evaluation of members, connections, and...

1

Evaluation of Members, Connections, and Subsystems

Tim Ingham

Outline

Member capacity– Axial– Flexural– Ductility

Connection capacity– Riveted– Gusset plates– Net section

Displacement capacity of subsystems

Capacity demand analysis

2

Axial Capacity, per AASHTO

Tension

lagshear for factor reduction section of areanet

section of area grosssteel ofstrength tensile

steel ofstrength yield where

==

==

=

=

=

UA

Af

f

UAfP

AfP

n

g

u

y

gun

gyn

Axial Capacity, per AASHTO

Compression

Ef

rKl

AfP

AfP

y

gyn

gyn

2

where

88.0 then 2.25 If

66.0 then 2.25 If

⎟⎠⎞

⎜⎝⎛=

=>

=≤

πλ

λλ

λ λ

modulus elastic thegyration of radius the

member theoflength unbraced thefactorlength effective the

where

===

=

ErlK

3

Axial Capacity, Slender Cross-Section

Flanges of boxes

Flanges of I-shapes

λsλpλrRatioMember

10141

−yFtb

tb

yF65

yF52

ry FF −238

yF190

yF110

A cross-section is slender if λ>λr for one of its components. The component will buckle before reaching the yield strain of the material.


Follow the procedure described in Appendix B of the Load and Resistance Factor Design Specification for Structural Steel Buildings

4

For Un-Stiffened Elements

leg theof thicknesstheleg theof width the

where

176 if 20000

1760.95 if 00437.0415.1

2

==

≤

⎟⎠⎞

⎜⎝⎛

=

<<−=

tb

tb

ftbf

Q

ftb

ff

tbQ

yy

s

yyys

For Stiffened Elements

element in the stress the widthactual the

widtheffective the)(with

9.641326)()(

==

=

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−==

fbfb

ftbb

tfb

fbfQ

e

ea

5


yfQ

crcr

gyn

gyQ

n

as

ea

ffQf

AfP

AfQP

QQQ

bttfb

fQ

cr λ

λ

λλ

λ

)(66.0)(

solution iterativean requiresequation first the

88.0 then 2.25Q If

66.0 then 2.25Q Ifiscapacity n compressio nominal The

)()(

=

=>

=≤

=

=∑

∑

Example: Slender Cross-Section

Mathcad Document

6

Built-Up Members

Net Section of Built-Up Member

7

Splices in Built-Up Members

Splices in Built-Up Members

Gross section capacity

Net section capacity– Of splice plates– Of member– Could occur through

intermediate plates

Load transfer capacity, through rivets or bolts

If member ductility demands exceed unity, then splices should be 25% stronger than the member

8

Laced Members

Laced Members

Not treated in modern codes

Tensile capacity– Use usual methodology– Don’t count the laces in gross

area

Compressive capacity– Use usual methodology

• Member stability• Section stability, based on

width-to-thickness ratios• With a modification

– Consider also the stability of the member between the points of attachment of the laces

9

Compressive Capacity of Laced Members

( )

gyration of radius themember theoflength unbraced the

factorlength effective thewhere

1.1 ,40For

3001 ,40For 2

===

=′≤

+=′>

rlK

KKrKl

rKLKK

rKl

Proportions of Laced Members

Per the AASHTO Standard Specifications, the slenderness of longitudinal elements must satisfy

}32,40{min

min rKl

ra ≤

10

Ill-Proportioned Laced Member

Must consider interaction of local and global buckling

See Bazant & Cedolin, “Stability of Structures: Elastic, Inelastic, Fracture, and Damage Theories,”Oxford University Press, 1991.

}32,40{min

min rKl

ra >

Stability of Laced Members

11

Actual Member

0

200

400

600

800

1000

1200

1400

1600

0.00 0.10 0.20 0.30

Time, s

Rea

ctio

n, k

ip

Full Lacing

Yield Strength

of Member

With Partial Lacing

0

200

400

600

800

1000

1200

1400

1600

0.00 0.10 0.20 0.30

Time, s

Rea

ctio

n, k

ip

Full Lacing Partial Lacing

Yield Strength

of Member

No convergence beyond this point

12

Shear Strength of Laced Members

For a single plane of single lacing

Check for external shears plus 2% of compressive capacity

f( )

properties sbar' theconsidersimplicitly andbar lacing a of

capacity ecompressiv theis )(where

22

221

lP

TLPTL

TV

n

n ++

=

Shear Strength of Laced Members

For a single plane of double lacing

The 0.7 factor accounts for the stabilizing effect of one lace on the other

( )

properties sbar' theconsidersimplicitly andbar lacing a of

capacity ecompressiv theis )(where

7.02 22

221

lP

TLPTLTV

n

n ++

=

13

Flexural Capacity

( )

modulussection themodulus plastic the

where

whereand element, gcontrollin on the based is where

with section,-crosscompact -non aFor

with section,-crosscompact For

==

=

⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

−−=

≤≤

=

<

SZ

SfM

MMMM

ZfM

yr

pr

prppn

rp

yp

p

λ

λλλλ

λλλ

λλ

Flexural Capacity

The components of the cross-sections of built-up and laced members that are subjected to tensile stress should satisfy net section requirements if the members are to reliably develop the plastic moment. Otherwise, use the elastic capacity.

Rivet hole creates a net section in the angle

14

Compactness Levels

Significant; section will achieve a rotational ductility of about 8 to 10

Ductile≤ λs

Modest; section will achieve a rotational ductility of 4

Compact≤ λp

Little; yields before local buckling, but section won’t reach the plastic moment

Non-compact≤ λr

None; local buckling occurs before yield

Slender> λr

DuctilityTermWidth-to-Thick. Ratio

Damage Levels

No damage– Loads don’t exceed the nominal capacity, per code

Minimal damage– Essentially elastic performance, characterized by

• Minor inelastic response• No apparent permanent deformations• Inconsequential yielding of secondary members

15

Damage Levels

Repairable damage– Can be repaired with a minimum risk of losing functionality, i.e.,

without closing the bridge, characterized by• Yield of members, although replacement should not be

necessary• Small permanent offsets, not interfering with functionality

Significant damage– Minimal risk of collapse, but may require closure to repair,

characterized by• Yield of members, possibly requiring replacement• Permanent offset of the structure

Flexural Ductility – Definitions

θθ θθ

isrotation yield the, and moment, yield the

moment plastic thewhere

from rotation, plastic thecapacityrotation the

where

yy

p

yy

pp

p

h

p

h

M

M

MM

R

θ

θθ

θθ

θθ

=

=

=

==

=

16

Flexural Ductility – Definitions

urecontraflexofpoint themoment to

maximum ofpoint thefrom distance theof 10% say,

length, hinge plastic theinertia ofmoment the

modulus elastic thewhere

===

=

p

py

y

LIE

LEIM

θ

θθ θθ

Flexural Ductility – New Members

111No11.42Minimal12.14Repairable138Significantλ ≤ λrλ ≤ λpλ ≤ λsDamage Level

Controlling Width-To-Thickness Ratio

1.0and valuesbulatedbetween ta

einterpolatlinearly 0.15.0For

5.0 force axial that theAssumes

ygyg

yg

FAPFA

FAP

<<

<

17

Flexural Ductility – Laced Members

Based on only a few tests of the cyclic behavior of laced members

May also be based on finite element analysis

Not to exceed values for new members, based on width-to-thickness ratios

Reduce capacity with axial force by the same rule as for new members

Check shear on laces

Check net section of components

NoMinimalRepairableSignificantDamage Level

11.31.62Ductility

Flexural Ductility – Laced Member Tests

Lee, K., & Bruneau, M., "Seismic Vulnerability Evaluation of Axially Loaded Steel Built-up Laced Members", MCEER-04-0007, Multidisciplinary Center for Earthquake Engineering Research, June, 2004

Tested specimens with various b/t and Kl/r ratios

18


Uang, C.M. & Kleiser, M., “Cyclic Performance of Latticed Members for San Francisco-Oakland Bay Bridge,” Report No. SSRP-97/01, Division of Structural Engineering, University of California at San Diego

Tested members for seismic retrofit of the SFOBB


Dietrich, A. & Itani, A., “Cyclic Behavior of Laced and Perforated Steel Members on the San Francisco-Oakland Bay Bridge,”Report No. CCEER 99-9, University of Nevada, Reno, December, 1999


19

Flexural Ductility – Perforated Members

Based on one test of the cyclic behavior of perforated members

May also be based on inelastic finite element analysis

Not to exceed the values for new members, based on width-to-thickness ratios

Reduce capacity with axial force by the same rule as for new members

Check net section of components


11.42.13Ductility

Flexural Ductility – Perf. Member Tests

Dietrich, A. & Itani, A., “Cyclic Behavior of Laced and Perforated Steel Members on the San Francisco-Oakland Bay Bridge,”Report No. CCEER 99-9, University of Nevada, Reno, December, 1999


20

Axial Ductility – New Members

Values assume a compact section, i.e., λ ≤ λp

Values for significant damage are based on Dowrick, “Earthquake Resistant Design,” 2nd edition, John Wiley & Sons

Values are for X or Z-braces / V or K-braces

1.0 / 1.01.0 / 1.01.0 / 1.0No1.3 / 1.11.5 / 1.21.7 / 1.4Minimal1.8 / 1.12.3 / 1.52.7 / 2.1Repairable2.4 / 1.23.5 / 1.84.5 / 3.0Significant< 120< 80< 40Damage Level

ksi 36Member yf

rKl

Axial Ductility – Laced and Perf. Members

Based on– Lee, K., & Bruneau, M.– Uang, C.M. & Kleiser, M.– Dietrich, A. & Itani, A.

Not to exceed the values for new members based on controlling slenderness ratio of the member

11.31.62Laced


11.42.13Perforated

21

Riveted Connections

Could also use the yield strength of rivets for higher performance and less damage

Check bearing strength per the AASHTO LRFD

rivet theof area nominal theplanesshear ofnumber the

shear singlein rivet theofstrengh ultimate the80.0

where

===

=

⋅=⋅

r

r

rrn

AmF

mAFR

φ

φφ

Bolted and Welded Connections

Bolted Connections– Design per the AASHTO LRFD– If member ductility demands exceed unity, then connections

should be 25% stronger than the members they connect

Welded Connections– Design per the AASHTO LRFD– If member ductility demands exceed unity, then connections

should be 25% stronger than the members they connect– Partial penetration welds

• Don’t use in regions subjected to inelastic deformation• Elsewhere, provide 150% of the required strength, but not less

than 75% of the member strength

22

Gusset Plates

Shall be designed to Fy for combined force and moment

Use Thorton’s methodology to calculate forces and moments

Shall be designed to Fu/√3 for uniform shear and to 0.74 Fu/√3 for flexural shear

Buckling shall be investigated using a truss analogy; with K=0.65

Gusset Plates

23

Gusset Plates

Gusset Plates

Unsupported edges should satisfy

Otherwise, add a stiffener

plategusset theof thicknesstheedge theoflength dunsupporte the

where

6.1

==

≤

tl

fE

tl

y

24

Gusset Plates

Added stiffeners should satisfy

( )

plategusset theof thicknessthelength) edge the(1/2 plategusset of width tributarythe

axis)own its(about stiffener theof inertia ofmoment thewhere

14483.1 ,2.9max 42

===

⎟⎠⎞⎜

⎝⎛ −≥

tbI

ttbI

s

s

Gusset Plates

Should be 25% stronger than the members they connect, if member ductility demands exceed unity

25

Fracture of the Net Section

possible force maximum the and force, seismic elastic the

member theofstrength yield the

),,min(sectionnet theacross ed transferrforcemember offraction the

member theofstrength tensileminimum the

member theof area gross theareanet effective the

where

2.1

max

max

==

=

===

==

≥

FF

f

FFfAD

f

AA

fAD

AA

eq

y

eqyg

u

g

e

ugg

e

α

α

Displacement Capacity of Systems

Portal Frames

Sway Frames

Support Towers

26

Sway Frame

Support Towers

27

Pushover Analysis

Modeling Requirements

Column– Plastic hinging element (with yield moment dependent on axial

force)– Include axial forces acting on columns

Braces– Phenomenological model– Physical model– Finite element model

• Utilizes plastic hinging elements• Requires geometrically nonlinear analysis

28

Phenomenological Model

Ikeda, Mahin, & Dermitzakis

Physical Model

Ikeda & Mahin

29

Finite Element Model

Force-Displacement Response

δ δ

30

Force-Displacement Response

δδ

Member Versus System Ductility

31

Pushover Analysis

A concentrated load at the top of the structure

A uniform load

A uniform acceleration, or mass proportional load

A concentrated displacement at the top of the structure

A displacement pattern derived from the displaced shape corresponding to the peak drift of the

A displacement pattern equal to one or more relevant mode shapes

Capacity/Demand Analysis

Demand Systemor Member Capacity Systemor Member =DC

32

Force Capacity/Demand Analysis

DemandCalculatedStrengthSection Net

Demand CalculatedStrength Spliceand/or Connection

Demand CalculatedStrength eCompressiv

Demand CalculatedStrength Tensile

==

==

==

==

DNDC

DSDC

DCDC

DTDC

N

S

C

T

Displacement Capacity/Demand Analysis

Demandnt Displaceme SubsystemCapacitynt Displaceme Subsystem

DemandDuctility n DeformatioMember CapacityDuctility n DeformatioMember

=

=

DC

DC

33

Threshold Values of C/D Ratio

1.01.01.0Conn. or Splice0.670.81.0Buckling1.01.01.0Fracture0.670.81.0YieldingSec., BracingMain, RedundantMain, Non-Red.Failure Mode

Member Type

Force Capacity/Demand < 1 ?

If only slightly less– A minor amount of yielding is probably acceptable– Capacity calculation is likely to be conservative– These ideas are reflected in the threshold values

If significantly less– Inelastic analysis warranted

• Get ductility demands of the member• Evaluate force redistribution to other members

34

Case 1: C/DT < 1

The tensile capacity is less than the demand– Inelastic response is implied– Fracture of the net section should be avoided– Connections and splices should be 25% stronger than the

member

Retrofit it may not be required if the ratio is above the threshold value, i.e., if C/DT > γT

Case 2: C/DT < γT

The tensile capacity is less than the (threshold) demand

Retrofit should be pursued to increase the tensile strength of the member

Revaluate strengthened member– Net section– Splices and connections

35

Case 3: C/DC < γC and γT < C/DT

The compressive capacity is less than the (threshold) demand

The tensile capacity exceeds the (threshold) demand

Retrofit should be pursued to increase the compressive strength of the member

Revaluate strengthened member– Net section– Splices and connections

Case 4: γC < C/DC and γT < C/DT

The compressive capacity exceeds the (threshold) demand

The tensile capacity exceeds the (threshold) demand

Evaluate connections and splices

36

Case 4a: C/DS < γS

The connection or splice capacity is less than the (threshold) demand

Strengthen connection

Case 4b: γS < C/DS

The connection or splice capacity exceeds the (threshold) demand

No retrofit is required, if the tensile capacity exceeds the demand, i.e., 1 < C/DT

Otherwise (C/DT < 1), the member response is inelastic– Case 1 applies– Connections and splices should be 25% stronger than the

member

37

Interaction of Axial Force and Flexure

AASHTO LRFD interaction equations

2.0 if 0.198

2.0 if 0.10.2

≥≤⎟⎟⎠

⎞⎜⎜⎝

⎛++

<≤⎟⎟⎠

⎞⎜⎜⎝

⎛++

r

u

ry

uy

rx

ux

r

u

r

u

ry

uy

rx

ux

r

u

PP

MM

MM

PP

PP

MM

MM

PP

AASHTO Inverted

5 if 0.1

89

89

5 if 0.12

2

≤≥

⎟⎟⎠

⎞⎜⎜⎝

⎛

++⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

+⎟⎟⎠

⎞⎜⎜⎝

⎛

>≥

⎟⎟⎠

⎞⎜⎜⎝

⎛

++⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

+⎟⎟⎠

⎞⎜⎜⎝

⎛

r

r

rxuyryux

ryrx

u

r

rxuyryux

ryrx

u

r

u

r

rxuyryux

ryrx

u

r

rxuyryux

ryrx

u

r

PP

MMMMMM

PP

MMMMMM

PP

PP

MMMMMM

PP

MMMMMM

PP

38

Capacity Demand Ratio

5 if

89

89

5 if 2

2

≤

⎟⎟⎠

⎞⎜⎜⎝

⎛

++⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

+⎟⎟⎠

⎞⎜⎜⎝

⎛

=

>

⎟⎟⎠

⎞⎜⎜⎝

⎛

++⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

+⎟⎟⎠

⎞⎜⎜⎝

⎛

=

r

r

rxuyryux

ryrx

u

r

rxuyryux

ryrx

u

r

u

r

rxuyryux

ryrx

u

r

rxuyryux

ryrx

u

r

PP

MMMMMM

PP

MMMMMM

PP

DC

PP

MMMMMM

PP

MMMMMM

PP

DC

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