4 Stress and Strain Dr. A.B. Zavatsky MT07 Lecture 1 Tensin and Cmpressin Nrmal stress and strain f a prismatic bar Mechanical prperties f materials Elasticity and plasticity Hke s law Strain energy and strain energy density issn s rati Nrmal stress rismatic bar: straight structural member having the same (arbitrary) crss-sectinal area A thrughut its length Axial frce: lad directed alng the axis f the member Free-bdy diagram disregarding weight f bar Examples: members f bridge truss, spkes f bicycle wheels, clumns in buildings, etc.
A L L+ Define nrmal stress as the frce divided by the riginal area A perpendicular r nrmal t the frce ( / A ). Greek letters (delta) and (sigma) When bar is stretched, stresses are tensile (taken t be psitive) If frces are reversed, stresses are cmpressive (negative) Example: rismatic bar has a circular crss-sectin with diameter d 50 mm and an axial tensile lad 10 kn. Find the nrmal stress. A 4 / π ( π d 4) 3 ( 10 10 ) 3 ( 50 10 ) N m 5. 099 10 6 N m Units are frce per unit area N / m a (pascal). One a is very small, s we usually wrk in Ma (mega-pascal, a x 10 6 ). Nte that N / mm Ma. 5.093 Ma 3 N N 10 mm N 6 6 10 a 10 mm mm m m Ma
Nrmal strain A L Define nrmal strain as the change in length divided by the riginal length L ( / L ). Greek letter (epsiln) L+ When bar is elngated, strains are tensile (psitive). When bar shrtens, strains are cmpressive (negative). Example: rismatic bar has length L.0 m. A tensile lad is applied which causes the bar t extend by 1.4 mm. Find the nrmal strain. L 1. 4 10. 0 3 m m 0. 0007 Units: nne, althugh smetimes quted as μ (micrstrain, x 10-6 ) r % strain 0. 0007 7 10 700 μ 4 4 ( 7 10 )( 10 10 ) 700 10 6 0.0007 0. 07% strain
Limitatins f the thery fr prismatic bars Axial frce must act thrugh the centrid f the crss-sectin. Otherwise, the bar will bend and a mre cmplicated analysis is needed. a b b a The stress must be unifrmly distributed ver the crss-sectin. Material shuld be hmgeneus (same thrughut all parts f the bar). Defrmatin is unifrm. That is, we assume that we can chse any part f the bar t calculate the strain. ( / ) ( L / ) L ( / 3) ( L / 3) etc.
Stress Cncentratins If the stress is nt unifrm where the lad is applied (say a pint lad r a frce applied thrugh a pin r blt), then there will be a cmplicated stress distributin at the ends f the bar (knwn as a stress cncentratin ). If we mve away frm the ends f the bar, the stresses becme mre unifrm and /A can be used (usually try t be at least as far away as the largest lateral dimensin f the bar, say ne diameter). Mechanical prperties f materials: tensile tests Either crsshead r actuatr mves up and dwn t apply lads. crsshead actuatr Apply lads under cmputer cntrl Lg vltage readings frm lad cell and extensmeter (r crsshead/actuatr) t cmputer extensmeter grip specimen Standardizatin f specimen size and shape and f test prcedure (ASTM, BSI, ISO) fixed base lad cell Output plts f frce versus extensin, but slpe f curve, maximum values, etc depend n specimen size
Stress-strain diagram fr tensin Structural steel in tensin (nt t scale) Ultimate stress Yield stress rprtinal limit Fracture Linear regin erfect plasticity r yielding Strain hardening Necking Structural steel (als called mild steel r lw-carbn steel; an irn ally cntaining abut 0.% carbn). Static (slw) lading. Linear elasticity & Hke s law When a material behaves elastically and als exhibits a linear relatinship between stress and strain, it is said t be linearly elastic. Hke s Law (ne dimensin) E where E mdulus f elasticity, units a E is the slpe f the stress-strain curve in the linear regin. Fr a prismatic bar made f linearly elastic material, E E A L L EA
Tables f mechanical prperties (Hwatsn, Lund, Tdd HLT) Stress-strain diagram fr cmpressin Tensile stress Cmpressive strain E Tensile strain If we lad a crystalline material sample in cmpressin, the frce-displacement curve (and hence the stress-strain curve) is simply the reverse f that fr lading in tensin at small strains (in the elastic regin). E Cmpressive stress The tensin and cmpressin curves are different at larger strains (the cmpressin specimen is squashed; the tensin specimen enters the plastic regin).
Elasticity and lasticity Static lading (gradually increases frm zer, with n dynamic r inertial effects due t mtin) and slw unlading. Within the elastic regin, the curves fr lading and unlading are the same. Elastic limit y Lading Unlading Lading Unlading Elastic lastic Elastic y erfectly plastic The stress-strain curve need nt be linear in the elastic regin. The stress-strain curve fr structual steel (and sme ther metal allys) can be idealized as having a linear elastic regin and a perfectly plastic regin. Elasticity and lasticity Static lading and slw unlading. ast the elastic limit, the curves fr lading and unlading are different. The un-lading curve is parallel t the (initial) elastic lading curve. Lading Elastic limit Unlading Relading y Lading Unlading Relading Residual strain Elastic recvery Residual strain Elastic recvery After unlading, there is a certain amunt f elastic recvery and sme residual strain, that is, a permanent elngatin f the specimen. Upn relading, the unlading curve is fllwed.
Strain energy rismatic bar subjected t a static lad. mves thrugh a distance and hence des wrk. Lad L+ d The wrk dne by the lad is equal t the area belw the lad-displacement diagram: dw W Δ 0 ( d )( d ) d d Displacement This wrk W prduces strains, which increase the energy f the bar itself. The strain energy U ( W) is defined as the energy absrbed by the bar during the lading prcess. Units are N m r J (jules). Lad Linearly elastic behaviur A U W area OAB 1 B O Displacement Fr a linearly elastic bar, L EA, s U L EA EA L Fr a linearly elastic spring, k, s U k k
Elastic and inelastic strain energy Lad A B O D Displacement During lading alng curve OAB, the wrk dne is the area under the curve (OABCDO). If lading is past the elastic limit A, the bar will unlad alng line BD, with permanent elngatin OD remaining. The elastic strain energy (area BCD) is recvered during unlading. C Inelastic strain energy (area OABDO) is lst in the prcess f permanently defrming the bar. Strain energy density Strain energy density u is the strain energy per unit vlume f material. The units are J / m 3 N m / m 3 N / m a Fr a prismatic bar f initial length L and initial crsssectinal area A : u U vlume ( L EA ) ( A L ) EA ( EA L ) ( A L ) E L Using / A and / L gives: E u E
If the material fllws Hke s Law ( E ), then u is the area under the stress-strain diagram. Stress a b Strain 1 u area ab E E issn s rati When a prismatic bar is stretched, it nt nly gets lnger, it gets thinner. Greek letter ν (nu) L+ S there is a tensile strain in the axial directin and a cmpressive strain in the ther tw (lateral) directins. Define issn s rati as: lateral strain ν axial strain lateral axial If axial strain is tensile (+), lateral strain is cmpressive (-). If axial strain is cmpressive (-), lateral strain is tensile (+). S issn s rati is a psitive number.
Fr mst metals and many ther materials, ν ranges frm 0.5 0.35. The theretical upper limit is 0.5 (rubber cmes clse t this). issn s rati hlds fr the linearly elastic range in bth tensin and cmpressin. When behaviur is nn-linear, issn s rati is nt cnstant. Limitatins Fr the lateral strains t be the same thrughut the entire bar, the material must be hmgeneus (same cmpsitin at every pint). The elastic prperties must be the same in all directins perpendicular t the lngitudinal axis (therwise we need mre than ne issn s rati). Generalized Hke s Law y z Apply x, get x, y -ν x, z -ν x Apply y, get y, x -ν y, z -ν y x x Apply z, get z, x -ν z, y -ν z z y Fr an istrpic linearly elastic material, / E in the x, y, and z directins. Use superpsitin t get the verall strains: x x ν y ν z E x y z x ν ν E E E 1 ( ) and similarly fr y and z. x x ν y ν z E