Mechanical Engineering Chapter 3 Stress And Deformation Analysis Part 1

11

CHAPTER 3

STRESS AND DEFORMATION ANALYSIS

DirectTensionandCompression

ͳǤ ߪ ൌ ܨȀܣǢܣ ൌ ߨሺͳͺଶെͳʹଶሻȀͶൌ૚૝૚Ǥ૝ܕܕ૛

͸Ǥ ߪ ൌ ܲȀܣǢܣ ൌ ሺͳʹ݉݉ሻଶൌ ͳͶͶଶǢߪ ൌͷͲͲͲȀͳͶͶଶൌ ૜૝Ǥૠۻ۾܉ሺۯܔܔሻ

ߜൌ௉௅

ா஺ ൌሺହ଴଴଴ேሻሺଵ଺ହ଴௠௠ሻ

ሺாேȀ௠௠మሻሺଵସସ௠௠మሻൌହ଻ଶଽଶ

ா

͹Ǥ ߜ ൌ ܲܮȀܧܣǢܲ ൌ ఋா஺

௅

ͳͲǤ ߪ ൌ ܲȀܣǢܣ ൌ ܲȀߪ௔௟௟௢௪

ͳͳǤ ߠ ൌͳͷιǢܨோൌଵହ଴଴௟௕

ଶ௦௜௡ఏ ൌଵହ଴଴௟௕

ଶ௦௜௡ሺଵହሻൌ ʹͺͻͺ



ܨோܨோ

ܨ ൌͳͷͲͲ



ͳʹǤ 

ܦ

ܴ

஺

ൌܣܤݏ݅݊ߠൌܣܤሺͲǤͺሻ

ܣ

ܤ

ܣ

15

 ǣሺͷǡ͸ሻ ǣ

ܣܦǡܦܧǡܧܨെʹ

ሺͲǤͶͺͶሻൌͲǤͻ͸ͺ

ଶ

ߪ

஺஽

ൌߪ

஽ா

ൌ͸ͲͲͲȀͲǤͻ͸ͺ ൌ൅૟૚ૢૡܘܛܑ

ଶ

ߪ

஺஻

ൌെͳͲͲͲͲȀʹǤͶʹൌ െ૝૚૜૛ܘܛܑ

ͳͶǤ ܣ ൌ ሺʹǤ͸ͷሻሺͳǤͶͲሻ൅ʹሾሺͳǤͶͲሻሺͲǤͷሻȀʹሿൌ ͶǤͶͳ

ଶ



ଶ

DirectShearStres

s



ͳ͸Ǥ ൌ ͲǤͷͲǢ

ܣ

௦

ൌʹሺߨܦ

ଶ

ȀͶሻൌʹߨሺͲǤͷͲሻ

ଶ

ȀͶ ൌ ͲǤ͵ͻʹ͹

ଶ



͹ͷ

ͷͲ

ܣ

ܨ

஺஼



௬

ܥʹͷͲͲ

16

ܤ

௬ൌʹͷͲͲ

ͳ͹Ǥ ͵Ǧͻǣൌܨோൌ ͳͷͲͲȀʹݏ݅݊ߠ

ͳͺǤ ͻͳ͹Ǥߠ ൌͳͷι

ͳͻǤ ͵Ǧ͹ǣǢܣ௦ൌܾήܮൌሺͳʹሻሺͶͷሻൌ ͷͶͲଶ

ʹͲǤ ȂǤ͵ǦʹͲǣܣ௦ൌሺ݌݁ݎ݅݉݁ݐ݁ݎሻݐ ൌ ൣʹǤͷͲ൅ʹǤͲͲ൅ͳǤͷͲ൅ξͲǤͷଶ൅ʹǤͷଶ൧ሺͲǤͲ͸Ͳሻ

ʹͳǤ Ȃ͵ǦʹͳǤൌ͸Ͳ൅ʹሺ͵Ͳሻ൅ʹሺ͹Ǥͷሻ൅͵ቂగሺଵହሻ

ଶቃൌ ʹͲͷǤ͹

17

Torsion

ʹ͵Ǥ ߠ ൌ்௅

ீ௃ǣܶ ൌ ͺͲͲή ൌͺͲͲൈͳͲଷήǣܩ ൌͺͲ ൌͺͲൈͳͲଷȀଶ

గ஽ర

ʹͶǤ ߬ ൌ ்

௓೛ൌ்

గ஽యȀଵ଺ ൌ଼଼Ǥ଴୪ୠή୧୬

஠ሺ଴Ǥସ଴୧୬ሻయȀଵ଺ ൌૠ૙૙૜ܘܛܑ

ʹͷǤ ܶ ൌ͸͵ͲͲͲሺܲሻȀ݊ ൌ ͸͵ͲͲͲሺͳͳͲሻȀͷ͸Ͳൌ ͳʹ͵͹ͷή

்

ʹ͸Ǥ ܶ ൌܲȀ݊ ൌ ଶ଼ൈଵ଴య୒ή୫Ȁୱ

ସହ୰ୟୢȀୱ ൌ͸ʹʹή

ʹ͹Ǥ ߠ ൌ்௅

ீ௃ǣܬ ൌ గ൫஽రିௗర൯

ଷଶ ൌగൣସ଴రିଷ଴ర൧

ଷଶ ସൌͳǤ͹ͳͺൈͳͲହସ

18

݄ ൌ ͲǤ͸Ͳ

NoncircularMembersinTorsion

ʹͺǤ ǣܽ ൌʹͷǢܳ ൌ ͲǤʹͲͺܽଷൌ ͵ʹͷͲଷ

ʹͻǤ ݄Ȁݎ ൌ ͲǤ͸ͲȀͳǤͷͲൌͲǤͶͲ

͵ͲǤ ܽ ൌ ʹǤͲǢܾ ൌͶǤͲǢݐ ൌ ͲǤͳͲͻ

ሺܽെݐሻൌ ͳǤͺͻǢሺܾെݐሻൌ ͵ǤͺͻͳǢܮ ൌ͸Ǥͷ ൌ͹ͺ

ଶ௧ሺ௔ି௧ሻమሺ௕ି௧ሻమ



͵ǦͳͲ

Beams

͵ͳǤ ߪ ൌ ܯȀܵǣܵ ൌ ܯȀߪ௔௟௟௢௪

ܵൌ

ଷ଺଴଴୪ୠή୤୲

ଵ଼଴଴଴୪ୠȀ୧୬మൈଵଶ୧୬

୤୲ ൌʹǤͶͲଷ

 ܣ ൌ ͸Ǥʹͷଶ

 ሻ 

ଷȀ͸

ܣ

͵



ͳʹͲͲ

ͳʹͲͲ

ܤ

ܥ

ܦ

ͳʹͲͲ

Ͷ



͵



Ͳ

ܸ

ሺ



ሻ

ܽ

20

ሻܵͶൈ͹Ǥ͹

ܵൌ͵ǤͲͶ

ଷǢܣ ൌ ʹǤʹ͸ଶ

 ሻ ܥͳͷൈͶͲ

 ሻ ͶǦͶͲ

͵ʹǤ ሻ ൌ ܣൈܮ ൌ ሺʹǤͷ݅݊ሻଶൈͳʹͲ ൌ ͹ͷͲଷǢܹ ൌ ሺͲǤʹͺ͵Ȁଷሻሺ͹ͷͲଷሻ

͵͵Ǥ ሺሻǢͳͶǦͳǣܽ ൌ ͵͸Ǣܮ ൌͳʹͲǢܲ ൌ ͳʹͲͲǢܧ ൌ͵ͲൈͳͲ଺

 ሻ ܫ ൌܽସȀͳʹ ൌ ሺʹǤͷͲሻସȀͳʹ ൌ͵Ǥʹ͸ସǣ࢟࢓ࢇ࢞ ൌଶǤଶ଼ଵ

ଷǤଶ଺ ൌ૙Ǥૠ૙૚ܑܖ

ሻܫൌܾ݄

ሻܫൌ

గ஽ర

଺ସ ൌగሺଷ଴଴ሻర

଺ସ ൌ͵Ǥͻͺସǣ࢟࢓ࢇ࢞ ൌ૙Ǥ૞ૠ૝ܑܖǢ࢟࢈ൌ૙Ǥ૝૟ૢܑܖ

͵ͶǤ 





ͲǤͶ ͲǤͺ

ͺͲͲ ͶͲͲ ͸ͲͲ ʹͲͲ

ͲǤͶ

ͲǤ͵ ͲǤ͵

ܣ

ܤܥ

ܣ

22

 ǣȂǣܦ௢ൌ ͶͷǢܦ௜ൌ ͶͲǢܣ ൌ ͵͵͵Ǥͺଶ

ܵൌ͵͵͸ͳଷ

͵͸Ǥ ሻ ȂሺሻȂͳͶǦʹǣܫ ൌ͹Ǥʹ͵ସͳͷെͳ͹

 ሻ ȂȂͳͶǦ͵

ݕ

஺ൌെ௉௫భ

మ௕

ଵଶாூ௅యሺ͵ܥଵെܥ

ଶݔଵሻ

ݔ

ଵൌͶ ൌͶͺǢܽ ൌ ͸ൌ ͹ʹǢܾ ൌͶൌ Ͷͺ

͵͹Ǥ ߪൌܯȀܵǢܵൌܯȀߪ

௔௟௟௢௪

ଶ

଼଴଴଴୪ୠή୧୬

 ǦǤ

 ͵ܫൈͳǤ͸͵͹Ǣܵ ൌ ͳǤͶͻଷǢܫ ൌ ʹǤʹͶସ

ͶͲͲ ͶͲ͵ͲʹͲ ͷͲͲ

െͳͲͲ

ܴ

ଵ

ܸ

ሺሻ

Ͳ

24

͵ͻǤ



ͶͲǤ



͸Ͳ

Ͷ

ͳʹ ͳʹ

 ͸Ͳ

ܯ ൌʹͶͲή

ͳʹͳʹ



ܴଵൌͳͲ

25

ͶͳǤ 



ͶʹǤ



ܯ ൌʹͶͲή

ͳʹ ͳʹ

ʹͶͲή ܴଶൌͳͲ

Ͳ



ሺήሻ

െͳʹͲ

ͷͲͲ ʹͲͲ

ͳ͹ͷ

͵ͻ͵͹ͷ

Ͳ Ͳ

ʹʹͷ

ܯ

ሺήሻ



26

ܸ

ሺሻ Ͳ

ܴுൌͶͷͲ

െͳͺ͹Ǥͷ

െͳͳʹͷͲ

ଶൌͳͺ͹.5N

ܯ ൌͶͷͲሺʹͷሻൌ ͳͳʹͷͲή

Ͷ͵Ǥ



ͶͶǤ





ͷͲͲ ͵ͲͲ

ܸ

ሺሻ



ͻ͵͹ͷͲ ͻ͵͹ͷͲ

Ͳ



ܯ

ሺήሻ Ͳ

27

ͶͷǤ



ܯ ൌͳͺͲሺ͸Ǥ͹ͷሻൌ ͳʹͳͷή



ܸ

Ͷ͸Ͳ

ͳʹͲ

ͳͳ̶ ͳʹ̶ ͷǤͷ̶

ʹ͸Ͷ ͳʹͲ

Ͷ͸Ǥ 

ǣ

ൌʹǤ͸ͶെͳǤͺͲ ൌͲǤͺͶ ՝

ͶͺǤ



ܣ

ܯ ൌ ʹͲͲሺͶሻ൅͸ͲͲሺͶሻൌ ͵ʹͲͲή

 ൌ ܶ ൌ ʹͲͲሺͺሻ൅͸ͲͲሺͺሻ ൌ ͸ͶͲͲήሺ



ሻ

Ȃ



ͶͲͲ

ͳʹ ͳ͸

ܣ

ܥ

ܤ



ܯ

Ͳ

30

ͷͲǤ



CombinedNormalStresses

ͷͳǤܣൌʹͲ

ሺͺͲሻൌͳ͸ͲͲଶ



ܸ

ሺሻ

Ͳ

െͲǤ͸ͷͲ

െͳǤʹͷ

ͳǤͳͺ

ܯ ൌͳǤʹͷሺͲǤʹʹሻൌͲǤʹ͹ͷή

ൌ ͳǤʹͷሺͲǤ͵Ͳሻൌ ͲǤ͵͹ͷή 

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