Fourth Edition CHAPTER MECHANICS OF MATERIALS Ferdinand P Beer E Russell Johnston, Jr John T DeWolf Lecture Notes: J Walt Oler Texas Tech University Torsion Xoắn © 2006 The McGraw-Hill Companies, Inc All rights reserved Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Contents Introduction Statically Indeterminate Shafts Torsional Loads on Circular Shafts Sample Problem 3.4 Net Torque Due to Internal Stresses Design of Transmission Shafts Axial Shear Components Stress Concentrations Shaft Deformations Torsion of Noncircular Members Shearing Strain Thin-Walled Hollow Shafts Stresses in Elastic Range Example 3.10 Normal Stresses Torsional Failure Modes Sample Problem 3.1 Angle of Twist in Elastic Range © 2006 The McGraw-Hill Companies, Inc All rights reserved 3- Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Tải xoắn trục tròn • Quan tâm đến ứng suất biến dạng trục tròn chịu xoắn • Động thực moment xoắn T lên trục • Trục truyền moment xoắn đến máy phát • Máy phát tạo moment xoắn T’ = T ngược chiều © 2006 The McGraw-Hill Companies, Inc All rights reserved 3- Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Moment xoắn ứng suất bên • Tập hợp ứng suất cắt nội lực moment xoắn nội lực, ngược chiều với moment xoắn tác dụng, T dF dA • Mặt dầu biết moment xoắn ứng xuất cắt gây ra, phân bố ứng suất khơng • Sự phân bố ứng suất phức tạp – phải xem xét biến dạng trục • Khơng giống ứng suất pháp tải dọc trục, phân bố ứng suất cắt moment xoắn giả thuyết đồng © 2006 The McGraw-Hill Companies, Inc All rights reserved 3- Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Brief – Tóm tắt Xét bên trái mc 1-1 AB Lập phương trình cân m 11 kN m t * x TAB TAB kN m t * x Thanh ngàm hai đầu (siêu tĩnh) * Thay ngàm A B TA TB = Có hai quan hệ m TA 90 TB AB © 2006 The McGraw-Hill Companies, Inc All rights reserved 3- Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Brief – Tóm tắt * Ứng suất cắt mcn T kN cm J with c J 12 c TAB dx rad GJ A * Điều kiện cứng J c14 rad L rad m * Góc xoắn tương đối đoạn ACB * Góc xoắn B c4 2 * Góc xoắn tỉ đối m © 2006 The McGraw-Hill Companies, Inc All rights reserved C B C rad B * Điều kiện bền kN cm 3- Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Thành phần cắt dọc trục • Moment xoắn tác dụng lên trục tạo nên ứng suất cắt mặt vng góc với trục • Để cân mặt chứa trục trục chịu xoắn phải xuất ứng suất cắt giá trị • Trục làm từ mãnh chịu xoắn xuật thành phần ứng suất cặt dọc trục • Các mãnh trượt tương có hai moment xoắn giá trị ngược chiều tác dụng © 2006 The McGraw-Hill Companies, Inc All rights reserved 3- Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Biến dạng trục • Quan sát hình: góc xoắn trục tỉ lệ với moment xoắn tác dụng chiều dài trục T L • Khi chịu xoắn, tiết diện trục tròn phẳng khơng méo • Tiết diện trục tròn rỗng đặc phẳng khơng méo trục tròn đối xứng trục • Tiết diện trục khơng tròn bị méo chịu xoắn © 2006 The McGraw-Hill Companies, Inc All rights reserved 3- Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Shearing Strain • Xét phần tử trục Khi tải xoắn tác dụng, phần tử biến dạng thành hình bình hành • Khi hai mặt bên phần tử phẳng, biến dạng cắt góc xoắn • Do L or L • Biến dạng cắt tỉ lệ với góc xoắn bán kính max c L and max G max G © 2006 The McGraw-Hill Companies, Inc All rights reserved c max c max L L Gc 3- Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Ứng suất phạm vi đàn hồi • Nhân phương trình vừa với modul cắt G, G c G max Từ định luật Hooke, G , c max Ứng suất cắt biến đồi tuyến tính với bán kính mặt cắt J 12 c • Nhắc lại tổng moment ứng suất nội lực moment xoắn tác dụng mặt cắt, T dA max dA max J c c J c4 2 c14 • Như ứng suất cắt, max © 2006 The McGraw-Hill Companies, Inc All rights reserved Tc T and J J - 10 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Example 3.10 SOLUTION: • Xác định dòng chảy trượt q qua thành ống • Tìm ứng suất cắt tương ứng thành ống Độ dày thành ống ko đổi, q 1.335 kip in t 0.160 in 8.34 ksi Độ dày thành ống thay đổi A 3.84 in.2.34 in. 8.986 in q T 24 kip - in kip 335 A 8.986 in.2 in AB AC 1.335 kip in 0.120 in AB BC 11.13 ksi BD CD 1.335 kip in 0.200 in BC CD 6.68 ksi © 2006 The McGraw-Hill Companies, Inc All rights reserved - 46 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Problem 3.1-2 • The solid spindle AB has a diameter ds = 1.5 in and is made of a steel with an allowable shearing stress of 12 ksi, while sleeve CD is made of a brass with an allowable shearing stress of ksi Determine the largest torque T that can be applied at A • The solid spindle AB is made of a steel with an allowable shearing stress of 12 ksi, and sleeve CD is made of a brass with an allowable shearing stress of ksi Determine (a) the largest torque T that can be applied at A if the allowable shearing stress is not to be exceeded in sleeve CD, (b) the corresponding required value of the diameter ds of spindle AB © 2006 The McGraw-Hill Companies, Inc All rights reserved - 47 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Problem 3.3-4 • Under normal operating conditions, the electric motor exerts a 12-kip.in torque at E Knowing that each shaft is solid, determine the maximum 1 ft 12 in 1 kip 1000 lb shearing stress in (a) 1 ksi 1000 psi shaft BC, (b) shaft CD, 1 psi lb (c) shaft DE in 1 MPa N / mm2 T lb.in J in psi c in T kip.in J in ksi c in • Solve Prob 3.4, assuming that a 1-in.-diameter hole has been drilled into each shaft © 2006 The McGraw-Hill Companies, Inc All rights reserved - 48 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Problem 3.5-6 • The allowable shearing stress is 15 ksi in the 1.5-in.-diameter steel rod AB and ksi in the 1.8-in.-diameter brass rod BC Neglecting the effect of stress concentrations, determine the largest torque that can be applied at A • The allowable shearing stress is 15 ksi in the steel rod AB and ksi in the brass rod BC Knowing that a torque of magnitude T = 10 kip.in is applied at A, determine the required diameter of (a) rod AB, (b) rod BC © 2006 The McGraw-Hill Companies, Inc All rights reserved - 49 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Problem 3.7-8 • The solid rod AB has a diameter dAB = 60 mm The pipe CD has an outer diameter of 90 mm and a wall thickness of mm Knowing that both the rod and the pipe are made of steel for which the allowable shearing stress is 75 MPa, determine the largest torque T that can be applied at A • The solid rod AB has a diameter dAB = 60 mm and is made of a steel for which the allowable shearing stress is 85 MPa The pipe CD, which has an outer diameter of 90 mm and a wall thickness of mm, is made of an aluminum for which the allowable shearing stress is 54 MPa Determine the largest torque T that can be applied at A © 2006 The McGraw-Hill Companies, Inc All rights reserved - 50 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Problem 3.9-10 • A torque of magnitude T 1000 N ? m is applied at D as shown Knowing that the diameter of shaft AB is 56 mm and that the diameter of shaft CD is 42 mm, determine the maximum shearing stress in (a) shaft AB, (b) shaft CD • A torque of magnitude T 1000 N ? m is applied at D as shown Knowing that the allowable shearing stress is 60 MPa in each shaft, determine the required diameter of (a) shaft AB, (b) shaft CD © 2006 The McGraw-Hill Companies, Inc All rights reserved - 51 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Problem 3.11 • The electric motor exerts a 500 N m-torque on the aluminum shaft ABCD when it is rotating at a constant speed Knowing that G = 27 GPa and that the torques exerted on pulleys B and C are as shown, determine the angle of twist between (a) B and C, (b) B and D © 2006 The McGraw-Hill Companies, Inc All rights reserved - 52 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Problem 3.12 • The aluminum rod AB (G = 27 GPa) is bonded to the brass rod BD (G = 39 GPa) Knowing that portion CD of the brass rod is hollow and has an inner diameter of 40 mm, determine the angle of twist at A © 2006 The McGraw-Hill Companies, Inc All rights reserved - 53 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Problem 3.13-14 • The solid spindle AB has a diameter ds = 1.5 in and is made of a steel with G = 11.2x106 psi and τall = 12 ksi, while sleeve CD is made of a brass with G = 5.6x106 psi and τall = ksi Determine the largest angle through which end A can be rotated 3.13 : max TAB TCD 3.14 : max CD TCD 1 CD TCD +Max T TAB ; TCD +Max T TCD 1 ; TCD A AD AB CD A AD AB CD • The solid spindle AB has a diameter ds = 1.75 in and is made of a steel with G = 11.2x106 psi and τall = 12 ksi, while sleeve CD is made of a brass with G = 5.6x106 psi and τall = ksi Determine (a) the largest torque T that can be applied at A if the given allowable stresses are not to be exceeded and if the angle of twist of sleeve CD is not to exceed 0.3750, (b) the corresponding angle through which end A rotates © 2006 The McGraw-Hill Companies, Inc All rights reserved - 54 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Problem 3.15 • Two shafts, each of 7/8-in diameter, are connected by the gears shown Knowing that G = 11.2x106 psi and that the shaft at F is fixed, determine the angle through which end A rotates when a 1.2 kip.in torque is applied at A © 2006 The McGraw-Hill Companies, Inc All rights reserved - 55 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Problem 3.16 • Two solid shafts are connected by gears as shown Knowing that G = 77.2 GPa for each shaft, determine the angle through which end A rotates when TA = 1200 N.m © 2006 The McGraw-Hill Companies, Inc All rights reserved - 56 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Problem 3.17 • A hole is punched at A in a plastic sheet by applying a 600-N force P to end D of lever CD, which is rigidly attached to the solid cylindrical shaft BC Design specifications require that the displacement of D should not exceed 15 mm from the time the punch first touches the plastic sheet to the time it actually penetrates it Determine the required diameter of shaft BC if the shaft is made of a steel with G 77 GPa and tall 80 MPa © 2006 The McGraw-Hill Companies, Inc All rights reserved - 57 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Problem 3.18 • The steel jacket CD has been attached to the 40-mm-diameter steel shaft AE by means of rigid flanges welded to the jacket and to the rod The outer diameter of the jacket is 80 mm and its wall thickness is mm If 500 N.mtorques are applied as shown, determine the maximum shearing stress in the jacket TL JG TJ cJ JJ Ts GJ s L T T T G s J L TJ GJ J L © 2006 The McGraw-Hill Companies, Inc All rights reserved - 58 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Problem 3.19 • The solid cylinders AB and BC are bonded together at B and are attached to fixed supports at A and C Knowing that the modulus of rigidity is 3.7x106 psi for aluminum and 5.6x106 psi for brass, determine the maximum shearing stress (a) in cylinder AB, (b) in cylinder BC T L TAB L B / BC BC GJ GJ TAB TCD T B / AB © 2006 The McGraw-Hill Companies, Inc All rights reserved - 59 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Problem 3.20-21 T A TL AB TCD B GJ ' B' 1.50 TAB A B' • Two solid steel shafts are fitted with flanges that are then connected by bolts as shown The bolts are slightly undersized and permit a 1.58 rotation of one flange with respect to the other before the flanges begin to rotate as a single unit Knowing that G = 11.2x106 psi, determine the maximum shearing stress in each shaft when a torque of T of magnitude 420 kip.ft is applied to the flange indicated 3.21 The torque T is applied to flange B 3.22 The torque T is applied to flange C ' " " T '' T TAB TAB TCD A " B " © 2006 The McGraw-Hill Companies, Inc All rights reserved - 60 ... hư hỏng xoắn • Vật liệu dẻo thường hư hỏng chịu cắt Vật liệu dòn chịu xoắn • Khi chịu xoắn, vật liệu dẻo đứt mặt phẳng có ứng suất cắt cực đại: mặt phẳng vng góc trục • Khi chịu xoắn, vật liệu... • DeWolf Tải xoắn trục tròn • Quan tâm đến ứng suất biến dạng trục tròn chịu xoắn • Động thực moment xoắn T lên trục • Trục truyền moment xoắn đến máy phát • Máy phát tạo moment xoắn T’ = T ngược... MATERIALS Beer • Johnston • DeWolf Góc xoắn • Khi đầu mút D trục AD cố định, góc xoắn AD tính góc xoắn ϕA đầu mút A • Khi hai đầu mút BE quay, góc xoắn BE độ lệch góc xoắn ϕB ϕE TL E / B E B