3 11 Recitation 11 November 25 2003 If there s anything you d like covered please let me know Also please let me know good times to hold a review for the final I was thinking perhaps Thursday or Friday after classes end Let me know if this would be appropriate By email is fine Today Review vocabulary for mechanical properties of Materials Go over stress strain relationships plasticity Example Problems if time Stress strain diagrams The relationship between loads and deflection stress strain in a structure of a member can be obtained from experimental loaddeflection stress strain curves 1 P dL 2 P 2 P 2 P P L Bending test Shear Test P dL 2 T T P Tension test Torsion test P Compression test The most common tests are tension test for ductile materials steel compression test for brittle materials concrete 2 Tension Test Yield Stress y Stress at which a slight increase in stress will result in appreciably increas in strain without increase in stress f C True Stress strain using actual area to calculate B Ultimate stress u Failure stress f Elastic Limit The upper stress level at which the material behaves elastically C Stress strain using original area to calculate A Proportional Limit PL The upper stress limit that strain varies linearly with stress Material follows Hooke s Law Elastic Necking Yield strain y 10 40 times elastic strain Yielding Strain Hardening Material can resist more load increase Necking Plastic Behaviour Material will deform permanently and will NOT return to its orginal shape upon unloading The Elastic Behaviour Material deformation that occurs is called will return to its orginal plastic deformation shape if material is loaded and unloaded within this range Stress strain diagram for ductile materials Hooke s Law E E is the modulus of elasticity Esteel 200 GPa Econcrete 29 GPa 21 29 GPa 3 Ductile Materials Materials that can be subjected to large strains before rupture Have high percent elongation L Lo Percent elongation f 100 Lo Have high percent reduction in area A Af Percent reduction in area o 100 Ao Have capacity to absorb energy If structure made of ductile materials is overloaded it will present large deformation before failing Some ductile materials do not exhibit a well defined yield point we will use offset method to define a yield strength Some ductile materials do not have linear relationship between stress and strain we call them nonlinear materials f y 0 002 or 0 2 offset Elastic plastic Materials Stress strain for structural steel will consist of elastic and perfectly plastic region We call this kind of material elastoplastic material Analysis of structures on the basis of elastoplastic diagram is called elastoplastic analysis or plastic analysis 4 Bilinear stress strain diagram having different slopes is sometimes used to approximate the general nonlinear diagrams This will include the strain hardening Perfectly plastic y ning rd e a h ain Str Nonlinear Linearly elastic Linearly elastic y Brittle Materials Materials that do not exhibit yielding before failure Some materials will show both ductile and brittle behaviours e g steel with high carbon content will demonstrate brittle behaviours while steel with low carbon content will be ductile or steel subjects to low temperature will be brittle while those in the high temperature environment will be ductile Creep Deformation which increases with time under constant load examples rubber band concrete bridge deck sagging between supports due to self weight therefore the deck is constructed with an upward camber o to t P In several situations creep will associate with high temperature If creep becomes important creep strength will be used in design 5 Relaxation Loss of stress with time under constant strain Another manifestation of creep o Prestressed wire Creep strength t Cyclic loading and fatigue Fracture after many cycles of loading If material is loaded into the plastic region upon unloading elastic strain will be recovered but plastic strain remains Elastic region Loading Unloading Permanent set Elastic recovery 6 Strain energy Energy stored internally throughout the volume of a material which is deformed by an external load k x F F Fo Work x xo Consider a linear spring having stiffness k If we apply a force F the spring will stretch x The relationship between F and x is F kx If we apply a force from zero to Fo and the spring stretches to the amount of xo the work done is the average force magnitude times the displacement i e 1 W Fo xo 2 From the conservation of energy this work done must be equivalent to the internal work or strain energy stored within the spring when it is deformed 7 If an infinitesimal element of elastic material is subjected to a normal stress then tensile force on the element will be dz dF dxdy The change in its length is dz dy dx The work done which equals to the strain energy stored in the element is 1 1 dU dxdy dz or dU dV 2 2 The total strain energy stored in a material will be U dV V The strain energy per unit volume or the strain energy density is u dU 1 dV 2 If the material is linear elastic E Hooke s Law holds the strain energy density will be 1 1 2 u 2 E 2 E If the stress reaches the proportional limit the strain energy density is called the modulus of resilience ur PL 1 ur PL PL 2 ur PL 8 The total strain energy density which stored in the material just before it fails is called the modulus of toughness ut ut Two Example Problems 9 Answer to 1 10 Answer to 2 11 More on crazing and shear deformations and zones next time Phenomena in amorphous polymers as discussed yesterday 12