Stress Transformation: Lecture (important Figures)

https://drive.google.com/file/d/0B0hvU4t2vA-CU1NrMkpxdmJSQUE/view?usp=sharing

Regarding Make up Test of ME-213

Those Students who need to appear in the Make-Up Test for ME-213, may kindly discuss the schedule with me till Saturday (8/11/14) so that It may not have any clash with other Make Up Test Schedule. Otherwise it will be notified at any day in the upcoming week.

Please inform the back log students who are in the III, IV, V year of graduation and appearing in this Course.

Stress Transformation: Plane Stress

Plane State of Stress

A class of common engineering problems involving stresses in a thin plate or on the free surface of a structural element, such as the surfaces of thin-walled pressure vessels under external or internal pressure, the free surfaces of shafts in torsion and beams under transverse load, have one principal stress that is much smaller than the other two. By assuming that this small principal stress is zero, the three-dimensional stress state can be reduced to two dimensions. Since the remaining two principal stresses lie in a plane, these simplified 2D problems are called plane stressproblems. Assume that the negligible principal stress is oriented in the z-direction. To reduce the 3D stress matrix to the 2D plane stress matrix, remove all components with z subscripts to get, where txy = tyx for static equilibrium. The sign convention for positive stress components in plane stress is illustrated in the above figure on the 2D element.

Coordinate Transformations

The coordinate directions chosen to analyze a structure are usually based on the shape of the structure. As a result, the direct and shear stress components are associated with these directions. For example, to analyze a bar one almost always directs one of the coordinate directions along the bar's axis. Nonetheless, stresses in directions that do not line up with the original coordinate set are also important. For example, the failure plane of a brittle shaft under torsion is often at a 45° angle with respect to the shaft's axis. Stress transformation formulas are required to analyze these stresses. The transformation of stresses with respect to the {x,y,z} coordinates to the stresses with respect to {x',y',z'} is performed via the equations, where q is the rotation angle between the two coordinate sets (positive in the counterclockwise direction). This angle along with the stresses for the {x',y',z'} coordinates are shown in the figure below,

Principal Stresses and Principal Directions

The normal stresses (sx' and sy') and the shear stress (tx'y') vary smoothly with respect to the rotation angle q, in accordance with the coordinate transformation equations. There exist a couple of particular angles where the stresses take on special values. First, there exists an angle qp where the shear stress tx'y' becomes zero. That angle is found by setting tx'y' to zero in the above shear transformation equation and solving for q (set equal to qp). The result is, The angle qp defines the principal directions where the only stresses are normal stresses. These stresses are called principal stresses and are found from the original stresses (expressed in the x,y,z directions) via, The transformation to the principal directions can be illustrated as:

Maximum Shear Stress Direction

Another important angle, qs, is where the maximum shear stress occurs. This is found by finding the maximum of the shear stress transformation equation, and solving for q. The result is, The maximum shear stress is equal to one-half the difference between the two principal stresses, The transformation to the maximum shear stress direction can be illustrated as:

Generalized Hookes Law (Anisotropic material)

Recalling One-dimensional Hooke's Law

Robert Hooke, who in 1676 stated, "The power (sic.) of any springy body is in the same proportion with the extension." announced the birth of elasticity. Hooke's statement expressed mathematically is, where F is the applied force (and not the power, as Hooke mistakenly suggested), u is the deformation of the elastic body subjected to the force F, and k is the spring constant (i.e. the ratio of previous two parameters).

Generalized Hooke's Law (Anisotropic Form)

Cauchy generalized Hooke's law to three dimensional elastic bodies and stated that the 6 components of stress are linearly related to the 6 components of strain. The stress-strain relationship written in matrix form, where the 6 components of stress and strain are organized into column vectors, is,   ,      s = C·e or,   ,      e = S·s where C is the stiffness matrix, S is the compliance matrix, and S = C-1. In general, stress-strain relationships such as these are known as constitutive relations. In general, there are 36 stiffness matrix components. However, it can be shown that conservative materials possess a strain energy density function and as a result, the stiffness and compliance matrices are symmetric. Therefore, only 21 stiffness components are actually independent in Hooke's law. The vast majority of engineering materials are conservative. Please note that the stiffness matrix is traditionally represented by the symbol C, while S is reserved for the compliance matrix.

Mid Semester marks: A2MB (revised on 3 Nov)

Section	S.No.	Fac. No	En No.	NAME	Marks(25)
A2MBG1	1	13MEB264	GE8041	SAMEER_HASAN	13
A2MBG1	2	13MEB265	GE8050	AMAN GARG	11.5
A2MBG1	3	13MEB266	GE8074	JUNAID_AKHTAR	15.5
A2MBG1	4	13MEB267	GE8133	SHANE_ALAM	14
A2MBG1	5	13MEB268	GE8080	SANAT_YAR_KHAN	17
A2MBG1	6	13MEB269	GE8090	MD._TAUFIQUE_KHAN_	13
A2MBG1	7	13MEB275	GH5160	HASSAN_SHAKIL_AHMAD	10
A2MBG1	8	13MEB305	GE6573	MD._NAZIBULLAH_ANSARI	14
A2MBG1	9	13MEB315	GE1427	MOHAMMAD_HARIS	13
A2MBG1	10	13MEB321	GE1323	KHURSHEED_AFROZ_ANSARI	17.5
A2MBG1	11	13MEB322	GG0544	PIYUSH_SENGAR	11.5
A2MBG1	12	13MEB323	GE3281	MANISH_KUMAR_DIXIT	14.5
A2MBG1	13	13MEB325	GH4474	ARMAN_HADI	14.5
A2MBG1	14	13MEB326	GH4468	MAHBOOB_AHMED	12.5
A2MBG1	15	13MEB327	GG0517	MD._RASHID_KHAN	13.5
A2MBG1	16	13MEB332	GE3422	SOHAIL_AKHTAR	10.5
A2MBG1	17	13MEB335	GE3477	CHIRAG_SINGHAL	12
A2MBG1	18	13MEB337	GE6587	ASHUTOSH_PANDEY	10.5
A2MBG1	19	13MEB339	GH4492	SHADAB_HUSSAIN	19
A2MBG1	20	13MEB342	GE0195	KAVISH_AHMAD	18
A2MBG1	21	13MEB344	GH4501	MD._AZAHARUDDIN_ANSARI	15
A2MBG1	22	13MEB352	GE6474	MOHD._SABIR_SIDDIQUI	10
A2MBG1	23	13MEB356	GH4528	SUMIT_VASHISHTHA	16.5
A2MBG1	24	13MEB357	GE1445	ANISHWET_VARSHNEY	17.5
A2MBG1	25	13MEB360	GH4535	UTKARSH_SHARMA	13
A2MBG2	26	13MEB361	GE1332	ARSHAD_KHAN	11.5
A2MBG2	27	13MEB364	GH4541	ATIF_AHMED	abs
A2MBG2	28	13MEB367	GE6561	MOHD_MUSTAFA_SALEEM	9.5
A2MBG2	29	13MEB368	GE1690	DEEPAK_KUMAR	abs
A2MBG2	30	13MEB372	GE1627	MANOJ_CHAUDHARY	10.5
A2MBG2	31	13MEB375	GH4569	HARSH_GUPTA	20.5
A2MBG2	32	13MEB403	GH4575	SHAHID_RAZA	18.5
A2MBG2	33	13MEB404	GH4582	MD._SAIF_AHMAD_	17
A2MBG2	34	13MEB407	GH4586	MOHD._DANISH_ANSARI	19.5
A2MBG2	35	13MEB408	GH4592	MOHD._MOIZ	14
A2MBG2	36	13MEB409	GH4599	ASHU_GUPTA	19
A2MBG2	37	13MEB410	GG1663	MOHD._SHAVEZ_KHAN	7
A2MBG2	38	13MEB415	GH4590	MUSTAFA_ASY	16.5
A2MBG2	39	13MEB417	GG0543	MOHD_ADNAN	19.5
A2MBG2	40	13MEB419	GG0604	AKASH_BANSAL	21
A2MBG2	41	13MEB421	GH4603	EMAD_AHMAD_KHAN	7.5
A2MBG2	42	13MEB422	GE1341	ASIF_ANWAR	13
A2MBG2	43	13MEB423	GH4609	SHABIR_HUSSAIN	21
A2MBG2	44	13MEB427	GE3403	SANCHI_SINGH_PATEL	11
A2MBG2	45	13MEB430	GH4620	YOGESH_VISHWAKARMA	19.5
A2MBG2	46	13MEB438	GE1635	ZAIN_AHMAD_KHAN	16
A2MBG2	47	13MEB451	GG0560	ARSALAN_GUZEL	13.5
A2MBG2	48	13MEB464	GH6665	PRANAV_PARASHAR	16
A2MBG2	49	13MEB503	GH6675	IMTIYAZ_ALI	16
A2MBG2	50	13MEB516	GH8049	NASHIT_JALAL	19.5
A2MBG2	51	13MEB548	GG0568	ASHWIN_KUMAR	16.5
BL		09MEB407	GD640		6.5
BL		11MEB057	GG0023		15.5

abs = absent

Mid Semester Marks: A2MA (revised on 3 Nov)

Section	S.No.	Fac. No	En No.	NAME	Marks(25)
A2MAG1	1	13MEB003	GE3460	AMIT_KUMAR_SINGH	13
A2MAG1	2	13MEB005	GE6566	PARITOSH_PRAKHAR	12
A2MAG1	3	13MEB018	GE3275	KHIZR_MOHAMMAD_KHAN	15
A2MAG1	4	13MEB023	GG0763	ANKUR_MITTAL	23
A2MAG1	5	13MEB025	GE1311	SAFIUL_HAQUE	11.5
A2MAG1	6	13MEB026	GG0471	MOHD_YAQZAN	17.5
A2MAG1	7	13MEB030	GE1331	PRAKUL_VARSHNEY	10.5
A2MAG1	8	13MEB031	GG0457	SAURAV_KUMAR	16
A2MAG1	9	13MEB032	GE0079	MUZZAMMIL_AHMAD	15.5
A2MAG1	10	13MEB033	GH3386	NEERAJ_KAUSHIK	14
A2MAG1	11	13MEB035	GG0468	MD._ABRARUL_HAQUE	16.5
A2MAG1	12	13MEB037	GE6482	MOHD._SALIM	12.5
A2MAG1	13	13MEB041	GE3468	ASHISH_VERMA	19
A2MAG1	14	13MEB044	GG0694	MOHD_NADEEM__ASHRAF_ANSARI	22
A2MAG1	15	13MEB046	GE6558	ADIL_EQBAL	14.5
A2MAG1	16	13MEB047	GE1822	SONVEER_SINGH	18.5
A2MAG1	17	13MEB059	GG0529	HUMZA_SIDDIQUE	17.5
A2MAG1	18	13MEB063	GH1715	AQUIB_SHAMIM	13.5
A2MAG1	19	13MEB065	GE1338	SAIFUR_RAHMAN	13
A2MAG1	20	13MEB072	GG0512	NOMAN_BADER	9
A2MAG1	21	13MEB073	GH3398	GULZAR_AHMAD	17
A2MAG1	22	13MEB074	GH3396	MOHAMMED_HAIDER	12.5
A2MAG1	23	13MEB076	GH5080	OMAR_SEEMAB_KHAN	13.5
A2MAG1	24	13MEB077	GH5081	AADIL_SUHAIL_AHMAD	11
A2MAG1	25	13MEB083	GH5128	ASIM_IQBAL_SIDDIQUI	21
A2MAG2	26	13MEB084	GH5130	ASHISH_KUMAR_SINGH	13
A2MAG2	27	13MEB102	GG0572	SABIR_HUSSAIN	17.5
A2MAG2	28	13MEB111	GH3495	AFSAL__P.T.	11
A2MAG2	29	13MEB116	GH3491	WAHIUDDIN_ALI	10
A2MAG2	30	13MEB120	GH3499	SHUBHAM_SHAHI	17
A2MAG2	31	13MEB122	GG0768	MOHD._AMIR	14
A2MAG2	32	13MEB123	GE6518	MIRZA_ANAS_BEG	24.5
A2MAG2	33	13MEB125	GH3500	DEVENDRA_SINGH	22.5
A2MAG2	34	13MEB127	GH4253	MD._ALTAB_SARKAR	16.5
A2MAG2	35	13MEB128	GH4316	SHAHNWAZ_AHMED	16.5
A2MAG2	36	13MEB131	GE6531	SAIF_ALAM_ANSARI	23
A2MAG2	37	13MEB136	GH4341	GANI_RAHIM	11.5
A2MAG2	38	13MEB143	GH4364	RUMAN_HASAN	18.5
A2MAG2	39	13MEB145	GH4389	ASHAB_AHMAD_ANSARI	13
A2MAG2	40	13MEB156	GE7109	KANAK_SINGHAL	22
A2MAG2	41	13MEB164	GE7142	SUMAIYA	16
A2MAG2	42	13MEB205	GH4403	MIR_MONIRUL_ALAM	11
A2MAG2	43	13MEB206	GH4363	MANAZIR_HAIDER	9.5
A2MAG2	44	13MEB210	GH4413	AQUIB_FAIYAZ	12.5
A2MAG2	45	13MEB213	GE1677	ABHISHEK_SINGH	13.5
A2MAG2	46	13MEB216	GG0514	SYED_YUSUF_ALI	15
A2MAG2	47	13MEB217	GH4417	ALEX_JOHN	12.5
A2MAG2	48	13MEB220	GE6552	ABU_SALIM	10
A2MAG2	49	13MEB223	GE1448	RAHUL_GOEL	11
A2MAG2	50	13MEB236	GH6659	MUSFERA_SIDDIQUA_JAVED	12.5
A2MAG2	51	13MEB263	GE8037	SAIFUL_WALI_KHAN	18
A2MAG2	52	12MEB141	GE3277	MAHAD_KHAN	abs

abs = absent