CTC 422 Design of Steel Structures Beams Flexure
CTC 422 Design 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 / 2 tf • Web width / thickness = h / tw
Compactness • Classification is given in Table B 4. 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 ≤ 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. F 2 -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. F 2 -2) • For unbraced length Lb > Lr • • Failure will be by elastic lateral torsional buckling Rapid reduction in Mn (AISC Eq. F 2 -3)
Bending Strength of -compact Shapes Non • 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 F 3 • Sections with noncompact webs • • Nominal moment capacity, Mn < Mp Calculate Mn using provisions of Code Section F 4
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 • Also lists Ix, and Shear Capacity Φv Vnx • Table lists Zx, Lp, Lr, and Moment Capacity, Φb Mp • 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, Φb. Mnx, 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 Φb. Mnx ≥ 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
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