# 2. Rigid bar ABC supports a weight of W = 50 kn. Bar ABC is pinned at A and supported at B by rod (1). What is the axial force in rod (1)?

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1 IDE 110 S08 Test 1 Name: 1. Determine the internal axial forces in segments (1), (2) and (3). (a) N 1 = kn (b) N 2 = kn (c) N 3 = kn 2. Rigid bar ABC supports a weight of W = 50 kn. Bar ABC is pinned at A and supported at B by rod (1). What is the axial force in rod (1)? N 1 = kn 3. Determine the axial force in bar AB. N AB = kn

2 4. A torque of T = 32 kn-m is transmitted between two flanged shafts by means of six bolts. Determine the shear force in each bolt. V bolt = kn 5. For the pin connection shown, determine the normal stress acting on the gross area, the normal stress acting on the net area, the shear stress in the pin, and the bearing stress in the steel plate at the pin. (a) σ gross = MPa (b) σ net = MPa (c) τ pin = MPa (d) σ b = MPa

3 6. A metal rod having two different diameters is loaded by an axial force P. Segments (1) and (2) are circular in cross section with diameters of 40 mm and 25 mm, respectively. If the normal stress in (1) is 65 MPa, what is the normal stress in segment (2)? σ 2 = MPa 7. The five-bolt connection must support an applied load of P = 600 kn. If the diameter of each bolt is 35 mm, determine the average shear stress in each bolt. τ = MPa 8. The three-bolt connection must support an applied load of P = 600 kn. If the average shear stress in each bolt must be limited to 210 MPa, determine the minimum bolt diameter that may be used in the connection. d = mm

4 9. The steel pipe column has an outside diameter of 8 in. and wall thickness of 0.5 in. The timber beam is 11 in. wide, and the upper plate has the same width. The load imposed on the column by the timber beam is 80 kips. Determine the length L of the rectangular upper bearing plate if the average bearing stress between the steel plate and the wood beam is not to exceed 500 psi. L = in. 10. Rigid bar ABCD is supported by two vertical bars. There is no strain in the vertical bars before load P is applied. After load P is applied, the axial strain in bar (2) is 700 μm/m. Determine the axial strain in bar (1). ε 1 = μm/m 11. A thin rectangular plate is uniformly deformed as shown. Determine the shear strain γ xy at P. γ xy = με

5 12. Two gage marks are placed exactly 80 mm apart on a 18-mm-diameter solid metal rod. When an axial tension force of 90 kn is applied to the rod, the distance between the gage marks in precisely measured as mm. Determine the modulus of elasticity of the metal. E = GPa 13. A 1.0-in. thick rectangular alloy bar is subjected to a tensile load P by pins at A and B. The width of the bar is w = 4.0 in. Strain gages bonded to the specimen measure the following strains in the longitudinal (x) and transverse (y) directions: ε x = 2100 με and ε y = -650 με. Determine Poisson s ratio for this specimen. ν = 14. A tension-test specimen with a diameter of in. and an 8.0-in gage length was tested to fracture. Two graphs of the collected data are given on the next page. Determine the following. (a) the modulus of elasticity = Msi (b) the proportional limit = ksi (c) the ultimate strength = ksi (d) the yield strength (0.20% offset) = ksi

6 This graph is zoomed to show initial data Stress (psi) Strain (in/in) This graph shows all of the collected data Stress (psi) Strain (in/in)

7 TRIGONOMETRY sin θ = opp / hyp cos θ = adj / hyp tan θ = opp / adj STATICS Symbol Meaning Equation Units x, y, z centroid position y = Σy i A i / ΣA i in, m I moment of inertia I = Σ(I i + d i 2 A i ) in 4, m 4 J N V M J solid circular shaft polar = πd 4 /32 moment J of inertia hollow circular shaft = π(d 4 o - d 4 i )/32 normal force shear force bending moment equilibrium V = -w(x) dx M = V(x) dx ΣF = 0 ΣM (any point) = 0 in 4, m 4 lb, N lb, N in-lb, Nm lb, N in-lb, Nm MECHANICS OF MATERIALS Topic Symbol Meaning Equation Units σ axial = N/A σ, sigma normal stress τ cutting = V/A σ bearing = F b /A b axial ε, epsilon normal strain ε axial = ΔL/L o = δ/l o ε transverse = Δd/d in/in, m/m γ, gamma shear strain γ = change in angle, γ = cθ rad E Young's modulus, modulus of elasticity σ = Eε (one-dimensional only) G shear modulus, modulus of rigidity G = τ/γ = E / 2(1+ν) ν, nu Poisson's ratio ν = -ε'/ε δ, delta deformation, elongation, deflection in, m δ = NL o /EA + αδtl o α, alpha coefficient of thermal expansion (CTE) in/inf, m/mc F.S. factor of safety F.S. = actual strength / design strength

8 Topic Symbol Meaning Equation Units τ, tau shear stress τ torsion = Tc/J torsion φ, phi angle of twist φ = TL/GJ rad, degrees θ, theta angle of twist per unit length, rate of twist θ = φ / L rad/in, rad/m P power P = Tω r 2 T 1 = r 1 T 2 watts = Nm/s hp=6600 in-lb/s ω, r 1 ω angular speed, speed of rotation 1 = r ω 2 2 rad/s omega f frequency ω = 2πf Hz = rev/s K stress concentration factor τ max = KTc/J flexure σ, sigma σ, sigma τ, tau normal stress σ beam = -My/I composite beams, n = E B /E A σ A = -My / I T σ B = -nmy / I T shear stress τ beam = VQ/Ib where Q = Σ(y bar i A i ) q shear flow q = V beam Q/I = nv fastener /s v or y beam deflection v = M(x) dx 2 / EI in, m Topic Equations Units stress transformation planar rotations σ u +σ y )/2 + (σ x )/2 cos(2θ) + τ xy sin(2θ) σ v +σ y )/2 - (σ x )/2 cos(2θ) - τ xy sin(2θ) τ uv = -(σ x )/2 sin(2θ) + τ xy cos(2θ) principals and max in-plane shear tan(2θ p ) = 2τ xy / (σ x ), θ s = θ p ± 45 o σ 1,2 +σ y )/2 ± sqrt { [ (σ x )/2 ] 2 + τ xy 2 } τ max = sqrt { [ (σ x )/2 ] 2 + τ xy 2 } = (σ 1 -σ 2 )/2 σ avg +σ y )/2 = (σ 1 +σ 2 )/2 strain transformation planar rotations ε u = (ε x )/2 + (ε x )/2 cos(2θ) + γ xy /2 sin(2θ) ε v = (ε x )/2 - (ε x )/2 cos(2θ) - γ xy /2 sin(2θ) γ uv /2 = -(ε x )/2 sin(2θ) + γ xy /2 cos(2θ) ε z = -ν (ε x + ε y ) / (1-ν) principals and max in-plane shear tan(2θ p ) = γ xy / (ε x ), θ s = θ p ± 45 o ε 1,2 = (ε x )/2 ± sqrt { [ (ε x )/2 ] 2 + (γ xy /2) 2 } γ max /2 = sqrt { [ (ε x )/2 ] 2 + (γ xy /2) 2 } ε avg = (ε x )/2 in/in, m/m Hooke's law 1D strain to stress σ = Eε 2D strain to stress σ x = E(ε x +νε y ) / (1-ν 2 ) σ y = E(ε y +νε x ) / (1-ν 2 ) τ xy = Gγ xy = Eγ xy / 2(1+ν) 2D stress to strain ε x -νσ y ) / E ε y = (σ y -νσ x ) / E ε z = -ν(ε x ) / (1-ν) γ xy = τ xy /G = 2(1+ν)τ xy / E in/in, m/m pressure σ spherical = pr/2t σ cylindrical, hoop = pr/t σ cylindrical, axial = pr/2t maximum principal stress theory σ 1,2 < σ yp σ radial, outside = 0 σ radial, inside = -p failure theories maximum shear stress theory τ max < 0.5 σ yp

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### The problem of transmitting a torque or rotary motion from one plane to another is frequently encountered in machine design. CHAPER ORSION ORSION orsion refers to the twisting of a structural member when it is loaded by moments/torques that produce rotation about the longitudinal axis of the member he problem of transmitting

### Mechanics of Solids. Mechanics Of Solids. Suraj kr. Ray Department of Civil Engineering Mechanics Of Solids Suraj kr. Ray (surajjj2445@gmail.com) Department of Civil Engineering 1 Mechanics of Solids is a branch of applied mechanics that deals with the behaviour of solid bodies subjected

### Stress and Strain ( , 3.14) MAE 316 Strength of Mechanical Components NC State University Department of Mechanical & Aerospace Engineering (3.8-3.1, 3.14) MAE 316 Strength of Mechanical Components NC State Universit Department of Mechanical & Aerospace Engineering 1 Introduction MAE 316 is a continuation of MAE 314 (solid mechanics) Review

### Torsion of Shafts Learning objectives Torsion of Shafts Shafts are structural members with length significantly greater than the largest cross-sectional dimension used in transmitting torque from one plane to another. Learning objectives Understand

### BE Semester- I ( ) Question Bank (MECHANICS OF SOLIDS) BE Semester- I ( ) Question Bank (MECHANICS OF SOLIDS) All questions carry equal marks(10 marks) Q.1 (a) Write the SI units of following quantities and also mention whether it is scalar or vector: (i)

###  Torsion. [7.1] Torsion. [7.2] Statically Indeterminate Torsion.  Torsion Page 1 of 21 Torsion Page 1 of 21  Torsion [7.1] Torsion [7.2] Statically Indeterminate Torsion  Torsion Page 2 of 21 [7.1] Torsion SHEAR STRAIN DUE TO TORSION 1) A shaft with a circular cross section is

### Solution: The strain in the bar is: ANS: E =6.37 GPa Poison s ration for the material is: Problem 10.4 A prismatic bar with length L 6m and a circular cross section with diameter D 0.0 m is subjected to 0-kN compressive forces at its ends. The length and diameter of the deformed bar are measured

### 5. What is the moment of inertia about the x - x axis of the rectangular beam shown? 1 of 5 Continuing Education Course #274 What Every Engineer Should Know About Structures Part D - Bending Strength Of Materials NOTE: The following question was revised on 15 August 2018 1. The moment

### Strength of Materials Prof S. K. Bhattacharya Department of Civil Engineering Indian Institute of Technology, Kharagpur Lecture - 18 Torsion - I Strength of Materials Prof S. K. Bhattacharya Department of Civil Engineering Indian Institute of Technology, Kharagpur Lecture - 18 Torsion - I Welcome to the first lesson of Module 4 which is on Torsion

### 2. Polar moment of inertia As stated above, the polar second moment of area, J is defined as. Sample copy GATE PATHSHALA - 91. Polar moment of inertia As stated above, the polar second moment of area, is defined as z π r dr 0 R r π R π D For a solid shaft π (6) QP 0 π d Solid shaft π d Hollow shaft, " ( do Chapter 5 CENTRIC TENSION OR COMPRESSION ( AXIAL LOADING ) 5.1 DEFINITION A construction member is subjected to centric (axial) tension or compression if in any cross section the single distinct stress Name: Date: Solid Mechanics Homework nswers Please show all of your work, including which equations you are using, and circle your final answer. Be sure to include the units in your answers. 1. The yield

### Module 5: Theories of Failure Module 5: Theories of Failure Objectives: The objectives/outcomes of this lecture on Theories of Failure is to enable students for 1. Recognize loading on Structural Members/Machine elements and allowable

### SRI CHANDRASEKHARENDRA SARASWATHI VISWA MAHAVIDHYALAYA SRI CHANDRASEKHARENDRA SARASWATHI VISWA MAHAVIDHYALAYA (Declared as Deemed-to-be University under Section 3 of the UGC Act, 1956, Vide notification No.F.9.9/92-U-3 dated 26 th May 1993 of the Govt. of

### PURE BENDING. If a simply supported beam carries two point loads of 10 kn as shown in the following figure, pure bending occurs at segment BC. BENDING STRESS The effect of a bending moment applied to a cross-section of a beam is to induce a state of stress across that section. These stresses are known as bending stresses and they act normally

### M. Vable Mechanics of Materials: Chapter 5. Torsion of Shafts Torsion of Shafts Shafts are structural members with length significantly greater than the largest cross-sectional dimension used in transmitting torque from one plane to another. Learning objectives Understand

### The University of Melbourne Engineering Mechanics The University of Melbourne 436-291 Engineering Mechanics Tutorial Four Poisson s Ratio and Axial Loading Part A (Introductory) 1. (Problem 9-22 from Hibbeler - Statics and Mechanics of Materials) A short

### A concrete cylinder having a a diameter of of in. mm and elasticity. Stress and Strain: Stress and Strain: 0. 2011 earson Education, Inc., Upper Saddle River, NJ. ll rights reserved. This material is protected under all copyright laws as they currently 8 1. 3 1. concrete cylinder having a a diameter of of 6.00 ME 270 3 rd Sample inal Exam PROBLEM 1 (25 points) Prob. 1 questions are all or nothing. PROBLEM 1A. (5 points) IND: In your own words, please state Newton s Laws: 1 st Law = 2 nd Law = 3 rd Law = PROBLEM Name Bioen 326 Midterm 2013 Page 1 Bioen 326 2013 MIDTERM Covers: Weeks 1-4, FBD, stress/strain, stress analysis, rods and beams (not deflections). Rules: Closed Book Exam: Please put away all notes and