:

21

11

ee

e

SSS

MM+

= 21

22

ee

e

SSS

MM+

=

Where: Se1 & Se2 are the effective section moduli for purlins 1 & 2 respectively about the X-X axis (in3)

From statics, the shear in each set of bolts is:

2

21

2

11

LwLM

V −= 2

12

1

22

LwL

MV −=

Where: 21

11 II

Iww

+≅

21

22 II

Iww

+≅

In general, we could eliminate the second term in the equation for bolt shear with only a small amount of conservatism.

w

Purlin 1 Purlin 2

L1 L2



B. Purlin-to-Frame Connections - Purlin-to-Frame connections are normally made with ASTM A325 1/2" diameter bolts and/or purlin clips. The following is a summary of standard purlin-to-frame connections:

Note: See Design Manual Section 5 for purlin strut and strut connection design.

a. 6½" & 8½" Purlins - (2) ½" A325 bolts purlin flange-to-frame connection. Bolts are checked for Shear forces derived from axial strut loads, Tension forces derived from uplift loads, and for combined Shear and Tension forces for the two bolt purlin to frame connection per AISI Section E3.4. Two bolt Bearing per AISI Section E3.3 is also checked. Bearing checks are made on the thinner purlin member for continuous lapped purlin to frame connections. b. 11½" Purlins - (6) ½" A325 bolts w/ purlin clip connection: - (2) bolts purlin flange-to-frame connection - (2) bolts purlin clip-to-frame connection - (2) bolts purlin clip-to-purlin web connection

Bolts are checked for Shear forces derived from axial strut loads, Tension forces derived from uplift loads, and for combined Shear and Tension forces for the four bolt purlin to frame connection per AISI Section E3.4. Four bolt Bearing per AISI Section E3.3 is also checked. Bearing checks are made on the thinner purlin member for continuous lapped purlin to frame connections.

c. 6½" & 8½" Eave Purlins - 6½" and 8½" Eave member connections to End Frames are made with (2) A325 bolts with details, design and design progression identical to "a" above. Eave member connections to Interior Frames are made with (8) A325 bolts similar to "b" above except that GC4 clips are used and each member end is attached to the clip. Design and design progression is identical to "d" above assuming a three-bolt strut connection. Exception: High Side Eave Purlins are attached to all frame locations using (2) A325 bolts


VP DESIGN MANUAL Section:4.1.C BUILDINGS, 10/15/01 Rev. 0 Page 1 of 3

INC.

APPENDIX C

COMMENTARY



INC.

Comparison of the provisions of this section with current VP Command The design procedures and progression given in this section of the Design Manual do not correspond 100% with the methodology currently programmed into VP Command. The following is an outline of the differences. It is not necessary to revise the designs produced by VP Command. However, if manual designs are produced the procedures in this manual should be followed.

Design Manual VP Command Paragraph B. Design Assumptions: The design assumptions in the lap region shown in this section reflect the recommendations found in the 1996 AISI Specification. Cb should be taken as unity.

Currently VP Command assumes that the purlin is laterally braced at the support, end of lap and inflection point. The value of Cb is calculated based on the moment diagram per the pre-1996 edition of the AISI specification. The 1996 specification has revised the equation for Cb.

R-Factors:

VP Command currently contains the gravity load R-factors from table 3-2. These R-factors are only used when compliance with the 1996 AISI is specified. Otherwise the values in table 3-1 are used. The uplift R-factors when insulation is present given in table 3-2 are not yet in VP Command. The uplift values used in VP Command currently are based on the prescriptive values given by the 1996 AISI specification. Since the test results in table 3-2 correlate fairly well with the AISI prescribed R-value for simple Z-purlins it is believed that use of the prescribed values for continuous Z-purlins is satisfactory until such time as the programming can be done.

Web Crippling: The web crippling provisions given in the Design Manual are based on what we know will be published in a future edition of the AISI specification. These provisions produce somewhat higher allowable web crippling forces than the current provisions being used in VP Command. The main impact of the change is to exterior reaction allowables.

VP Command is currently calculating web crippling allowables on the basis of an adjusted pre-1996 AISI value as follows: Pd = 1.2 Pa Where: Pa = AISI allowable value per section C3.4-1 The 1.2 adjustment factor was based on VP research using purlins that were bolted to their supports. The 1996 edition of the AISI specification now contains a multiplier of 1.3 to adjust for Z-sections that are bolted to their supports. Therefore, VP Command is currently quite conservative relative to the current AISI and the Design Manual.



INC.

Design Manual VP Command

Combined bending & web crippling: In the Design Manual we are invoking future AISI provisions based on the fact that these provisions have already passed the AISI balloting process.

Current VP Command employs provisions based on VP research and testing. The following formula is applied. 0.884(P/Pa) + (M/Max0) ≤ 1.66 This provision does not produce far different results from the provisions contained in the manual.

Shear: This manual invokes the shear provisions in the 1996 AISI specification. These provisions are slightly different than those in the previous edition of the specification and produce slightly higher allowable shear forces in some cases.

VP Command is using the pre-1996 AISI shear provisions, which are slightly more conservative than the 96’ provisions.

Purlin lap bolt shear calculation: The purlin lap bolt shear calculation given in this manual reflects the analysis assumptions used in design of the purlins in the lap region.

The current VP Command method designs the lap bolts for the member end shear of the single purlin at the end of the lap taken directly from the structural analysis results.

Example: The example in this section reflects current VP Command and is not 100% consistent with the provisions of the main section as described above. It will be updated along with VP Command.

Note: All of the differences between this document and VP Command will be rectified as soon as possible. In the mean time it is not necessary to modify VP Command designs.


VP DESIGN MANUAL Section: 4.1.EX BUILDINGS, 06/24/98 Rev. 0 Page 1 of 9 INC.

APPENDIX EX

8 ½" Z - PURLIN DESIGN EXAMPLE

FOR

THREE SPAN CONTINUOUS PURLIN SYSTEM



THREE SPAN CONTINUOUS Z-PURLIN DESIGN EXAMPLE

GENERAL DESIGN PARAMETERS The following design example is intended to illustrate the design of continuous Z purlins using VP Command. Given: 1. Three span continuous 8 ½" Z purlin system 2. Spans @ 24'/ 25' / 24' w/ 1' endwall extension 3. Exterior Spans, t = 0.082" 4. Interior Spans, t = 0.059" 5. Roof covering attached to purlins. 6. Building geometry is : 60' wide x 75' long, eave height 20' and roof slope 1:12 . Required: 1. Check the design against AISI for gravity loads.2. Check the design against AISI for wind loads. Design Assumptions: 1. The purlins are connected within the lapped portions in a manner that achieves full continuity between the individual purlin members. 2. The continuous beam analysis to establish shear and moments assumes continuous members in which the Ix values within lapped portions is the sum of the individual members. 3. The strength within the lapped portions is assumed to be the sum of the strengths of individual sections. 4. The attachment of roof covering to the purlin provides continuous lateral support of the top flange.

5. For gravity loads, the compression (bottom) flange at and near the interior supports is assumed to be fully braced between support and the end of lap. The inflection point is also assumed to be a brace point. 6. For uplift loads, the compression (top) flange at and near the interior supports is assumed to be fully braced between support and the end of lap by roof panels. The inflection point is also assumed to be a brace point. Member Section Properties: Exterior Bay t = 0.082 in. Sf = 3.03 in.3 Sxe = 3.03 in.3 Iy = 1.90 in.4 Fy = 55 ksi r = 0.3125 in. Interior Bay t 0.059 in. Sf 2.13 in.3 Sxe 1.78 in.3 Iy 1.16 in.4 Fy 55 ksi r 0.3125 in. Load Combinations: Dead + Live Dead + Wind



GRAVITY SNOW LOADING

The following loads are based on the 1986 MBMA for 30-psf-ground snow. Snow Loads: Roof Snow = .7(30psf) = 21 psf @ 5'-0" purlin space: Purlin Load = 21 psf (5') = 105.00 plf

Dead Loads: Roof Panel Weight: w = 1.065 psf @ 5'-0" purlin space: Purlin Load = 1.065 psf (5') = 5.325 plf Purlin Weight: w1 = 4.12 plf ( for t = 0.082" purlin) w2 = 2.89 plf ( for t = 0.059" purlin)

24'-0" 25'-0" 24'-0"1'-0" 1'-0"t=0.082" t=0.059" t=0.082"

1'-6" 1'-6"2'-6" 2'-6"

105 plf snow + 1.1 plf panel dead + purlin dead

1.66k

1.66k

1.42k

1.42k

1.45k

1.45k

1.10k

1.10k1.03k

1.03k

1.20k 1.20k3.10k 3.10k

57.54 "-k

56.41 "-k

57.54 "-k

56.41 "-k

84.88 "-k 84.88 "-k

19.62 "-k

46.52 "-k 46.52 "-k0.69 "-k 0.69 "-kMoment

Shear

+

+

-

-

5.33' 7.08' Inflection



GRAVITY SNOW LOADING: Bending Capacity

End Span: a. @ Maximum Positive Moment [ AISI C3.1.1a]t1 = 0.082" ; M = + 57.54 "- k Roof covering provides continuous top flange support. Ma1 = Sxe Fy / 1.67 =(3.03)(55)/1.67 = 99.79 "- k Stress Ratio = 57.54/99.79 = 0.58 (OK) b. @ End of Purlin Lap [ AISI C3.1.2 ] t1 = 0.082" ; M = - 56.41 "- k Compression flange is unbraced between inflection point and end of purlin lap. Check for lateral buckling strength. L = 64" - 18" = 46" Iyc = 0.952; Cb = 1.75 Me = p2 E Cb d Iyc / 2 L2 Me = p2 (E)(1.75)(8.5)(.952)/2(49)2 = 974.25 "- k My = Sf Fy = (3.03)(55) = 166.65 "- k 2.78 My = 463.29 "- k ; Me > = 2.78 My @ Mc = My = 166.65 "- k Mn = (Sxe/Sf) Mc = (3.03/3.03) 166.65 Mn = 166.65 "- k Ma = Mn / 1.67 = 99.79 "- k Stress Ratio = 56.41/99.79 = 0.57 (OK) c. @ Frame Support [ AISI C3.1.2 ] Lapped purlins: t1 = 0.082" , t2= 0.059" M = - 64.88 "- k Compression flange is unbraced between frame support and end of purlin lap. Unbraced lengths are 18"and 30". By Inspection, lateral buckling strength does not govern. Ma1 = Sxe Fy /1.67 = (3.03)(55)/1.67 = 99.79 "- k Ma2 = Sxe Fy /1.67 = (1.78)(55)/1.67 = 58.62 "- k Ma = 58.62 "- k + 99.79 "- k = 158.41 "- k Stress Ratio = 84.88/158.41 = 0.54 (OK)

Interior Span: d. @ Max. Positive Moment [ AISI C3.1.1a] t2= 0.059" ; M = + 19.67 "- k Roof covering provides continuous top flange support. Ma2 = Sxe Fy / 1.67 = (1.78)(55)/1.67 Ma2 = 58.62 "- k Stress Ratio = 19.67/58.62 = 0.34 (OK) e. @ End of Left Purlin Lap [ AISI C3.1.2 ] t2 = 0.059" ; M = - 56.7 "- k Compression flange is unbraced between inflection point and end of purlin lap. Check for lateral buckling strength. L = 85" - 30" = 55" Iyc = 0.578; Cb = 1.75 Me = p2 E Cb d Iyc / 2 L2 Me = p2 (E)(1.75)(8.5)(.578)/2(55)2 =413.76 "- kMy = Sf Fy = (2.12)(55) = 116.60 "- k 2.78 My = 324.15 "- k ; Me > = 2.78 My @ Mc = My = 116.60 "- k Mn = (Sxe/Sf) Mc = (1.78/2.12)116.60 "- k Mn = 97.90 "- k Ma = Mn / 1.67 = 58.62 "- k Stress Ratio = 46.52/58.62 = 0.79 (OK) f. @ End of Right Purlin Lap [ AISI C3.1.2 ] t2 = 0.059" ; M = - 46.52 "- k Compression flange is unbraced between inflection point and end of purlin lap. Check for lateral buckling strength. Same as left lap: Stress Ratio = 46.52/58.62 = 0.79 (OK)



GRAVITY SNOW LOADING: Shear, Bending & Shear Capacity

Shear: [ AISI C3.2] t1 = 0.082" ; h1 = 8.5 - 2(0.3125 + 0.082) = 7.71" 1.38[E kv/Fy]½ = 1.38[(29500)(5.34)/55]½ = 73.9 h/t = 94.04 > 73.9 Va1 = 0.53 E kv t3 / h = 0.53(E)(5.34)(.082)3/7.71Va1 = 5.97 k t2 = 0.059" ; h2 = 8.5 - 2(0.3125 + 0.059) = 7.76" h/t = 131.18 > 73.9 Va2 = 0.53 E kv t3 / h = 0.53(E)(5.34)(.059)3/7.76Va2 = 2.21 k End Span: a. @ End of Purlin Lap t1 = 0.082" , V = 1.45 k , Va1 = 5.97 k Stress Ratio = 1.45/5.97 = 0.24 (OK) b. @ Exterior Support t1 = 0.082" , V = 1.03 k , Va1 = 5.97 k Stress Ratio = 1.03/5.97 = 0.17 (OK) c. @ Interior Support Lapped purlins: t1 = 0.082" , t2 = 0.059" Va = Va1 + Va2 = 5.97 k + 2.21 k = 8.18 k V = 1.66 k Stress Ratio = 1.66/8.18 = 0.20 (OK) Interior Span: d. @ End of Purlin Laps t1 = 0.059 , V = 1.1 k , Va2 = 2.21 k Stress Ratio = 1.1/2.21 = 0.50 (OK)

Combined Bending & Shear: [ AISI C3.3] (M/Ma)2 + (V/Va)2 < 1 End Span: a. @ End of Right Purlin Lap t1 = 0.082" , V = 1.45 k , Va1 = 5.97 k M = - 56.41 "-k ; Ma = 99.79 "-k (56.41/99.79)2 + (1.45/5.97)2 = .38 < 1 (OK) b. @ Exterior Support t1 = 0.082" , V = 1.03 k , Va1 = 5.97 k M = - 0.69 "-k ; Ma = 99.79 "-k (0.69/99.79)2 + (1.03/5.97)2 = .06 < 1 (OK) c. @ Interior Support Lapped purlins: t1 = 0.082" , t2 = 0.059" V = 1.66 k ; Va = 8.18 k M = - 84.88 "-k ; Ma = 158.41 "-k (84.88/158.41)2 + (1.66/8.18)2 = .33 < 1 (OK) Interior Span: d. @ End of Purlin Laps t1 = 0.059 , V = 1.1 k , Va2 = 2.21 k M = - 46.52 "-k ; Ma = 58.62 "-k (46.52/58.62)2 + (1.1/2.21)2 = .88 < 1 (OK)



GRAVITY SNOW LOADING: Web Crippling, Bending, & Web Crippling

Web Crippling: [ AISI C3.4] t1 = 0.082" h = 7.71" ; h/t = 7.71/0.082 = 94.02 N = 5" ; N/t = 5/0.082 = 60.98 k = Fy/33 = 55/33 = 1.6667 C1 = 1.22-.22k = 0.8533 C2 = 1.06-.06(R/t) = 0.8313 C3 = 1.33-.33k = 0.7800 C4 = 0.5 < [1.15-.15(R/t)] < 1.0 = 0.5784 Ch = 1 Interior Reactions: Pa1 = t2kC1C2Ch[291-.4(h/t)][0.75+0.011(N/t)] Pa1 = 2.86 k Exterior Reactions: Pa1 = t2kC3C4Ch[179-.33(h/t)][0.71+0.015(N/t)] Pa1 = 1.22 k Similarly for t2 = 0.059" Interior Reaction: Pa2 = 1.47 k a. @ Exterior Support t1 = 0.082" P = 1.2 k , Pa1 = 1.22 k Stress ratio = 1.2/(1.2)1.22 = .82 < 1 (OK) b. @ Interior Support Lapped purlins: t1 = 0.082" , t2 = 0.059" P = 3.1 k , Pa = 1.47 k + 2.86 k = 4.33 k Stress ratio = 3.1/(1.2)4.33 = .60 < 1 (OK)

Combined Bending & Web Crippling: (M/Ma) + 0.884 (P/Pa) < 1.66 [ VP Equation ] a. @ Exterior Support t1 = 0.082" , P = 1.2 k , Pa1 = 1.22 k M = - 61.3 "-k ; Ma = 99.79 "-k (61.3/99.79) + 0.884(1.2/1.22) = 1.48 < 1.66 Stress Ratio = 1.48/1.66 = .89 (OK) b. @ Interior Support Lapped purlins: t1 = 0.082" , t2 = 0.059" P = 3.1 k ; Pa = 4.33 k M = - 84.88"-k ; Ma = 158.41 "-k (84.88/158.41) + 0.884(3.1/4.33) = 1.17 <1.66 Stress Ratio = 1.17/1.66 = .70 (OK)



UPLIFT WIND LOADING

The following loads are based on the 1986 MBMA code for 90-mph wind speeds. Basic wind pressure = 17.97 psf Edge strip width, Z = 6'-0" @ 1 ft. purlin extension: MBMA edge zone "s" A = 5 ft. x 1 ft. = 5 ft.2, GCp = 1.7 w = 1.7 ( 17.97 psf) = 30.55 psf @ 5'-0" purlin space: Purlin Load = 30.55 psf (5') = 152.75 plf

@ Edge zone portions of end bay purlin : MBMA edge zone "s" ( 0 - 5 ft.) A = 5 ft. x 24 ft. = 120 ft.2 , GCp = 1.4 w = 1.4 ( 17.97 psf) = 25.16 psf @ 5'-0" purlin space: Purlin Load = 25.16 psf (5') = 125.79 plf @ Remaining interior zones :MBMA zone "r" A > 120 ft.2 , GCp = 1.4 w = 1.15 ( 17.97 psf) = 20.67 psf Purlin Load = 20.67 psf (5') = 103.33 plf

24'-0" 25'-0" 24'-0"1'-0" 1'-0"t=0.082" t=0.059" t=0.082"

1'-6" 1'-6"2'-6" 2'-6"

1.39k

1.39k

1.19k

1.19k

1.30k

1.30k

0.98k

0.98k1.02k

1.02k

-1.12k -1.12k-2.57k -2.57k

52.84 "-k

50.37 "-k

52.84 "-k

50.37 "-k

74.14 "-k 74.14 "-k

16.61 "-k

42.19 "-k 42.19 "-k0869 "-k 0.86 "-kMoment

Shear

-

-

+

+

5.25' 7.17' Inflection

103 plf Wind Uplift

Panel & Purlin Dead Loads

126 plf153 plf

6'-0"edge strip



UPLIFT WIND LOADING: Bending Capacity

End Span: a. @ Max. Negative Moment [ AISI C3.1.1 ] t1 = 0.082" ; M = - 52.84 "- k Compression flange unbraced, use R factor. Ma1 = [0.7 Sxe Fy / 1.67] 4/3 Ma1 = [(0.7)(3.03)(55)/1.67] 4/3 = 93.14 "- k Stress Ratio = 52.84/93.14 = 0.57 (OK) b. @ End of Purlin Lap [ AISI C3.1.2 ] t1 = 0.082" ; M = 50.37 "- k End of lap is between inflection points. R factor not applicable. Roof covering provides continuous compression (top) flange support. Ma1 = [Sxe Fy / 1.67] 4/3 Ma1 = [(3.03)(55)/1.67] 4/3 = 133.05 "- k Stress Ratio = 50.37/133.05 = 0.38 (OK) c. @ Interior Frame Support [ AISI C3.1.2 ] Lapped purlins: t1 = 0.082" , t2= 0.059" M = 74.14 "- k Interior support is between inflection points. R factor not applicable. Roof covering provides continuous compression (top) flange support. Ma1 = 133.05 "- k Ma2 = [Sxe Fy /1.67] 4/3 Ma2 = [(1.78)(55)/1.67] 4/3 = 78.16 "- k Ma = 133.05 "- k + 78.16 "- k = 211.21 "- k Stress Ratio = 74.14/211.21 = 0.35 (OK)

Interior Span: d. @ Max. Negative Moment [ AISI C3.1.1 ] t2 = 0.059" ; M = - 16.61 "- k Compression flange unbraced, use R factor. Ma2 = [0.7 Sxe Fy / 1.67] 4/3 Ma2 = [(0.7)(1.78)(55)/1.67] 4/3 = 54.71 "- k Stress Ratio = 16.61/54.71 = 0.30 (OK) e. @ End of Left Purlin Lap [ AISI C3.1.2 ] t2 = 0.059" ; M = 42.19 "- k End of lap is between inflection points. R factor not applicable. Roof covering provides continuous compression (top) flange support. Ma2 = [Sxe Fy / 1.67] 4/3 Ma1 = [(1.78)(55)/1.67] 4/3 = 78.16 "- k Stress Ratio = 42.19/78.19 = 0.54 (OK) f. @ End of Right Purlin Lap [ AISI C3.1.2 ] t2 = 0.059" ; M = - 46.52 "- k Same as left lap: Stress Ratio = 46.52/58.62 = 0.79 (OK)



UPLIFT WIND LOADING: Shear, Bending & Shear

General: [ AISI C3.2 ] t1 = 0.082", h1 = 8.5 -2(0.3125 + 0.082) = 7.71" 1.38[Ekv/Fy]½ =1.38[(29500)(5.34)/55]½ = 73.9 h/t = 94.04 > 73.9 Va1 = 0.53 E kv t3 / h = 0.53(E)(5.34)(.082)3/7.71Va1 = 5.97 k (4/3) = 7.96 k t2 = 0.059" ; h2 = 8.5 - 2(0.3125 + 0.059) = 7.76" h/t = 131.18 > 73.9 Va2 = 0.53 E kv t3 / h = 0.53(E)(5.34)(.059)3/7.76Va2 = 2.21 k (4/3) = 2.95 k End Span: a. @ End of Purlin Lap t1 = 0.082" , V = 1.30 k , Va1 = 7.96 k Stress Ratio = 1.30/7.96 = 0.16 (OK) b. @ Exterior Support t1 = 0.082" , V = 1.02 k , Va1 = 7.96 k Stress Ratio = 1.02/7.96 = 0.13 (OK) c. @ Interior Support Lapped purlins: t1 = 0.082" , t2 = 0.059" Va = Va1 + Va2 = 7.96 k + 2.95 k = 10.91 k V = 1.39 k Stress Ratio = 1.39/10.91 = 0.13 (OK) Interior Span: d. @ End of Purlin Laps t1 = 0.059 , V = 0.98 k , Va2 = 2.95 k Stress Ratio = 0.98/2.95 = 0.33 (OK)

General: [ AISI C3.2 ] (M/Ma)2 + (V/Va)2 < 1 End Span: a. @ End of Purlin Lap t1 = 0.082" , V = 1.30 k , Va1 = 7.96 k M = 50.37 "-k ; Ma = 133.05 "-k (50.37/133.05)2 + (1.30/7.96)2 = .17 < 1 (OK) b. @ Exterior Support t1 = 0.082" , V = 1.02 k , Va1 = 7.96 k M = 0.86 "-k ; Ma = 133.05 "-k (0.86/133.05)2 + (1.03/7.96)2 = .02 < 1 (OK) c. @ Interior Support Lapped purlins: t1 = 0.082" , t2 = 0.059" V = 1.39 k ; Va = 10.91 k M = 74.14 "-k ; Ma = 211.21 "-k (74.14/211.21)2 + (1.39/10.91)2 = .14 < 1 (OK) Interior Span: d. @ End of Purlin Laps t1 = 0.059 , V = 0.98 k , Va2 = 2.95 k M = 42.19 "-k ; Ma = 78.16 "-k (42.19/78.16)2 + (0.98/2.95)2 = .40 < 1 (OK)


VP DESIGN MANUAL Section: 4.1.S1 BUILDINGS, 06/24/98 Rev. 0 Page 1 of 6 INC.

APPENDIX S1

SUPPLEMENTARY INFORMATION

EXAMPLE CALCULATION OF SECTION PROPERTIES

FOR 8½" Z-PURLIN, t = 0.073"



Verification of Full Section Properties

Element 3 & 4: t = 0.073", I.R. = 0.3125" w = 2.5" - (0.3125" + 0.073") [ 1 + Tan(25°)] = 1.934738" L(3) = L(4) = w/2 = 0.9674" Ycg(3) = Ycg(3) = t/2 = 0.365" Io(3) = Io(3) = 0.0 Element 11 & 12: L(11) = L(12) = w/2 = 0.9674" Ycg(11) = Ycg(12) = 8.5" - t/2 = 8.4635" Io(11) = Io(12) = 0.0 Element 6, 7, 8, & 9: Web Flat, w = 8.5" - 2 (0.3125" + 0.073") = 7.7290" L(6) =L(7) =L(8) =L(8) = w/4 = 1.9323" Ycg(6) = ( 0.3125" + 0.073") + 1.9323" / 2 = 1.3516" Ycg(7) = ( 0.3125" + 0.073") + 1.9323 + 1.9323" / 2 = 3.2839" Ycg(8) = ( 0.3125" + 0.073") + (2)1.9323 + 1.9323" / 2 = 5.2161" Ycg(9) = ( 0.3125" + 0.073") + (3)1.9323 + 1.9323" / 2 = 7.1484" Io(6) = Io(7) = Io(8) = Io(9) = L3 t / 12 = (1.9323")3 (0.073") / 12 = 0.0439 in4

D = 0.9206"

d

50 o

11 12 13

14

10 i.r. = 5/16"9

8

6

7

5 4 3

1 2

b = 2.50"w = 1.93"

h = 8.5" t = 0.073" Fy = 55 ksi

Reference Ycg

Reference Xcg +-+ -



Element 5 & 10: L(5) = L(10) = 1.5708 ( 0.3125" + 0.073"/2) = 0.5482" c = 0.637( 0.3125" + 0.073"/2 ) = 0.2223" Ycg(5) = (0.3125" + 0.073") - c = 0.1632" Ycg(10) = 8.5" -(0.3125" + 0.073") + c = 8.3368" Io(5) = Io(10) = 0.149 R3 t = 0.149 (0.3125" + 0.073"/2)3 (0.073") = 0.0005 in4 Element 2 & 13: L(2) = L(10) = Rθ = (0.3125" + 0.073"/2) ( 50° π / 180° ) = 0.3046" c1 = R sin(θ) / θ = (0.3125" + 0.073"/2) Sin( 50°) / [ 50°π / 180° ] = 0.3064" Ycg(2) = (0.3125" + 0.073") - c1 = 0.0791" Ycg(13) = 8.5" -(0.3125" + 0.073") + c1 = 8.4209" Io(2) = Io(13) = [ .5(θ + Sinθ Cosθ) - (Sinθ)2 / θ ] R3 t = 0.00003129 in4 = 0 Element 1 & 14: D = 0.9206" d = 0.9206" -( 0.3125" + 0.073" ) Tan (25°) = 0.74084" L(1) = L(14) = 0.74084"

50o

12 13

14

. YsA2

A1

A1 = Tan(25°) ( 0.3125" + 0.073"/2 ) = 0.16274" A2 = A1 Sin(50°) = 0.12467" Ys = t/2 + L2 = 0.12467" + 0.073"/2 = 0.16117" Ycg(14) = 8.5" - Ys - [ L(14) / 2 ] Sin(50°) = 8.0551" Ycg(1) = Ys + [ L(1) / 2 ] Sin(50°) = 0.4449" Io(1) = Io(14) = Sin(50°) L3 t / 12 = Sin(50°) (0.74084")3 (0.073) / 12 = 0.0015 in4 Evaluating Yna, Ix, Sx: Yna = [Σ (Ycg L )] / [Σ ( L )] = 62.8391 / 14.7857 = 4.25" Ix = Σ Io + t Σ [L ( Yna - Ycg )2 ] = 0.1796 + 0.073(155.1308) = 11.5041 in4 Sx = Ix / Yna = 11.5041 in4 / 4.25" = 2.7068 in3



Verification of Effective Section Properties for Flexural Strength, Se

Effective Width of Top Compression Flange: (elements 11 and 12) Per AISI B.4 Ft = 55 ksi S = 1.28 ( E / f )½ = 1.28 ( 29500 / 55 )½ = 29.64419 w = 2.5" - (0.3125" + 0.073") [ 1 + Tan(25°)] = 1.934738" t = 0.073" w/t = 1.934738/0.073 = 26.50326 S/3 = 9.8814 S/3 < w/t < S : use Case II of AISI B4.2 Stiffener Length, d: d = 0.9206" -( 0.3125" + 0.073" ) Tan (25°) = 0.74084" Moment of Inertia of full stiffener, Is: Is = d3 t sin(θ) /12 = (0.74084")3 (0.073) sin(25°) /12 = 0.0014515 in4

Adequate moment of inertia of stiffener, Ia Ia = 399 t4 [(w/t)/S] -0.33 3 = 399 (0.073")4 [(26.50326)/(29.64419)] -0.33 3 = 0.002033 in4 Evaluate K: n = ½ C2 = Is/Ia = 0.0014515/0.002033 = 0.71386 C1 = 2 - C1 = 1.28614 D/w = 0.9206"/1.934738" = 0.47583

D = 0.9206"

d

50o

11 12 13

14

10 i.r. = 5/16"9

8

6 7

5 4 3

1 2

b = 2.50"

Ft = 55 ksi Compression

F1 = 50.12 ksi

F2 = -47.78 ksi

neutral axis

Yt = 4.3424"

b1 = 1.9785"

b2 = 1.9785"

w = 1.93"

3.9570"

3.7720"

F3 = 52.96 ksi

h = 8.5" t = 0.073" Fy = 55 ksi

Fb = -52.66 ksi



k = [ 4.82 - 5(D/w) ] ( Is / Ia )½ + 0.43 = [ 4.82 - 5(0.47583 ] ( 0.0014515 / 0.002033 )½ + 0.43 k = 2.49230 k < 5.25 - 5(D/w) = 5.25 - 5(0.47583) = 2.87085 OK Evaluating Flange effective width : ( AISI eq. B2.1-4,3,2) λ = [1.052 / k½ ] (w/t) ( f / E )½ = [1.052 / (2.49230)½ ] (26.50326) ( 55 / 29500 )½ λ = 0.76259 > 0.673 ρ = ( 1 - 0.22/λ ) / λ = 0.93302 b = ρ w = 0.93302 ( 1.934738") = 1.80516" note: "b" is the total effective length of the flange, distributing according to AISI figure B4-2 results in: Element # 11 Effective Length = C1 (b) /2 = 1.28614(1.80516) / 2 = 1.16084" Element # 12 Effective Length = C2 (b) /2 = 0.71386(1.80516) / 2 = 0.64431" Centroid from bottom of both elements: Ycg = 8.5" - t/2 = 8.5" - 0.073"/2 = 8.4635" Effective Length of Lip Stiffener: ( element 14 ) Evaluate Stress F3 at Top of Lip:

50o

12 13

14

. YsF3 = 52.96 ksiFt = 55.00 ksi

A2

A1

A1 = Tan(25°) ( 0.3125" + 0.073"/2 ) = 0.16274" A2 = A1 SIn(50°) = 0.12467" Ys = t/2 + A2 = 0.12467" + 0.073"/2 = 0.16117" F3 = [(4.3424" - 0.16117") / (4.3424")] 55 ksi = 52.95865 ksi Evaluate Effective Lip Length, d's : d = 0.74084" F3 = 52.95865 ksi k = 0.43 λ = [1.052 / k½ ] (w/t) ( f / E )½ = [1.052 / (0.43)½ ] (0.74084/0.073) ( 52.95865 / 29500 )½ λ = 0.68983 > 0.673 ρ = ( 1 - 0.22/λ ) / λ = 0.98732 d's = ρ d = 0.98732 ( 0.74084") = 0.73145" Evaluate the Reduced Effective Lip Length, ds : ds = d's ( Is / Ia) = 0.73145" ( 0.0014515 / 0.002033 ) = 0.52223" Element # 14 Effective Length = ds = 0.52223" note: ds is located per AISI figure B4-2 Centroid from bottom: Ycg = 8.5" - Ys - (ds/2)sin(50°) = 8.5" - 0.16117" - (0.52223"/2)sin(50°) = 8.1388" Io(14) = Sin(50°) L3 t / 12 = Sin(50°) (0.52223")3 (0.073) / 12 = 0.0005 in4



Effective Length of Web Compression Elements 8 & 9: Per AISI B2.3 F1 = 50.1173 ksi (top of web) F2 = -47.7765 ksi (bottom of web) ψ = F2 / F1 = 50.1173 / -47.7765 = -0.95329 k = 4 + 2( 1 - ψ )3 + 2(1 - ψ ) = 4 + 2( 1 + 0.95329 )3 + 2(1 + 0.95329 ) = 22.81152 w(web) = 8.5" - 2( 0.3125" + 0.073") = 7.7290" w/t = 7.7290/0.073 = 105.8767 λ = [1.052 / k½ ] (w/t) ( f / E )½ = [1.052 / (22.81152)½ ] (105.8767) ( 50.1173 / 29500 )½ λ = 0.96122 > 0.673 ρ = ( 1 - 0.22/λ ) / λ = 0.80223 be = ρ w = 0.80223 ( 7.7290") = 6.2004" b1 = be / ( 3-ψ ) = 6.2004"/ ( 3 + 0.95329 ) = 1.5684" ψ < -0.236 b2 = be/2 = 3.1002" b1 + b2 = 4.6687" Compression portion of the web = 4.3424" - 0.3125" - 0.073" = 3.9569" b1 + b2 > 3.9569" therefor; b1 = b2 = 3.9569"/2 = 1.9785" note that b1 and b2 correspond to web elements 8 and 9. Centroid from bottom: Ycg(9) = 8.5" - 0.3125" - 0.073" - b1/2 = 7.1253" Ycg(8) = 8.5" - 0.3125" - 0.073" -b1 - b2/2 = 5.1468" Io(9) = Io(8) = L3 t /12 = (1.9785")3 (0.073") /12 = 0.0471 in4 Length of Web Tension Elements 6 & 7: Web elements 6 and 7 are in tension and are filly effective. w (tension part of web) = 8.5" - 4.3424" - (0.3125" + 0.073") = 3.7720" L(6) = L(7) = 3.7720"/2 = 1.8860" Ycg(6) = ( 0.3125" + 0.073" ) + 1.8860"/2 = 1.3285" Ycg(7) = ( 0.3125" + 0.073" ) + 1.8860" + 1.8860"/2 = 3.2145 Iox(6) = Iox(7) = L3 t /12 = (1.8860")3 (0.073") /12 = 0.0408 in4 Evaluating Yna, Ix, Sx: Yna (from bottom) = [Σ (Ycg L )] / [Σ ( L )] = 60.0244 / 14.4374 = 4.1576" Yna (from top) = 8.5" - 4.1576" = 4.3424" Ix = Σ Io + t Σ [ L ( Yna - Ycg )2 ] = 0.1788 + 0.073(149.8699) = 11.1193 in4 Sx = Ix / Yna(from top) = 11.1193 in4 / 4.3424" = 2.5606 in3



APPENDIX S1

SUPPLEMENTARY INFORMATION

EXAMPLE CALCULATION OF SECTION PROPERTIES

FOR 8½" Z-PURLIN, t = 0.073"



Verification of Full Section Properties

Element 3 & 4: t = 0.073", I.R. = 0.3125" w = 2.5" - (0.3125" + 0.073") [ 1 + Tan(25°)] = 1.934738" L(3) = L(4) = w/2 = 0.9674" Ycg(3) = Ycg(3) = t/2 = 0.365" Io(3) = Io(3) = 0.0 Element 11 & 12: L(11) = L(12) = w/2 = 0.9674" Ycg(11) = Ycg(12) = 8.5" - t/2 = 8.4635" Io(11) = Io(12) = 0.0 Element 6, 7, 8, & 9: Web Flat, w = 8.5" - 2 (0.3125" + 0.073") = 7.7290" L(6) =L(7) =L(8) =L(8) = w/4 = 1.9323" Ycg(6) = ( 0.3125" + 0.073") + 1.9323" / 2 = 1.3516" Ycg(7) = ( 0.3125" + 0.073") + 1.9323 + 1.9323" / 2 = 3.2839" Ycg(8) = ( 0.3125" + 0.073") + (2)1.9323 + 1.9323" / 2 = 5.2161" Ycg(9) = ( 0.3125" + 0.073") + (3)1.9323 + 1.9323" / 2 = 7.1484" Io(6) = Io(7) = Io(8) = Io(9) = L3 t / 12 = (1.9323")3 (0.073") / 12 = 0.0439 in4

D = 0.9206"

d

50 o

11 12 13

14

10 i.r. = 5/16"9

8

6

7

5 4 3

1 2

b = 2.50"w = 1.93"

h = 8.5" t = 0.073" Fy = 55 ksi

Reference Ycg

Reference Xcg +-+ -



Element 5 & 10: L(5) = L(10) = 1.5708 ( 0.3125" + 0.073"/2) = 0.5482" c = 0.637( 0.3125" + 0.073"/2 ) = 0.2223" Ycg(5) = (0.3125" + 0.073") - c = 0.1632" Ycg(10) = 8.5" -(0.3125" + 0.073") + c = 8.3368" Io(5) = Io(10) = 0.149 R3 t = 0.149 (0.3125" + 0.073"/2)3 (0.073") = 0.0005 in4 Element 2 & 13: L(2) = L(10) = Rθ = (0.3125" + 0.073"/2) ( 50° π / 180° ) = 0.3046" c1 = R sin(θ) / θ = (0.3125" + 0.073"/2) Sin( 50°) / [ 50°π / 180° ] = 0.3064" Ycg(2) = (0.3125" + 0.073") - c1 = 0.0791" Ycg(13) = 8.5" -(0.3125" + 0.073") + c1 = 8.4209" Io(2) = Io(13) = [ .5(θ + Sinθ Cosθ) - (Sinθ)2 / θ ] R3 t = 0.00003129 in4 = 0 Element 1 & 14: D = 0.9206" d = 0.9206" -( 0.3125" + 0.073" ) Tan (25°) = 0.74084" L(1) = L(14) = 0.74084"

50o

12 13

14

. YsA2

A1

A1 = Tan(25°) ( 0.3125" + 0.073"/2 ) = 0.16274" A2 = A1 Sin(50°) = 0.12467" Ys = t/2 + L2 = 0.12467" + 0.073"/2 = 0.16117" Ycg(14) = 8.5" - Ys - [ L(14) / 2 ] Sin(50°) = 8.0551" Ycg(1) = Ys + [ L(1) / 2 ] Sin(50°) = 0.4449" Io(1) = Io(14) = Sin(50°) L3 t / 12 = Sin(50°) (0.74084")3 (0.073) / 12 = 0.0015 in4 Evaluating Yna, Ix, Sx: Yna = [Σ (Ycg L )] / [Σ ( L )] = 62.8391 / 14.7857 = 4.25" Ix = Σ Io + t Σ [L ( Yna - Ycg )2 ] = 0.1796 + 0.073(155.1308) = 11.5041 in4 Sx = Ix / Yna = 11.5041 in4 / 4.25" = 2.7068 in3



Verification of Effective Section Properties for Flexural Strength, Se

Effective Width of Top Compression Flange: (elements 11 and 12) Per AISI B.4 Ft = 55 ksi S = 1.28 ( E / f )½ = 1.28 ( 29500 / 55 )½ = 29.64419 w = 2.5" - (0.3125" + 0.073") [ 1 + Tan(25°)] = 1.934738" t = 0.073" w/t = 1.934738/0.073 = 26.50326 S/3 = 9.8814 S/3 < w/t < S : use Case II of AISI B4.2 Stiffener Length, d: d = 0.9206" -( 0.3125" + 0.073" ) Tan (25°) = 0.74084" Moment of Inertia of full stiffener, Is: Is = d3 t sin(θ) /12 = (0.74084")3 (0.073) sin(25°) /12 = 0.0014515 in4

Adequate moment of inertia of stiffener, Ia Ia = 399 t4 [(w/t)/S] -0.33 3 = 399 (0.073")4 [(26.50326)/(29.64419)] -0.33 3 = 0.002033 in4 Evaluate K: n = ½ C2 = Is/Ia = 0.0014515/0.002033 = 0.71386 C1 = 2 - C1 = 1.28614 D/w = 0.9206"/1.934738" = 0.47583

D = 0.9206"

d

50o

11 12 13

14

10 i.r. = 5/16"9

8

6 7

5 4 3

1 2

b = 2.50"

Ft = 55 ksi Compression

F1 = 50.12 ksi

F2 = -47.78 ksi

neutral axis

Yt = 4.3424"

b1 = 1.9785"

b2 = 1.9785"

w = 1.93"

3.9570"

3.7720"

F3 = 52.96 ksi

h = 8.5" t = 0.073" Fy = 55 ksi

Fb = -52.66 ksi



k = [ 4.82 - 5(D/w) ] ( Is / Ia )½ + 0.43 = [ 4.82 - 5(0.47583 ] ( 0.0014515 / 0.002033 )½ + 0.43 k = 2.49230 k < 5.25 - 5(D/w) = 5.25 - 5(0.47583) = 2.87085 OK Evaluating Flange effective width : ( AISI eq. B2.1-4,3,2) λ = [1.052 / k½ ] (w/t) ( f / E )½ = [1.052 / (2.49230)½ ] (26.50326) ( 55 / 29500 )½ λ = 0.76259 > 0.673 ρ = ( 1 - 0.22/λ ) / λ = 0.93302 b = ρ w = 0.93302 ( 1.934738") = 1.80516" note: "b" is the total effective length of the flange, distributing according to AISI figure B4-2 results in: Element # 11 Effective Length = C1 (b) /2 = 1.28614(1.80516) / 2 = 1.16084" Element # 12 Effective Length = C2 (b) /2 = 0.71386(1.80516) / 2 = 0.64431" Centroid from bottom of both elements: Ycg = 8.5" - t/2 = 8.5" - 0.073"/2 = 8.4635" Effective Length of Lip Stiffener: ( element 14 ) Evaluate Stress F3 at Top of Lip:

50o

12 13

14

. YsF3 = 52.96 ksiFt = 55.00 ksi

A2

A1

A1 = Tan(25°) ( 0.3125" + 0.073"/2 ) = 0.16274" A2 = A1 SIn(50°) = 0.12467" Ys = t/2 + A2 = 0.12467" + 0.073"/2 = 0.16117" F3 = [(4.3424" - 0.16117") / (4.3424")] 55 ksi = 52.95865 ksi Evaluate Effective Lip Length, d's : d = 0.74084" F3 = 52.95865 ksi k = 0.43 λ = [1.052 / k½ ] (w/t) ( f / E )½ = [1.052 / (0.43)½ ] (0.74084/0.073) ( 52.95865 / 29500 )½ λ = 0.68983 > 0.673 ρ = ( 1 - 0.22/λ ) / λ = 0.98732 d's = ρ d = 0.98732 ( 0.74084") = 0.73145" Evaluate the Reduced Effective Lip Length, ds : ds = d's ( Is / Ia) = 0.73145" ( 0.0014515 / 0.002033 ) = 0.52223" Element # 14 Effective Length = ds = 0.52223" note: ds is located per AISI figure B4-2 Centroid from bottom: Ycg = 8.5" - Ys - (ds/2)sin(50°) = 8.5" - 0.16117" - (0.52223"/2)sin(50°) = 8.1388" Io(14) = Sin(50°) L3 t / 12 = Sin(50°) (0.52223")3 (0.073) / 12 = 0.0005 in4



Effective Length of Web Compression Elements 8 & 9: Per AISI B2.3 F1 = 50.1173 ksi (top of web) F2 = -47.7765 ksi (bottom of web) ψ = F2 / F1 = 50.1173 / -47.7765 = -0.95329 k = 4 + 2( 1 - ψ )3 + 2(1 - ψ ) = 4 + 2( 1 + 0.95329 )3 + 2(1 + 0.95329 ) = 22.81152 w(web) = 8.5" - 2( 0.3125" + 0.073") = 7.7290" w/t = 7.7290/0.073 = 105.8767 λ = [1.052 / k½ ] (w/t) ( f / E )½ = [1.052 / (22.81152)½ ] (105.8767) ( 50.1173 / 29500 )½ λ = 0.96122 > 0.673 ρ = ( 1 - 0.22/λ ) / λ = 0.80223 be = ρ w = 0.80223 ( 7.7290") = 6.2004" b1 = be / ( 3-ψ ) = 6.2004"/ ( 3 + 0.95329 ) = 1.5684" ψ < -0.236 b2 = be/2 = 3.1002" b1 + b2 = 4.6687" Compression portion of the web = 4.3424" - 0.3125" - 0.073" = 3.9569" b1 + b2 > 3.9569" therefor; b1 = b2 = 3.9569"/2 = 1.9785" note that b1 and b2 correspond to web elements 8 and 9. Centroid from bottom: Ycg(9) = 8.5" - 0.3125" - 0.073" - b1/2 = 7.1253" Ycg(8) = 8.5" - 0.3125" - 0.073" -b1 - b2/2 = 5.1468" Io(9) = Io(8) = L3 t /12 = (1.9785")3 (0.073") /12 = 0.0471 in4 Length of Web Tension Elements 6 & 7: Web elements 6 and 7 are in tension and are filly effective. w (tension part of web) = 8.5" - 4.3424" - (0.3125" + 0.073") = 3.7720" L(6) = L(7) = 3.7720"/2 = 1.8860" Ycg(6) = ( 0.3125" + 0.073" ) + 1.8860"/2 = 1.3285" Ycg(7) = ( 0.3125" + 0.073" ) + 1.8860" + 1.8860"/2 = 3.2145 Iox(6) = Iox(7) = L3 t /12 = (1.8860")3 (0.073") /12 = 0.0408 in4 Evaluating Yna, Ix, Sx: Yna (from bottom) = [Σ (Ycg L )] / [Σ ( L )] = 60.0244 / 14.4374 = 4.1576" Yna (from top) = 8.5" - 4.1576" = 4.3424" Ix = Σ Io + t Σ [ L ( Yna - Ycg )2 ] = 0.1788 + 0.073(149.8699) = 11.1193 in4 Sx = Ix / Yna(from top) = 11.1193 in4 / 4.3424" = 2.5606 in3

Purlin Design-VP Design Manual

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