# A beam is a structural member that is subjected primarily to transverse loads and negligible

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1 Chapter. Design of Beams Flexure and Shear.1 Section force-deformation response & Plastic Moment (M p ) A beam is a structural member that is subjected primarily to transverse loads and negligible axial loads. The transverse loads cause internal shear forces and bending moments in the beams as shown in Figure 1 below. w P x V(x) M(x) Figure 1. Internal shear force and bending moment diagrams for transversely loaded beams. These internal shear forces and bending moments cause longitudinal axial stresses and shear stresses in the cross-section as shown in the Figure below. ε σ df = σ bdy d y M(x) V(x) b Curvature = φ = ε/d F ε d / = + d / σ b dy σ M d / = + d / σ b dy y (Planes remain plane) Figure. Longitudinal axial stresses caused by internal bending moment. 1

2 Steel material follows a typical stress-strain behavior as shown in Figure 3 below. σ u σ ε y ε Figure 3. Typical steel stress-strain behavior. ε u If the steel stress-strain curve is approximated as a bilinear elasto-plastic curve with yield stress equal to, then the section Moment - Curvature (M-φ) response for monotonically increasing moment is given by Figure 4. M p B C D E Section Moment, M M y A ε y ε y 5ε y 10ε y ε y ε y 5ε y Curvature, φ A: Extreme fiber reaches ε y B: Extreme fiber reaches ε y C: Extreme fiber reaches 5ε y D: Extreme fiber reaches 10ε y E: Extreme fiber reaches infinite strain Figure 4. Section Moment - Curvature (M-φ) behavior. 10ε y A B C D E

3 In Figure 4, M y is the moment corresponding to first yield and M p is the plastic moment capacity of the cross-section. - The ratio of M p to M y is called as the shape factor f for the section. - For a rectangular section, f is equal to 1.5. For a wide-flange section, f is equal to 1.1. Calculation of M p : Cross-section subjected to either + or - at the plastic limit. See Figure 5 below. A 1 Plastic centroid. A 1 A y 1 y A (a) General cross-section (b) Stress distribution (c) Force distribution F = σ A 1 y A 1 = A y σ A 1 y = A / = 0 A M = σ y (y 1 + y ) Where, y = centroid of = centroid of A 1 A (d) Equations Figure 5. Plastic centroid and M p for general cross-section. The plastic centroid for a general cross-section corresponds to the axis about which the total area is equally divided, i.e., A 1 = A = A/ - The plastic centroid is not the same as the elastic centroid or center of gravity (c.g.) of the cross-section. 3

4 - As shown below, the c.g. is defined as the axis about which A 1 y 1 = A y. y y 1 c.g. = elastic N.A. A 1, y 1 About the c.g. A 1 y 1 = A y A, y For a cross-section with at-least one axis of symmetry, the neutral axis corresponds to the centroidal axis in the elastic range. However, at M p, the neutral axis will correspond to the plastic centroidal axis. For a doubly symmetric cross-section, the elastic and the plastic centroid lie at the same point. M p = x A/ x (y 1 +y ) As shown in Figure 5, y 1 and y are the distance from the plastic centroid to the centroid of area A 1 and A, respectively. A/ x (y 1 +y ) is called Z, the plastic section modulus of the cross-section. Values for Z are tabulated for various cross-sections in the properties section of the LRFD manual. φ M p = 0.90 Z F y - See Spec. F1.1 where, M p = plastic moment, which must be 1.5 M y for homogenous cross-sections M y = moment corresponding to onset of yielding at the extreme fiber from an elastic stress distribution = F y S for homogenous cross-sections and = F yf S for hybrid sections. Z = plastic section modulus from the Properties section of the AISC manual. S = elastic section modulus, also from the Properties section of the AISC manual. 4

5 Example.1 Determine the elastic section modulus, S, plastic section modulus, Z, yield moment, M y, and the plastic moment M p, of the cross-section shown below. What is the design moment for the beam cross-section. Assume 50 ksi steel. 1 in. F in. W t w = 0.5 in. 16 in. F 1.0 in. 15 in. A g = 1 x ( ) x x 1.0 = in A f1 = 1 x 0.75 = 9 in A f = 15 x 1.0 = 15.0 in A w = 0.5 x ( ) = 7.15 in distance of elastic centroid from bottom = y 9 ( / ) y = = in I x = / / / = 1430 in 4 S x = I x / ( ) = in 3 M y-x = F y S x = kip-in. = kip-ft. distance of plastic centroid from bottom = y p (y y p =.15 in. p ) = =

6 y 1 =centroid of top half-area about plastic centroid = = in y =centroid of bottom half-area about plas. cent. = = in. Z x = A/ x (y 1 + y ) = x ( ) = in 3 M p-x = Z x F y = x 50 = kip-in. = kip-ft. Design strength according to AISC Spec. F1.1= φ b M p = 0.9 x = kip-ft. Check = M p 1.5 M y Therefore, kip-ft. < 1.5 x = kip-ft. - OK! Reading Assignment 6

7 . Flexural Deflection of Beams Serviceability Steel beams are designed for the factored design loads. The moment capacity, i.e., the factored moment strength (φ b M n ) should be greater than the moment (M u ) caused by the factored loads. A serviceable structure is one that performs satisfactorily, not causing discomfort or perceptions of unsafety for the occupants or users of the structure. - For a beam, being serviceable usually means that the deformations, primarily the vertical slag, or deflection, must be limited. - The maximum deflection of the designed beam is checked at the service-level loads. The deflection due to service-level loads must be less than the specified values. The AISC Specification gives little guidance other than a statement in Chapter L, Serviceability Design Considerations, that deflections should be checked. Appropriate limits for deflection can be found from the governing building code for the region. The following values of deflection are typical maximum allowable total (service dead load plus service live load) deflections. Plastered floor construction L/360 Unplastered floor construction L/40 Unplastered roof construction L/180 In the following examples, we will assume that local buckling and lateral-torsional buckling are not controlling limit states, i.e, the beam section is compact and laterally supported along the length. 7

8 Example. Design a simply supported beam subjected to uniformly distributed dead load of 450 lbs/ft. and a uniformly distributed live load of 550 lbs/ft. The dead load does not include the self-weight of the beam. Step I. Calculate the factored design loads (without self-weight). w U = 1. w D w L = 1.4 kips / ft. M U = w u L / 8 = 1.4 x 30 / 8 = kip-ft. Step II. Select the lightest section from the AISC Manual design tables. From page of the AISC manual, select W16 x 6 made from 50 ksi steel with φ b M p = kip-ft. Step III. Add self-weight of designed section and check design w sw = 6 lbs/ft Therefore, w D = 476 lbs/ft = lbs/ft. w u = 1. x x 0.55 = kips/ft. Therefore, M u = x 30 / 8 = kip-ft. < φ b M p of W16 x 6. OK! Step IV. Check deflection at service loads. w = kips/ft. = 1.06 kips/ft. = 5 w L 4 / (384 E I x ) = 5 x (1.06/1) x (30 x 1) 4 / (384 x 9000 x 301) =.14 in. > L/360 - for plastered floor construction Step V. Redesign with service-load deflection as design criteria L /360 = 1.0 in. > 5 w L 4 /(384 E I x ) Therefore, I x > in 4 8

9 Select the section from the moment of inertia selection tables in the AISC manual. See page select W1 x 44. W1 x 44 with I x = 843 in 4 and φ b M p = 358 kip-ft. (50 ksi steel). Deflection at service load = = in. < L/360 - OK! Note that the serviceability design criteria controlled the design and the section Example.3 Design the beam shown below. The unfactored dead and live loads are shown in the Figure. 10 kips (live load) 0.67 k/ft. (dead load) 0.75 k/ft. (live load) 15 ft. 30 ft. Step I. Calculate the factored design loads (without self-weight). w u = 1. w D w L = 1. x x 0.75 =.004 kips / ft. P u = 1. P D P L = 1. x x 10 = 16.0 kips M u = w U L / 8 + P U L / 4 = = kip-ft. Step II. Select the lightest section from the AISC Manual design tables. From page of the AISC manual, select W1 x 44 made from 50 ksi steel with φ b M p = kip-ft. Self-weight = w sw = 44 lb/ft. 9

10 Step III. Add self-weight of designed section and check design w D = = kips/ft w u = 1. x x 0.75 =.0568 kips/ft. Therefore, M u =.0568 x 30 / = kip-ft. < φ b M p of W1 x 44. OK! Step IV. Check deflection at service loads. Service loads Distributed load = w = = kips/ft. Concentrated load = P = D + L = kips = 10 kips Deflection due to uniform distributed load = d = 5 w L 4 / (384 EI) Deflection due to concentrated load = c = P L 3 / (48 EI) Therefore, service-load deflection = = d + c = 5 x x / (384 x 9000 x 1 x 843) + 10 x / (48 x 9000 x 843) = = 1.49 in. Assuming unplastered floor construction, max = L/40 = 360/40 = 1.5 in. Therefore, < max - OK! 10

11 .3 Local buckling of beam section Compact and Non-compact M p, the plastic moment capacity for the steel shape, is calculated by assuming a plastic stress distribution (+ or - ) over the cross-section. The development of a plastic stress distribution over the cross-section can be hindered by two different length effects: (1) Local buckling of the individual plates (flanges and webs) of the cross-section before they develop the compressive yield stress. () Lateral-torsional buckling of the unsupported length of the beam / member before the cross-section develops the plastic moment M p. M M Figure 7. Local buckling of flange due to compressive stress (σ) The analytical equations for local buckling of steel plates with various edge conditions and the results from experimental investigations have been used to develop limiting slenderness ratios for the individual plate elements of the cross-sections. See Spec. B5 (page ), Table B5.1 ( ) and Page of the AISC-manual Steel sections are classified as compact, non-compact, or slender depending upon the slenderness (λ) ratio of the individual plates of the cross-section. 11

12 - Compact section if all elements of cross-section have λ λ p - Non-compact sections if any one element of the cross-section has λ p λ λ r - Slender section if any element of the cross-section has λ r λ It is important to note that: - If λ λ p, then the individual plate element can develop and sustain for large values of ε before local buckling occurs. - If λ p λ λ r, then the individual plate element can develop but cannot sustain it before local buckling occurs. - If λ r λ, then elastic local buckling of the individual plate element occurs. Compressive axial stress, σ Compact Non-Compact Slender Effective axial strain, ε Figure 8. Stress-strain response of plates subjected to axial compression and local buckling. Thus, slender sections cannot develop M p due to elastic local buckling. Non-compact sections can develop M y but not M p before local buckling occurs. Only compact sections can develop the plastic moment M p. All rolled wide-flange shapes are compact with the following exceptions, which are noncompact. 1

13 - W40x174, W14x99, W14x90, W1x65, W10x1, W8x10, W6x15 (made from A99) The definition of λ and the values for λ p and λ r for the individual elements of various crosssections are given in Table B5.1 and shown graphically on page For example, Section Plate element λ λ p λ r Wide-flange Flange b f /t f 0.38 E / Fy 0.38 E / FL Web h/t w 3.76 E / Fy 5.70 E / Fy Channel Flange b f /t f 0.38 E / Fy 0.38 E / FL Web h/t w 3.76 E / Fy 5.70 E / Fy Square or Rect. Box Flange (b-3t)/t 1.1 E / Fy 1.40 E / Fy Web (b-3t)/t 3.76 E / Fy 5.70 E / Fy In CE405 we will design all beam sections to be compact from a local buckling standpoint 13

14 .4 Lateral-Torsional Buckling The laterally unsupported length of a beam-member can undergo lateral-torsional buckling due to the applied flexural loading (bending moment). (a) M (b) M M M Figure 9. Lateral-torsional buckling of a wide-flange beam subjected to constant moment. 14

15 Lateral-torsional buckling is fundamentally similar to the flexural buckling or flexuraltorsional buckling of a column subjected to axial loading. - The similarity is that it is also a bifurcation-buckling type phenomenon. - The differences are that lateral-torsional buckling is caused by flexural loading (M), and the buckling deformations are coupled in the lateral and torsional directions. There is one very important difference. For a column, the axial load causing buckling remains constant along the length. But, for a beam, usually the lateral-torsional buckling causing bending moment M(x) varies along the unbraced length. - The worst situation is for beams subjected to uniform bending moment along the unbraced length. Why?.4.1 Lateral-torsional buckling Uniform bending moment Consider a beam that is simply-supported at the ends and subjected to four-point loading as shown below. The beam center-span is subjected to uniform bending moment M. Assume that lateral supports are provided at the load points. P P L b Laterally unsupported length = L b. If the laterally unbraced length L b is less than or equal to a plastic length L p then lateral torsional buckling is not a problem and the beam will develop its plastic strength M p. L p = 1.76 r y x E / Fy - for I members & channels (See Pg ) 15

16 If L b is greater than L p then lateral torsional buckling will occur and the moment capacity of the beam will be reduced below the plastic strength M p as shown in Figure 10 below. Z x F y = M p M n = M p L b L M n = M p ( M p M r ) L r L p p S x (F y 10) =M r Moment Capacity, M n M n = π EI L b y π EC GJ + L b w L p L Lrr Unbraced length, L b Figure 10. Moment capacity (M n ) versus unsupported length (L b ). As shown in Figure 10 above, the lateral-torsional buckling moment (M n = M cr ) is a function of the laterally unbraced length L b and can be calculated using the equation: M n = M cr = π L b E I y π E G J + L b I y C w where, M n = moment capacity L b = laterally unsupported length. M cr = critical lateral-torsional buckling moment. E = 9000 ksi; G = 11,00 ksi I y = moment of inertia about minor or y-axis (in 4 ) J = torsional constant (in 4 ) from the AISC manual pages. C w = warping constant (in 6 ) from the AISC manual pages. 16

17 This equation is valid for ELASTIC lateral torsional buckling only (like the Euler equation). That is it will work only as long as the cross-section is elastic and no portion of the crosssection has yielded. As soon as any portion of the cross-section reaches the yield stress F y, the elastic lateral torsional buckling equation cannot be used. - L r is the unbraced length that corresponds to a lateral-torsional buckling moment M r = S x (F y 10). - M r will cause yielding of the cross-section due to residual stresses. When the unbraced length is less than L r, then the elastic lateral torsional buckling equation cannot be used. When the unbraced length (L b ) is less than L r but more than the plastic length L p, then the lateral-torsional buckling M n is given by the equation below: - If L p L b L r, then L M n = M p ( M p M r ) L b r L L p p - This is linear interpolation between (L p, M p ) and (L r, M r ) - See Figure 10 again. 17

18 .4. Moment Capacity of beams subjected to non-uniform bending moments As mentioned previously, the case with uniform bending moment is worst for lateral torsional buckling. For cases with non-uniform bending moment, the lateral torsional buckling moment is greater than that for the case with uniform moment. The AISC specification says that: - The lateral torsional buckling moment for non-uniform bending moment case = C b x lateral torsional buckling moment for uniform moment case. C b is always greater than 1.0 for non-uniform bending moment. - C b is equal to 1.0 for uniform bending moment. - Sometimes, if you cannot calculate or figure out C b, then it can be conservatively assumed as 1.0. C b =.5 M max 1.5 M + 3 M A max + 4 M B + 3 M c where, M max = magnitude of maximum bending moment in L b M A = magnitude of bending moment at quarter point of L b M B = magnitude of bending moment at half point of L b M C = magnitude of bending moment at three-quarter point of L b The moment capacity M n for the case of non-uniform bending moment - M n = C b x {M n for the case of uniform bending moment} M p - Important to note that the increased moment capacity for the non-uniform moment case cannot possibly be more than M p. - Therefore, if the calculated values is greater than M p, then you have to reduce it to M p 18

19 M p Moment Capacity, M n M r C b = 1.5 C b = 1. C b = 1.0 L p L r.5 Beam Design Unbraced length, L b Figure 11. Moment capacity versus L b for non-uniform moment case. Example.4 Design the beam shown below. The unfactored uniformly distributed live load is equal to 3 kips/ft. There is no dead load. Lateral support is provided at the end reactions. w L = 3 kips/ft. 4 ft. Lateral support / bracing Step I. Calculate the factored loads assuming a reasonable self-weight. Assume self-weight = w sw = 100 lbs/ft. Dead load = w D = = 0.1 kips/ft. Live load = w L = 3.0 kips/ft. Ultimate load = w u = 1. w D w L = 4.9 kips/ft. Factored ultimate moment = M u = w u L /8 = kip-ft. 19

20 Step II. Determine unsupported length L b and C b There is only one unsupported span with L b = 4 ft. C b = 1.14 for the parabolic bending moment diagram, See values of C b shown in Figure. Step III. Select a wide-flange shape The moment capacity of the selected section φ b M n > M u (Note φ b = 0.9) φ b M n = moment capacity = C b x (φ b M n for the case with uniform moment) φ b M p - Pages in the AISC-LRFD manual, show the plots of φ b M n -L b for the case of uniform bending moment (C b =1.0) - Therefore, in order to select a section, calculate M u /C b and use it with L b to find the first section with a solid line as shown in class. - M u /C b = 354.4/1.14 = kip-ft. - Select W16 x 67 (50 ksi steel) with φ b M n =357 kip-ft. for L b = 4 ft. and C b =1.0 - For the case with C b = 1.14, φ b M n = 1.14 x 357 = kip-ft., which must be φ b M p = 491 kip-ft. OK! Thus, W16 x 67 made from 50 ksi steel with moment capacity equal to kip-ft. for an unsupported length of 4 ft. is the designed section. Step IV. Check for local buckling. λ = b f / t f = 7.7; Corresponding λ p = 0.38 (E/Fy) 0.5 = 9.19 Therefore, λ < λ p - compact flange λ = h/t w = 34.4; Corresponding λ p = 3.76 (E/F y ) 0.5 = 90.5 Therefore, λ < λ p Compact section. - compact web - OK! This example demonstrates the method for designing beams and accounting for C b > 1.0 0

21 Example.5 Design the beam shown below. The concentrated live loads acting on the beam are shown in the Figure. The beam is laterally supported at the load and reaction points. w sw = 0.1 kips/ft. 30 kips 30 kips 1 ft. 8 ft. 10 ft. Lateral support / bracing 30 ft. Step I. Assume a self-weight and determine the factored design loads Let, w sw = 100 lbs/ft. = 0.1 kips/ft. P L = 30 kips P u = 1.6 P L = 48 kips w u = 1. x w sw = 0.1 kips/ft. The reactions and bending moment diagram for the beam are shown below. 48 kips 48 kips w sw = 0.1 kips/ft. A B C D 1 ft. 8 ft. 10 ft kips 53 kips A B C D kip-ft. 54 kip-ft. Step II. Determine L b, C b, M u, and M u /C b for all spans. 1

22 Span L b C b M u M u /C b (ft.) (kip-ft.) (kip-ft.) AB BC (assume) CD It is important to note that it is possible to have different L b and C b values for different laterally unsupported spans of the same beam. Step III. Design the beam and check all laterally unsupported spans Assume that span BC is the controlling span because it has the largest M u /C b although the corresponding L b is the smallest. From the AISC-LRFD manual select W1 x 68 made from 50 ksi steel (page ) Check the selected section for spans AB, BC, and CD Span L b (ft.) φ b M n for C b = 1.0 from C b φ b M n for C b value col. 3 x col. 4 φ b M p limit AB kip-ft BC CD kip-ft. Thus, for span AB, φ b M n = 600 kip-ft. > M u for span BC, φ b M n = 57.0 kip-ft. > M u For span CD, φ b M n = 600 kip-ft. > M u - OK! -OK! -OK!

23 Step IV. Check for local buckling λ = b f / t f = 6.0; Corresponding λ p = 0.38 (E/Fy) 0.5 = 9.19 Therefore, λ < λ p - compact flange λ = h/t w = 43.6; Corresponding λ p = 3.76 (E/F y ) 0.5 = Therefore, λ < λ p - compact web Compact section. - OK! This example demonstrates the method for designing beams with several laterally unsupported spans with different L b and C b values. Example.6 Design the simply-supported beam shown below. The uniformly distributed dead load is equal to 1 kips/ft. and the uniformly distributed live load is equal to kips/ft. A concentrated live load equal to 10 kips acts at the mid-span. Lateral supports are provided at the end reactions and at the mid-span. w D = 1.0 kips/ft. w L =.0 kips/ft. 10 kips A B C 1 ft. 1 ft. Step I. Assume the self-weight and calculate the factored design loads. Let, w sw = 100 lbs/ft. = 0.1 kips/ft. w D = = 1.1 kips/ft. w L =.0 kips/ft. w u = 1. w D w L = 4.5 kips/ft. P u = 1.6 x 10 = 16.0 kips 3

24 The reactions and the bending moment diagram for the factored loads are shown below. w u = 4.5 kips/ft. 16 kips B 1 ft. 1 ft. 6.4 kips 6.4 kips x M(x) = 6.4 x x / Step II. Calculate L b and C b for the laterally unsupported spans. Since this is a symmetric problem, need to consider only span AB L b = 1 ft.; C b =.5 M max 1.5 M max + 3 M + 4 M A B + 3 M c M(x) = 6.4 x 4.5 x / Therefore, M A = M(x = 3 ft.) = kip-ft. M B = M(x = 6 ft.) = 9.08 kip-ft. M C = M(x = 9ft.) = kip-ft M max = M(x = 1 ft.) = kip-ft. - quarter-point along L b = 1 ft. - half-point along L b = 1 ft. -three-quarter point along L b = 1 ft. - maximum moment along L b =1ft. Therefore, C b = 1.37 Step III. Design the beam section M u = M max = kip-ft. L b = 1.0 ft.; C b = 1.37 M u /C b = 41.44/1.37 = kip-ft. - Select W1 x 48 made from 50 ksi with φ b M n = 3 kip-ft. for L b = 1.0 ft. and C b =1.0 - For C b = 1.37, φ b M n = _ k-ft., but must be < or = φ b M p = 398 k-ft. 4

25 - Therefore, for C b =1.37, φ b M n = 398 k-ft. < M u Step IV. Redesign the section - Select the next section with greater capacity than W1 x 48 - Select W18 x 55 with φ b M n = 345 k-ft. for L b = 1 ft. and C b = 1.0 For C b = 1.37, φ b M n = 345 x 1.37 = k-ft. but must be φ b M p = 40 k-ft. Therefore, for C b = 1.37, φ b M n = 40 k-ft., which is < M u (41.44 k-ft), (NOT OK!) - Select W 1 x 55 with φ b M n = 388 k-ft. for L b = 1 ft. and C b = 1.0 For C b 1.37, φ b M n = 388 x 1.37 = k-ft., but must be φ b M p = 473 k-ft. Therefore, for C b = 1.37, φ b M n = 473 k-ft, which is > M u (41.44 k-ft). (OK!) Step V. Check for local buckling. λ = b f / t f = 7.87; Corresponding λ p = 0.38 (E/F y ) 0.5 = 9.19 Therefore, λ < λ p - compact flange λ = h/t w = 50.0; Corresponding λ p = 3.76 (E/F y ) 0.5 = Therefore, λ < λ p - compact web Compact section. - OK! This example demonstrates the calculation of C b and the iterative design method. 5

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