MECHANICAL ENGINEERING

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1 1 SSC-JE SFF SELECION COMMISSION MECHNICL ENGINEERING SUDY MERIL Cassroom Posta Correspondence est-series16 Rights Reserved

2 C O N E N 1. SIMPLE SRESSES ND SRINS 3-3. PRINCIPL SRESS ND SRIN SRIN ENERGY ND HEORIES OF FILURE HIN ND HICK CYLINDERS ND SPHERES SHER FORCE ND BENDING MOMEN SRESSES IN BEMS DEFLECION OF BEMS ORSION OF SHFS ND SPRINGS COLUMNS ND SRUS Cassroom Posta Correspondence est-series16 Rights Reserved

3 CHPER SIMPLE SRESSES ND SRINS SRESS (): It is the interna resistance offered by a body against the deformationnumericay, it is given as force per unit area. Stress on eementary area, i.e. F df im( /) N m d his unit is caed Pa(Pasca) In case of norma stress df aways (perpendicuar) to area d.

3 3 CHPER SIMPLE SRESSES ND SRINS SRESS (): It is the interna resistance offered by a body against the deformationnumericay, it is given as force per unit area. Stress on eementary area, i.e. F df im( /) N m d his unit is caed Pa(Pasca) In case of norma stress df aways (perpendicuar) to area d. Pasca is a sma unit in practice. hese units are generay used 1kPa 1 Pa 1 N/m 3 3 1MPa 1 Pa 1 N/m GPa 1 Pa 1 N/m 1. Norma Stress:It may be tensie or compressive depending upon the force acting on the materia. ensie and compressive stresses are caed direct stresses. When,, When,, ensie Compressive he other types of norma stress is bending norma stress. Bending stress are ineary distributed from zero at neutra axis to maximum at surface. In bending, the cross-sectiona area rotates about transverse axis and the axis about which the cross-sectiona area rotates is caed neutra axis hence in bending, neutra axis is aways transverse axis. Cassroom Posta Correspondence est-series16 Rights Reserved

4 4. Shear Stress (): It is the intensity of shear resistance aong a surface (Let X-X). Shear force ( N /) m Shear rea In case of shear stress force aways parae to the sheared area i.e. P is parae to sheared area in figure. 3. Conventiona or Engineering Stress ( ): It is defined as the ratio of oad (P) to the origina area of crosssection ( ): P 4. rue Stress (): It is defined as the ratio of oad (P) to the instantaneous area of cross-section (): P or,(1) Where = strain (1) Initia voume = Fina voume SRINS (): It is defined as the change in ength per unit ength. It is a dimensioness quantity. i. e. change in ength d origina ength ( + ) d P P 1. Conventiona or Engineering strain: It is defined as the change in ength per unit origina ength. Where, Cassroom Posta Correspondence est-series16 Rights Reserved

5 = Deformed ength e.g. from above figure. = Origina ength 5 d. Natura Strain: It is defined as the change in ength per unit instantaneous ength. d d n n(1) n n d so, n(1) 1 e e 1 Voume of the specimen is a assumed to be constant during pastic deformation L L -Vaid ti neck formation. 3. Shear Strain (): It is the strain produced under the action of shear stresses. d Shear Strain = tan For sma strain, tan From figure, CC or BDD d tan ransverse dispacement Distance from ower face d CC Shear strain cause deformation in shape but voume remains same. 4. Superficia strain ( s ): It is defined as the change in area of cross section per unit origina area. s Cassroom Posta Correspondence est-series16 Rights Reserved

6 Where, = Fina area = Origina area 6 5. Voumetric Strain ( v ): It is defined as the change in voume per unit origina voume. V V V V Where, V = Fina voume V = Origina voume Stress andstrain are tensor (neither vector nor scaar) of nd order. Voumetric strain V x y z Voumetric strain for various shapes: z, z y, y h x, x x, x b (i) Rectanguar body: V = bh on partia differentiation V()(.)( b.h.) b h h b V b h V V b h V x y z Note: x, y, are the strain corresponding to the stresses z x, y, in x-direction, y-direction, z-direction z respectivey x y z V (1 ) POISSON Ratio E =.5 For rubber (ii) For cyindrica body: V = y, y 4 d z, z Cassroom Posta Correspondence est-series16 Rights Reserved

7 V d d d 7 V d V V d V d (iii) For spherica body V 3 d d 4 V r 3 3 d = r Gauge Length: It is that portion of the test specimen over which extension or deformation is measured. i.e. this ength is used in cacuating strain vaue. 1 Poisson s ratio or : Vaue of v varies between ( 1 to.5) m he ratio of the atera strain to ongitudina strain is caed the Poisson s ratio. Latera strain Longitudina strain or d d For a given materia, the vaue of is constant throughout the ineary eastic range. For most of the metas the vaue of ie between.5.4 varies from ( 1 to.5) Note: for ductie materia is greater than for britte metas. abe Materia Vaue of Remarks Cork Foam Rubber Concrete C.I Used in botte to withstand pressure For cork v For rubber v.5 For concrete v.1. Cassroom Posta Correspondence est-series16 Rights Reserved

8 Isotropic Materia: hese materias have same eastic properties in a directions. No. of independent eastic constants =, i.e. if any of eastic constants is known then other can be derived. Orthotropic Materia: he number of independent eastic constants is 9. 8 nisotropic materias: hese materias don t have same eastic properties in a directions. Eastic moduii wi vary with additiona stresses appearing. here is a couping between shear stress and norma stress for an isotropic materia. he number of independent eastic constants is 1. Hooke s Law:It states that when a materia is oaded such that the intensity of stress is within a certain imit, the ratio of the intensity of stress to the corresponding strain is a constant which is characteristics of that materia. i. e. Stress Constant E Strain i.e., E Where, E = Young s Moduus (N/m ) Or Moduus of Easticity For stee, vaue of E = 1 GPa (1 GPa = 1 3 N/mm ) 1 For auminum, vaue of E = 73 GPa E rd E stee 3 For Pastic, vaue of E = 1 GPa 14 GPa Note : s fexibiity increases, vaue of young s moduus decreases. It is resistance to eastic strain. Shear Moduus of Easticity OR Moduus of Rigidity (G or C): It is defined as the ratio of shearing stress to shearing strain. Shear stress G or C i. e. G Shear strain Buk Moduus (K): It is defined as the ratio of uniform stress intensity to voumetric strain within the eastic imits. Stress K Voumetric Strain Note:Eastic constant reationship (i) E (1) C v, where, v = Poisson s ratio. (ii) E 3(1 K ) v Cassroom Posta Correspondence est-series16 Rights Reserved

9 (iii) 3K C v 6K C (iv) 9KC E 3K C 9 SRESS-SRIN DIGRM: 1. Ductie materia (Mid Stee): u yu y e p Stress Eastic definition c bd a e Uniform Pastic definition f g Non uniform Pastic definition Eastic Pastic or residua strain Stress, Figure: ypica stress-strain diagram for a ductie materia Point a Limit of proportionaity: Up to this point a, Hooke s aw isobeyed; oa is a straight ine. Stress corresponding to this point is caed proportiona imit stress, p Upto point a, Hooke s aw is obeyed and stress is proportiona to strain. herefore, O is straight ine. Point is caed imit of proportionaity. Point b Eastic imit point: ab is not a straight ine but upto point b the materia remains eastic. Stress corresponding to this point is caed eastic imit stress, e. Eastic imit > Proportiona imit. Generay, point a and b coincides. Point c upper yied point: t this point the cross-sectiona area starts decreasing.it initiates pastic deformation. Point c Lower yied point. his is point of which, minimum stress is required to maintain pastic behavior. Point d Lower yied point: t this point the specimen eongates by a considerabe amount without any increase in stress. he vaue of stress at this point is 5 N / mm for mid stee. y Cassroom Posta Correspondence est-series16 Rights Reserved

he vaue of strain at yied stress is about.1 or.1% Lower yied point d is observed, if rate of oading is sow. Upper yied point c is observed, if rate of oading is fast.

10 he vaue of strain at yied stress is about.1 or.1% Lower yied point d is observed, if rate of oading is sow. Upper yied point c is observed, if rate of oading is fast. Portion de represents pastic yieding : -ypica vaue of strain is.14 or 1.4% i.e. strain in range de is at east 1 times the strain at the yied point. Portion ef represents strain hardening : Strain increases fast with strain, ti the utimate oad is reached. Point f Utimate stress: Corresponding strain is % for mid stee. It is the maximum stress to which the materia can be subjected in a simpe tensie test. t this point necking of materia begins. Point g Breaking Stress: - Corresponding strain is caed fracture strain. It is about 5% for mid stea. Concept of reduced area (R): q = f o o Reduction of area is more a measure of deformation required to produce faiure and its chief contribution resuts from necking process. here is a compicate state of stress in necking condition. R is the most sensitive ductiity parameter and is usefu in detecting quaity changes in materias. 1 Comparison of Engineering and true stress strain curve: he true stress-strain curve is aso known as fow curve. rue stress-strain curve gives a true indication of deformation characteristics because it is based on the instantaneous dimension of specimen. In engineering stress-strain curve, the stress drops down after necking since it is based on the origina area. In true stress strain curve, the stress however increases after necking since the cross section area of the specimen decreases rapidy after necking. he fow curve of many metas in the region of uniform pastic deformation can be expressed by simpe power aw. K() where, K is the strength co-efficient, n is time stress. n is the strain hardening coefficient. n = for perfecty pastic soid n = 1 In eastic soid For most metas.1 <n<.5 if force is tensie, since area decreases. rue Nomina if force is compressive, since area increase. rue Nomina Cassroom Posta Correspondence est-series16 Rights Reserved

11 11 # Reation between utimate tensie strength and true stress at maximum oad. P max Utimate tensie strength u rue stress at maximum oad = () o u P max Pmax Pmax L o o L P max P max o o rue strain at max oad () n or e Eiminating we get Here, Pmax o P max () P max u P max o e () u u e is the max force. = origina cross section area = instantaneous cross section area Based on the above theory two exampes has been provided. o o Cassroom Posta Correspondence est-series16 Rights Reserved

1 Exampe 1. Ony eongation no neck formation. In the tension test of rod shown initiay it was 4mm and L = 15 mm. Determine the true strain using changes in both ength and area. Soution: Here ol o L i.

3 4 ductie materia is tested such that necking occurs then the fina gauge ength is L = 14 mm and the fina minimum cross section area is 35 mm L o 1 mm.

12 1 Exampe 1. Ony eongation no neck formation. In the tension test of rod shown initiay it was 4mm and L = 15 mm. Determine the true strain using changes in both ength and area. Soution: Here ol o L i.e., 5 1 = 4 15 Exampe : true strain can be cacuated both by area and ength formua as foows. o o d 15 n.3 1 o 5 n n.3 4 ductie materia is tested such that necking occurs then the fina gauge ength is L = 14 mm and the fina minimum cross section area is 35 mm L o 1 mm. Determine the true strain using change in both ength and area. So. Check L o o mm i.e. L o o L = = L Necking occurs and force appied is tensie. o 5 n n d 14 n.336(wrong) 1 o o 5mm and L o 1mm. 5 mm 5 mm no neck formation mm fter the appication of oad its though the rod shown initiay was of area 3 Inference: fter necking gauge ength gives error but area and diameter can be used for the cacuation of true strain at and before fracture. It Eongation with neck formation.. Britte Materia (Cast Iron): o 5 mm and Figure: ypica stress-strain diagram for a ductie materia Cassroom Posta Correspondence est-series16 Rights Reserved

Proof stress: It is given corresponding to.% of strain.

13 13 In these materias, eongation and reduction in area of the specimen is very sma. he yied point is not marked at a. he straight portion of the diagram is very sma. Proof stress: It is given corresponding to.% of strain. ine parae to inear portion of curve is drawn passing through.% strain: E E E ota Pastic Eastic Concept of Eastic and Pastic strain by graph: Cassroom Posta Correspondence est-series16 Rights Reserved

ELASTICITY PREVIOUS EAMCET QUESTIONS ENGINEERING

ELASTICITY PREVIOUS EAMCET QUESTIONS ENGINEERING. If the ratio of engths, radii and young s modui of stee and brass wires shown in the figure are a, b and c respectivey, the ratio between the increase