Terminology• Fatigue: damaging process under repeated loading• Fatigue limit: alternating stress level below which no failure occurs for an unlimited number of cycles• Fatigue life: number of stress cycles to failure for a defined cyclic stress• Fatigue strength: alternating stress level corresponding to fatigue failure for a defined number of cycles

Why fatigue is getting more important now• Ratio between variable load and self-weight increases, thus greater stress range (this is because self weight is reducing due to higher strength material, while load remain same. Thus variable load is now a higher proportion of total load• Increase of load intensity compared to design• New applications• Probabilistic design fatigue check

Concrete versus SteelConcrete SteelMonolithic• Composite material• Many micro cracks• Notch insensitive• High scatter of properties• Emphasis on material

• Connections• Homogeneous material• One macro crack• Notch sensitive• Low scatter of properties• Emphasis on connections

Risk of Fatigue DamageRisk is Increased by Secondary effects: crack width increase may resultin corrosion)Risk is Reduced by • Concrete strength increase after 28-days • Redistribution of internal forces

Sources of Variable Stresses in concreteActions • Traffic, cranes, machines • Wind, wavesRestrained/Imposed deformations • Shrinkage / swelling • TemperatureFrost-thaw cycles

Number of Load CyclesLow-cycle high amplitude fatigue < 103 CyclesHigh-cycle low amplitude fatigue > 103 Cycles

Influencing Factors• Internal 1. Dimensions 2. concrete composition 3. reinforcement• External 1. Loading 2. Frequency 3. rest periods• Environmental 1. hardening conditions (temp., RH) 2. working life 3. corrosive, in/under water, ….

Concrete Material Aspects• Properties 1. Composition 2. preparation and hardening• Structure 1. inhomogeneous (dimensions coarse aggregate compared to sectional dimensions) 2. micro cracks (restrained volume changes; temperature, shrinkage) These aspect of Concrete material Results in - high scatter - relatively notch insensitive- continuing hydration (self-healing) - crack growth at many spots

Stresses at Fatigue FailureNotched elastic-plastic member• Mostly yield stress is reached• Failure after repeated yield strain• only stress rangeBrittle member• Failure at fatigue strength• Both maximum stress and stress range

Deformation under Repeated LoadingWith number of load repetitions:• reduced stiffness• reduced energy dissipation• increased deformation

• Ultimate strain independent from load level

Fatigue Damage Accumulation longitudinal strain • acoustic emission • ultrasonic pulse

velocity • Palmgren-Miner hypothesis

Fracture Mechanics

Crack Growth History Crack propagation life “Np” is to be determined. Assessment of crack propagation life:How many cycles from initial defect to critical crack size? We notice 3 stages

I – Crack initiation stageII – Crack propagation (with stable crack growth)III – Crack propagation (without stable crack growth)

0.15 mm is a typical limit for crack initiation

Input: •Initial defect (from inspection or assumed length from welding defect) •Critical crack size (through thickness, plastic or brittle failure) •Stress range •Number of load cycles •Geometry of (welded) connection •Crack growth material parameters

Theory - stress in crack tip Fracture of an elastic material: consider a 2-dimensional, infinitely wide plate of elastic material with central crack. Griffith used the linear elastic stress analysis solution for the stresses around an elliptical hole in a plate subject to uniform tension. He allowed the ellipse to degenerate to a crack and derived an expression for the energy released when an element of material at the end of the crack fractured to give incremental extension of the crack. He then suggested that, if the energy released was greater than the surface tension or cohesive force energy which had been holding the element together, then the situation was unstable, and continued unstable crack extension (i.e. fracture) would occur.

Experiments glass (Griffith)

Fracture stress decreases with √a where C

is measure of surface energy: C=√ 2 E . γ eπ

where γ e is

surface tension. Thus there will be no fracture if

The term is dependent only on the applied stress and crack size, and defines the gradient of stress with inverse square root of distance away from the singularity at the crack tip. The

term was defined by Irwin as the stress intensity factor and given the symbol K. It should be noted that K is not a stress concentration factor, and that K has dimensions and units of stress

x .

Energy release rateUnloading over circle (approximation). • Energy

release rate G in circle: G=π a2 σ2

ESurface energy increases (new surface with surface tension γ): 4 a . γ e

Total energy in plate is

EFFECTS OF MODE OF LOADING

The description of the stress intensity factor given earlier is based on the simple case of an infinite plate with a central crack of length 2a subject to remote tension stress. This mode of loading is known as Mode 1 and the stress intensity factor resulting from this loading is strictly K1. There are two other forms of loading which produce a similar effect

of a stress singularity because forces cannot be transmitted across the free surfaces of a crack. These forms are shear loadings parallel to the crack surfaces either in the plane of the plate, also known as edge sliding, (Mode II stress intensity factor KII), or perpendicular to the plane of the plate, also known as skew sliding or antiplane strain, (Mode III stress intensity factor KIII). These three different forms of loading are shown in Figure 2. In practice in structural components there may be combinations of the different modes to consider.

Plastic zoneIn real materials used for structural purposes, such as structural steels, the infinite stresses predicted by elastic theory at a crack tip are relieved by the occurrence of yielding. A first approximation to the size of the plastic zone at a crack tip is given by finding the distance ry from the crack tip at which the elastic stress level is equal to the yield strength.

Crack growth rateThe proposal that the rate of crack propagation per cycle should be controlled by the range of the stress intensity factor for the cycle was first made by P C Paris as part of his research work. The general relationship now known universally as the Paris Law is as follows:

= C(K)m (1) or

where

da/dN is crack growth rate per cycleC and m are material constants,∆ K is the range of stress intensity factor at the crack tip

Paris law(1979): for subcritical crack size (K<Kc) there is a relation between the SIF range (ΔK) and the crack growth rate (=da/dN)

Experimental work to investigate the relationship between fatigue crack growth rate and range of stress intensity factor can be presented on a graph of da/dN against DK. In general, such a graph on log-log scales shows three regions. At the bottom end there is a threshold region of DK below which cracks do not propagate. This threshold value, DKm, is dependent on both mean stress and environmental conditions. At the top end of the graph the rate of fatigue crack propagation may be increased if the upper end of the applied stress intensity factor range approaches the material fracture toughness. In between these regions the graph is generally linear on logarithmic scales. By taking logarithms of both sides of Equation (1) it can be seen that it predicts that log da/dN should be proportional to log DK, so that the slope of the straight line is the constant m, and the position of the line is determined by the constant C.

A, m and KΔ th are material dependent parameters (steel, aluminium)

•A, m and Kth are Δ independent of the geometry of the detail!

•A, and Kth depend on the residual stress levelΔ

•There is a possibility of taking account of the residual stress level

•Practice: assume high residual stresses (>0.5 fy)

da/dN is the rate of crack growth per cycle

K (or SIF) is the range of stress intensity factor Δat the crack tip

Kmat = fracture toughness

ΔKth or ΔKtr = threshold level

R dependent laws only for base material

Crack growth testsSeveral types of standard specimens:•SEN4B (4 point bending)

•SEN3B (3 point bending)

•Compact tension (CT)

•SENT (Single Edge Notch Tension)

•Etc.

Crack growth monitoring:•Camera or microscope (visual)

•Alternate current Potential Drop (ACPD)

•Strain gauges

•Crack marking

•Clip gauge

Corrosion crack growth rate•A corrosive environment such as seawater influences the crack growth rate.

•The crack growth may be retarded by the corrosion product crack closure)

•The crack growth may be accelerated by hydrogen embrittlement.

•Furthermore, corrosion pits may act as fatigue crack starting points.

  What is the difference between the stress concentration factor and stress intensity factorThe stress concentration factor is a number that raises stress locally due to factors such asholes and change in cross section. In the latter case, the sharper the radius at he crosssection change, the higher the stress concentration. Typically, these factors range from 1to 3 and sometimes more.Stress intensity factor is a bit different; it is an inherent property of the material that istested and defined for cracks or flaws. For cracks and flaws, the radius is very small,approaching zero for sharp corners, and stress concentration factors become very veryhigh, approaching infinity. In this case we use the measured stress intensity factor andequations of fracture mechanics to calculate allowable stresses.

Stress concentration factors are due to geometrial changes of cross sections and regardless of the load condition such as bending,stretching,shearing,.... Therefore, you can find these factors mentioned in tables or figures in handbooks .No matter of what kind of load condition, you can choose the one corresponding to your geometry during your design.

However , when there is a crack in your model, the stress intensity factor comes into design which not only is dependent to geometry also extremely to load condition and that's why you can't find these factors easily in handbooks. They are determined experimentally according to each part geometry and load condition.