Chapter 2: Strain. Chapter 3: Torsion. Chapter 4: Shear and Moment Diagram

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2 Chapter 2: Strain Chapter 3: Torsion Chapter 4: Shear and Moment Diagram

3 Strain - Is defined as change in length per unit length, simply put it is unit deformation L

4 Stress and Strain Exist concurrently in nature; if a body is under stress, it also exhibits strain. Considering a typical tensile strength test on a steel reinforcing bar, the following diagram is produced

5 Stress, STRESS VERSUS STRAIN Ultimate Strength Actual Rupture Strength Yield Point Rupture Strength Elastic Limit Proportional Limit O Strain,

6 For the strain to be assumed constant and for the average value to be representative of the whole, the following conditions must be met: 1. Specimen must be of constant cross section. 2. Material must be homogeneous. 3. Load must be axial and constant.

7 Hooke s Law: Axial Deformation Stress, Yield Point Ultimate Strength Actual Rupture Strength Rupture Strength O Elastic Limit Proportional Limit Strain, Within the region up to the elastic limit, a material is said to behave elastically. Beyond which, the material behaves plastically.

8 Within the proportional limit, stress and strain varies linearly, that is, stress is proportional to strain. Derivation:

9 An aluminum bar having a cross-sectional area of 160 mm 2 carries the axial loads at the positions shown in the figure. If E = 70 x 10 3 MPa, compute the total deformation of the bar. Assume that the bar is suitably braced to prevent buckling. 35 kn 15 kn 30 kn 10 kn 0.8 m 1.0m 0.6m

10 An aluminum bar having a cross-sectional area of 160 mm 2 carries the axial loads at the positions shown in the figure. If E = 70 x 10 3 MPa, compute the total deformation of the bar. Assume that the bar is suitably braced to prevent buckling. 10 kn 15 kn 30 kn 35 kn 0.8 m 1.0m 0.6m

11 An aluminum bar is fastened between a steel rod and a bronze rod as shown. Axial loads are applied at the positions indicated. Assume that the assembly is suitably braced to prevent buckling and that E s = 200 x 10 3 MPa, E a = 70 x 10 3 MPa, and E b = 83 x 10 3 MPa. Find the value of the force acting on each of the rod so that it will not exceed a maximum overall deformation of 2 mm or a stress of 120 MPa, 140 MPa and 80 MPa, respectively. 3P P 4P 2P Bronze A = 450 mm 2 Aluminum A = 600 mm 2 Steel A = 300 mm 2 0.6m 1.0m 0.8m

12 The rigid bar AB, attached to two vertical rods as shown is horizontal before the load P is applied. If the load P = 50 kn, determine its vertical movement. Steel L = 3m A = 300 mm 2 E = 200 GPa Aluminum L = 4m A = 500 mm 2 E = 70 GPa A 2m 3m B P = 50 kn

13 Two aluminum rods AB and BC, hinged to rigid supports, are pinned together at B to carry a vertical load P = 20 kn. If each rod has a cross-sectional area of 400 mm 2 and E = 70 x 10 3 MPa, compute the elongation of each rod and the horizontal and vertical displacements of point B. Assume =30 0 = 30 o and B 3m A 2m P = 20 kn C

14 A round bar of length 10m tapers uniformly from a diameter 100 mm at one end to a smaller diameter 30 mm at the other. Determine the elongations caused by an axial tensile load P = 50 kn. 30mm P 100 mm

15 A rod is composed of three segments and carries the axial loads P1 = 120 kn and P2 = 50 kn. Determine the stress in each material if the walls are rigid. Bronze A = 2400 mm 2 E = 83 GPa P1 Aluminum A = 1200 mm 2 E = 70 GPa P2 Steel A = 600 mm 2 E = 200 GPa 0.6m 0.4m 0.3 m

16 A rod is composed of three segments and carries the axial loads P1 = 100 kn and P2 = 60 kn. Determine the stress in each material if the left wall yields 0.60 mm. Bronze A = 2400 mm 2 E = 83 GPa P1 Aluminum A = 1200 mm 2 E = 70 GPa P2 Steel A = 600 mm 2 E = 200 GPa 0.6m 0.4m 0.3 m

17 A rigid beam with negligible mass is pinned at one end and supported by two rods. The beam was initially horizontal before the load P was applied. Find the vertical movement of P if P = 120 kn. A Steel L = 3m A = 300 mm 2 E = 200 GPa 3m 2m Aluminum L = 4m A = 500 mm 2 E = 70 GPa 1m P = 120 kn

18 Three rods, each with an area of 300 mm 2, jointly support the load of 10 kn, as shown. Assuming there was no slack or stress in the rods before the load was applied, find the stress in each rod. E st = 200 GPa and E br = 83 GPa. Steel 3m Bronze Bronze kn

19 A rigid block of mass M is supported by three symmetrically spaced rods as shown in the figure. Each copper rod has an area of 900 mm 2 ; E = 120 GPa; and the allowable stress is 70 MPa. The steel rod has an area of 1200 mm 2 ; E = 200 GPa; and the allowable stress is 140 MPa. Determine the largest mass M which can besupported. M Copper 160 mm Steel 240 mm Copper 160 mm

20 A rigid platform in the figure has negligible mass and rests on two aluminum bars, each 250 mm long. The center bar is steel and is mm long. Find the stress in the steel bar after the center load P = 400 kn is applied. Each aluminum bar has an area of 1200 mm 2 and E = 70 GPa. The steel bar has an area of 2400 mm 2 and E = 200 GPa. P = 400 kn Aluminum 250 mm Steel mm Aluminum 250 mm

21 As shown in the figure, a rigid beam with negligible mass is pinned at O and supported by two rods, identical except for length. If P = 30 kn. Find the (a) load in A, (b) load in B and if rod A elongates by 2 mm, (c) how much elongation of rod B? (d) Vertical movement of P. 2m 2m 1.5m P = 30 kn O L = 1.5 m A L = 2 m B