1. Notes on AISI Design Methods for Sheathing Braced Design of Wall Studs in Compression B.W. Schafer report to AISI-COFS Design Methods Committee April 2008 Civil Engineering at JOHNS HOPKINS UNIVERSITY

2. Overview Part of the AISI-COFS funded project on the design of sheathed walls with dis-similar sheathing. Presentation developed from a report of the same name Project updates, including full report available at www.ce.jhu.edu/bschafer/sheathedwalls Report covers –Design methods via 1962, 1980-2004, 2007 AISI –Examination of sheathing stiffness ‘k’ –Initial summary of known demands and limit states on a sheathed wall stud in compression

3. AISI Design Methods 1962 AISI Design Manual Winter’s method discrete spring model 1980 to 2004 AISI Spec. Simaan and Peköz shear diaphragm model 2007 AISI-COFS Wall Stud Standard (S211) “simplified” discrete spring model

4. Basic notation I 1, r 1, P cr1 : strong-axis buckling perpendicular to the plane of the wall I 2, r 2, P cr2 : weak-axis buckling parallel to the plane of the wall e: initial imperfection

5. 1962 AISI Specification “The safe load-carrying capacity of a stud may be computed on the basis that the wall material or sheathing (attached to the stud) furnishes adequate lateral support to the stud in the plane of the wall, provided the wall material and its attachments to the stud comply with the following requirements:” 1962 AISI Design Manual

6. 1962 AISI Specification “The safe load-carrying capacity of a stud may be computed on the basis that the wall material or sheathing (attached to the stud) furnishes adequate lateral support to the stud in the plane of the wall, provided the wall material and its attachments to the stud comply with the following requirements:” fastener spacing, a, must be less than 1962 AISI Design Manual

7. 1962 AISI Specification “The safe load-carrying capacity of a stud may be computed on the basis that the wall material or sheathing (attached to the stud) furnishes adequate lateral support to the stud in the plane of the wall, provided the wall material and its attachments to the stud comply with the following requirements:” fastener spacing, a, must be less than 1962 AISI Design Manual

8. fastener-sheathing stiff- ness, k, must be at least: 1962 AISI Specification “The safe load-carrying capacity of a stud may be computed on the basis that the wall material or sheathing (attached to the stud) furnishes adequate lateral support to the stud in the plane of the wall, provided the wall material and its attachments to the stud comply with the following requirements:” fastener spacing, a, must be less than 1962 AISI Design Manual

9. fastener-sheathing stiff- ness, k, must be at least: 1962 AISI Specification “The safe load-carrying capacity of a stud may be computed on the basis that the wall material or sheathing (attached to the stud) furnishes adequate lateral support to the stud in the plane of the wall, provided the wall material and its attachments to the stud comply with the following requirements:” fastener spacing, a, must be less than fasteners must have at least this much strength: 1962 AISI Design Manual

10. Fastener spacing limit #1 fastener spacing, a, must be less than Design basis: a max1 requires that weak-axis buckling of the stud, including contributions from the wall stiffness k, that is developed from fasteners at spacing a, is greater than or equal to the squash load of the column. 1962 AISI Design Manual

11. Fastener spacing limit #1 fastener spacing, a, must be less than Design basis: a max1 requires that weak-axis buckling of the stud, including contributions from the wall stiffness k, that is developed from fasteners at spacing a, is greater than or equal to the squash load of the column. Winter (1960) solution for column on a stiff, continuous, foundation convert continuous foundation stiffness, , to discrete springs, k, and set P cr2 =Af y solve for a, you get 1962 AISI Design Manual

12. Fastener spacing limit #2 fastener spacing, a, must be less than Design basis: a max2 requires that weak-axis buckling of the stud over a length of 2a (twice the fastener spacing) must be greater than the strong-axis buckling over the entire length. 1962 AISI Design Manual

13. Fastener spacing limit #2 fastener spacing, a, must be less than Design basis: a max2 requires that weak-axis buckling of the stud over a length of 2a (twice the fastener spacing) must be greater than the strong-axis buckling over the entire length. solve for a, you get Equating P cr1 and P cr2 implies Insure weak-axis buckling over 2a does not control 1962 AISI Design Manual

14. fastener-sheathing stiff- ness, k, must be at least: minimum stiffness k Design basis: k min is the same as the a max1 design check. sub in for E and , solve for k (from Winter 1960) 1962 AISI Design Manual

15. fastener strength fasteners must have at least this much strength: Design basis: forces developed in an imperfect, but stiff continuous foundation under the design load P, should be carried by the fasteners, plus Winter adds some interesting empirical corrections. Winter (1960) defines forces developed in a stiff con- tinuous foundation supporting an imperfect column 1962 AISI Design Manual

16. fastener strength (continued) fasteners must have at least this much strength: again, Winter uses the solution for buckling of a column on a stiff continuous foundation, this time to find the ideal stiffness  id (k id ): and which results in after substitution and rearranging, we find this ends, rational derivation and Winter introduces empiricism 1962 AISI Design Manual

17. fastener strength (continued) fasteners must have at least this much strength: 1.assume k act =k id only under first denominator term 2.remove 2 under the radical in first term as well 3.remaining k act = k results in Brace forces resulting from Example 11 of 1962 design manual note k act =80k id

18. fastener strength (continued) fasteners must have at least this much strength: 1.assume k act =k id only under first denominator term 2.remove 2 under the radical in first term as well 3.remaining k act = k results in 2% “rule” Winter’s 1962 F min equation Winter’s 1962 equation without empirical modification Brace forces resulting from Example 11 of 1962 design manual note k act =80k id 1962 AISI Design Manual

19. Summary of 1962 Specification The fastener-sheathing stiffness must insure the following condition is met The fastener-sheathing strength must insure the following condition is met In addition to insure adequate performance in the face of potential defects If the above conditions are met P cr = P cr1 (strong-axis). where 1962 AISI Design Manual

20. Critique of 1962 Specification The fastener-sheathing stiffness must insure the following condition is met The fastener-sheathing strength must insure the following condition is met In addition to insure adequate performance in the face of potential defects If the above conditions are met P cr = P cr1 (strong-axis). useful, but arbitrary, does not even insure that strong axis controls couched in something theoretical, but in the end empirical, not wholly consistent with current approaches arbitrary, realistic? Underlying theory not directly applicable, no torsional-flexural buckling check, to my knowledge none of us have ever even done a k test!? 1962 AISI Design Manual

21. 1980 to 2004 AISI Specification Also known as the Simaan and Peköz method, or the shear diaphragm model, or the “D4 method”. Existed from 1980 to 2004 in Spec. Section D4. Abandoned in favor of a return to Winter’s method, more or less. Why was the method abandoned? What can we learn from the “mistakes”? 1980 to 2004 AISI Spec.

22. Practical limitations of D4 method “The design expressions are complex. The design expressions do not give credit to the presence of supplementary steel bridging which is typically installed in order to align members and to provide necessary structural integrity during erection and in the completed structure. Provided there is adequate steel bridging, the imperfect sheathing approach in Section D4 (a) can produce a lower capacity than an all steel approach. The most popular sheathing, gypsum wallboard, is seen by some as too moisture and load cycle sensitive to act as a reliable structural brace for the service life of a structure. Other restrictions in Section D4 (a) are for the most part impractical for typical use.” (Trestain 2002). 1980 to 2004 AISI Spec.

23. Theoretical limitation of the D4 Method For the most part Winter’s method may conceptually be understood as a column supported by discrete springs: The D4 method has never been presented with an equivalent mechanical model, instead it is always described as a summation of energies (column bending + shear distortion of diaphragm), what is underneath the hood? 1980 to 2004 AISI Spec.

24. Consider the shear energy term 1980 to 2004 AISI Spec. shear energy “shear” angle

25. Consider the shear energy term 1980 to 2004 AISI Spec. shear energy “shear” angle equivalence with rotational spring foundation:

26. Consider the shear energy term 1980 to 2004 AISI Spec. shear energy “shear” angle equivalence with rotational spring foundation: mechanically equivalent model to shear diaphragm considering discrete fasteners

27. D4 method summary The method was abandoned for numerous practical reasons, and now can be seen to have a serious theoretical limitation However, lots of great and complicated mechanics in the D4 method. Torsional-flexural buckling is treated thoroughly (even for dis- similar and one-sided sheathing). The role of shear in deforming the sheathing is real, but have to be careful with how that actually braces the stud 1980 to 2004 AISI Spec.

28. 2007 AISI-S211 Wall Stud Standard “Wall stud assemblies using a sheathing braced design shall be designed assuming that identical sheathing is attached to both sides of the wall stud and connected to the bottom and top horizontal members of the wall to provide lateral and torsional support to the wall stud in the plane of the wall.” “Both ends of the stud shall be connected to restrain rotation about the longitudinal stud axis and horizontal displacement perpendicular to the stud axis.” Further, in B1.2(b) it is prescribed that the global buckling load of a stud, with fasteners spaced distance “a” apart shall be determined ignoring any sheathing contribution (i.e. k = 0) over a distance of 2a, i.e.: 2007 AISI-COFS Wall Stud Standard (S211)

29. P cr by AISI-S211-07 It is assumed that the sheathing provides enough stiffness that P cr ignoring the sheathing over a length equal to twice the fastener spacing, a, is always less than P cr considering the sheathing: buckling across a defective fastener buckling of the stud engaging all fasteners 2007 AISI-COFS Wall Stud Standard (S211)

30. P cr by AISI-S211-07 It is assumed that the sheathing provides enough stiffness that P cr ignoring the sheathing over a length equal to twice the fastener spacing, a, is always less than P cr considering the sheathing: buckling across a defective fastener buckling of the stud engaging all fasteners 2007 AISI-COFS Wall Stud Standard (S211)

31. P cr by AISI-S211-07 It is assumed that the sheathing provides enough stiffness that P cr ignoring the sheathing over a length equal to twice the fastener spacing, a, is always less than P cr considering the sheathing: validity of assumption depends on the stiffness k, as k  0 definitely not a valid assumption buckling across a defective fastener buckling of the stud engaging all fasteners 2007 AISI-COFS Wall Stud Standard (S211)

32. AISI-S211-07 Commentary Wall Stud Standard provides only minor guidance, primarily wall stud design is left to rational analysis. However, the commentary provides one such rational analysis method, relying primarily on the 2% rule for fastener demands. Let us revisit the classic derivation to better understand the implications of the 2% rule. 2007 AISI-COFS Wall Stud Standard (S211)

33. 2% rule in AISI-S211-07 Commentary Consider the basic derivation (supplementing the commentary) 2007 AISI-COFS Wall Stud Standard (S211)

34. 2% rule in AISI-S211-07 Commentary Consider the basic derivation (supplementing the commentary) 2007 AISI-COFS Wall Stud Standard (S211) Finally solving for the bracing force: note, 1%P, not 2%P, but brace force is a function of k and do can be higher or lower

35. Comparison of 1962 and 2007 methods 1962 AISI Design Manual 2007 AISI-COFS Wall Stud Standard (S211) Analyze any designPrescribed failure mode

36. Looking towards new methods..

37. Limit states (abridged) Stud –Global buckling (F, T, FT – including sheathing) should the stud be checked assuming defective fastener? –Local buckling (probably ignore sheathing) –Distortional buckling (probably including sheathing) should the stud be checked assuming a defective fastener? Connections –Stud-fastener-sheathing connection –Track-fastener-sheathing connection –Stud-to-track connection Sheathing Construction loads (requires all-steel check)

38. Limit states (abridged) Stud –Global buckling (F, T, FT – including sheathing) should the stud be checked assuming defective fastener? –Local buckling (probably ignore sheathing) –Distortional buckling (probably including sheathing) should the stud be checked assuming a defective fastener? Connections –Stud-fastener-sheathing connection –Track-fastener-sheathing connection –Stud-to-track connection Sheathing Construction loads (requires all-steel check) state of development...

39. Up to date work on bracing a stud

40. Sheathing k w x y

41. w x y

42. When sheathing k matters? If k sheathing <10 k fastener... probably matters sheathing fastener k fastener

43. Getting past flexure... Support of the stud exists for other DOF as well, and in some cases preliminary characterization has been done

44. Overview Part of the AISI-COFS funded project on the design of sheathed walls with dis-similar sheathing. Presentation developed from a report of the same name Project updates, including full report available at www.ce.jhu.edu/bschafer/sheathedwalls Report covers –Design methods via 1962, 1980-2004, 2007 AISI –Examination of sheathing stiffness ‘k’ –Initial summary of known demands and limit states on a sheathed wall stud in compression All for now, Thank you...

45. Current Modeling

46. Stress distribution at Peak Load (Gray color represents yielding - connections)

47. Lab – General View 1

48. Lab – CFS specimens for scale