CTC 422 Design of Steel Structures

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CTC 422Design of Steel Structures

• Beams - Flexure

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Objectives of Structural Design

• Structure is adequate to support loads which will
  be applied during its life
• Strength provided strength required
• Structure will meet serviceability requirements
• Deflection
• Vibration
• Structure will meet functional requirements
• Structure will meet economic requirements

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Beam Design

• Student Objectives
• Analyze a beam to calculate load, shear, moment
  and deflection and to determine if a given beam
  is adequate
• Design (select) a beam to safely to support a
  load considering moment, shear and deflection

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Beam Design

• Beam
• A structural member which carries loads applied
  perpendicular to its longitudinal axis
• These loads cause shear and bending (moment)
• Different terms used for beams depending on
  application or location
• Girder, stringer, joist, lintel, spandrel,
  purlin, girt
• Behavior of all is the same.
• All are beams

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Load and Resistance Factor Design - LRFD

• Design strength Required strength
• FRn Ru
• For bending
• Fb Mn Mu
• Where
• Mn Nominal moment strength
• Fb Strength reduction factor for bending 0.9
• Mu Required moment strength based on factored
  loads

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Load and Resistance Factor Design - LRFD

• Nominal moment capacity, Mn, depends on the
  failure mechanism of the beam
• Beam can fail by
• Full yielding of the cross-section
• Lateral torsional buckling (LTB)
• Can be inelastic or elastic buckling
• Flange local buckling (FLB)
• Web local buckling (WLB)
• Failure mechanism is related to
• Lateral bracing of the beam
• Whether or not the beam cross-section is compact

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Failure Mechanism and Nominal Moment Capacity, Mn

• If beam remains stable up to its full plastic
  moment capacity
• Failure is by yielding of the full section
• Mn Mp
• Instability could be overall beam instability
• Lateral torsional buckling (elastic or inelastic)
• Prevented by adequate lateral bracing of the
  beams compression flange
• Instability could also be local instability
• Flange local buckling or web local buckling
• Dependent on width / thickness ratios of
  compression elements
• Compactness, non-compactness or slenderness of
  section

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Compactness

• Structural shapes are classified as compact,
  non-compact, or slender
• Compact
• Section reaches its full strength (yield) before
  local buckling occurs
• Strength of section is governed by material
  strength
• Non-compact
• Only a portion of the cross-section reaches its
  full strength (yield) before local buckling
  occurs
• Slender
• Cross-section does not yield before local
  buckling occurs
• Strength is governed by buckling
• Compactness, non-compactness, or slenderness is a
  property of the cross-section itself
• A function of the width / thickness ratios of its
  flanges and its web
• Flange width / thickness bf / 2tf
• Web width / thickness h / tw

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Compactness

• Classification is given in Table B4.1
• Notation
• ? width / thickness ratio
• ?p upper limit for compact category
• ?r upper limit for non-compact category
• If ? ?p and the flange is continuously attached
  to the web, the shape is compact
• If ?p ? ?r, the shape is non-compact
• If ? gt ?r, the shape is slender
• Category is based on the worst width / thickness
  ratio
• Example If web is compact and flange is
  non-compact, section is classified as non-compact
• Most standard W, M, S, and C sections are compact
• A few are non-compact because of their flanges,
  but none are slender

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Bending Strength of Compact Shapes

• Moment strength of a compact shape is a function
  of, Lb, the unbraced length of its compression
  flange
• Lb distance between points braced against
  lateral displacement of compression flange
• Lp limiting laterally unbraced length for limit
  state of yielding
• Lr limiting laterally unbraced length for limit
  state of inelastic lateral torsional buckling
• Compression flange may be braced by
• Perpendicular framing
• Steel roof deck or floor deck
• Concrete slab
• Cross-bracing

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Bending Strength of Compact Shapes

• If the compression flange is continuously braced
  (Lb Lp)
• Failure will be by yielding at full plastic
  moment
• Nominal moment capacity, Mn Mp Fy Zx (AISC
  Eq. F2-1)
• Design strength Fb Mn Fb Mp
• For unbraced length Lb gt Lp
• Failure will be by inelastic lateral torsional
  buckling
• Nominal moment capacity, Mn lt Mp
• At Lb Lp, Mn 0.7 Fy Sx
• For Lp lt Lb lt Lr , linear interpolation from Mn
  Mp to Mn 0.7 Fy Sx (AISC Eq. F2-2)
• For unbraced length Lb gt Lr
• Failure will be by elastic lateral torsional
  buckling
• Rapid reduction in Mn (AISC Eq. F2-3)

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Bending Strength of Non-compact Shapes

• Most standard W, M, S, and C sections are compact
• A few are non-compact because of their flanges,
  but none are slender
• Shapes with noncompact flanges are listed in User
  note on page 16.1-49
• Sections with compact webs and noncompact (or
  slender) flanges
• Nominal moment capacity, Mn lt Mp
• Calculate Mn using provisions of Code Section F3
• Sections with noncompact webs
• Nominal moment capacity, Mn lt Mp
• Calculate Mn using provisions of Code Section F4

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Design Aids Braced Beams

• Table 3-2, W-Shapes Selection by Zx
• Applies to wide flange shapes with Fy 50 ksi
• Applies mainly to sections which are adequately
  braced (Lb Lp)
• Can be used for unbraced length up to Lb Lr
• Best to use this table only if fully braced
• Table lists Zx, Lp, Lr, and Moment Capacity, Fb
  Mp
• Also lists Ix, and Shear Capacity Fv Vnx
• Non-compact sections indicated by the footnote
  f
• Moment capacity in table has been adjusted for
  non-compactness
• Sections in table are grouped by weight
• Lightest section in group is in bold
• Choose this section if there is no depth
  restriction

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Design Aids Unbraced Beams

• Table 3-10, Available Moment vs. Unbraced Length
• Applies to wide flange shapes with Fy 50 ksi
• Also applies to channel shapes with Fy 36 ksi
• Table is a plot of available flexural strength,
  FbMnx, versus unbraced length Lb
• Bending Coefficient in Table conservatively taken
  as Cb 1
• See Table 3-1 for values of Cb
• Choose beam that has available moment strength
  FbMnx Mu at an unbraced length Lb Design Lb
• Choose a beam above and to right of (Lb, Mu)
• Solid line Beam chosen is lightest section
  available for the given combination of Mu and Lb
• Dashed line A lighter section is available

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Design Aids Channels

• Braced Channels
• Table 3-8, Maximum Total Uniform Load C Shapes
• Applies to channel shapes with Fy 50 ksi
• Applies only to sections which are adequately
  braced (Lb Lp)
• Best to use this table only if fully braced
• Table lists Zx, Lp, Lr, and Moment Capacity, Fb
  Mp
• Also lists Shear Capacity Fv Vnx
• Unbraced Channels
• Table 3-10, Available Moment vs. Unbraced Length
• Applies to channel shapes with Fy 36 ksi