DESIGN OF EAMS AND SHAFTS! asis for eam Design! Stress Variations Throughout a Prismatic eam! Design of pristmatic beams! Steel beams! Wooden beams! Design of Shaft! ombined bending! Torsion 1
asis for eam Design - The effects of an internal axial force are often neglected in design. - Application of shear and flexure formulas are limited to beams made of a homogeneous material that has linear-elastic behavior. - The cross-sectional area must have an axis of symmetry in the plane of the loading. - The design does provide an adequate means of obtaining both a safe and economical design. 2
Stress Variations Throughout a Prismatic eam P w A R A - 2 m 2 m R V M R A R A R A - P - M max x -R x - 5 4 3 2 1 V M τ 4 τ 3 τ 2 Shear stress distribution (τ) σ 5 σ 4 σ 5 σ 2 ending stress distribution (σ) 3
- 5 4 3 2 1 V M τ 4 τ 3 τ 2 τ max σ 5 σ 4 τ σ 2 σ 5 5 y x σ 5 σ 1 x (-σ 5, 0) σ 2 σ y (0, 0) 4 y x σ 4 x (-σ 4, τ 4 ) σ 1 τ max τ max τ σ 2 σ τ 4 τ max τ y (0, -τ 4 ) x (0, τ 3 ) 3 y x σ 1 σ 2 σ τ 3 y (0, -τ 3 ) 4
- 5 4 3 2 1 V M σ 5 τ 4 σ 4 τ 3 τ 2 2 σ1σ τ τ max x (σ 2, τ 2 ) 2 y x σ 2 σ 1 σ 2 σ τ 2 y (0, -τ 2 ) τ max τ τ max 1 y x σ 1 σ 1 y (0, 0) σ 2 σ x (σ 1, 0) τ max 5
Prismatic eam Design Section Modulus (S) σ Mc I M I / c M S S req' d M σ allow 6
Example 1 A beam is to be made of steel that has an allowable bending stress of σ allow 150 MPa and an allowable shear stress of τ allow 100 MPa. Select an appropriate W shape that will carry the loading shown. 200 kn 100 kn 2 m 2 m 2 m 7
200 kn 100 kn 50 kn 250 kn 2 m 2 m 2 m V(kN) 50 M(kN m) -150 100-100 - -200 x(m) x (m) Allowable normal stress: σ allow 150 MPa σ S req d M S M σ allow (200x10 3 ) 1333x10 3 mm 3 150x10 6 Using the table in Appendix, the following beams are adequate: W 610x82 S 1870x10 3 mm 3 W 460x97 S 1910x10 3 mm 3 W 460x89 S 1770x10 3 mm 3 W 460x82 S 1610x10 3 mm 3 W 460x74 S 1460x10 3 mm 3 W 410x85 S 1510x10 3 mm 3 Use W 460x74 because the beam having the least weight. 8
200 kn 100 kn W 460x74 : A 9460 mm 2, d 457 mm, t w 9.02 mm, b f 190 mm, t f 14.5 mm, I 333x10 6 mm 4, 50 kn 250 kn 2 m 2 m 2 m V(kN) 100 50 - x(m) 221.25 14.5 214 9.02 190 457-150 heck allowable shear stress: τ allow 100 MPa Q (221.25)(190x14.5) (214/2)(214x9.02) τ max 816.08x10-6 m 3 VQ It V(816.08x10-6 ) (333x10-6 )(0.00902) 32.6 MPa <100 MPa, O.K. 271.7V (271.7 )(150x10 3 ) 9
200 kn 100 kn heck shear and normal stress at 50 kn 250 kn 2 m 2 m 2 m 200 kn m σ max 137.2 MPa τ max V(kN) 50 M(kN m) -150 100-100 - -200 30.44 MPa 128.5 MPa 214 150 kn Β x(m) σ, MPa τ I 333x10 6 mm 4 x (m) - ending: σ σ - Shear: τ τ My V max Q It (150x10 3 )[(0.2213)(0.19x0.0145)] (333x10-6 )(0.00902) I σ τ (-200x10 3 )(0.214) (333x10-6 ) -128.5 MPa 9.02 221.25 190 Q 14.5 30.44 MPa 10
30.44 MPa y x 128.5 MPa σ average σ x σ y -128.530-64.27 MPa 2 2 R σ x - σ y ( ) (τ xy ) 2 2 (64.27) 2 (30.44) 2 ± 71.11 MPa σ avg -64.27 MPa (-64.27, 71.11) τ σ - R average (-135.4, 0) 2 (-128.5, -30.44) x R O y (0, 30.44) 1 (6.84, 0) σ σ average R τ max 71.11 MPa < 100 MPa, O.K. σ max 135.4 MPa < 150 MPa, O.K. (-64.27, -71.11) (σ x - σ y )/2 (-128.5)/2 64.27 11
omments Q ya 221.25 14.5 214 9.02 457 Q (221.25)(190x14.5) (214/2)(214x9.02) 816083.71 mm 3 816.08x10-6 m 3 190 τ max VQ It V(816.08x10-6 ) (333x10-6 )(0.00902) 271.7V 221.25 14.5 214 9.02 457 τ avg V max A web V (0.457-2x0.0145)(0.00902) 190 259.03V 4.7 % off 221.25 14.5 214 9.02 190 457 τ avg V max A web V (0.457)(0.00902) 242.59V 10.7 % off Note: dimension in mm I 333x10 6 mm 4, 12
Example 2 The laminated wooden beam shown supports a uniform distributed loading of 12 kn/m. If the beam is to have a height-to-width ratio of 1.5, determine its smallest width. The allowable bending stress is σ allow 9 MPa and the allowable shear stress is τ allow 0.6 MPa. Neglect the weight of the beam. 12 kn/m 1.5a A 1 m 3 m a 13
12 kn/m A 1 m 32 kn 3 m V (kn) 20 - -12 M (kn m) 16 kn a 16 12 x 0 x1.33 m x (m) - -16 1.5a heck allowable shear stress τ allow 0.6 MPa VQ τ It V τ 1.5 A 20x10 0.6x10 3 6 1.5 1.5(a 2 ) 10.67 a 0.183 m 183 mm - x (m) Use a 185 mm -6 14
12 kn/m A 1 m 32 kn 3 m V (kn) 20 16 kn 16 12 x 0 1.5a 277.5 a 185 mm heck allowable normal stress σ allow 9 MPa - -12 x1.33 m - -16 x(m) σ max Mc I (10.67x10 3 )(278/2) (1/12)(185)(278 3 ) M (kn m) 4.48 MPa 10.67 4.48 MPa < 9 MPa, O.K. - x (m) Use a 185 mm -6 15
Example 3 The wooden T-beam shown is made from two x 30 mm boards. If the allowable bending stress is σ allow 12 MPa and the allowable shear stress is τ allow 0.8 MPa, determine if the beam can safely support the loading shown without having the deflection more than L/240. The spacing of nails specified to hold the two boards together if each nail can resist 1.50 kn in shear are: between use 125 mm, and D use 250 mm. Take E 12 GPa, and the formula for deflection at the the mid-span: 4 3 5wL PL υ ( ) 768EI 48EI w0.5 kn/m P1.5 kn 30 mm 2 m 2 m D 30 mm 16
0.5 kn/m 1.5 kn 1.5 kn 2 m 2 m V (kn) 1.5 D 1 kn 0.5-1.0 - -1.0 x (m) M (kn m) 2.0 x (m) 17
Section property y ' y 30 mm y ya A (100 )( 200 30 ) (215 )( 200 2(200 30 ) 30 ) 30 mm 157.5 mm y ' 230 157.5 72.5 mm I Σ ( I Ad 2 ) ( I Ad 2 ) ( I Ad 2 ) web flang [(1/12)(30)(200 3 ) (30x200)(157.5-100) 2 ] [(1/12)(200)(30 3 ) (200x30)(215-157.5) 2 ] 60.125x10 6 mm 4 60.12x10-6 m 4 18
0.5 kn/m 1.5 kn - Stress distribution at point 1.5 kn 2 m 2 m D 1 kn 0.309 V (kn) 1.5 1.5 kn τ -, MPa 0.5-1.0 - -1.0 (60.12x10-6 )(0.03) x(m) heck allowable shear stress: τ allow 0.8 MPa τ max V max Q It (1.5x10 3 )[(0.07875)(0.03x0.1575)] 72.5 157.5 200 30 200 0.309 MPa 0.309 MPa < 0.8 MPa, O.K. 30 I 60.12x10-6 m 4 19
0.5 kn/m 1.5 kn - Stress distribution at point 1.5 kn 2 m 2 m M(kN m) 2.0 D 1 kn x (m) 20 kn m 0.206 1 kn τ, MPa 5.24 σ, MPa 200 heck allowable normal stress: σ allow 12 MPa σ max Mc I (2x103 )(0.1575) (60.12x10-6 ) 72.5 157.5 30 200 2 kn m 5.24 MPa 5.24 MPa < 12 MPa, O.K. 30 I 60.12x10-6 m 4 20
w0.5 kn/m P1.5 kn υ L 4 5wL 768EI 1.5 k N L/22 m L/22 m 3 PL ( ) 48EI D 72.5 1 kn 157.5 200 30 200 30 I 60.12x10-6 m 4 heck maximum deflection: υ allow L/240 υ L 4 5wL 768EI 3 PL ) ( 48EI 4 5(.5)4 ) 9 768(12 10 )(60.12 10 0.00115 0.00277 0.00392 m, ( ) < 4 240 ) 3 (1.5)4 9 48(12 10 )(60.12 10 ( 6 6 0.0167 m, O. K. ) ) 21
0.5 kn/m 1.5 kn 1.5 kn 2 m 2 m V(kN) 72.5 30 1.5 157.5 200 0.5 x(m) - 30-1.0-1.0 I 60.125x10-6 m 4 Nail spacing: s 125 mm, and s D 250 mm; (F s ) allow 1.5 kn - Segment D 1 kn F nail τa shear VQ ( It ) 200 (s x0.03) τ -, MPa τ F nail (1.5x10 3 )[(0.0575)(0.2x0.03)] (0.125x0.03) (60.12x10-6 )(0.03) 1.076 kn < 1.5 kn, O.K. 22
0.5 kn/m 1.5 kn 1.5 kn 2 m 2 m - Segment D D 1 kn V (kn) 1 m 72.5 30 1.5 1 157.5 200 0.5 x(m) - 30-1.0-1.0 I 60.125x10-6 m 4 Nail spacing: s 125 mm, and s D 250 mm; (F s ) allow 1.5 kn F nail τa shear VQ ( It ) 200 (s D x0.03) τ -, MPa τ F nail (1x10 3 )(0.0575)(0.2x0.03) (60.12x10-6 )(0.03) (0.25x0.03) 1.434 kn < 1.5 kn, O.K. 23
Example 4 From the beam shown, determine the largest load P that can be applied to the beam. Requirements specified: σ allow 12 MPa τ allow 0.8 MPa Nail spacing @ 150 mm φ 3 mm τ allow 200 MPa P 2 m 2 m D 72.5 mm 30 mm 157.5 mm I 60.12x10-6 mm 4 24
V (N) M (N m) P P/2 2 m 2 m P/2 P/2 -P/2 P 30 mm 72.5 mm D P/2 157.5 mm I 60.125x10-6 mm 4 x (m) -P/2 x (m) Q neck A neck y 30 (0.03 0.2)(72.5 ) 2 3 0.345 m Maximum shear force on Nail: Nail spacing @ 150 mm, φ 3 mm, τ allow 200 MPa F s (τ allow ) nail (A nail ) (200x10 6 )(πx0.0015 2 ) 1414 N VQ 1414 ( )( t s) It P ( )(0.345) 1414 2 )(0.150) 6 (60.125 10 ) P 3286 N 25
V (N) M (N m) P 2 m 2 m P/2 P/2 -P/2 P 30 mm 72.5 mm D 157.5 mm I 60.125x10-6 mm 4 x (m) -P/2 x (m) Q max A max y 0.1575 (0.03 0.1575)( ) 2 3 3 0.372 10 m ending: σ allow 12 MPa Shear : τ allow 0.8 MPa σ Mc I P(0.1575) 60.125 10 6 12 10 6 P 4580 N VQ τ It P 3 ( )(0.372 10 ) 6.8 10 2 (60.125 10 )(0.03) 0 6 P 3879 N 26
σ allow 12 MPa τ allow 0.8 MPa Nail spacing @ 150 mm φ 3 mm τ allow 200 MPa P 2 m 2 m D Summary: Nail, shear 3.28 kn, use 3 kn eam, bending 4.58 kn eam, shear 3.88 kn 27
Example 5 From the beam shown, determine the largest load w that can be applied to the beam. onditions specified: Wood σ allow 12 MPa in tension σ allow 8 MPa in compression τ allow 0.8 MPa For each nail, (F nail ) allow 1.5 kn @ 100-mm spacing w 72.5 mm 30 mm 4 m 2 m D 157.5 mm I 60.12x10-6 mm 4 28
w 1.5w 4 m 4.5w 2 m D 72.5 mm 157.5 mm 30 mm M (N m) 1.125w I 60.12x10-6 mm 4 x (m) -2w eam : ending : σ allow 12 MPa in tension, σ allow 8 MPa in compression - ompression - Tension 6 (2w)(0.1575) 6 ( 1. 125w)( 0. 1575) 8 10 12 10 6 6 60.12 10 60. 125 10 w 1.527 kn/m w 4.071 kn/m 6 (1.125w)(0.725) 8 10 6 (2w)(0.725) 6 12 10 60.12 10 6 60.12 10 w 5.897 kn/m w 4.975 kn/m 29
w 1.5w 4 m 4.5w 2 m D 72.5 mm 157.5 mm 30 mm V (N) 1.5w 2 w I 60.12x10-6 mm 4 x (m) M (N m) 1.125w -2.5w x (m) -2w eam: Shear @, τ allow 0.8 MPa VQ τ It 0.1575 (2.5w)(0.1575 0.03 ) 6 0.8 10 2 6 (60.125 10 )(0.03) w 1.55 kn/m Shear on nail : (F nail ) allow 1.5 kn spacing 100 mm VQ F s (τ allow ) nail (A s ) ( )( t s) It (2.5w)(0.2 0.03 0.0575) 1.5 ( t)(0.1) 6 (60.125 10 )( t) w 1.04 kn/m 30
σ allow 12 MPa in tension σ allow 8 MPa in compression τ allow 0.8 MPa (F nail ) allow 1.5 kn Nail spacing 100 mm w 4 m 2 m D Summary: eam bending, compression 1.53 kn/m tension 4.07 kn/m eam shear, 1.55 kn/m Nail shear 1.04 kn/m, use 1 kn/m 31