Monther Dwaikat

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68402 Structural Design of Buildings II61420
Design of Steel Structures62323 Architectural
Structures II
Design of Beam-Columns

• Monther Dwaikat
• Assistant Professor
• Department of Building Engineering
• An-Najah National University

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Beam-Column - Outline

• Beam-Columns
• Moment Amplification Analysis
• Second Order Analysis
• Compact Sections for Beam-Columns
• Braced and Unbraced Frames
• Analysis/Design of Braced Frames
• Analysis/Design of Unbraced Frames
• Design of Bracing Elements

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Design for Flexure LRFD Spec.

• Commonly Used Sections
• I shaped members (singly- and doubly-symmetric)
• Square and Rectangular or round HSS

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Beam-Columns
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Beam-Columns
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Beam-Columns
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Beam-Columns
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Beam-Columns
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Beam-Columns
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Beam-Columns

• Likely failure modes due to combined bending and
  axial forces
• Bending and Tension usually fail by yielding
• Bending (uniaxial) and compression Failure by
  buckling in the plane of bending, without torsion
• Bending (strong axis) and compression Failure
  by LTB
• Bending (biaxial) and compression (torsionally
  stiff section) Failure by buckling in one of the
  principal directions.
• Bending (biaxial) and compression (thin-walled
  section) failure by combined twisting and
  bending
• Bending (biaxial) torsion compression
  failure by combined twisting and bending

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Beam-Columns

• Structural elements subjected to combined
  flexural moments and axial loads are called
  beam-columns
• The case of beam-columns usually appears in
  structural frames
• The code requires that the sum of the load
  effects be smaller than the resistance of the
  elements
• Thus a column beam interaction can be written as
  
• This means that a column subjected to axial load
  and moment will be able to carry less axial load
  than if no moment would exist.

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Beam-Columns

• AISC code makes a distinct difference between
  lightly and heavily axial loaded columns

AISC Equation
AISC Equation
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Beam-Columns

• Definitions
• Pu factored axial compression load
• Pn nominal compressive strength
• Mux factored bending moment in the x-axis,
  including second-order effects
• Mnx nominal moment strength in the x-axis
• Muy same as Mux except for the y-axis
• Mny same as Mnx except for the y-axis
• ?c Strength reduction factor for compression
  members 0.90
• ?b Strength reduction factor for flexural
  members 0.90

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Beam-Columns

• The increase in slope for lightly axial-loaded
  columns represents the less effect of axial load
  compared to the heavily axial-loaded columns

Unsafe Element
Pu/fcPn
Safe Element
0.2
Mu/fbMn
These are design charts that are a bit
conservative than behaviour envelopes
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Moment Amplification

• When a large axial load exists, the axial load
  produces moments due to any element deformation.
• The final moment M is the sum of the original
  moment and the moment due to the axial load. The
  moment is therefore said to be amplified.
• As the moment depends on the load and the
  original moment, the problem is nonlinear and
  thus it is called second-order problem.

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Moment Amplification

• Second-order Moments, Pud and Pu?

Moment amplification in column braced against
sidesway Mu Mnt Pud
Moment amplification in unbraced column Mu Mlt
Pu?
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Moment Amplification

• Using first principles we can prove that the
  final moment Mmax is amplified from M0 as

• The amplification factor B can be

Where
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Second Order Analysis
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Second Order Analysis
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Second Order Analysis
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Second Order Analysis
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Second Order Analysis
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Second Order Analysis
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Second Order Analysis
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Compact Sections for Beam-Columns

• The axial load affects the ratio for compactness.
  When the check for compactness for the web is
  performed while the web is subjected to axial
  load the following ratios shall be

Flange limit is similar to beams
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Braced and Unbraced Frames

• Two components of amplification moments can be
  observed in unbraced frames
• Moment due to member deflection (similar to
  braced frames)
• Moment due to sidesway of the structure

Unbraced Frames
Member deflection
Member sidesway
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Unbraced and Braced Frames

• In braced frames amplification moments can only
  happens due to member deflection

Braced Frames
Sidesway bracing system
Member deflection
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Unbraced and Braced Frames

• The AISC code approximate the effect by using two
  amplification factors B1 and B2

AISC Equation
AISC Equation

• Where
• B1 amplification factor for the moment occurring
  in braced member
• B2 amplification factor for the moment occurring
  from sidesway
• Mnt and Pnt is the maximum moment and axial force
  assuming no sidesway
• Mlt and Plt is the maximum moment and axial force
  due to sidesway
• Pr is the required axial strength

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Unbraced and Braced Frames

• Braced frames are those frames prevented from
  sidesway.
• In this case the moment amplification equation
  can be simplified to

AISC Equation

• KL/r for the axis of bending considered
• K 1.0

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Unbraced and Braced Frames

• The coefficient Cm is used to represent the
  effect of end moments on the maximum deflection
  along the element (only for braced frames)

• When there is transverse loading on the beam
  either of the following case applies

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Unbraced and Braced Frames

• AISC requires stability bracing to have
• Specific strength to resist the lateral load
• Specific axial stiffness to limit the lateral
  deformation.

Braced Frames
Unbraced Frames

• Where Pu is the sum of factored axial load in the
  braced story
• Pbr is bracing strength and bbr is braced or
  unbraced frame stiffness (f 0.75)

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Unbraced and Braced Frames

• Unbraced frames can observe loading sidesway
• In this case the moment amplification equation
  can be simplified to

BMD
AISC Equation
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Unbraced and Braced Frames

• A minimum lateral load in each combination shall
  be added so that the shear in each story is given
  by

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Analysis of Unbraced Frames
is the sum of factored axial loads on all columns
in floor
is the drift due to the unfactored horizontal
forces
is the story height
story shear produced by unfactored horizontal
forces
is the drift index (is generally between 1/500 to
1/200)
is the sum of Euler buckling loads of all columns
in floor
is the factored axial load in the column
RM
can be conservatively taken as 0.85
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Ex. 5.1- Beam-Columns in Braced Frames

• A 3.6-m W12x96 is subjected to bending and
  compressive loads in a braced frame. It is bent
  in single curvature with equal and opposite end
  moments and is not loaded transversely. Use Grade
  50 steel. Is the section satisfactory if Pu
  3200 kN and first-order moment Mntx 240 kN.m
• Step I From Section Property Table
• W12x96 (A 18190 mm2, Ix 347x106 mm4, Lp
  3.33 m, Lr 14.25 m, Zx 2409 mm3, Sx 2147
  mm3)

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Ex. 5.1- Beam-Columns in Braced Frames

• Step II Compute amplified moment
• - For a braced frame let K 1.0
• KxLx KyLy (1.0)(3.6) 3.6 m
• - From Column Chapter ?cPn 4831 kN
• Pu/?cPn 3200/4831 0.662 gt 0.2 ? Use eqn.
• - There is no lateral translation of the frame
  Mlt 0
• ? Mux B1Mntx
• Cm 0.6 0.4(M1/M2) 0.6 0.4(-240/240)
  1.0
• Pe1 ?2EIx/(KxLx)2 ?2(200)(347x106)/(3600)2
  52851 kN

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Ex. 5.1- Beam-Columns in Braced Frames
Mux (1.073)(240) 257.5 kN.m Step III
Compute moment capacity Since Lb 3.6 m
Lp lt Lblt Lr
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Ex. 5.1- Beam-Columns in Braced Frames
Step IV Check combined effect
? Section is satisfactory
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Ex. 5.2- Analysis of Beam-Column

• Check the adequacy of an ASTM A992 W14x90 column
  subjected to an axial force of 2200 kN and a
  second order bending moment of 400 kN.m. The
  column is 4.2 m long, is bending about the strong
  axis. Assume
• ky 1.0
• Lateral unbraced length of the compression flange
  is 4.2 m.

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Ex. 5.2- Analysis of Beam-Column

• Step I Compute the capacities of the beam-column
• ?cPn 4577 kN ?Mnx 790 kN.m
• ?Mny 380 kN.m
• Step II Check combined effect

OK
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Design of Beam-Columns

• Trial-and-error procedure
• Select trial section
• Check appropriate interaction formula.
• Repeat until section is satisfactory

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Design of Unbraced Frames

• Design can be performed using the following
  procedure
• Use a procedure similar to that of braced frames
• To start the design assume B1 1.0 and compute
  B2 by assuming the ratio
• Compute Mu and perform same procedure used for
  braced frames

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Ex. 5.3- Analysis-External Column

• Check the exterior column of an unbraced frame
  shown in the figure for the following load
  combination. All columns are 3.8 m long and all
  beams are 9 m long. Assume A992 steel.

W24x76
W14x90
For this frame
W24x76
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Ex. 5.3- Analysis-External Column

• Step I Calculate Kx and Ky
• Effective length, Ky ,
• assumed braced frame

W24x76
W14x90
W24x76
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Ex. 5.3- Analysis-External Column

• Step II Calculate ?Pn and p

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Ex. 5.3- Analysis-External Column

• Step III Determine second-order moments-No
  translation, Mnt

Due to lack of information, assume Cm 1.0
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Ex. 5.3- Analysis-External Column

• Step IV Determine second-order moments -
  Translation, Mlt.
• Dont know all columns in story, thus assume the
  frame will have a deflection limit
• For this frame
• Thus,

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Ex. 5.3- Analysis-External Column

• Step V Second-order moment
• Step VI Check combined effect

OK

• Thus, the W14x90, Fy 344 MPa will work for this
  loading case.
• Now it should be checked for any other load case,
  such as 1.2D1.6L

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Ex. 5.4 Design-Beam Column

• Select a W shape of A992 steel for the
  beam-column of the following figure. This member
  is part of a braced frame and is subjected to the
  service-load axial force and bending moments
  shown (the end shears are not shown). Bending is
  about the strong axis, and Kx Ky 1.0. Lateral
  support is provided only at the ends. Assume that
  B1 1.0.

PD 240 kN
PL 650 kN
MD 24.4 kN.m
ML 66.4 kN.m
4.8 m
MD 24.4 kN.m
ML 66.4 kN.m
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Ex. 5.4 Design-Beam Column

• Step I Compute the factored axial load and
  bending moments
• Pu 1.2PD 1.6PL 1.2(240) 1.6(650) 1328
  kN.
• Mntx 1.2MD 1.6ML 1.2(24.4) 1.6(66.4)
  135.5 kN.m.
• B1 1.0 ? Mux B1Mntx 1.0(135.5) 135.5
  kN.m
• Step II compute ?Mnx, ?Pn
• The effective length for compression and the
  unbraced length for bending are the same KL
  Lb 4.8 m.
• The bending is uniform over the unbraced length ,
  so Cb1.0
• Try a W10X60 with ?Pn 2369 kN and ?Mnx 344
  kN.m

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Ex. 5.4 Design-Beam Column

• Step III Check interaction equation
• Step IV Make sure that this is the lightest
  possible section.
• ? Try W12x58 with ?Pn 2247 kN and ?Mnx
  386 kN.m
• ? Use a W12 x 58 section

OK
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Design of Base Plates

• We are looking for design of concentrically
  loaded columns. These base plates are connected
  using anchor bolts to concrete or masonry
  footings
• The column load shall spread over a large area of
  the bearing surface underneath the base plate

AISC Manual Part 16, J8
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Design of Base Plates

• The design approach presented here combines three
  design approaches for light, heavy loaded, small
  and large concentrically loaded base plates

Area of Plate is computed such that
n
m
B
0.8 bf
where
If plate covers the area of the footing
0.95d
N
If plate covers part of the area of the footing

• The dimensions of the plate are computed such
  that m and n are approximately equal.

A1 area of base plate
A2 area of footing
fc compressive strength of concrete used for
footing
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Design of Base Plates
Thickness of plate
However ? may be conservatively taken as 1
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Ex. 5.5 Design of Base Plate

• For the column base shown in the figure, design a
  base plate if the factored load on the column is
  10000 kN. Assume 3 m x 3 m concrete footing with
  concrete strength of 20 MPa.

N
0.95d
W14x211
0.8bf
B
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Ex. 4.7- Design of Base Plate

• Step I Plate dimensions
• Assume thus
• Assume m n
• N 729.8 mm say N 730 mm
• B 671.8 mm say B 680 mm

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Ex. 4.7- Design of Base Plate

• Step II Plate thickness

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Ex. 4.7- Design of Base Plate

• Selecting the largest cantilever length
• use 730 mm x 670 mm x 80 mm Plate

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Eccentrically Loaded Columns

• For eccentrically loaded columns
• Compute dimensions such that stress (q) is less
  than concrete compressive strength.
• Compute thickness so that the ultimate moment on
  the plate equals the full plastic moment
  multiplied by ?, where ? 0.9.
• no tension e eccentricity
• Mu ultimate moment per (mm) width on the
  plate