MECHANICS OF MATERIALS
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1 CHATER MECHANICS OF MATERIAS Ferdinand. Beer E. Russell Johnston, Jr. John T. DeWolf Energy Methods ecture Notes: J. Walt Oler Teas Tech niversity 6 The McGraw-Hill Copanies, Inc. All rights reserved.
2 MECHANICS OF MATERIAS Energy Methods Strain Energy Strain Energy Density Elastic Strain Energy for Noral Stresses Strain Energy For Shearing Stresses Saple roble. Strain Energy for a General State of Stress Ipact oading Eaple.6 Eaple.7 Design for Ipact oads Work and Energy nder a Single oad Deflection nder a Single oad Saple roble.4 Work and Energy nder Several oads Castigliano s Theore Deflections by Castigliano s Theore Saple roble.5 6 The McGraw-Hill Copanies, Inc. All rights reserved. -
3 MECHANICS OF MATERIAS Strain Energy A unifor rod is subjected to a slowly increasing load The eleentary work done by the load as the rod elongates by a sall d is d d eleentary work which is equal to the area of width d under the loaddeforation diagra. The total work done by the load for a deforation, d total work strain energy which results in an increase of strain energy in the rod. In the case of a linear elastic deforation, kd k 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 3
4 MECHANICS OF MATERIAS Strain Energy Density To eliinate the effects of size, evaluate the strainenergy per unit volue, V A ε d u σ dε strain energy density The total strain energy density resulting fro the deforation is equal to the area under the curve to ε. As the aterial is unloaded, the stress returns to zero but there is a peranent deforation. Only the strain energy represented by the triangular area is recovered. Reainder of the energy spent in deforing the aterial is dissipated as heat. 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 4
5 MECHANICS OF MATERIAS Strain-Energy Density The strain energy density resulting fro setting ε ε R is the odulus of toughness. The energy per unit volue required to cause the aterial to rupture is related to its ductility as well as its ultiate strength. If the stress reains within the proportional liit, ε Eε σ u Eε dε E The strain energy density resulting fro setting σ σ Y is the odulus of resilience. σ u Y Y odulus of E resilience 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 5
6 MECHANICS OF MATERIAS Elastic Strain Energy for Noral Stresses In an eleent with a nonunifor stress distribution, d u li u dv total strain energy V V dv For values of u < u Y, i.e., below the proportional liit, σ dv elastic strain energy E nder aial loading, d AE σ A dv A d For a rod of unifor cross-section, AE 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 6
7 MECHANICS OF MATERIAS Elastic Strain Energy for Noral Stresses For a bea subjected to a bending load, σ M y dv dv E EI Setting dv da d, σ M y I A M y EI dad M EI A y da d M EI d For an end-loaded cantilever bea, M d EI 6EI 3 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 7
8 MECHANICS OF MATERIAS Strain Energy For Shearing Stresses For a aterial subjected to plane shearing stresses, u γ y τ y d γ y For values of τ y within the proportional liit, u y τ Gγ y τ y γ y G The total strain energy is found fro u dv τ y dv G 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 8
9 MECHANICS OF MATERIAS Strain Energy For Shearing Stresses For a shaft subjected to a torsional load, τ y T ρ dv dv G GJ Setting dv da d, τ y Tρ J A T ρ GJ T GJ d dad T GJ A ρ da d In the case of a unifor shaft, T GJ 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 9
10 MECHANICS OF MATERIAS Saple roble. SOTION: Deterine the reactions at A and B fro a free-body diagra of the coplete bea. Develop a diagra of the bending oent distribution. a) Taking into account only the noral stresses due to bending, deterine the strain energy of the bea for the loading shown. b) Evaluate the strain energy knowing that the bea is a W45, 4 kips, ft, a 3 ft, b 9 ft, and E 9 6 psi. Integrate over the volue of the bea to find the strain energy. Apply the particular given conditions to evaluate the strain energy. 6 The McGraw-Hill Copanies, Inc. All rights reserved. -
11 MECHANICS OF MATERIAS Saple roble. SOTION: Deterine the reactions at A and B fro a free-body diagra of the coplete bea. b RA RB a Develop a diagra of the bending oent distribution. b M M a v 6 The McGraw-Hill Copanies, Inc. All rights reserved. -
12 MECHANICS OF MATERIAS Saple roble. Over the portion AD, M b Over the portion BD, M a v Integrate over the volue of the bea to find the strain energy. a b M M d + EI EI a dv b d + EI EI EI a b 6EI b a 3 3 a b kips 44 in. ( 4 kips) ( 36 in) ( 8 in) 3 4 a 36 in. b 8 in. 6 9 ksi 48 in 44 in 3 E 9 ksi I 48 in 4 3 b a a b 6EI ( )( )( ) 3.89 in kips d ( a + b) 6 The McGraw-Hill Copanies, Inc. All rights reserved. -
13 MECHANICS OF MATERIAS Strain Energy for a General State of Stress reviously found strain energy due to uniaial stress and plane shearing stress. For a general state of stress, u ( σ ε + σ ε + σ ε + τ γ + τ γ + τ γ ) y y z z y With respect to the principal aes for an elastic, isotropic body, u u u v d E [ ( )] σ + σ + σ ν σ σ + σ σ + σ σ a b c a b b c c a u v + u d v a b c 6 E G ( σ + σ + σ ) due to volue change y [ ( ) ( ) ( ) ] σ σ + σ σ + σ σ due to distortion a b b c c a Basis for the aiu distortion energy failure criteria, u σ Y d < d Y ( u ) Y for a tensile test tspecien 6G yz yz z z 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 3
14 MECHANICS OF MATERIAS Ipact oading To deterine the aiu stress σ Consider a rod which is hit at its end with a body of ass oving with a velocity v. Rod defors under ipact. Stresses reach a aiu value σ and then disappear. - Assue that the kinetic energy is transferred entirely to the structure, v - Assue that the stress-strain diagra obtained fro a static test is also valid under ipact loading. Maiu value of the strain energy, σ dv E For the case of a unifor rod, σ E v E V V 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 4
15 MECHANICS OF MATERIAS Eaple.6 SOTION: Due to the change in diaeter, the noral stress distribution is nonunifor. Find the static load which produces the sae strain energy as the ipact. Evaluate the aiu stress Body of ass with velocity v hits the end of the nonunifor rod BCD. Knowing that the diaeter of the portion BC is twice the diaeter of portion CD, deterine the aiu value of the noral stress in the rod. resulting fro the static load 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 5
16 MECHANICS OF MATERIAS Eaple.6 Find the static load which produces the sae strain energy as the ipact. ( ) ( ) AE AE 4AE 5 6 AE Evaluate the aiu stress resulting SOTION: fro the static load Due to the change in diaeter, the noral stress distribution is σ A nonunifor. 6E v 5 A σ σ V dv EE EE 8 v E 5 A 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 6
17 MECHANICS OF MATERIAS Eaple.7 SOTION: The noral stress varies linearly along the length of the bea and across a transverse section. Find the static load which produces the sae strain energy as the ipact. A block of weight W is dropped fro a height h onto the free end of the cantilever bea. Deterine the aiu value of the stresses in the bea. Evaluate the aiu stress resulting fro the static load 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 7
18 MECHANICS OF MATERIAS Eaple.7 SOTION: The noral stress varies linearly along the length of the bea and across a transverse section. Wh σ σ V dv EE EE Find the static load which produces the sae strain energy as the ipact. For an end-loaded cantilever bea, 3 6EI 6 EI 3 Evaluate the aiu stress resulting fro the static load σ M I c 6 c I 6WhE ( ) ( ) I c I c E 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 8
19 MECHANICS OF MATERIAS Design for Ipact oads Maiu stress reduced by: unifority of stress low odulus of elasticity with high yield strength For the case of a unifor rod, σ E V For the case of the nonunifor rod, σ V σ 4A 6 E 5 A ( / ) + A( / ) 5A 8 E V 6 E ( ) if it f t I c ( ) ( 4 ) ( ) I / c πc / c πc V high volue σ / For the case of the cantilever bea 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 9 σ 4 4 E V 4 4
20 MECHANICS OF MATERIAS Work and Energy nder a Single oad Strain energy ay also be found fro the work of the single load, d For an elastic deforation, reviously, we found the strain energy by integrating the energy d kd k density over the volue. For a unifor rod, Knowing the relationship between u dv σ dv force and displaceent, E ( A) AE Ad E AE AE AE 6 The McGraw-Hill Copanies, Inc. All rights reserved. -
21 MECHANICS OF MATERIAS Work and Energy nder a Single oad Strain energy ay be found fro the work of other types of single concentrated loads. Transverse load Bending couple Torsional couple y dy y M dθ M θ 3 3EI 3 6EI θ M M EI M EI φ T dφ T φ T T JG T JG 6 The McGraw-Hill Copanies, Inc. All rights reserved. -
![l BC F + BD AE BD [ 3 3 ( ) ( ) ].6 +.8 AE l.](/docs-images/81/83251009/images/22-4.jpg "364 AE Equating work and strain energy,.")
22 MECHANICS OF MATERIAS Deflection nder a Single oad If the strain energy of a structure due to a single concentrated load is known, then the equality between the work of the load and energy ay be used to find the deflection. Fro the given geoetry,.6l. BC BD 8 Fro statics, F +.6 F. BC BD 8 l Strain energy of the structure, F BC AE l BC F + BD AE BD [ 3 3 ( ) ( ) ] AE l.364 AE Equating work and strain energy,.364 AE l y B. 78 AE y B 6 The McGraw-Hill Copanies, Inc. All rights reserved. -
23 MECHANICS OF MATERIAS Saple roble.4 SOTION: Find the reactions at A and B fro a free-body diagra of the entire truss. Apply the ethod of joints to deterine the aial force in each eber. Mebers of the truss shown consist of sections of aluinu pipe with the cross-sectional areas indicated. sing E 73 Ga, deterine the vertical dfl deflection of fthe point itee caused dby the load. Evaluate the strain energy of the truss due to the load. Equate the strain energy to the work of and solve for the displaceent. 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 3
24 MECHANICS OF MATERIAS Saple roble.4 SOTION: Find the reactions at A and B fro a free-body diagra of the entire truss. A 8 A B y 8 Apply the ethod of joints to deterine the aial force in each eber. F DE 7 8 F AC FCD F CE F DE F AB 5 4 F CE 8 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 4
25 MECHANICS OF MATERIAS Saple roble.4 Evaluate the strain energy of the Equate the strain energy to the work by truss due to the load. and solve for the displaceent. i i i F F A E E A E i ( ) 97 i i y E y E y E 97 E ( 3)( 3) y E The McGraw-Hill Copanies, Inc. All rights reserved. - 5
26 MECHANICS OF MATERIAS Work and Energy nder Several oads Deflections of an elastic bea subjected to two concentrated loads, + α + α + α + α Copute the strain energy in the bea by evaluating the work done by slowly applying followed by, ( ) α + α + α Reversing the application sequence yields ( ) α + α + α Strain energy epressions ust be equivalent. It follows that α α (Mawell s s reciprocal theore). 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 6
27 MECHANICS OF MATERIAS Castigliano s Theore Strain energy for any elastic structure subjected to two concentrated loads, ( ) α + α + α Differentiating with respect to the loads, α α + α + α Castigliano s theore: For an elastic structure subjected to n loads, the deflection j of fthe point of application of j can be epressed as j and θ j φ j M 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 7 j j T j
28 MECHANICS OF MATERIAS Deflections by Castigliano s Theore Application of Castigliano s theore is siplified if the differentiation with respect to the load j is perfored before the integration or suation to obtain the strain energy. In the case of a bea, M d j EI EI j j M M d For a truss, n n Fi i j A i ie j i Fi i A E i F i j 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 8
29 MECHANICS OF MATERIAS Saple roble.5 SOTION: Mebers of the truss shown consist of sections of aluinu pipe with the cross-sectional areas indicated. sing E 73 Ga, deterine the vertical deflection of the joint C caused by the load. For application of Castigliano ss theore, introduce a duy vertical load Q at C. Find the reactions at A and B due to the duy load fro a free-body diagra of the entire truss. Apply the ethod of joints to deterine the aial force in each eber due to Q. Cobine with the results of Saple roble.4 to evaluate the derivative with respect to Q of the strain energy of the truss due to the loads and Q. Setting Q, evaluate the derivative which is equivalent to the desired displaceent at C. 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 9
30 MECHANICS OF MATERIAS Saple roble.5 SOTION: Find the reactions at A and B due to a duy load Q at C fro a free-body diagra of the entire truss. A 3 Q Ay Q B Apply the ethod of joints to deterine the aial force in each eber due to Q. Q F F F F CE DE AC ; F CD Q 3 AB ; F BD 4 Q 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 3
31 MECHANICS OF MATERIAS Saple roble.5 Cobine with the results of Saple roble.4 to evaluate the derivative with respect to Q of the strain energy of the truss due to the loads and Q. F F y C 463 Ai E Q E i i i ( Q ) Setting Q, evaluate the derivative which is equivalent to the desired displaceent at C. y C ( 3 ) N 73 9 a y C.36 6 The McGraw-Hill Copanies, Inc. All rights reserved. - 3
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