환경재료역학 Chapter 2: Tension, compression and shear

충남대학교 환경소재공학과

그린건축연구실(CNUTim)

2.1 Introduction

2.1.1 Mechanics of materials?

재료역학(mechanics of materials)

– 여러가지 형태의 하중을 받고 있는 고체의 거동을 연구하는 응용역학

– 하중에 의한 응력, 변위(변형) 및 변형율을 구함

– 안전한 설계를 위하여

생활에 응용

2.1.2 Normal stress and strain

• Prismatic bar: a straight structural member having constant cross section throughout its length

• Cross section: the section taken perpendicular to its longitudinal axis

• Stress: the intensity of force(force per unit area) – σ(N/m2 = Pa)(kgf/m2)(lbf/in2)

• Strain: elongation per unit length (deformation/ length) – ε(no unit)

• Tension: the axial forces producing a uniform stretching of the bar

• Tensile stress: the resulting stress when the bar is stretched by the bar

:P Tensile stress

Aσ =

• Compression: the axial forces producing a uniform compression of the bar

• Compressive stress: the resulting stress when the bar is compressed by the force, P

• Normal stress: the stress which acts in a direction perpendicular to the cut surface – Tensile and compressive stresses

• Shear stress: the stress acting parallel to the surface

:P Compressive stressA

σ = −

:P Shear stressA

τ =

• Sign convention: define tensile stress as positive and compressive stress as negative

• Unit: SI(International system of units) – Stress: Pa – Force: N = kg·m/sec2

– k(kilo) = 103, M(mega) = 106, G(giga) = 109

• USCS(US Customary system) – Stress: psi = lbf/in2, ksi = kips/in2

• Acceleration of gravity – 9.807m/sec2 = 32.2ft/sec2 = 386.1in/sec2

• Weight of a 1 kg mass – W = mg = (1kg)(9.807m/sec2) = 9.807N

• Weight of 1 lb mass – W = mg = (1lb)(32.2ft/sec2) = 32.2 lb-ft/sec2

• Conversion of units

20.4536 9.807 / sec1 (1 )( )( ) 4.4481

kg mldf lbf Nlb f

= =

1 0.4536lb kg=

1,000 4.4481 (1 )( )( ) 4.4481 1

lbf Nkip kip kNkip lbf

= =

22

1 1,000 4.448 11 ( )( )( )( ) 6.8951 1 0.0254

kip lbf N inksi MPain kip lbf m

= =

2 22

1,000 1 1 0.00254 121 (1 )( )( )( )( ) ( ) 0.000214.448 1,000 1 1

N lbf kip m inkPa k ksfm k N lbf in ft

= =

22 2

1 1 2.541 (1 )( )( )( ) 0.01420.4536 1,000 1

kgf kgf lbf kip cm ksicm cm kgf lbf in

= =

• Stress distribution at the end: assume a uniform distribution over the cross section at any point within the bar that is at least a distance d away from the end

• Tensile strain

• Compressive strain

• Normal strain – The strain associated with normal stress

– Dimensionless

• Uniaxial stress and strain – The bar is prismatic

– The load acts through the centroid of the cross section

– The material is homogeneous

Lδε =

Lδε −

=

• Ex)

Cross section: 20mmx40mm

3

2 2

70 70 10 87.5120 40 800 ( )1000

P kN N MPamA mm mm mmmm

σ ×= = = =

×

61.2 429 1010002.8 ( )1

mmmmL m

m

δε −= = = ×

• Extensometer – Measures deformation during loading – Mechanical gauges or electrical resistance strain gauges

• Static test – Load is applied slowly

• Dynamic test – Rate of loading is very high and must be measured

• Nominal stress – Stress calculated by using initial cross sectional area

• True stress – Stress calculated by using actual cross sectional area

• Nominal strain – Strain calculated by using initial gauge length

• True strain(natural strain) – Strain calculated by using actual length

• Testing machine – Produce load-deformation curve – Covert to stress-strain diagram(characteristics of the material)

2.2 Force and Deformation

2.2.1 Force(하중) – 구조물에 작용하는 외부의 힘

(1) 응력에 따른 분류 : 인장, 압축, 전단, 휨(굽힘), 비틀림

(2) 이론식에 따른 분류 : 축하중, 휨하중, 모멘트, 토크

(3) 분포상태에 따른 분류 : 집중, 분포

2.2.2 Stress(응력) – 하중에 대한 재료내부의 저항력으로, 하중을 단면적으로 나눈 값 (단위면적당의 하중)

– 수직응력(σ), 전단응력(τ)

설계를 하는데 있어 가장 중요한 것은 제품이나 구조물이 안전한 강도를 보유하도록 응력을 규정된 값 이하가 되도록

형상이나 구조를 설계하는 것입니다.

변위(displacement)

- 물체가 운동을 하였을 때의 위치 변화

변형(deformation)

- 물체가 힘을 받았을 때 내부 구성 요소간의 상대적인 변위

변형율(Strain)

- 단위 길이당 변형량

2.2.3 Displacement(변위), Deformation(변형)

아래의 연속과정에서 변위와 변형은 ?

변형량은 달라도

변형율은 같다!!!

2.2.4 Strain

– 재료의 초기길이(L)에 대한 변형량(δ )의 비 (δ /L, 단위길이당 변형량)

– 길이변형율(ε), 전단변형율(γ)

Stress-strain diagram

Characteristic properties of material

- B : proportional region

- C : yield point, yield stress

- D : strain hardening

- E : ultimate stress

- G : fracture

Hooke’s law Applied within elastic limit

where E = elastic modulus

2.2.5 Stress-strain diagram

Eσ ε=

• Standard tests are performed by using a universal testing machine

1) Standard testing method – Korea Industrial standard(KS)

– American Society for Testing and Materials(ASTM)

– American Standard Association(ASA)

– National Bureau od Standard(NBS)

– British Standard(BS)

– Japan Industrial Standard(JIS)

– Japan Agricultural Standard(JAS)

2.2.6 Strength test

2) Tension test – Specimen

• Metal, PVC,…: cylindrical

• Wood:

• KS F 2207, ASTM D143(tension test): Homework

– Ex) Stress-strain diagram for structural steel

σmax = σult = ultimate stress σyield = yield stress = yield point offset yield stress σprop = ptoportional limit 1: linear region 2: perfect plasticity or yielding 3: strain hardening 4: necking CE’: true stress-strain curve where stress is calculated by using actual cross sectional area

• Ductility – The property of the materials that undergo large strains

before failure

– Structural steel, aluminum alloy, copper,…

• Offset yield stress – The method to determine yield point for materials having

large plastic deformation and indistinct yield point

• Percent elongation

• Percent reduction

• Brittle material – The material that fails at relatively low values of strain

– Concrete, stone, cast iron, glass, ceramic,…

– Wood has intermediate property between brittle and ductile

(100) (%)f o

o

L LL−

=

(100) (%)o f

o

A AA−

=

3) Compression test – Different shape of stress-strain curve from tension

Compressive stress-strain diagram Compressive stress-strain diagram for a ductile material for a brittle material

2.3 Elasticity and plasticity

• Loading – applying loads

• Unloading – removing loads

• Elasticity – The property of a material by which the material returns to

its original dimensions after unloading

Residual strain = Permanent strain Deformation is called the permanent set

• Elastic limit(proportional limit) – The stress at the upper limit of the elastic region

• Plasticity – The characteristic of a material by which it undergoes

inelastic strains beyond the stress at the proportional limit

• Plastic flow – Large plastic deformation developed in a ductile material

• Reloading

Proportional limit or yield point in the first loading stage: E in the second loading or reloading stage: E’ In the pre-stressed material yield point is increased