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# Mechanics of Deformable Bodies 057 019

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This 11 page Class Notes was uploaded by Augustine Schmitt on Friday October 23, 2015. The Class Notes belongs to 057 019 at University of Iowa taught by Colby Swan in Fall. Since its upload, it has received 143 views. For similar materials see /class/228005/057-019-university-of-iowa in General Engineering at University of Iowa.

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Date Created: 10/23/15

05701900B Mechanics of Deformable Bodies College of Engineering Instructor CC Swan University of Iowa Period 33 Spring Semester 2009 Textbook Section 1267 Topic Analysis of Statically Indeterminate Beams Objectives 1 Review of beam theory thus far 2 Key idea for indeterminate beams 3 Demonstrate new concept with examples 1 Review of beam differential equations wxd VEI 4V 1 o For beams having linear elastic material behaviors dx d and undergoing small deformations the differential Vx E dsv 2 equations that relate a general applied loading wx dx 5amp3 on a beam to the shear distribution Vx the dg div moment distribution Mx the slope distribution Mx EIE Efdx z 3 9x and the deflection distribution Vx are as dv shown 6x a 4 When the differential equations are integrated constants of integration are introduced These constants are determined from the boundary conditions or support conditions ofthe beam me waer c 5 Mx EVOEWE c 6 6x awry C 7 me 36W C 8 For complicated loading conditions wx the algebra associated with the solving the fourth order differential equations can become somewhat complicated and messy For this reason singularity functions were introduced in Section 123 to simplify the process of integrating the differential equations 05701900B Mechanics of Deformable Bodies College of Engineering Instructor CC Swan University of Iowa 2 Application to Statically Indeterminate Beams Statically indeterminate beams are Sum those that have redundant supports 3 quotMquot For such beams the magnitude of the support reactIons cannot be I a I determlned slmply from the equatlons 1 IQ of statics X 4i l 4 m 3 m 3 m Fortunately it is fairly straightforward to analyze indeterminate beams given what we now know The loading on an indeterminate beam can be specified as wx and this will include all of the reaction forces acting on the beam whether they are known or unknown For each redundant support there will be a corresponding kinematic constraint on either the slope or deflection of the beam When these additional kinematic constraints are taken into account the redundant reaction forces are can be determined in much the same manner as the constants of integration 05701900B Mechanics of Deformable Bodies Instructor CC Swan Example 1 12107 Determine the moment reactions at the supports A and B El is constant College of Engineering University of Iowa P 1 B Hi FE 05701900B Mechanics of Deformable Bo dies College of Engineering Instructor CC Swan University oroWa Example 2 127109 Determine the reactions at the supports and then sketch the shear and moment diagrams El is constant 05701900B Mechanics of Deformable Bodies College ofEngineen39ng lnstmctor CC Swan University of Iowa Example 3 Forthe beams shownthe Young39s modulus Ew2107 kPa and the crosssections have dimensions h290mm and b70mm Solve for the support reactions at A B G and H 057019BBB Mechanics of Deformable Bodies College of Engineering Instructor CC Swan University of Iowa Period 20 Spring Semester 2009 Topic States of Stress Caused by Combined Loadings Textbook Chapter 8 Objectives To this point we have looked at stresses in structural members due to axial loading torsion bending and transverse shear Often these types of loads act simultaneously on structural members The ability to assess the state of stress from combined loadings is important and will be treated in both Chapters 8 and 9 1 In the current period we will first look at stresses in thinwalled pressure vessels 2 Then we will look at stresses from combined loads 1 Thinwalled pressure vessels Key assumptions a Pressure inside of the vessel is greater than that outside b Thickness ofthe walls is small relative to the radius of curvature rt 210 P Cylindrical Vessels 39r ii GitX L t if pk 2 if r r 1 l Age 0 i hoop stress 039l axial stress 039Z 2 057019BBB Mechanics ofDeformable Bodies College of EngineeIing Instructor CC Swan University of Iowa b Spherical Vessels 2 State of Stress Caused by Combined Loadings Apply the principal of superposition to find each load39s contribution to specific stress components at points of interest Key Assumption Linear relationship exists between applied loadings and the stress components This implies a Linear elastic material behavior b Small deformation 057019BBB Mechanics ofDeformable Bodies College ofEngineeIing Instructor CC Swan University of Iowa Examp e 1 86 The openended polyvinyl chloride pipe has an inner diameter of 4in and thickness of 02in If it carries flowingwater at 60 psi pressure determine the state of stress in the walls of the pipe 87 If the flow of water within the pipe is stopped due to the closing of a valve determine the state of stress in the walls ofthe pipe Neglect the weight ofthe water Assume the supports only exert vertical forces on the pipe 057019BBB Mechanics ofDeformable Bodies College ofEngineeIing Instructor CC Swan University of Iowa Example 2 836 The cylinder of negligible weight rests on a smooth floor Determine the eccentric distance ey at which the load can be placed so that the normal stress at point A is zero 057019BBB Mechanics ofDeformable Bodies Instructor CC Swan College of EngineeIing University of Iowa Example 3 870 The wall hanger has a thickness of 025 in and is used to support the vertical reactions ofthe beam that is loaded as shown Ifthe load is transferred uniformly to each strap of the hanger determine the state of stress at point Cand D of the strap at B Assume the vertical reaction F at this and acts in the center and on the edge ofthe bracket as shown 2 kipfl 0572019zBBB Mechanics of Deformable Bodies College of Engineering Instructor CC Swan University of Iowa Example 3 870 The wall hanger has a thickness of 025 in and is used to support the vertical reactions of the beam that is loaded as shown If the load is transferred uniformly to each strap of the hanger determine the state of stress at point C and D of the strap at B Assume the vertical reaction F at this end acts in the center and on the edge of the bracket as shown 2 I 539 i I 3 i g 2 Mn 0 Pg vo ezow sz 39 x2144 R 8 a 4 g D I 3 1 MAE m 39 M 1 24 44 2w WW 1 A I lav 3 f 5 S L39 a HXzm 0333 a e H 2 G7 E 5 i i4 i 5 b A le We a 2 awry C5 l M a i c w 7 t W 0 1 6 A f A 7449234

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