rigid frames: compression & buckling twelve

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Rigid Frames 1 Architectural Structures

lecture

twelve

rigid frames:

compression & buckling

A

RCHITECTURAL

S

TRUCTURES

:

F

ORM,

B

EHAVIOR, AND

D

ESIGN

A

RCH 331

HÜDAVERDİ TOZAN

S

PRING 2013

(2)

Rigid Frames

• rigid frames have no

pins

• frame is all one body

• joints transfer

moments and shear

• typically statically

indeterminate

• types

– portal

– gable

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Rigid Frames

(4)

Rigid Frames

– moments get redistributed

– deflections are smaller

– effective column lengths are shorter

– very sensitive to settling

(5)

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Moment Redistribution

• continuous slabs & beams with uniform

loading

– joints similar to fixed ends, but can rotate

• change in moment to center =

– M

_max

for simply supported beam

8

2

(6)

Rigid Frames

• resists lateral

loadings

• shape depends on

stiffness of beams

and columns

• 90° maintained

(7)

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Rigid Frames

• staggered truss

– rigidity

(8)

Rigid Frames

• connections

– steel

– concrete

(9)

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Braced Frames

• pin connections

• bracing to prevent lateral movements

(10)

Braced Frames

• types of bracing

– knee-bracing

– diagonal

– X

– K or chevron

– shear walls

diagonal

X

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Shear Walls

(12)

Compression Members

• designed for strength & stresses

• designed for serviceability & deflection

• need to design for stability

– ability to support a

specified load

without sudden or

unacceptable

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Column Buckling

• axially loaded columns

• long & slender

– unstable equilibrium =

buckling

(14)

Modeling

• can be modeled with a spring at mid-height

• • when moment

from deflection

exceeds the

spring capacity

... “boing”

• critical load P

(15)

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Effect of Length

(16)

Buckling Load

• related to deflected shape (P



)

• shape of sine wave

• Euler’s Formula

• smallest I governs

 

2

2 L

EI

P

_critical





(17)

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Critical Stress

• short columns

• slenderness ratio = L

_e

/r (L/d)

• radius of gyration =

A

I

r



a

actual

critical

F

A

P

f





weak axis

 

2

2 













r

L

E

L

A

EAr

A

P

f

e

critical



2 e

2 critical

r

L

EA

P

















(18)

Critical Stresses

• when a column gets stubby, F

_y

will limit the

load

• real world has loads

with eccentricity

• C

_c

for steel and

allowable stress

y

c

e

F

E

C

r

L

_

_

2 

2

(19)

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Effective Length

• end conditions affect shape

(20)

Bracing

• bracing affects shape of buckle

in one direction

(21)

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Centric & Eccentric Loading

• centric

– allowable stress from strength or buckling

• eccentric

(22)

Combined Stresses

– axial + bending

– design

I

Mc

A

P

f

_max





.

max

S

F

f

F

f



_cr



cr

e

P

M





(23)

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Stress Limit Conditions

– ASD interaction formula

– with biaxial bending

1

a a

F

f

b b

F

f

0 .

1 



b

a

F

f

F

f

0 .

1 



by

bx

a

F

f

F

f

F

f

interaction diagram

(24)

Stress Limit Conditions

– in reality, as the column flexes,

the moment increases

– P-



effect

0 .

1 )

(







bx

b

a

F

factor

ion

Magnificat

f

F

f

P

(25)

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Rigid Frame Analysis

• members see

– shear

– axial force

– bending

• V & M diagrams

– plot on “outside”

(26)

Rigid Frame Analysis

– need support reactions

– free body diagram each member

– end reactions are equal and opposite on

next member

– “turn” member

like beam

(27)

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Rigid Frame Analysis

– FBD & M

• opposite end

reactions at joints

M+

(28)

Rigid Frame Design

• loads and combinations

– usually uniformly distributed gravity loads

– worst case for largest moments...

(29)

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Rigid Frame Design

• frames & floors

– rigid frame can have slab floors or slab

with connecting beams

• other

– slabs or plates

(30)

Rigid Frame Design

• floors – plates & slabs

– one-way behavior

• side ratio > 1.5

• “strip” beam

– two-way behavior

(31)

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Rigid Frame Design

• columns in frames

– ends can be “flexible”

– stiffness affected by beams

and column = EI/L

– for the joint

• l

_c

is the column length of each column

• l

_b

is the beam length of each beam

• measured center to center

b

c

l

EI

l

EI

G







(32)

Rigid Frame Design

• column effective length, k

A

(33)

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Tools – Multiframe

(34)

Tools – Multiframe

• frame window

– define frame members

• or pre-defined frame

– select points, assign supports

– select members,

assign section

– load window

– select point or member,

add point or distributed

loads

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• to run analysis choose

– Analyze menu

• Linear

• plot

– choose options

• results

– choose

options