Beam columns
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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 Formula. REQUIRED CAPACITY P r P c

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Beam-Columns

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Beam columns

Beam-Columns


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 formula

Interaction Formula

REQUIRED CAPACITY

Pr Pc

Mrx Mcx

Mry Mcy


Axial capacity p c

Axial Capacity Pc


Axial capacity p c1

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


Axial capacity p c2

Axial Capacity Pc

LRFD


Axial capacity p c3

Axial Capacity Pc

ASD


Moment capacities

Moment Capacities


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 cy1

Moment Capacity Mcx or Mcy

REMEMBER TO ACCOUNT FOR LOCAL BUCKLING IF APPROPRIATE


Moment capacity m cx or m cy2

Moment Capacity Mcx or Mcy

LRFD

ASD


Axial demand

Axial Demand


Axial demand p r

Axial Demand Pr

LRFD

ASD

factored

service


Demand

Demand


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 amplification1

additional moment causes additional deflection

Second Order Effects & Moment Amplification

Consider

Mmax = Mo + PD


Design codes

Design Codes

AISC Permits

Second Order Analysis

or

Moment Amplification Method

Compute moments from 1st order analysis

Multiply by amplification factor


Braced vs unbraced frames

Braced vs. Unbraced Frames

Eq. C2-1a


Braced vs unbraced frames1

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 frames

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 frames1

Braced Frames

a = 1 for LRFD

= 1.6 for ASD


Braced frames2

Braced Frames

Cm coefficient accounts for the shape of the moment diagram


Braced frames3

Braced Frames

Cm For Braced & NO TRANSVERSE LOADS

M1: Absolute smallest End Moment

M2: Absolute largest End Moment


Braced frames4

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 frames1

Unbraced Frames


Unbraced frames2

Unbraced Frames


Unbraced frames3

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 frames4

Unbraced Frames

Used when shape is known

e.g. check of adequacy

Used when shape is NOT known

e.g. design of members


Unbraced frames5

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


Homework

Homework

  • 6.2-1

  • 6.2-2

  • 6.5-2

  • 6.5-6

  • 6.6-1


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