F. P. Beer et al., Meccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 2014 McGraw-Hill Education (Italy) srl

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2 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reactions: Σ = 0: bp = 0 = Pb Σ = 0: ap = 0 = Pa From to B: 0 < x < a Σ Fy = 0: Pb V = 0 Pb Σ J = 0: x = 0 V = Pb = Pbx From B to : a < x < Pa Σ Fy = 0: V + = 0 V Pa Σ K = 0: + ( x) = 0 Pa = Pa( x) = Pab t section B: = PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

3 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reactions: Σ B = 0: w + w = 0 = Σ = 0: w B w = 0 B = Free body diagram for determining reactions. Over whole beam, 0 < x < Place section at x. Replace distributed load by equivalent concentrated load. w Σ Fy = 0: wx V = 0 V w = x w x Σ J = 0: x + wx + = 0 w = ( x x ) aximum bending moment occurs at x =. w = x ( x) w max = 8 PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

4 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl From to B (0 < x < a) : F = 0: wx V = 0 y x J = 0: ( wx) + = 0 From B to ( a < x < ) : V = wx wx = F = 0: wa V = 0 V = wa y a a J = 0: ( wa) x + = 0 = wax PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

5 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl wx 0 Σ Fy = 0: x V = 0 t x =, wx 0 x Σ J = 0: x + = 0 V wx 0 = wx 0 = w w V max = 0 0 V = 0 w = w 0 = max PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

6 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reactions: = = wa From to B: 0 < x < a = 0: wa wx V = 0 F y x J = 0: wax + ( wx) + = 0 V = w( a x) = w ax x From B to : a < x < a = 0: wa wa V = 0 F y a J = 0: wax + wax + = 0 From to : a < x < F = 0: V w( x) + wa = 0 y V = 0 = wa V = w( x a) x J = 0: w( x) + wa( x) = 0 = wa[( x) ( x) ] PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

7 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl alculate reactions after replacing distributed load by an equivalent concentrated load. Reactions are = = w ( a ) From to B: 0 < x < a Σ Fy = 0: w( a) V = 0 V = w( a) Σ = 0: w( a) + = 0 From B to : a < x < a x a b = = w ( ax ) PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed, Place section cut at x. Replace distributed load by equivalent concentrated load. Σ Fy = 0: w( a) w( x a) V = 0 V w = x x a J = 0: w( a) x + w( x a) 0 + = [( ) ( ) ] = w a x x a

8 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl PROBE 5. (ontinu From to : a < x < t x =, Σ Fy = 0: V + w( a) = 0 Σ J = 0: + w( a)( x) = 0 w V = ( a) = w ( a )( x ) a max = w 8 PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

9 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl t B, V = 00N, = 0 t E +, F = 0 : V 00 = 0 V = 00N y = 0 : (0.5)(00) = 0 t +, E = 45 N m F = 0 : V = 0 y (a) V = 00 N = 0 : + (0.)(500) (0.55)(00) = 0 = 45 N m t +, F = 0 : V = 0 V = 00N y = 0 : (0.5)(00) + (0.55)(500) (0.75)(00) = 0 (b) = 7.5 N m PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

10 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl PROBE 5.8 (ontinued) t, F = 0 : V = 0 V = 00 N y = 0 : (0.)(00) (0.55)(00) + (0.85)(500) (.05)(00) = 0 = 7.5 N m PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

11 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reactions: = 0 : + ()(4) ()(40) = 0 = 8kN = 8kN = 0: ()(4) ()(40) = 0 = 7kN = 7kN to. 0 < x < m F = 0: 8 x V = 0 V = ( 8 x)kn y x J = 0: 8 x ( x) = 0 = ( 8x x )kn m to B. m < x < m F = 0: V 40= 0 y V = 40kN = 0: ( x)(40) = 0 K = (40x 0) kn m From the diagrams, (a) V = 40.0kN max (b) = 40.0kN m max PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

12 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reaction at : B = 0: R + (0.550)(75) + (0.00)(75) = 0 R = 85 N lso, R 5 N B = to : V = 85 N to : V = 0 N to B : V = 5 N t and B, = 0 Just to the left of, Σ = 0: (0.5)(85) + = 0 =.5 N m Just to the right of, Σ = 0: (0.5)(85) + (0.050)(75) + = 0 = 7.50 N m Just to the left of, = 0 N m Just to the right of, Σ = 0: (0.50)(85) + (0.00)(75) + = 0 Σ = 0: + (0.5)(5) = 0 =.5 kn (a) (b) V max = 85.0 N max =.5 N m PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

13 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Over the whole beam, Σ F = 0:.5w.5.5 = 0 w = kn/m y to : 0 x < 0. m Σ F = 0: x V = 0 V = ( x) kn y x Σ J = 0: ( x) + = 0 = ( x ) kn m t, x = 0. m V = 0. kn, = kn m to : 0. m < x <. m = 90 N m Σ F = 0: x.5 V = 0 V = (x.5) kn y x Σ J = 0: ( x) + (.5)( x 0.) + = 0 t the center of the beam: x = 0.75 m V = 0 = 0.5 kn m =.5 N m = ( x.5x ) kn m +, t x = 0. m, V = 0.9 kn (a) aximum V = 0.9 kn = 900 N (b) aximum =.5 N m PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

14 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reaction at : B = 0: (.0)() + (.5)() + (.8)(4.5)(.5) = 0 = 7.05 kn Use as free body. Σ = 0: (7.05)(.5) + (.8)(.5)(0.75) = 0 = 8.55 kn m = N m I c = bh = (80)(00) = 80 0 mm 4 4 = 80 0 m = (00) = 50 mm = 0.50 m c ( )(0.50) σ = = = Pa I 80 0 σ = 7. Pa PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

15 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Use B as free body. = 0: (00)() = 0 = 00 lb ft = 4. 0 lb in For rectangular section, I = bh = (4)(8) = 70.7 in c = h = 4 in. c (4. 0 )(4) σ = = =.05 0 psi σ =.0 ksi I 70.7 PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

16 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Use portion B as free body. = 0 : + ()(.)(.05) + (8)(.) = 0 =.45 kn m =.45 0 N m For W0 0 : S = mm = m Normal stress: σ =.45 0 = S = Pa σ = 7.7 Pa PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

17 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reactions: By symmetry, = B = 0 : = B = 80 kn F y Using left half of beam as free body, J = 0: (80)() + (0)(.) + (50)(0.4) + = 0 = 04 kn m = 04 0 N m For W0 5 : S = mm = m Normal stress: 04 0 σ = = = 9. 0 Pa S σ = 9. Pa PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

18 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Use entire beam as free body. B = 0: (.)() + (.)(50) + (0.8)(50) = 0 = 7 kn Use portion as free body. = 0: (.4)(7) + (.)() = 0 = 09. kn m For W40, S = 90 0 mm Normal stress: σ = = S N m 90 0 m = Pa σ = 9.5 Pa PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

19 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reactions: = 0 : 4 4 (4)()() = 0 = 8 kn F = 0 : 8 + (4)() = 0 = 7 kn y to : 0 < x < m to : m < x < 4m to B: 4m < x < m Fy = 0: V 8= 0 V = 8 kn J = 0: + 8x = 0 = ( 8 x) kn m Fy = 0: V 8= 0 V = 8 kn J = 0: + 8x 4= 0 = ( 8x + 4)kN m F y = 0: V 4( x) = 0 V = ( 4x + 44) kn max = 5 kn m = 5 0 N m J = 0: x 4( x) = 0 = ( x) kn m For S50 5 section, S = 48 0 mm 5 0 N m Normal Stress: σ = = =. 0 Pa S 48 0 m σ =. Pa PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

20 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Statics: onsider portion B and BE separately. = B = E = 0 Portion BE: E = 0: (9)(.) + (48)(.) () + (0)(.5) = 0 E = 48kN = 5kN t midpoint of B: Fy = 0: V = 0 = 0 : = (9)(.) (9)(0.) = 57. kn m Just to the left of : F = 0 : V = 9 48 = 44 kn y = 0 : = (9)(0.) (48)(0.) = 7 kn Just to the left of : Fy = 0 : V = 0 5 = + 04 kn = 0 : = (5)(.5) = + 84 kn m PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

21 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl PROBE 5. (ontinued) From the diagram: = 84 kn m = 84 0 N m max For W0 0 rolled steel shape, S x = mm = m Stress: σ = m S max σ m 84 0 = = Pa σ = 99.5 Pa m PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

22 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reaction at : = 0 : (5)(.)(.) = 0 B = 5 kn to : 0 < x <.m to B:.m < x < 4.8m F = 0: 5 V = 0 V = 5kN y = 0: x = 0 J = (0x 40) kn m Fy = 0 : 5 5( x.) V = 0 V = ( 5x + 75)kN K = 0: x x. + (5)( x.) = 0 = (.5x + 75x 7) kn m Normal stress: For W00., S = 98 0 mm σ = = = S 98 0 m N m Pa σ = 5.9 Pa PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

23 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl By symmetry, = B Σ F = 0: B = 0 y = B = kn Shear: to : V = kn + to : V = kn + to E : V = kn + E to B: V = kn Bending moment: t, Σ = 0: ()() = 0 = kn m t, +Σ = 0: ()() + (8)() = 0 4kN m By symmetry, = kn m at E. = kn m max = 4 kn m occurs at E. E For W0.8, = 80 0 mm = 80 0 m S x Normal stress: σ max 4 0 max = = Sx 80 0 = Pa σ max = 4.9 Pa PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

24 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl By symmetry, = B Σ F = 0: 8 + W 8 + B = 0 y = B = 8 0.5W Bending moment at : Σ = 0: (8 0.5 W)() + = 0 Bending moment at : = (8 0.5 W) kn m Σ = 0: (8 0.5 W)() + (8)() + = 0 = (8 W) kn m Equate: = W 8 = 8 0.5W (a) W = 0.7 kn =.7 kn m =.7 kn m =.7.0 N m =.7 kn m max For W0.8 rolled steel shape, W = 0.7 kn S x = 80 0 mm = 80 0 m (b) σ max.7 0 max Sx 80 0 = = = Pa σ = 9.5 Pa max PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

25 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl P = 480 N Q = 480 N Reaction at : Σ = 0: a + 480( a 0.5) 480( a) = 0 70 = 90 N a Bending moment at : Σ = 0: = 0 0 = 0.5 = 480 N m a Bending moment at : Σ = 0: 480( a) = 0 = 480( a) N m (a) (b) Equate: For rectangular section, 0 = 480( a) = 480 a a = 0.80 m a = 8 mm = 8. N = 4. N m = 4. N m S = bh S = ()() = 48 mm = 48 0 m σ 9 4. = = = Pa S.48 0 max max 9 σ = 99. Pa max PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

26 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl P = 480 N Q = 0 N Reaction at : Σ = 0: a + 480( a 0.5) 0( a) = 0 50 = 800 N a Bending moment at : Σ = 0: = 0 80 = 0.5 = 400 N m a Bending moment at : Σ = 0: 0( a) = 0 = ( a) N m (a) Equate: 80 = 0 0a = 400 a 0a + 80a 80 = 0 a = m,.087 m (b) Reject negative root. =.04 N = N m = N m For rectangular section, S = bh ()(8) 48 mm 48 0 S = = = m σ = = = Pa S 48 0 max max 9 max a = 89 mm σ = 89.5 Pa PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

27 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Weight density: γ = ρg et = total length of beam. W = ρg = b ρg Reactions at and : = = W Bending moment at : W Σ = 0: + = 0 W = 8 Bending moment at center of beam: W W W Σ E = 0: = = 4 For a square section, Normal stress: W b ρ g max = = 8 8 S = b bρg/8 ρg σ = = = S b / b Solve for b: b = ρ g σ ata: =. m ρ = 780 kg/m g = 9.8 m/s (a) σ = 0 0 Pa (b) σ = 50 0 Pa (a) (b) (.) (780)(9.8) b = =. 0 m ()(0 0 ) (.) (780)(9.8) b = =. 0 m ()(50 0 ) b =. mm b =. mm PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

28 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Σ = 0: bp = 0 = Pb Σ = 0: ap = 0 = Pa t +, V = = Pb = 0 to B : 0 < x < a t +, B x 0 w = 0 wdx = 0 V V = 0 a apb Pba B = Vdx = dx = 0 0 Pb Pa V = P = P = B + to : a < x < B Pb V = Pba = x a w = 0 wdx = 0 V V = 0 B Pa Pab B = Vdx = ( a) = a Pab Pba Pab = B = = 0 Pa V = max Pab = PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

29 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl w Σ B = 0: + w = 0 = w Σ = 0: B w = 0 B = dv w dx = x 0 V V = wdx = wx V = V wx = wx d V dx = x xw = Vdx = wx dx 0 0 wx wx = V w = wx wx wx = + w = ( x x ) aximum occurs at x =, where V d = = 0 dx w max = 8 PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

30 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Over B: V = 0 = 0 x 0 V = wdx = wx d = V = wx dx x wx = Vdx = 0 x 0 wx = t B: x = a VB = wa wa B = Over B: w = 0 dv 0 V constant VB dx = = = V = wa d V wa dx = = x x B a a = Vdx = wax = wa ( x a ) = wa( x a) wa = wax a PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

31 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl w = x w0 V V = 0, = 0 x wx 0 wx 0 V = = 0 dv W0x = w = dx d dx w0 x = V = V wx 0 = wx = V dx = dx x x wx 0 = PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

32 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reactions: = = wa to B: 0 < x < a w = w V = = wa, = 0 x 0 V V = wdx = wx V = w( a x) d V wa wx dx = = x = Vdx = ( wa wx ) dx 0 0 VB = 0 B = wa x = wax wx B to : a < x < a V = 0 d V 0 dx = = x a = V dx = 0 B = B = wa PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

33 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl PROBE 5.8 (ontinued) x a to : V V = wdx = w[ x ( a)] x x a a = V dx = [ wx ( a )] dx x = w ( a) x x a x ( a) = w ( a) x + ( a) x ( a) = w ( a) x + V = w[ x ( a)] x ( a) = wa w ( a) x + PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

34 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reactions. = = w ( a ) t. V = = w ( a ), = 0 to B. 0 < x < a w = 0 V V = wdx= 0 B VB = V = w ( a ) 0 a a a B = V dx = w ( a ) dx 0 0 B = w ( aa ) B to. a < x < a w = w x V V = wdx= w( x a) B a V = w( a) w( x a) = w( x) d V w( x) dx = = x x ( B a ) a = V dx = w x x ( wx x a a ) = + = w ( aa ) + wx ( x a+ a) ( ) = wx x a PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

35 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl PROBE 5.9 (ontinued) t. x = a V = w( a) = ( a) a to. V = V = w ( a ) = 0 t x =, a max = w 8 PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

36 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl = 0 : (0.)(00) (0.55)(00) = 7.5 N m + (0.85)(500) (.05)(00) = 0 F = 0 : V = 0 y V = Shear: to : 00 N V = 00 N to : V = = 00 N to E: V = = 00 N E to B: V = = 00 N reas under shear diagram: to : Vdx = (00)(0.) = 0 N m to : Vdx = ( 00)(0.5) =.5 N m to E: Vdx = ( 00)(0.) = 90 N m Bending moments: = 7.5 N m E to B: Vdx = (00)(0.5) = 45 N m = + V dx = = 7.5 N m = + V dx = = 45 N m E E = + V dx = = 45 N m B E E = + V dx = = 0 (a) aximum V = 00 N PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed, (b) aximum = 7.5 N m

37 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reactions: = 0 : + ()()() (40)() = 0 = 8kN = 0 : ()()() (40)() = 0 = 7kN Shear diagram: V = 8kN to : 0 < x < m w = kn/m V V = wdx= dx= 4 kn V 0 0 = 4 8 = kn to B: V B = + 7 = 40 kn reas of shear diagram: to : Vdx = ( 8 )() = 40 kn m to B: Vdx = ()(40) = 40 kn m Bending moments: = 0 = + Vdx = 0 40 = 40kN m = + Vdx = = 0 B (a) aximum V = 40.0 kn (b) aximum = 40.0 kn m PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

38 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reaction at : Reaction at B: B =.75 kn Beam B and loading: (See sketch.) reas of load diagram: to : (.4)(.5) = 8.4 kn to B: (0.)(.5) =. kn Shear diagram: Σ B = 0:.0 + (.5)(.0)(.5) + (.5)() = 0 =.75 kn V V V + Over to, V =.75.5x V B =.75 kn = =.5 kn =.5 = 4.5 kn = 4.5. =.75 kn t G, V =.75.5x = 0 x =.98 m reas of shear diagram: G G to G: G to : to B: (.98)(.75) =.5089 kn m (0.474)(.5) = kn m (0.)( ) =.4 kn m PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

39 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl PROBE 5.44 (ontinued) Bending moments: = 0 G + B = =.5089 kn m = =. kn m =..7 =.4 kn m =.4.4 = 0 (a) (b) V max =.75 kn max =.5 kn m PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

40 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl = 0: B + ()(4) + (0.5)(4) = 0 = kn = 0: B ()(4) (.5)(4) = 0 B = kn Shear diagram: to : V = kn to : V = 4 = kn to B: V = 4 = kn reas of shear diagram: to : Vdx = ()() = kn m to : Vdx = ()( ) = kn m to E: Vdx = ()( ) = kn m Bending moments: = 0 = 0 + = kn m + + B = + 4 = kn m = = 4kN m = 4 + = kn m = = 0 (a) (b) V =.00 kn max max =.00 kn m PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

41 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl By symmetry, = B. Σ F = 0: + B (4.5)(.8) = 0 y = B = 7.05 kn Shear diagram: to : V = 7.05 kn w =.8 kn/m t, V = 7.05 (.8)(.5) = 4.5 kn t +, V = 4.5 =.5 kn + to : w =.8 kn/m t, V =.5 (.5)(.8) =.5 kn t +, V =.5 = 4.5 kn + to B: w =.8 kn t B, V = 4.5 (.5)(.8) = 7.05 kn raw the shear diagram: V = 0 at point E, the midpoint of. reas of the shear diagram: to : to E: E to : to B: ( )(.5) = 8.55 kn m (.5)(0.75) = kn m (.5)(0.75) kn m = ( )(.5) 8.55 kn m = PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

42 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl PROBE 5.4 (ontinued) Bending moments: E B = 0 = = 8.55 kn m = = kn m = = 8.55 kn m = = 0 = N m For a rectangular section, (80)(00) S = bh = =. 0 mm =. 0 m aximum normal stress at : σ = = S. 0 = Pa σ = 7. Pa PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

43 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reactions: By symmetry, = B. = 0 : = B = 80 kn F y Shear diagram: to : V = 80 kn to : V = 80 0 = 50 kn to E: V = = 0 reas of shear diagram: to : Vdx = (80)(0.8) = 4 kn m to : Vdx = (50)(0.8) = 40 kn m to E: Vdx = 0 Bending moments: = 0 = = 4kN m = = 04 kn m = = 04 kn m E = 04 kn m = 04 0 N m max For W0 5, S = mm = m Normal stress: 04 0 σ = = = 9. 0 Pa S σ = 9.Pa PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

44 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl (a) k =. aximum occurs at x =. wx 0 kw0( x) wx 0 w= = ( + k) kw. dv wx 0 = w= kw0 ( + k) dx wx 0 0 V = kw x ( + k) + V = 0 at x= 0 = 0 d wx 0 = V = kw 0 x ( + k ) dx kw0x w0x = ( + k) + = 0 at x = 0 = 0 kw0x ( + k) w0x = wx 0 = w0 x V wx 0 wx 0 = w 0 max = (b) k =. t V = 0 at x = ( ) ( ) w w w x =, = = = w t x =, = 0 V wx 0 wx 0 = 4 0 wx 0 wx = 4 4 w 0 = max 7 PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

45 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl dv dx = w= w x d V = w0 + = dx x = w + x+ 0 x 0 = 0 at x = 0 = 0 = 0 at x= 0= w0 + = w0 (a) (b) x 0 0 V = w + w x 0 0 = w + w x d max occurs when V 0. xm 0 V = w0 ( x )/ = w0 ( x x / ) dx = = = xm = max = w 0 = 0.04w max 0 PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

46 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl dv π x = w= w0 sin dx w 0 π x d V = cos + = π dx 0 w π x = sin + x+ π = 0at x = 0 = 0 = 0at x= 0= = 0 w 0 π x (a) V = cos π d V 0 at x dx = = = w 0 π x = sin π (b) w 0 max = π π sin w 0 max = π PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

47 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl = 0 : ()() ()(4)() + 4B = 0 B = 5.5 kn = 0 : (5)() + ()(4)() 4 = 0 B = 8.5 kn Shear: to : V = kn + : V = =.5kN B: V =.5 ()(4) = 5.5 kn ocate point where V = 0. reas of the shear diagram: d 4 d = d = d =.7 m 4 d =.8 m to : Vdx = (.0)() =.0kN m to : Vdx = (.7)(.5) = kn m to B: Bending moments: = 0 Vdx = (.8)( 5.5) = kn m = 0.0 =.0kN m = = kn m = = 0 B aximum = kn m = N m PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

48 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl PROBE 5.55 (ontinued) For pipe: co = do = (0) = 80 mm, ci = di = (40) = 70 mm ( o i ) π π I = c c = (80) (70) =.5 0 mm S I.5 0 = = =.40 0 mm =.40 0 m c 80 o Normal stress: σ = = = 0. 0 Pa S.40 0 σ = 0. Pa PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

49 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl w = 0 = 0: 4 R + ()(50) ()(50) = 0 = 0: R 4 R ()(50) ()(50) = 0 R = 50 kn = 50 kn Shear: to : V = 50 kn V = 50 kn to : V = = 00 kn to B: V = = 50 kn reas of shear diagram: to : Vdx = (50)() = 00 kn m to : Vdx = ( 00)() = 400 kn m to B: Vdx = (50)() = 00 kn m Bending moments: = 0 = + Vdx = = 00 kn m = + Vdx = = 00 kn m = + Vdx = = 0 B aximum = 00 kn m = 00 0 N m For W40 4 rolled steel section, S x = 00 0 mm = 00 0 m σ m max 00 0 = = =.4 0 Pa S 00 0 x σ =.4 Pa m PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

50 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reaction: = 0 : (80)(.)() = 0 B = 7kN Shear: V = to : 7kN V = 7 kn V = 7 (80)(.) = 5 kn to : ocate point where V = 0: V( x) = 80x + 7 = 0 x = 0.95 m reas of shear diagram: V = 5 kn to : Vdx = (.)(7) = 9. kn m to E: E to : Vdx = (0.95)(7) =. kn m Vdx = (0.5)( 5) =.9 kn m to B: Vdx = (.)( 5) =.4 kn m Bending moments: = 0 kn m = =. kn m =. +. = 7. kn m E = 7..9 = 50.4 kn m = = kn m B For W50 80, S = 98 0 mm Normal stress: σ 7. 0 N m max = = = Pa S 98 0 m σ = 8.5 Pa m PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

51 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl (a) Replace distributed load by equivalent concentrated load at the centroid of the area of the load diagram. For the triangular distribution, the centroid lies at x =. W = w0 (a) Fy = 0: R W = 0 R = w0 = 0:0= 0 equilibrium V = 0, = 0, at x = < x <, dv = w = w x dx 0 0 d = V = w x + = w x dx 0 wx 0 wx = + = Just to the left of, V w0( /) = = w0 9 5 Just to the right of, V = w0 + R = w0 9 8 Note sign change. aximum occurs at. 4 aximum = w For square cross section, I = a c = a w0 ( /) 4 = = w 0 8 (b) σ m c max 4 w 8 w w = = = = I 8 a 7 a a w 0 σ m = (.5 a) PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

52 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl (8)() mm 4 I = = c= d = 8 mm I S = = mm = m c t, = Sσ = ( )( 5.9) =.5 N m t, = Sσ = ( )( 9.9) =.5 N m Σ = 0:. (0.)(400) 0.P 0.5Q= 0 0.P+ 0.5Q = 8.5 () Σ = 0:.5 (0.05)(00) 0.P 0.5Q= 0 0.P+ 0.5Q= 0.5 () Solving () and () simultaneously, P = 500 N Q = 50 N Reaction force at : R = 0 R = 50 N m V = 50 N V = 50 =.5 N m =.5 N m =.5 N m V max = 50 N max = N m PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

53 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl (a) t, (4)(0) 4 0 mm 4 0 mm I = = c= I S = = mm = m = Sσ c = (4.4 0 )(55 0 ) = 79 N m t F, F = (4.4 0 )(7.5 0 ) = 540 N m Using free body FB, Σ = 0: B= 0 F 540 B = = 800 N 0. Using free body EFB, Σ = 0: 79 Q+ (0.8)(800) = 0 Q = 0 N Using entire beam, Σ = 0: 0. P (0.7)(0) + (.)(800) = 0 P = 40 N = 00 N Shear diagram and its areas: to : Σ F = 0: = 0 V = 00 N = (0.)(00) = 70 N m + to E : V = = 0 N = (0.5)(0) = 80 N m E + to B: V = 0 0 = 800 N = (0.5)( 800) = 900 N m Bending moments: E EB y (b) E B = 0 = = 70 N m = = 900 N m = = 0 Normal stress. max = 900 N m σ 900 = = =.5 0 Pa S σ =.5 Pa max max max PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

54 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reactions: onstruct shear and bending moment diagrams: Σ = 0:.4 + (.)(.8) + (0.8)(.) = 0 =.4 kn Σ = 0: (0.8)(.8) (.)(.) +.4 = 0 = kn max =.4 kn m=.4 0 N m σ all = Pa = 0 Pa max.4 0 Smin = = σ all 0 = 00 0 m = 00 0 mm (40) S = bh = h = 00 0 ()(00 0 ) h = 40 = 0 0 mm h = 7. mm PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

55 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reactions: F = 0: (.)(8) = 0 y =. kn = 0: (.8)(.)(8) = 0 = 8.88 kn m Shear diagram: V = V =. kn B V =. (.)(8) = 0 reas of shear diagram: to B : (.)(.) = 5.9 kn m Bto : (.)(.) =.9 kn m Bending moments: = 8.88 kn m = =.9 kn m B = = 0 = 8.88 kn m = N m max σ = max S max S max N m = = = 40 0 m = 40 0 mm σ 0 Pa max For a rectangular section, S = bh S (40 0 ) h = = = 94mm b 5 h = 94 mm PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

56 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reactions: By symmetry, =. Fy = 0 : ()(.5) ()(.5) ()(.5) = 0 Shear diagram: = =.75 kips V =.75 kips V B =.75 ()(.5) = 4.5 kips V = 4.5 ()(.5) = 4.5 kips V = 4.5 ()(.5) =.75 kips ocate point E where V = 0 : By symmetry, E is the midpoint of B. reas of the shear diagram: to B : ()(4.5) + ()(.5) = 8 kip ft B to E : ()(4.5) =.75 kip ft E to : ()( 4.5) =.75 kip ft to : By antisymmetry, 8 kip ft PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

57 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl PROBE 5. (ontinu) Bending moments: = 0 = = 8 kip ft B = = 4.75 kip ft E = = 8 kip ft = 8 8 = 0 σ max max max (4.75 kip ft)( in/ft) = S = = = 9.74 in S σ.750 ksi max For a rectangular section, S = bh S (9.74) h = = = 4.7 in. h = 4.7 in. b 5 PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

58 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl By symmetry, B = Shear: to B: V =.5 kn V + = = 9 kn B V = 9 ()() = 9 kn to : V = =.5 kn reas of the shear diagram: Σ F = 0: B ()() = 0 B= =.5 kn to B: Vdx = (0.)(.5) =.5 kn m B to E: E to : to : y Vdx = (.5)(9) =.75 kn m Vdx =.75 kn m Vdx =.5 kn m Bending moments: = 0 B = =.5 kn m E = = 8.5 kn m = =.5 kn m =.5.5 = 0 aximum = 8.5 kn m = N m σ all = Pa = 0 Pa max Smin = = = σ 0 m = mm all For a rectangular section, S = bh = (00) h ()( ) h = = mm h = 0 mm 00 PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

59 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl B = 0:.4 + (0.)(.)() = 0 =.7 kn = 0: (.8)(.)() +.4B = 0 B = 8. kn Shear: V V V B + B V =.7 kn =.7 (.4)() = 4.5 kn = =. kn =. (.)() = 0 ocate point where V = 0. d.4 d = d =.48 d = 0.9 m.4 d =.5 m reas of the shear diagram: to : to B: B to : Vdx = (0.9)(.7) =.5 kn m Vdx = (.5)( 4.5) =.75 kn m Vdx = (.)(.) =. kn m Bending moments: = 0 B = =.5 kn m =.5.75 =. kn m =. +. = 0 aximum =. kn m =. 0 N m σ all = Pa = 0 Pa S min σ all 0. 0 = = = 80 0 m = 80 0 mm For rectangular section, S = bh = b (50) = 80 0 ()(80 0 ) b = b = 48.0 mm 50 PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

60 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Shape S, ( 0 mm ) W w = + x ( )kn/m = + x dv = w = x dx V = x x + V = 0 at x = 0, = 0 d V = = x x dx = x x + = 0 at x = 0, = 0 = x x occurs at x = m. max max = ()() () = 80 kn m = 80 0 N m σ all = 0 Pa = 0 0 Pa S 80 0 = = =.5 0 m = 5 0 mm min σ all 0 0 W ightest acceptable wide flange beam: W50 W W W W PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

61 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl S min Section modulus σ = 0 pa all max 8 kn m = = = m σ 0 Pa all = mm Shape S, ( 0 mm ) W W W40 90 W W0 00 W Use W50 9 PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

62 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reaction: = 0 : 0 + (7.5)(0) + (5)(40) = 0 = 5kN Shear diagram: to B: V = 5 kn B to : V = 5 0 = 5 kn to : V = 5 40 = 5 kn reas of shear diagram: to B: (.5)(5) =.5 kn m B to : (.5)(5) =.5 kn m to : (5)( 5) = 75 kn m Bending moments: = 0 = =.5 kn m B = = 75 kn m = = 0 = 75 kn m = 75 0 N m max σ all = 0 Pa = 0 0 Pa Shape S x, ( 0 mm ) S S S S 75 0 = = = m min σ all 0 0 = mm ightest S-section: S PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

63 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reactions: By symmetry, B =. F = 0 : 70 + B (9)(45) + 70 = 0 y B = = 7.5kN Shear: V = 70 kn V = = 70 kn B V + = = 0.5 kn B V = 0.5 (9)(45) = 0.5 kn V + = = 70 kn V = 70 kn raw shear diagram. ocate point E where V = 0. E is the midpoint of B. reas of the shear diagram: to B: Vdx = ()( 70) = 0 kn m B to E: Vdx = (4.5)(0.5) = kn m E to : Vdx = (4.5)( 0.5) = kn m to : Vdx = ()(70) = 0 kn m PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

64 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl PROBE 5.7 (ontinu) Bending moments: = 0 = 0 0 = 0.5 kn m B = = 45.5 kn E = = 0 kn = = 0 = 45.5kN m = N m max σ all = 0Pa = 0 0 Pa Shape S( 0 mm ) S S S ightest S-shape S σ = S S = = =.55 0 m σ 0 0 = mm PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

65 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl = 0 : ()(.5) (.5)(.5) ()(.5) = 0 =.75 kn m max = =.75 kn m S S I max min σ all min min min c Smin c 4 4 ( c c ) min max max = c I min = (50) (.5 0 ) =.85 0 mm c c max.75 0 N m = = = 45 0 m = 45 0 mm 50 0 Pa I = I = = (50)(45 0 ) =.5 0 mm π = π π = mm t = c c = = 7.0 mm min max t = 9mm PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

66 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reactions. By symmetry, B = F = 0: B + 0.9w = 0 y Shear: V = 0 B = = 0.45w V = 0 0.w = 0.w B V = 0.w w = 0.5w B + V = 0.5w 0.5w = 0.5w V = 0.5w w = 0.w + V = 0.w 0.w = 0 reas: to B. (0.)( 0. w ) = 0.0 w entroid and moment of inertia: B to E (0.5)(0.5 w ) = 0.05 w Bending moments: = 0 B = 0 0.0w= 0.0w = 0.0w+ 0.05w= 0.05w E Part,mm y,mm y (0 mm ) d, mm. 4 4 d (0 mm ) I (0 mm ) Σ Y 0 0 = = 50 mm I = d + I = 0 0 mm PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

67 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl PROBE 5.8 (ontinued) Top: Bottom: Iy / = (0 0 )/0 = mm = m Iy / = (0 0 ) / ( 50) = 7. 0 mm = 7. 0 m Bending moment limits ( = σ Iy / ) and load limits w. Tension at B and : ompression at B and : Tension at E: ompression at E: 0.0 w = (80 0 )(45. 0 ) 0.0 w = ( 0 0 )(7. 0 ) 0.05 w = (80 0 )(7. 0 ) 0.05 w = ( 0 0)(45. 0 ) w = 8. 0 N/m w = N/m w = N/m w = N/m The smallest allowable load controls: w = N/m w = 7.8 kn/m PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

68 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl Reactions: By symmetry, B = Σ F = 0: B+ 0.9w= 0 y B = = 0.45w Shear: V = 0 V = 0 0.w= 0.w B V + = 0. w+ 0.45w= 0.5w B V = 0.5w 0.5w= 0.5w V + = 0.5w+ 0.45w= 0. w V = 0. w 0. w= 0 reas: to B: (0.)( 0. w ) = 0.0 w B to E: (0.5)(0.5 w ) = 0.05 w Bending moments: = 0 B = 0 0.0w= 0.0w = 0.0 w+ 0.05w= 0.05w entroid and moment of inertia: Part,mm y,mm E y,(0 mm) d,mm 4 d (0 mm ) 4 I,(0 mm) Σ Y = = 0mm I =Σ d +Σ I = 0 0 mm 400 PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

69 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl PROBE 5.8 (ontinued) Top: Bottom: Iy / = (0 0 ) / (50) = 7. 0 mm = 7. 0 m 8 Iy / = (0 0 ) / ( 0) = mm = m Bending moment limits ( = σ Iy / ) and load limits w. Tension at B and : ompression at B and : Tension at E: ompression at E: 0.0 w = (80 0 )(7. 0 ) 0.0 w = ( 0 0 )( ) 0.05 w = (80 0 )( ) 0.05 w = ( 0 0 )(7. 0 ) w = N/m w = N/m w =.4 0 N/m w = 4. 0 N/m The smallest allowable load controls: w = N/m w = 08.8 kn/m PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

70 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl For W40 74, S = 40 0 mm = 40 0 m σ all = 40Pa = 40 0 Pa all = Sσ all = (40 0 )(40 0 ) = N m = 04.4 kn m Reactions: By symmetry, = B, = + Σ F = 0: + B ()() = 0 y = B = 98 kn = 98 0 N +Σ Fy = 0: + l = 0 = = ( l) kn () Shear and bending moment in beam B: 0 < x< a, V = 98 x kn t, x a. Set all. = = max = 98x x kn m = 98a a kn m = a a 98 = 04.4 a 98a = 0 a = 4.75 m,.487 m (a) By geometry, l = a=.5 m l =.5 m From (), = = 0.5 kn raw shear and bending moment diagrams for beam. V = 0 at point E, the midpoint of. PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

71 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl PROBE 5. (ontinued) rea from to E: Vdx = (0.50) l 9.0 kn m = S E = 9.0 kn m = N m E min σ all 40 0 = = =.44 0 m =.44 0 mm Shape S (0 mm ) W W W W W (b) Use W0 44. PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

72 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl B = = 0 VB = V = (7.) w=. w rea B to E of shear diagram: (.)(. w ) =.48 w = 0+.48w=.48w E entroid and moment of inertia: Part (mm ) y (mm) y (mm ) d (mm) Σ d (mm ) I (mm ) Y = =.4 mm I =Σ d +Σ I = 0. 0 mm ocation y (mm) Iy / (0 mm ) Top Bottom also (0 m ) PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

73 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl PROBE 5. (ontinued) Bending moment limits: = σ Iy / Tension at E: (0 0 )( ) = 9. 0 N m ompression at E: ( 50 0 )(58. 0 ) = N m Tension at and : (0 0 )(58. 0 ) = N m ompression at and : (a) llowable load w: ( 50 0 )( ) =. 0 N m.48w= 9. 0 w= N/m w =.485 kn/m Shear at : V = ( a+.) w rea to B of shear diagram: av ( ) + VB = aa ( + 7.) w Bending moment at (also ): (b) istance a: = a ( a+ 7.) w aa ( + 7.)( ) = a + a = a =.95 m PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

74 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl B B = = 0 V = V = P rea B to E of shear diagram:.4 P E = P =.4 P = F entroid and moment of inertia: Part (mm ) y (mm) y (mm ) d (mm) Σ d (mm ) I (mm ) Y = =.4 mm I =Σ d +Σ I = 0. 0 mm ocation y (mm) Iy / (0 mm ) Top also (0 m ) Bottom PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

75 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl PROBE 5. (ontinued) Bending moment limits: = σ Iy / Tension at E and F: (0 0 )( ) = 9. 0 N m ompression at E and F: ( 50 0 )(58. 0 ) = N m Tension at and : (0 0 )(58. 0 ) = N m ompression at and : ( 50 0 )( ) =. 0 N m (a) llowable load P: Shear at :.4 P = 9. 0 P = N P = 4.0 kn V = P rea to B of shear diagram: Bending moment at : av = ap = ap = a (b) istance a: a =. 0 a =.7 m PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

76 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl ead load: = 48 ft a = 4 ft P = 4 kips P = kips W = 0.75 kip/ft R rea to E of shear diagram: = RB = (48)(0.75) = 8 kips (8)(8) = kip ft max = kip ft = 59 kip in at point E. ap (4)() ive load: u = = =.4 ft ( P + P) ()(0) x= u = 4.4 =. ft x+ a =. + 4 =. ft x a = 48. =.4 ft Σ B = 0: 48 R + (5.4)(4) + (.4)() = 0 R = 4.5 kips PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

77 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl PROBE 5. (ontinued) Shear: to : V = 4.5 kips to : V = = kips to B: V = kips rea: to : (.)(4.5) = 9.5 kip ft Bending moment: = 9.5 kip ft = 8 kip in esign: S γ + γ = ϕ = ϕσ S min U γ + γ = ϕσ U (.5)(59) + (.75)(8) = (0.9)(0) = 84. in U min Shape S (in ) W W7 84 Use W7 84. W W 0 7 W PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

78 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl = 48 ft a = 4 ft P = 4 kips P = kips W = 0.75 kip/ft See solution to Prob for calculation of the following: = 59 kip in = 8 kip in For rolled steel section W7 84, S = in llowable live load moment * : * U * ϕσ U = γ + γ = ϕ = ϕσus S γ γ (0.9)(0)() (.5)(59) =.75 = 47 kip in Ratio: * = 8 = = + Increase.%. PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

79 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl = m, w = 50 N/m = 0.5 kn/m w = 00N/m= 0.kN/m, P = kn ead load: rea to of shear diagram: Bending moment at : ive load: R = ()(0.5) =.8 kn (8)(.8) =. kn m. kn m =. 0 N m R = [()(0.) + ] = 7.8 kn Shear at : V = 7.8 (8)(0.) = kn rea to of shear diagram: Bending moment at : esign: (8)(7.8 + ) = 4. kn m 4. kn m = 4. 0 N m γ + γ = ϕ = ϕσ S U U U γ + γ (.)(. 0 ) + (.)(4. 0 ) S = = ϕσ (0.9)(50 0 ) = m = mm For a rectangular section, S = bh S ()( ) h = = h = 8 mm b 75 PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

80 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl P R = R B = P Σ J = 0: x+ = 0 Px = 0 Px S = = σ σ all all < x < For a rectangular cross section, Equating, S = bh all Px Px bh = h = σ σ b all P x (a) t x =, h= h0 = h= h0, 0 < x< σ b For x >, replace x by x. all (b) Solving for P, allbh0 σ ()(7 0 )(0.05)(0.00) P = = = 0 0 N P = 0 kn ()(0.8) PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

81 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl By symmetry, = B For 0 x, Σ Fy = 0: w0+ B= 0 = B = w0 4 wx 0 dv wx 0 wx 0 w= = w= V = dx t x = 0, V = w0: = w0 4 4 wx 0 wx d = V = w = + w x dx 4 4 t x = 0, = 0: = 0 (a) t x =, For constant strength, For a rectangular section, w ( 4 ) 0 = x w 0 = = 4 w = 0, S S = S = = = = ( x 4 x ) σ σ σ 0 0 all all all S0 0 S h S = bh S0 = bh0 = S0 h0 (b) ata: = 800 mm h0 = 00 mm b= 5 mm σ all = 7 Pa x 4x h= h0 S (5)(00).7 0 mm.7 0 m 0 = bh0 = = = all 0 0 = σ S = (7 0 )(.7 0 ) = 0 N m ()( 0 ) w = = = 5 0 N/m w 0 = 5 kn/m (0.800) PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

82 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl dv wx 0 = w = dx wx 0 d V = = dx wx 0 wx 0 = = wx 0 S = = σ σ all all For a rectangular cross section, Equating, S = bh wx 0 wx 0 bh = h = σ all σ b all t x =, h= h = 0 w 0 all σ b (a) ata: = 750 mm = 0.75 m, b= 0 mm = 0.00 m w 0 = = σ all = = 00 kn/m 00 0 N/m, 00 Pa 00 0 Pa h x h = 0 / (b) h 0 (00 0 )(0.75) = = m (00 0 )(0.00) h 0 = 7.7 mm PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

83 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl dv π x = w= w0 sin dx w 0 π x V = cos + π w 0 V = 0 at x = 0 = π d w 0 π x = V = cos dx π w 0 πx w 0 πx = x sin = x sin π π π π w π x S = = 0 x sin σall πσ all π For a rectangular cross section, Equating, S = bh w π x 0 bh = x sin πσ all π w 0 π h x x = sin πσ allb π / t x =, (a) ata: / 0 w 0 w h= h0 = =.78 πσ allb π σ allb x π x h= h0 sin π π = 750 mm = 0.75 m, b= 0 mm = 0.00 m w 0 = = σ all = = 00 kn/m 00 0 N/m, 00 Pa 00 0 Pa / x h=.59 h0 sin π π x / (b) h 0 (00 0 )(0.75) =.78 = m (00 0 )(0.00) h 0 = 97. mm PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

84 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl R P = RB = 0 < x < P Σ J = 0: x+ = 0 Px max x = or =. Bending moment diagram is two straight lines. t, S = bh = et be the point where the thickness changes. t, max max x S = bh =. 00 mm h 00 mm 4. S h x = = = = = x = 0. m S l =. x = 0.9 l =.800 m PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

85 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl 0.8 N R = RB = = 0.4 w Shear: to : V = 0.4 w to B: V = 0.4 w reas: to : (0.8)(0.4) w= 0. w to E: Bending moments: (0.4)(0.4) w= 0.08 w t, = 0.40 w to : = 0.40 wx t, S = bh = max = 0.40w et F be the point were the thickness changes. t F, S bh wx F = F F = 0.40 F F F 00 mm F 0.40 F = = = = = h 00 mm S h wx S w x F l = 0.5 m =. xf = 0.95 m l =.900 m PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

86 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl = = 0 RkN R B x Σ J = 0: 0 x+ (40 x) + = 0 = 0 x 0x kn m t center of beam, x = 4m = 0 kn m t, t center of beam, I = Ibeam + Iplate x = (8 l ) =.5 m = 95 kn m = 0 + (00)(7.5) + + (00)(7.5) 4 = mm 457 c = = mm I S = = mm = m c (a) Normal stress: 0 0 σ = = = 5. 0 Pa S σ = 5. Pa t, S = 40 0 mm = 40 0 m (b) Normal stress: 95 0 σ = = =. 0 Pa S 40 0 σ =. Pa PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,

87 F. P. Beer et al., eccanica dei solidi, Elementi di scienza delle costruzioni, 5e - isbn , 04 cgraw-hill Education (Italy) srl R = R B = w x Σ J = 0: wx+ wx + = 0 w = ( x x ) w = x ( x ) where w = 0 kn/m and =. m. For the tapered beam, h= a+ kx a = 0 mm 00 0 k = = mm/m For a rectangular cross section, Bending stress: S = bh = b ( a+ kx ) w x x σ = = S b ( a+ kx) dσ To find location of maximum bending stress, set = 0. dx σ ( + ) ( ) ( )( + ) = = 4 d w d x x w a kx x x x a kx k dx b dx ( a+ kx) b ( a+ kx) w( a+ kx)( x) k( x x ) = b ( a+ kx) wa+ kx ax kx kx + kx = b ( a+ kx) wa ax+ kx = 0 = b ( a+ kx) PROPRIETRY TERI. 0 The cgraw-hill ompanies, Inc. ll rights reserved. No part of this anual may be displayed,