b'YOUR TOOLKIT FOR BUILDING EXCELLENCETheaxialstiffnessofamember,aratioofforceEach joint member in this analysis is 6 inches long and is toaxialdeformation,whichisequivalenttorigidly attached to the member from which it extends and aspringconstant,isdefinedbyEquation2as simply attached or supported to the connecting member.The P/ = EA / L[2] weight of the framing is ignored.The material and geometry of the joint member is set to match the flexural stiffness, Where, EI, of the main member from which it extends.The slip modulus is calculated using Wood Handbook FPL (2010) P/ = axial stiffnessEquation 8-4 assuming five 0.148x3 common wire nailsE = elastic modulusat each joint.The depth and thickness of the joint member is calculated using Equations 7 and 8.Table 1 summarizesA = cross-sectional area of a member the geometry and structural properties of the frame.L = length of a member A j= (P/) jL j/ E[6]Even though a triangle is not an excellent representation ofh j= (12 I/A j ) 1/2[7]a circle (the ring), they both have a closed-loop geometryb j= A j /h j [8]with a single load path, and the relationship of the joints and the overall concept is applicable.This can be validatedwhere,by comparing two knee-brace models, one with three slipA j = cross section area of the joint member joints and the other with one equivalent slip joint.The frame on the left (Figure 2) is modeled with a slip at allh j= depth of the joint memberthree joints of the knee-brace triangle.The frame on the right has one slip joint at the bottom of the knee braceb j= thickness of the joint memberwith axial stiffness that is equivalent to the axial stiffnessI = moment of inertia of the joint member (equal of the three joints determined by merging Equations 1 andto moment of inertia of the main member)2 into Equation 3.Because a segment of the main member (post, knee brace, rafter) is removed in the process, theL j= length of the joint membernew joint member is defined to represent both the slippage of the fasteners and the axial stiffness of the segment ofSince knee braces are only loaded axially, and assuming pin-the original member that the joint member is replacingpin connection at each end, and ignoring the weight of the (Equation 4). The equivalent axial stiffness of the jointmembers, the b jand h jof the joint member can be calculated member in the one-joint model is defined by Equation 5.using Equation 9.This simplification is not applied to the (P/) j,eq= [ 1/(P/) j,1+ 1/(P/) j,2 + 1/(P/) j,3]-1[3] joint members at the top of the post and at the bottom of the rafter as these joint members require member-specific (P/) j= (EA/L) j= [ 1/(EA/L) m+ 1/(P/) slip] -1 [4] flexural stiffness to resist internal shear and bending forces. (P/) j,eq= (EA/L) j, eq= [ 1/(EA/L) j,1+ 1/(EA/L) j,2 + 1/(EA/L) j,3]-1 [5] h j , knee brace= b j , knee brace= A j ,knee brace [9]where,(P/) j= axial stiffness of the joint(EA/L) j = axial stiffness of the joint expressed using terms E, A, and L(EA/L) m= axial stiffness of the original frame member (segment) that is being replaced by the joint member(P/) slip= slippage modulus of the fasteners (group) used in the connection at the joint(P/) j,eq= axial stiffness of the equivalent joint for one- Figure 2: joint model Knee brace with slip at all joints (left) and with an equivalent slip at (EA/L) j,eq = axial stiffness of the equivalent joint for one- one joint (right)joint model expressed using terms E, A, and Lcontinued on page: 18 FRAME BUILDER - MAY2023 / 17'