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B

¦Ó A a 60¡ã A

¦Ó C Y ¦ÓBC B ¦Ó A ?

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BC a 60¡ã A

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600600?Y?0, ??a?a?sin2???a?a?sin2??BC?a?a?0 600300?X?0£¬???a?a?cos2???a?a?cos2??BC?a?a?0

½âµÃ£º?BC?3?£¬?BC?0 Ö÷Ó¦Á¦£º?1?

Y ¦ÓAB 3?£¬?3??£¨?tan600?2?3?£©??3? 3120¡ã X

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(20 points) A circular shaft (d=200 mm) with bending moment M = 96kN?m and torque T= 84kN?m is shown in the figure. Points A and B are the points on the surface of the shaft. Let E=200GPa, ? = 0.25 and ?allow = 160 MPa. (a) find ? and ? on the cross section at A; (b)draw stress element at A £¨or B£©(C) Exam the shaft with 4th £¨or 3th£©strength theory

Z M A M y T y T T x B Solution: £¨1£©£¨Aµã£©µ¥ÔªÌ壨3·Ö£© ÿ¸ö·½Ïò´í¿Û1·Ö δ±êÓ¦Á¦Öµ¿Û1·Ö ?

? ??MyM32M???122.23MPa £¨5·Ö£© IWz?d3??T?16T??53.5MPa £¨5·Ö£© 3IP?d?r4M2?0.75T2??153.36MPa??allow Safe (7 points)Wz

£¨2£©£¨Bµã£©µ¥ÔªÌ壨3·Ö£© ÿ¸ö·½Ïò´í¿Û1·Ö δ±êÓ¦Á¦Öµ¿Û1·Ö ? ? ??MyM32M????122.23MPa £¨5·Ö£© 3IWz?d??T?16T??53.5MPa £¨5·Ö£© IP?d3?r4M2?T2??162.42MPa??allow Unsafe (7 points)Wz 2008-2009Äê¶È¹¤³ÌÁ¦Ñ§Ë«Ó×Ô¶¯»¯07£© Ó¦Á¦×´Ì¬·ÖÎöÁ·Ï° ¹²10Ò³ µÚ5Ò³

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(25 points) A circular shaft (d=200 mm) with bending moment My and torque Mx is shown in the figure. Points A and B are on the surface of the shaft. From the test we know that the axial normal strain at A and the normal strain along 45? direction with axis at B are ?0 = 6?10-4 and ?45 = 4?10-4 respectively. Let E=200GPa, ? = 0.25 and ?allow = 160 MPa. (a) Draw the stress element at A and B£»£¨b£©Find the magnitudes of My and Mx; (c) Exam the shaft with 4th strength theory. z My A My y Mx B B 45 ? x Mx µ¥ÔªÌ壨3·Ö£© ÿ¸ö·½Ïò´í¿Û1·Ö

? or ? ?

???0E?120MPa £¨2·Ö£© M??w?120?106??d332?30??103N?m 45E ?45??1E???2E??E(1??) £¨6·Ö£© ???1???64?106Pa £¨1·Ö£© T??w63t?64?10??d316?32??10N?m £¨4·Ö£© 2 ?.75T2r4?M?0W?163.4MPa??allow Unsafe (5·Ö£©

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(15 points) The storage tank shown is made of a material

that has a uniaxial yield stress of 170MPa. If the tank has an outer diameter of 32cm and a wall thickness of 0.28cm, what is the maximum pressure the tank could carry without yielding according to the maximum shear stress (Tresca) yield criterion?

Solution

The critical points are on the outer surface of the tank as shown in the figure. (1pt)

A pD?0 2tpD?????0

4t????// ?/ So

?1????pD (2pts) 2tpD (2pts) ?2?????4t?3?0 (2pts)

According to the maximum shear stress (Tresca) yield criterion,

?1??3??Y (4pts)

2?t0.28pD??Y p?Y?2??170?2.975MPa (2pts) 2tD32

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a) È¡µ¥ÔªÌå

ÔÚÌùÓ¦±äƬ´¦£¨ÖáÍâ±íÃ棩ȡһµ¥ÔªÌ壨ÈçͼËùʾ£©£¬ÓÐ

M1M132M1 ??31W?d3?d32MM216M2 ??2??31Wt?d3?d16y ?45??45?????45?y ¦Ó ¦Ò x b) ½øÐÐÓ¦Á¦×´Ì¬·ÖÎö

½èÖúÓÚÓ¦Á¦Ô²½øÐзÖÎö£¬¸ù¾Ýµ¥ÔªÌåÓëÓ¦Á¦Ô²µÄ¶ÔÓ¦¹ØϵÓÐ

¦Ò? x ¦Á¦Òx?¦Òy2¦Òx?¦Òy?¦Òx?¦Òy2cos2¦Á?¦Óxysin2a¦Ó??45???2sin2??¦Óxycos2a

2????45???0???45???2????

c) Ó¦Óû¢¿Ë¶¨ÂÉ

B(?,??) (?45?,?45?) ??1? EO?A(?,???)

11??1??(?45?????45?)????E2EE1??1????0???2E(??45?,??45?) d) Çó½âÀ­Á¦ºÍŤ¾Ø ÓÉÉÏÃæµÄ·ÖÎöµÃµ½·½³Ì×é

32M1?E?0?

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M1??d3E?0?32=0.278(kN¡¤m)

?d3E1??M2?(?45???0?)=0.214(kN¡¤m)

16(1??)2