재료역학 chapter 1: tension, compression and...
Post on 12-Jun-2020
2 Views
Preview:
TRANSCRIPT
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
top related