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**Teaching Modules for Steel Instruction**

Tension Member THEORY SLIDES Developed by Scott Civjan University of Massachusetts, Amherst Tension Theory

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**Tension Members: Chapter D: Tension Member Strength**

Chapter B: Gross and Net Areas Chapter J: Block Shear Part 5: Design Charts and Tables Tension Spec 13th Ed

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**Gross and Net Areas: Criteria in Table B3.13**

Strength criteria in Chapter D: Design of Members for Tension Tension Spec 13th Ed

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**Fracture on Effective Net Area ft=0.75 (Wc=2.00)**

Yield on Gross Area ft=0.90 (Wc=1.67) Fracture on Effective Net Area ft=0.75 (Wc=2.00) Block Shear ft=0.75 (Wc=2.00) Tension Spec 13th Ed

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**Yielding on Gross Area Ag**

Tension Theory

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**Yield on Gross Area Pn=FyAg Equation D2-1 ft=0.90 (Wc=1.67)**

Ag= Gross Area Total cross-sectional area in the plane perpendicular to tensile stresses Tension Spec 13th Ed

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**Fracture on Effective Net Area Ae**

Tension Theory

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**Fracture on Effective Net Area**

Pn=FuAe Equation D2-2 ft=0.75 (Wc=2.00) Ae= Effective Net Area Accounts for any holes or openings, potential failure planes not perpendicular to the tensile stresses, and effects of shear lag Tension Spec 13th Ed

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**Fracture on Effective Net Area**

If holes are included in the cross section less area resists the tension force Bolt holes are larger than the bolt diameter In addition processes of punching or drilling holes can damage the steel around the perimeter Tension Theory

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**Fracture on Effective Net Area**

Holes or openings Section D3.2 Account for 1/16” greater than bolt hole size shown in Table J3.3 Accounts for potential damage in fabrication Tension Spec 13th Ed

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**Fracture on Effective Net Area**

An= Net Area Modify gross area (Ag) to account for the following: Holes or openings Potential failure planes not perpendicular to the tensile stresses Tension Spec 13th Ed

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**Fracture on Effective Net Area**

Design typically uses average stress values This assumption relies on the inherent ductility of steel Initial stresses will typically include stress concentrations due to higher strains at these locations Pu Highest strain locations yield, then elongate along plastic plateau while adjacent stresses increase with additional strain Eventually at very high strains the ductility of steel results in full yielding of the cross section Therefore average stresses are typically used in design Tension Theory

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**Fracture on Effective Net Area**

Similarly, bolts and surrounding material will yield prior to fracture due to the inherent ductility of steel Therefore assume each bolt transfers equal force Pu Tension Theory

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**Fracture on Effective Net Area**

The plate will fail in the line with the highest force (for similar number of bolts in each line) Each bolt line shown transfers 1/3 of the total force Pu 1/3Pu Pu/6 Pu Pu/6 Pu 2/3Pu Pu/6 Pu Net area reduced by hole area Pu Cross Section Bolt line 3 2 1 Tension Theory

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**Fracture on Effective Net Area**

The plate will fail in the line with the highest force (for similar number of bolts in each line) Each bolt line shown transfers 1/3 of the total force Bolt line 1 resists Pu in the plate Bolt line 2 resists 2/3Pu in the plate Bolt line 3 resists 1/3Pu in the plate Force in plate Net area reduced by hole area Pu 1/3 Pu 2/3 Pu Pu Cross Section Bolt line 3 2 1 Tension Theory

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**Fracture on Effective Net Area**

For a plate with a typical bolt pattern the fracture plane is shown Yield on Ag would occur along the length of the member Both failure modes depend on cross-sectional areas Pu Fracture failure across section at lead bolts Yield failure (elongation) occurs along the length of the member Tension Theory

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EXAMPLE Tension Spec 13th Ed

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**Fracture on Effective Net Area**

What if holes are not in a line perpendicular to the load? Need to include additional length/Area of failure plane due to non-perpendicular path g s Pu Additional strength depends on: Geometric length increase Combination of tension and shear stresses Combined effect makes a direct calculation difficult Tension Theory

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**Fracture on Effective Net Area**

Diagonal hole pattern Additional length of failure plane equal to s2/4g Section B3.13 and D3.2 s= longitudinal center-to-center spacing of holes (pitch) g= transverse center-to-center spacing between fastener lines (gage) g s Pu Tension Spec 13th Ed

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**Fracture on Effective Net Area**

An=Net Area An=Ag-#(dn)t+(s2/(4g))t #= number of holes intersected by failure plane dn= corrected hole diameter per B.3-13 t= thickness of tension member Other terms defined on previous slides Tension Spec 13th Ed

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**Fracture on Effective Net Area**

When considering angles When considering angles: Find gage (g) on page 1-46 “Workable Gages in Standard Angles” unless otherwise noted Tension Spec 13th Ed

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EXAMPLE Tension Spec 13th Ed

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**Shear Lag Fracture on Effective Net Area**

Accounts for distance required for stresses to distribute from connectors into the full cross section Largest influence when Only a portion of the cross section is connected Connection does not have sufficient length Tension Theory

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**Fracture on Effective Net Area**

Shear Lag affects members where: Only a portion of the cross section is connected Connection does not have sufficient length Tension Theory

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**Fracture on Effective Net Area**

Section Carrying Tension Forces Pu Distribution of Forces Through Section Fracture Plane l= Length of Connection Tension Theory

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**Fracture on Effective Net Area**

Effective Net Area in Tension Area not Effective in Tension Due to Shear Lag Pu Shear lag less influential when l is long, or if outstanding leg has minimal area or eccentricity Tension Theory

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**Fracture on Effective Net Area**

Ae= Effective Net Area Modify net area (An) to account for shear lag Ae= AnU Equation D3-1 U= Shear Lag Factor Reduction Or value per Table D3.1 = Connection eccentricity l= length where force transfer occurs (distance parallel to applied tension force along bolts or weld) Tension Spec 13th Ed

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**Fracture on Effective Net Area**

Pn=FuAe Equation D2-2 ft=0.75 (Wc=2.00) Ae= Effective Net Area Accounts for any holes or openings, potential failure planes not perpendicular to the tensile stresses, and effects of shear lag Tension Spec 13th Ed

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**Fracture on Effective Net Area**

Ae=Effective Net Area An=Net Area Ae≠An Due to Shear Lag Pu As the force is transferred from each bolt it spreads through the tension member. This is sometimes called the “flow of forces” Note that the forces from the left 4 bolts act on the full cross section at the failure plane (bolt line nearest load application) Boundary of force transfer into the plate from each bolt Tension Theory

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**Fracture on Effective Net Area**

Pu At the fracture plane (right bolts) forces have not engaged the entire plate. Now consider a much wider plate Portion of member carrying no tension Fracture Plane Effective length of fracture plane Describing this in terms of an extremely wide plate in sheet metal, or tin foil, will allow students to intuitively grasp the concept. Tension Theory

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**Fracture on Effective Net Area**

Pu This concept describes the Whitmore Section lw= width of Whitmore Section 30o 30o Whitmore Section applies to bolted or welded sections. Tension Theory

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EXAMPLE Tension Spec 13th Ed

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Block Shear Tension Theory

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**Failure Tears Out Block of Steel**

Block Shear Failure Tears Out Block of Steel Block Defined by Center Line of Holes Edge of Welds State of Combined Yielding and Fracture Failure Planes At Least One Each in Tension and Shear Tension Theory

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**Typical Examples in Tension Members**

Block Shear Typical Examples in Tension Members Angle Connected on One Leg W-Shape Flange Connection Plate Connection Tension Theory

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**Block Shear Angle Bolted to Plate Shear plane on Angle Pu**

Tension plane on Angle Shear plane on Plate Tension plane on Plate (Shorter Dimension Controls) Shear plane on Angle Pu Pu Note that angle tension plane goes towards the free edge rather than up into the other leg, as this would be less resistance. Similarly the Plate would fail vertically towards the closest edge. Tension Theory

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**Block Shear Angle Bolted to Plate Pu Block Failure from Angle Pu**

Block shear could occur on either the angle or the plate – lowest failure mode strength of all controls Block Failure From Plate Tension Theory

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**Flange of W-Shape Bolted to Plate**

Block Shear Flange of W-Shape Bolted to Plate Tension planes on W-Shape Shear planes on W-Shape Pu First look at the W-Shape, then the plate Tension Theory

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**Flange of W-Shape Bolted to Plate**

Block Shear Flange of W-Shape Bolted to Plate Pu Block Failure in W-Shape First look at the W-Shape, then the plate Tension Theory

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**Flange of W-Shape Bolted to Plate**

Block Shear Flange of W-Shape Bolted to Plate Shear planes on Plate Tension plane on Plate Tension planes on Plate Pu Pu Here 2 possible plate block shear modes could occur, the lower strength would control. Tension Theory

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**Flange of W-Shape Bolted to Plate**

Block Shear Flange of W-Shape Bolted to Plate Pu Block Failure in Plate Pu Block Failure in Plate Tension Theory

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**Angle or Plate Welded to Plate**

Block Shear Angle or Plate Welded to Plate Pu Weld around the perimeter Two possible block shear failures can be described by the perimeter of the welds – that with the lower strength would control. Two Block Shear Failures to Check Tension Theory

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**Angle or Plate Welded to Plate**

Block Shear Angle or Plate Welded to Plate Tension plane on Plate (Shorter Dimension Controls) Shear planes on Plate Shear plane on Plate Pu Pu Two possible block shear failures can be described by the perimeter of the welds – that with the lower strength would control. Tension Theory

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**Angle or Plate Welded to Plate**

Block Shear Angle or Plate Welded to Plate Pu Block Failure From Plate Pu Two possible block shear failures can be described by the perimeter of the welds – that with the lower strength would control. Tension Theory

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Block Shear Block Shear Rupture Strength (Equation J4-5) Smaller of two values will control ft=0.75 (Wc=2.00) Agv= Gross area subject to shear Anv= Net area subject to shear Ant= Net area subject to tension Ubs= 1 or 0.5 (1 for most tension members, see Figure C-J4.2) Tension Spec 13th Ed

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EXAMPLE Tension Spec 13th Ed

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Bearing at Bolt Holes Tension Theory

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**Bolts bear into material around hole**

Bearing at Bolt Holes Bolts bear into material around hole Direct bearing can deform the bolt hole an excessive amount and be limited by direct bearing capacity If the clear space to adjacent hole or edge distance is small, capacity may be limited by tearing out a section of base material at the bolt Tension Theory

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**Bearing at Bolt Holes Bolt Pu**

Bolt induces bearing stresses on the base material Tension Theory

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**Bearing at Bolt Holes Bolt Pu**

Which can result in excessive deformation of the bolt hole Tension Theory

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**Bearing at Bolt Holes Lc Bolt Pu**

When bearing stresses act on bolts that are near the edge of the material (Lc dimension is small) Tension Theory

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**Bearing at Bolt Holes Pu**

A block of material can tear out to the plate edge due to bearing Note, this is differentiated from block shear since there is no tension plane for the failure. Tension Theory

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**Bearing at Bolt Holes Lc Bolt Pu**

Similarly, when bearing stresses act on bolts that are closely spaced (Lc dimension is small) Tension Theory

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**Bearing at Bolt Holes Pu**

A block of material can tear out between the bolt holes due to bearing stresses Note, this is differentiated from block shear since there is no tension plane for the failure. Tension Theory

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**Bearing at Bolt Holes ft=0.75 (Wc=2.00)**

For standard, oversized and short-slotted hole or long slotted hole with slot parallel to the direction of loading: (Equation J3-6a) Bearing Limit Tearout Limit ft=0.75 (Wc=2.00) Lc= Clear distance in the direction of force t= thickness of connected material d= nominal bolt diameter Fu= Specified minimum tensile strength of the connected material Tension Spec 13th Ed

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**Bearing at Bolt Holes Other situations have similar design equations:**

For the similar case, but when deformation of the bolt hole is not a design consideration: (Equation J3-6b) For long-slotted hole with slot perpendicular to the direction of force: (Equation J3-6c) Tension Spec 13th Ed

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