Kiến trúc xây dựng - Chương 8: Mechanics of materials

MECHANICS OF MATERIALS Third Edition Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University CHAPTER © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Principle Stresses Under a Given Loading © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 2 Principle Stresses Under a Given Loading Introduction Principle Stresses in a Beam Sample Problem 8.1 Sample Problem 8.2 Design of a Transmission Shaft Sample Problem 8.3 Stresses Under Combined Loadings Sample Problem 8.5 © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 3 Introduction • In Chaps. 1 and 2, you learned how to determine the normal stress due to centric loads In Chap. 3, you analyzed the distribution of shearing stresses in a circular member due to a twisting couple In Chap. 4, you determined the normal stresses caused by bending couples In Chaps. 5 and 6, you evaluated the shearing stresses due to transverse loads In Chap. 7, you learned how the components of stress are transformed by a rotation of the coordinate axes and how to determine the principal planes, principal stresses, and maximum shearing stress at a point. • In Chapter 8, you will learn how to determine the stress in a structural member or machine element due to a combination of loads and how to find the corresponding principal stresses and maximum shearing stress © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 4 Principle Stresses in a Beam • Prismatic beam subjected to transverse loading It VQ It VQ I Mc I My mxy mx =−= =−= ττ σσ • Principal stresses determined from methods of Chapter 7 • Can the maximum normal stress within the cross-section be larger than I Mc m =σ © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 5 Principle Stresses in a Beam © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 6 Principle Stresses in a Beam • Cross-section shape results in large values of τxy near the surface where σx is also large. • σmax may be greater than σm © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 7 Sample Problem 8.1 SOLUTION: • Determine shear and bending moment in Section A-A’ • Calculate the normal stress at top surface and at flange-web junction. A 160-kN force is applied at the end of a W200x52 rolled-steel beam. Neglecting the effects of fillets and of stress concentrations, determine whether the normal stresses satisfy a design specification that they be equal to or less than 150 MPa at section A-A’. • Evaluate the shear stress at flange- web junction. • Calculate the principal stress at flange-web junction © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 8 Sample Problem 8.1 SOLUTION: • Determine shear and bending moment in Section A-A’ ( )( ) kN160 m-kN60m375.0kN160 = == A A V M • Calculate the normal stress at top surface and at flange-web junction. ( ) MPa9.102 mm103 mm4.90MPa2.117 MPa2.117 m10512 mkN60 36 = == = × ⋅== − c yσ S M b ab A a σ σ © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 9 Sample Problem 8.1 • Evaluate shear stress at flange-web junction. ( ) ( )( )( )( ) MPa5.95 m0079.0m107.52 m106.248kN160 m106.248 mm106.2487.966.12204 46 36 36 33 = × ×== ×= ×=×= − − − It QV Q A bτ • Calculate the principal stress at flange-web junction ( ) ( ) ( )MPa 150MPa9.169 5.95 2 9.102 2 9.102 2 2 22 2 1 2 1 max >= +⎟⎠ ⎞⎜⎝ ⎛+= ++= bbb τσσσ Design specification is not satisfied. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 10 Sample Problem 8.2 The overhanging beam supports a uniformly distributed load and a concentrated load. Knowing that for the grade of steel to used σall = 24 ksi and τall = 14.5 ksi, select the wide- flange beam which should be used. SOLUTION: • Determine reactions at A and D. • Find maximum shearing stress. • Find maximum normal stress. • Calculate required section modulus and select appropriate beam section. • Determine maximum shear and bending moment from shear and bending moment diagrams. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 11 Sample Problem 8.2 • Calculate required section modulus and select appropriate beam section. sectionbeam62select W21 in7.119 ksi24 inkip24 3max min × =⋅== all M S σ SOLUTION: • Determine reactions at A and D. kips410 kips590 =⇒=∑ =⇒=∑ AD DA RM RM • Determine maximum shear and bending moment from shear and bending moment diagrams. kips43 kips 2.12withinkip4.239 max max = =⋅= V VM © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 12 Sample Problem 8.2 • Find maximum shearing stress. Assuming uniform shearing stress in web, ksi14.5ksi 12.5 in 8.40 kips43 2 max max <=== webA Vτ • Find maximum normal stress. ( ) ksii45.1 in8.40 kips 2.12 ksi3.21 5.10 88.9ksi6.22 ksi6.22 27in1 inkip602873 2b 3 max === === =⋅== web b ab a A V c yσ S M τ σ σ ( ) ksi24ksi4.21 ksi45.1 2 ksi3.21 2 ksi3.21 2 2 max <= +⎟⎠ ⎞⎜⎝ ⎛+=σ © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 13 Design of a Transmission Shaft • If power is transferred to and from the shaft by gears or sprocket wheels, the shaft is subjected to transverse loading as well as shear loading. • Normal stresses due to transverse loads may be large and should be included in determination of maximum shearing stress. • Shearing stresses due to transverse loads are usually small and contribution to maximum shear stress may be neglected. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 14 Design of a Transmission Shaft • At any section, J Tc MMM I Mc m zym = +== τ σ 222where • Maximum shearing stress, ( ) 22 max 22 2 2 max 2 section,-crossannular or circular afor 22 TM J c JI J Tc I Mc m m += = ⎟⎠ ⎞⎜⎝ ⎛+⎟⎠ ⎞⎜⎝ ⎛=+⎟⎠ ⎞⎜⎝ ⎛= τ τστ • Shaft section requirement, all TM c J τ max 22 min ⎟⎠ ⎞⎜⎝ ⎛ + =⎟⎠ ⎞⎜⎝ ⎛ © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 15 Sample Problem 8.3 SOLUTION: • Determine the gear torques and corresponding tangential forces. • Find reactions at A and B. • Identify critical shaft section from torque and bending moment diagrams. Solid shaft rotates at 480 rpm and transmits 30 kW from the motor to gears G and H; 20 kW is taken off at gear G and 10 kW at gear H. Knowing that σall = 50 MPa, determine the smallest permissible diameter for the shaft. • Calculate minimum allowable shaft diameter. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 16 Sample Problem 8.3 SOLUTION: • Determine the gear torques and corresponding tangential forces. ( ) ( ) ( ) kN49.2mN199Hz802 kW10 kN63.6mN398 Hz802 kW20 kN73.3 m0.16 mN597 mN597 Hz802 kW30 2 =⋅== =⋅== =⋅== ⋅=== DD CC E E E E FT FT r TF f PT π π ππ • Find reactions at A and B. kN90.2kN80.2 kN22.6kN932.0 == == zy zy BB AA © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 17 Sample Problem 8.3 • Identify critical shaft section from torque and bending moment diagrams. ( ) mN1357 5973731160 222 max 22 ⋅= ++=⎟⎠ ⎞⎜⎝ ⎛ +TM © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 18 Sample Problem 8.3 • Calculate minimum allowable shaft diameter. m25.85m02585.0 m1014.27 2 shaft,circular solid aFor m1014.27 MPa50 mN 1357 363 36 22 == ×== ×=⋅=+= − − c c c J TM c J all π τ mm7.512 == cd © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 19 Stresses Under Combined Loadings • Wish to determine stresses in slender structural members subjected to arbitrary loadings. • Pass section through points of interest. Determine force-couple system at centroid of section required to maintain equilibrium. • System of internal forces consist of three force components and three couple vectors. • Determine stress distribution by applying the superposition principle. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 20 Stresses Under Combined Loadings • Axial force and in-plane couple vectors contribute to normal stress distribution in the section. • Shear force components and twisting couple contribute to shearing stress distribution in the section. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 21 Stresses Under Combined Loadings • Normal and shearing stresses are used to determine principal stresses, maximum shearing stress and orientation of principal planes. • Analysis is valid only to extent that conditions of applicability of superposition principle and Saint-Venant’s principle are met. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 22 Sample Problem 8.5 Three forces are applied to a short steel post as shown. Determine the principle stresses, principal planes and maximum shearing stress at point H. SOLUTION: • Determine internal forces in Section EFG. • Calculate principal stresses and maximum shearing stress. Determine principal planes. • Evaluate shearing stress at H. • Evaluate normal stress at H. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 23 Sample Problem 8.5 SOLUTION: • Determine internal forces in Section EFG. ( )( ) ( )( ) ( )( ) mkN3m100.0kN300 mkN5.8 m200.0kN75m130.0kN50 kN75kN50kN 30 ⋅=== ⋅−= −= −==−= zy x zx MM M VPV Note: Section properties, ( )( ) ( )( ) ( )( ) 463121 463 12 1 23 m10747.0m040.0m140.0 m1015.9m140.0m040.0 m106.5m140.0m040.0 − − − ×== ×== ×== z x I I A © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 24 Sample Problem 8.5 • Evaluate normal stress at H. ( )( ) ( )( ) ( ) MPa66.0MPa2.233.8093.8 m1015.9 m025.0mkN5.8 m10747.0 m020.0mkN3 m105.6 kN50 46 4623- =−+= × ⋅− × ⋅+×= −++= − − x x z z y I bM I aM A Pσ • Evaluate shearing stress at H. ( )( )[ ]( ) ( )( )( )( ) MPa52.17 m040.0m1015.9 m105.85kN75 m105.85 m0475.0m045.0m040.0 46 36 36 11 = × ×== ×= == − − − tI QV yAQ x z yzτ © 2002 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALS Third Edition Beer • Johnston • DeWolf 8 - 25 Sample Problem 8.5 • Calculate principal stresses and maximum shearing stress. Determine principal planes. °= °=== −=−=−= =+=+= =+== 98.13 96.272 0.33 52.172tan MPa4.74.370.33 MPa4.704.370.33 MPa4.3752.170.33 pp min max 22 max p CD CY ROC ROC R θ θθ σ σ τ °= −= = = 98.13 MPa4.7 MPa4.70 MPa4.37 min max max pθ σ σ τ