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Beam-Columns. A. B. P 1. C. D. P 2. E. F. Members Under Combined Forces. Most beams and columns are subjected to some degree of both bending and axial load. e.g. Statically Indeterminate Structures. Interaction Formulas for Combined Forces. e.g. LRFD

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members under combined forces

A

B

P1

C

D

P2

E

F

Members Under Combined Forces

Most beams and columns are subjected to some degree of both bending and axial load

e.g. Statically Indeterminate Structures

interaction formulas for combined forces
Interaction Formulas for Combined Forces

e.g. LRFD

If more than one resistance is involved consider interaction

basis for interaction formulas
Basis for Interaction Formulas

Tension/Compression & Single Axis Bending

Tension/Compression & Biaxial Bending

Quite conservative when compared to actual ultimate strengths

especially for wide flange shapes with bending about minor axis

aisc interaction formula chapter h
AISC Interaction Formula – CHAPTER H

AISC Curve

r = required strength

c = available strength

slide6
REQUIRED CAPACITY

Pr Pc

Mrx Mcx

Mry Mcy

axial capacity p c9
Axial Capacity Pc

Elastic Buckling Stress corresponding to the controlling mode of failure (flexural, torsional or flexural torsional)

Fe:

Theory of Elastic Stability (Timoshenko & Gere 1961)

Flexural Buckling

Torsional Buckling

2-axis of symmetry

Flexural Torsional Buckling

1 axis of symmetry

Flexural Torsional Buckling

No axis of symmetry

AISC Eqtn

E4-4

AISC Eqtn

E4-5

AISC Eqtn

E4-6

effective length factor
Effective Length Factor

Free to rotate and translate

Fixed on top

Free to rotate

Fixed on bottom

Fixed on bottom

Fixed on bottom

effective length of columns

Ic Lc

Ig Lg

Ig Lg

A

Ic Lc

B

Effective Length of Columns
  • Assumptions
  • All columns under consideration reach buckling Simultaneously
  • All joints are rigid
  • Consider members lying in the plane of buckling
  • All members have constant A

Define:

effective length of columns12
Effective Length of Columns

Use alignment charts (Structural Stability Research Council SSRC)

LRFD Commentary Figure C-C2.2 p 16.1-241,242

  • Connections to foundations
  • (a) Hinge
    • G is infinite - Use G=10
  • (b) Fixed
    • G=0 - Use G=1.0
moment capacity m cx or m cy
Moment Capacity Mcx or Mcy

REMEMBER TO CHECK FOR NON-COMPACT SHAPES

moment capacity m cx or m cy17
Moment Capacity Mcx or Mcy

REMEMBER TO ACCOUNT FOR LOCAL BUCKLING IF APPROPRIATE

axial demand p r
Axial Demand Pr

LRFD

ASD

factored

service

second order effects moment amplification

P

y

M

Second Order Effects & Moment Amplification

P

W

ymax @ x=L/2 = d

Mmax @ x=L/2 = Mo + Pd = wL2/8 + Pd

additional moment causes additional deflection

second order effects moment amplification24
Second Order Effects & Moment Amplification
  • Total Deflection cannot be Found Directly
  • Additional Moment Because of Deformed Shape
  • First Order Analysis
    • Undeformed Shape - No secondary moments
  • Second Order Analysis (P-d and P-D)
    • Calculates Total deflections and secondary moments
    • Iterative numerical techniques
    • Not practical for manual calculations
    • Implemented with computer programs
design codes
Design Codes

AISC Permits

Second Order Analysis

or

Moment Amplification Method

Compute moments from 1st order analysis

Multiply by amplification factor

derivation of moment amplification27
Derivation of Moment Amplification

Moment Curvature

P

M

2nd order nonhomogeneous DE

derivation of moment amplification28
Derivation of Moment Amplification

Boundary Conditions

Solution

derivation of moment amplification31
Derivation of Moment Amplification

Moment

Mo(x)

Amplification Factor

braced vs unbraced frames33
Braced vs. Unbraced Frames

Eq. C2-1a

Mnt = Maximum 1st order moment assuming no sidesway occurs

Mlt = Maximum 1st order moment caused by sidesway

B1 = Amplification factor for moments in member with no sidesway

B2 = Amplification factor for moments in member resulting from sidesway

braced frames36
Braced Frames

Pr = required axial compressive strength

= Pu for LRFD

= Pa for ASD

Pr has a contribution from the PD effect and is given by

braced frames37
Braced Frames

a = 1 for LRFD

= 1.6 for ASD

braced frames38
Braced Frames

Cm coefficient accounts for the shape of the moment diagram

braced frames39
Braced Frames

Cm For Braced & NO TRANSVERSE LOADS

M1: Absolute smallest End Moment

M2: Absolute largest End Moment

braced frames40
Braced Frames

Cm For Braced & NO TRANSVERSE LOADS

COSERVATIVELY Cm= 1

unbraced frames
Unbraced Frames

Eq. C2-1a

Mnt = Maximum 1st order moment assuming no sidesway occurs

Mlt = Maximum 1st order moment caused by sidesway

B1 = Amplification factor for moments in member with no sidesway

B2 = Amplification factor for moments in member resulting from sidesway

unbraced frames44
Unbraced Frames

a = 1.00 for LRFD

= 1.60 for ASD

= sum of required load capacities for all columns in the story under consideration

= sum of the Euler loads for all columns in the story under consideration

unbraced frames45
Unbraced Frames

Used when shape is known

e.g. check of adequacy

Used when shape is NOT known

e.g. design of members

unbraced frames46
Unbraced Frames

I = Moment of inertia about axis of bending

K2 = Unbraced length factor corresponding to the unbraced condition

L = Story Height

Rm = 0.85 for unbraced frames

DH = drift of story under consideration

SH = sum of all horizontal forces causing DH