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CE 203
Summary of lectures 4-8: Material Properties and Axial Deformations
Material properties are determined by structural testing. The simplest test is the axial load testing of a material specimen prepared according to certain specifications (e.g., ASTM). The main outcomes of the test are the axial deformation (change in length Dl) for applied load (measured in terms of internal force N). A plot of the axial strain e = Dl/l, and axial stress s= N/A gives the picture of the material behavior under axial effects.
Main regions of the s-e Diagram:
Linear or nonlinear elastic regions;
Yield plateau;
Strain hardening;
Fracture (failure) region.
Material Types:
Ductile: Modulus toughness is much larger than modulus of resilience.
Brittle: Modulus toughness is not much different from modulus of resilience.
Loading and Unloading Behavior: Loading behavior will follow the stress-strain diagram and may be linear elastic, nonlinear elastic or (inelastic). Unloading (reducing the stress) is assumed elastic, and due to loading unloading cycles some materials show increase in strength (Bauschinger s effect).
Hooke s law: For linear elastic range s= E e, where E is called Young s modulus. For steel Esteel= 210 GPa.
Elastic and Plastic Strains: When a material returns to its original shape completed, it is has been subjected to only elastic deformations (caused by elastic strains ee) elastic material. But if final geometry after load removal is not the same as the original geometry, there has been permanent deformation (caused by plastic strains ep ). Total strain etot = ee+ ep.
Factor of Safety FS: It is a numerical value of a measure of the ratio of a strength parameter (e.g.: tensile or compressive strength; shear strength; bearing strength) to computed corresponding values of applied loads or other external effects. For a safe design the value of FS is greater than 1.0 but for economical designs the value is usually in the range of 1.5 to 3.5.
FS = Strength Parameter/Load Effects of the same type.
Axial Deformation: it may be elongation (+ve) or contractions (-ve), and is always accompanied by
lateral deformations (-ve or +ve). Axial deformation is designated as ea and lateral deformation is designated as el. The deformations are mechanical and/or thermal.
Poisson s ratio: ratio much less than 1.0 and cannot be more than 0.5. It is designated as n= - el/ea.
Mechanical Displacements and Deformations: Displacements (u+du) and (u) will give axial deformation d(Dl) = the difference du. The axial strain e= du/dx0.
uf = u0 + +"e(x) dx=+"N(x)/[A(x).E(x)] dx
= uo + NL/AE.
is valid for a uniform bar segment with all values constant.
Otherwise, Dlmech = uf - u0 = S Ni Li/Ai Ei.
Thermal Deformations: Dlther= a l DT, where a is a coefficient of thermal expansion for a given material.
Total Deformations: Dltot = Dlmech + Dlther.
Axial Deformations: Combined Mechanical/Thermal Problems
Statically determinate problems:
Use the equations of statics to determine the normal forces.
The equation of deformation is written such that : TOTAL axial deformation at a point = mechanical deformation + thermal deformation.
Statically in-determinate problems:
Write the equations of statics to determine the normal forces. The equations are not enough to determine the unknowns.
Assume a FINAL position of deformation that does NOT violate the conditions of the problem supports and geometry.
Write the equation of deformation in terms of the normal forces and based on the initial and final position assumed. The equation is called the compatibility Equation.
Solve the equations from steps 1 and 3 simultaneously.
Typical Problem With Mechanical and Thermal Effects: Due to temperature rise DT.
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