Mechanics of Deformable Bodies PDF

Mechanics of Deformable Bodies Fundamental Concepts of Strain and Deformation ENGR. CHRISTOPHER G. CHAN Civil Engineering Department Fundamental Concepts of Strain and Deformation Simple Strain - is also known as unit def...

Mechanics of Deformable Bodies Fundamental Concepts of Strain and Deformation ENGR. CHRISTOPHER G. CHAN Civil Engineering Department Fundamental Concepts of Strain and Deformation Simple Strain - is also known as unit deformation. It is the ratio of the change in length caused by the applied force to the original length. Where: δ = is the deformation L = is the original length CEMDEF40-Mechanics of Deformable Bodies Fundamental Concepts of Strain and Deformation Stress-Strain Diagram Consider the tension test for steel as shown. Specimen under Tension Test Stress-Strain Diagram Watch this! Tensile Test of Steel CEMDEF40-Mechanics of Deformable Bodies Fundamental Concepts of Strain and Deformation Proportional Limit and Hooke’s Law. As seen in the stress-strain diagram is a straight line from the origin O to a point called the proportional limit. This plot is a manifestation of Hooke’s law as postulated by Robert Hooke in 1678. Stress is proportional to strain; that is, Where: 𝜎 = is the stress. 𝐸 = is the modulus of elasticity or Young’s modulus and is equal to the slope of the stress- strain diagram from O to proportional limit. Elastic Limit A material is said to be elastic if, after being loaded, the material returns to its original shape when the load is removed. The elastic limit is, as its name implies, the stress beyond which the material is no longer elastic. The permanent deformation that remains after the removal of the load is called the permanent set. The elastic limit is slightly larger than the proportional limit. However, because of the difficulty in determining the elastic limit accurately, it is usually assumed to coincide withCEMDEF40-Mechanics the proportional limit. of Deformable Bodies Fundamental Concepts of Strain and Deformation Elastic and Plastic Ranges The region in the stress-strain diagram from O to proportional limit is called the elastic range. The region from proportional limit to rupture strength is called the plastic range. Yield Point The point where the stress-strain diagram becomes almost horizontal is called the yield point, and the corresponding stress is known as the yield stress or yield strength. Beyond the yield point there is an appreciable elongation, or yielding, of the material without a corresponding increase in load. Offset For materials that do not have a well-defined yield point, yield Method stress is determined by the offset method. This method consists of drawing a line parallel to the initial tangent of the stress-strain curve; this line starts at a prescribed offset strain, usually 0.2% (𝜺 = 0.002). The intersection of this line with the stress-strain curve as shown, is called the yield point at 0.2% offset. CEMDEF40-Mechanics of Deformable Bodies Fundamental Concepts of Strain and Deformation Ultimate Stress The ultimate stress or ultimate strength, as it is often called, is the highest stress on the stress-strain curve. Rupture Stress The rupture stress or rupture strength is the stress at which failure occurs. It is also known as the breaking strength. Modulus of Resilience It is the work done on a unit volume of material as the force is gradually increased from point O to the proportional limit, in N.m/m3. It may be calculated as the area under the stress-strain curve from O to the elastic limit E. Resilience is the ability of material to absorb energy without creating a permanent distortion. Modulus of Toughness It is the work done on a unit volume of material as the forced is gradually increased from O to rupture strength, in N.m/m3.. It may be calculated as the area under the entire stress- strain curve (from O to rupture strength) CEMDEF40-Mechanics of Deformable Bodies Fundamental Concepts of Strain and Deformation Strain Hardening. When yielding has ended, an increase in load can be supported by the specimen, resulting in a curve that rises continuously but becomes flatter until it reaches a maximum stress referred to as the ultimate stress. CEMDEF40-Mechanics of Deformable Bodies Fundamental Concepts of Strain and Deformation Working Stress and Factor of Safety The working stress w is also called the allowable stress allow is the maximum safe axial stress used in design. In most designs, the working stress should be limited to values not exceeding the proportional limit so that the stresses remain in the elastic range (the straight-line portion of the stress-strain diagram). However, because the proportional limit is difficult to determine accurately, it is customary to base the working stress on either the yield stress s y or the ultimate stress ult, divided by the factor of safety (FS). Thus, CEMDEF40-Mechanics of Deformable Bodies Fundamental Concepts of Strain and Deformation Axial Deformation Simple stress: Simple Strain: Modulus of Elasticity: Understanding Modulus of Elas ticity Where: P = applied axial load Note: Compressive A = cross-section area forces will tend to shorten the member δ = axial deformation while tensile forces L = length of the memberwill tend to elongate or increase the length of the member. CEMDEF40-Mechanics of Deformable Bodies Fundamental Concepts of Strain and Deformation For any member where cross-sectional area is not uniform. For a rod of unit mass ρ suspended vertically from one end, the total elongation due to its own weight is, where: ρ = is the unit mass of the rod L = length of the rod M = total mass of the rod A = cross-sectional area of the rod g = 9.81 m/s2 CEMDEF40-Mechanics of Deformable Bodies Fundamental Concepts of Strain and Deformation Stiffness (k) Stiffness is the ratio of the steady force acting on an elastic body to the resulting displacement. Shearing Deformation Shearing forces cause shearing deformation. An element subject to shear does not change in length but undergoes a change in shape. The change in angle at the corner of an original rectangular element is called the shear strain and is expressed as CEMDEF40-Mechanics of Deformable Bodies Fundamental Concepts of Strain and Deformation The ratio of the shear stress τ and the shear strain γ is called the modulus of elasticity in shear or modulus of rigidity and is denoted as G, in MPa. The relationship between the shearing deformation and the applied shearing force is Where: γ= shearing strain δs= shearing deformation G = modulus of rigidity V = is the shearing force As = the shearing area CEMDEF40-Mechanics of Deformable Bodies Fundamental Concepts of Strain and Deformation Poisson’s Ratio ( ν) When a bar is subjected to a tensile loading there is an increase in length of the bar in the direction of the applied load, but there is also a decrease in a lateral dimension perpendicular to the load. The ratio of the sidewise deformation (or strain) to the longitudinal deformation (or strain) is called the Poisson's ratio and is denoted by ν. For most steel, it lies in the range of 0.25 to 0.3, and 0.20 for concrete. Poisson’s Ratio Where: εx is strain in the x-direction and εy and εz are the strains in the perpendicular direction. The negative sign indicates a decrease in the transverse dimension when εx is positive. CEMDEF40-Mechanics of Deformable Bodies Fundamental Concepts of Strain and Deformation Biaxial Deformation If an element is subjected simultaneously by tensile stresses, σx and σy, in the x and y directions, the strain in the x direction is σx/E and the strain in the y direction is σy/E. Simultaneously, the stress in the y direction will produce a lateral contraction on the x direction of the amount -ν εy or -ν σy/E. The resulting strain in the x direction will be Triaxial Deformation If an element is subjected simultaneously by three mutually perpendicular normal stresses σx, σy, and σz, which are accompanied by strains εx, εy, and εz, respectively, Note: Tensile stresses and elongation are taken as positive. Compressive stresses and contraction are taken as negative. CEMDEF40-Mechanics of Deformable Bodies Fundamental Concepts of Strain and Deformation Relationship Between E, G, and ν The relationship between modulus of elasticity E, shear modulus G and Poisson's ratio ν is: Bulk Modulus of Elasticity or Modulus of Volume Expansion, K The bulk modulus of elasticity K is a measure of a resistance of a material to change in volume without change in shape or form. It is given as where V is the volume and ΔV is change in volume. The ratio ΔV/V is called volumetric strain and can be expressed as CEMDEF40-Mechanics of Deformable Bodies Fundamental Concepts of Strain and Deformation Statically Indeterminate Members When the reactive forces or the internal resisting forces over a cross section exceed the number of independent equations of equilibrium, the structure is called statically indeterminate. These cases require the use of additional relations that depend on the elastic deformations in the members. CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Fundamental Concepts of Strain and Deformation CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies Sample Problems CEMDEF40-Mechanics of Deformable Bodies EXERCISE CEMDEF40-Mechanics of Deformable Bodies SOLUTION CEMDEF40-Mechanics of Deformable Bodies EXERCISE CEMDEF40-Mechanics of Deformable Bodies SOLUTION CEMDEF40-Mechanics of Deformable Bodies References ENGR. CHRISTOPHER G. CHAN Civil Engineering Department