CTC 422Design of Steel Structures

Beams - Flexure

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

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

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

Load and Resistance Factor Design - LRFD

• Design strength ≥ Required strength• ΦRn ≥ Ru

• For bending• Φb Mn ≥ Mu

• Where:• Mn = Nominal moment strength• Φb = Strength reduction factor for bending = 0.9

• Mu = Required moment strength based on factored loads

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

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 beam’s

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

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

Compactness

• Classification is given in Table B4.1• Notation:

• λ = width / thickness ratio• λp = upper limit for compact category• λr = upper limit for non-compact category

• If λ ≤ λp and the flange is continuously attached to the web, the shape is compact

• If λp ≤ λ ≤ λr, the shape is non-compact• If λ > λr, 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

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

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 Φb Mn = Φb Mp

• For unbraced length Lb > Lp• Failure will be by inelastic lateral torsional buckling• Nominal moment capacity, Mn < Mp

• At Lb = Lp, Mn = 0.7 Fy Sx

• For Lp < Lb < Lr , linear interpolation from Mn = Mp to Mn = 0.7 Fy Sx (AISC Eq. F2-2)

• For unbraced length Lb > Lr• Failure will be by elastic lateral torsional buckling• Rapid reduction in Mn (AISC Eq. F2-3)

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 < Mp • Calculate Mn using provisions of Code Section F3

• Sections with noncompact webs• Nominal moment capacity, Mn < Mp • Calculate Mn using provisions of Code Section F4

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, Φb Mp• Also lists Ix, and Shear Capacity Φv 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

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, ΦbMnx, 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 ΦbMnx ≥ 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

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, Φb Mp

• Also lists Shear Capacity Φv Vnx

• Unbraced Channels• Table 3-10, Available Moment vs. Unbraced Length

• Applies to channel shapes with Fy = 36 ksi