1Tension Theory Developed by Scott Civjan University of Massachusetts, Amherst.

1Tension Theory

Developed by Scott CivjanUniversity of Massachusetts, Amherst

Tension Members: Chapter D: Tension Member Strength Chapter B: Gross and Net Areas Chapter J: Block Shear Part 5: Design Charts and Tables

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Gross and Net Areas: Criteria in Table B3.13 Strength criteria in Chapter D: Design of

Members for Tension

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Yield on Gross Areat=0.90 (c=1.67)

Fracture on Effective Net Areat=0.75 (c=2.00)

Block Sheart=0.75 (c=2.00)

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Yielding on Gross Area Ag

5Tension Theory

Pn=FyAg Equation D2-1

Yield on Gross Area

t=0.90 (c=1.67)

Ag= Gross AreaTotal cross-sectional area in the plane perpendicular to tensile stresses

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Fracture on Effective Net Area Ae

7Tension Theory

Ae= Effective Net AreaAccounts for any holes or openings, potential failure planes not perpendicular to the tensile stresses, and effects of shear lag

Pn=FuAe Equation D2-2

t=0.75 (c=2.00)

Fracture on Effective Net Area

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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

9Tension Theory


Holes or openings

Section D3.2Account for 1/16” greater than bolt hole size shown in Table J3.3 Accounts for potential damage in fabrication

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An= Net AreaModify gross area (Ag) to account for the following:


Holes or openings

Potential failure planes not perpendicular to the tensile stresses

11

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Design typically uses average stress values


This assumption relies on the inherent ductility of steel

Pu

Initial stresses will typically include stress concentrations due to higher strains at these locations

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

12Tension Theory

Pu

Similarly, bolts and surrounding material will yield prior to fracture due to the inherent ductility of steel

Therefore assume each bolt transfers equal force


13Tension Theory

PuPu

Pu2/3Pu

Pu/6

Pu/6Pu1/3Pu

Pu/6

Pu/6

Pu/6

Pu/6Pu

0

Pu/6

Pu/6

Pu/6

Pu/6

Pu/6

Pu/6

The plate will fail in the line with the highest force (for similar number of bolts in each line)


Pu

Cross Section

Net area reduced by hole area

Each bolt line shown transfers 1/3 of the total force

123Bolt line14Tension Theory


PuPu

0 1/3Pu

2/3Pu

Cross Section

Net area reduced by hole area

Force in plate

Bolt line 1 resists Pu in the plateBolt line 2 resists 2/3Pu in the plateBolt line 3 resists 1/3Pu in the plate

123Bolt line15Tension Theory

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

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

16Tension Theory

Yield failure (elongation) occurs along the length of the member

EXAMPLE

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What if holes are not in a line perpendicular to the load?

Additional strength depends on:Geometric length increaseCombination of tension and shear stresses

Combined effect makes a direct calculation difficult

Need to include additional length/Area of failure plane due to non-perpendicular path


18Tension Theory

g

s

Pu


Diagonal hole patternAdditional length of failure plane equal to s2/4gSection 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

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An=Net Area

An=Ag-#(dn)t+(s2/(4g))t

#= number of holes intersected by failure planedn= corrected hole diameter per B.3-13t= thickness of tension memberOther terms defined on previous slides

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When considering angles

When considering angles: Find gage (g) on page 1-46

“Workable Gages in Standard Angles” unless otherwise noted

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EXAMPLE

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Accounts for distance required for stresses to distribute from connectors into the full cross section


Only a portion of the cross section is connected

Connection does not have sufficient length

Largest influence when

Shear Lag

23Tension Theory

Shear Lag affects members where:Only a portion of the cross section is connectedConnection does not have sufficient length


24Tension Theory


Distribution of Forces Through Section

Pu

l= Length of Connection

Section Carrying Tension Forces

Fracture Plane

25Tension Theory


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

26Tension Theory

Ae= Effective Net AreaModify net area (An) to account for shear lag


Or value per Table D3.1

l

xU 1

Ae= AnU Equation D3-1

U= Shear Lag Factor Reduction

= Connection eccentricity

l= length where force transfer occurs (distance parallel to applied tension force along bolts or weld)

x

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Ae= Effective Net AreaAccounts for any holes or openings, potential failure planes not perpendicular to the tensile stresses, and effects of shear lag

Pn=FuAe Equation D2-2

t=0.75 (c=2.00)


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Ae=Effective Net Area An=Net AreaAe≠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

29Tension Theory


Pu

At the fracture plane (right bolts) forces have not engaged the entire plate.

Fracture Plane

Portion of member carrying no tension

Effective length of fracture plane

30Tension Theory

Now consider a much wider plate

This concept describes the Whitmore Section


Pu lw= width of Whitmore Section

30o

30o

31Tension Theory

EXAMPLE

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Block Shear

33Tension Theory

State of Combined Yielding and Fracture

Block Shear

Failure Planes

Failure Tears Out Block of Steel

Block Defined by Center Line of Holes Edge of Welds

At Least One Each in Tension and Shear

34Tension Theory

Block Shear

Typical Examples in Tension Members

Angle Connected on One Leg

W-Shape Flange Connection

Plate Connection

35Tension Theory

Block Shear

Pu

Angle Bolted to Plate

Pu

Tension plane on Angle

Shear plane on PlateTension plane on Plate (Shorter Dimension Controls)

Shear plane on Angle

36Tension Theory

Block Failure from Angle

Block Shear

Pu

Angle Bolted to Plate

Block Failure From Plate

Pu

37Tension Theory

Block Shear

Pu

Tension planes on W-Shape

Shear planes on W-Shape

First look at the W-Shape, then the plate

38Tension Theory

Flange of W-Shape Bolted to Plate

Block ShearFlange of W-Shape Bolted to Plate

Pu

Block Failure in W-Shape

First look at the W-Shape, then the plate

39Tension Theory

Block ShearFlange of W-Shape Bolted to Plate

Pu

Pu

Shear planes on PlateTension plane on Plate

Shear planes on PlateTension planes on Plate

40Tension Theory

Block Shear

Pu

Block Failure in Plate

Pu

Block Failure in Plate

41Tension Theory

Flange of W-Shape Bolted to Plate

Block Shear

Pu

Angle or Plate Welded to Plate

Weld around the perimeter

Two Block Shear Failures to Check

42Tension Theory

Block Shear

Pu

Pu

Tension plane on Plate (Shorter Dimension Controls)

Shear planes on Plate

Tension plane on Plate

Shear plane on Plate

43Tension Theory


Block Shear

Pu

Pu

Block Failure From Plate

44Tension Theory


Block Shear

Block Shear Rupture Strength (Equation J4-5)

ntubsgvyntubsnvun AFUAF.AFUAF.R 6060

t=0.75 (c=2.00)

Agv= Gross area subject to shearAnv= Net area subject to shearAnt= Net area subject to tensionUbs= 1 or 0.5 (1 for most tension members, see Figure C-J4.2)

Smaller of two values will control

45

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EXAMPLE

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Bearing at Bolt Holes

47Tension Theory

Bolts bear into material around hole

Direct bearing can deform the bolt hole an excessive amount and be limited by direct bearing capacity

48Tension Theory

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



Pu

Bolt

49Tension Theory

Bolt induces bearing stresses on the base material


Pu

Bolt

50Tension Theory

Which can result in excessive deformation of the bolt hole


Pu

Bolt

51Tension Theory

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

Lc


Pu

52Tension Theory

A block of material can tear out to the plate edge due to bearing


Pu

Bolt

53Tension Theory

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

Lc


Pu

54Tension Theory

A block of material can tear out between the bolt holes due to bearing stresses


For standard, oversized and short-slotted hole or long slotted hole with slot parallel to the direction of loading:

(Equation J3-6a)uucn dtFtFLR 4.22.1

t=0.75 (c=2.00)

Lc= Clear distance in the direction of forcet= thickness of connected materiald= nominal bolt diameterFu= Specified minimum tensile strength of the connected material

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Bearing LimitTearout Limit


For the similar case, but when deformation of the bolt hole is not a design consideration:

(Equation J3-6b)uucn dtFtFLR 0.35.1

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For long-slotted hole with slot perpendicular to the direction of force:

(Equation J3-6c)uucn dtFtFLR 0.20.1

Other situations have similar design equations:

1Tension Theory Developed by Scott Civjan University of Massachusetts, Amherst.

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