MECN 4600 Inter - Bayamon Lecture Mechanical Measurement and Instrumentation MECN 4600 Professor: Dr. Omar E. Meza Castillo Department of Mechanical Engineering Inter American University of Puerto Rico Bayamon Campus

Lecture 11 MECN 4600 Inter - Bayamon 2 Tentative Lecture Schedule TopicLecture Basic Principles of Measurements Response of Measuring Systems, System Dynamics Error & Uncertainty Analysis 1, 2 and 3 Sensor & Transducers Basic Electronics, Signal Processing 4 Measurement of Pressure 5 Measurement of Temperature 6, 7, 8 Measurement of Stress-Strain 9 Measurement of Level 10 Measurement of Fluid Flow 11 Measurement of Time Constant 12

Lecture 11 MECN 4600 Inter - Bayamon Determination of Young’s Modulus Topic 11: Stress-Strain 3

Lecture 11 MECN 4600 Inter - Bayamon  To determine the Young’s Modulus 4 Course Objectives

Lecture 11 MECN 4600 Inter - Bayamon Stress-Strain  The experimental analysis of stress is accomplished by measuring the deformation of a part under load and inferring the existing state of stress from the measured deflections. Again, consider the rod in figure. 5 FNFN FNFN B B B B FNFN Cross-sectional A C

Lecture 11 MECN 4600 Inter - Bayamon Stress-Strain  If the rod has a cross-sectional area of A C, and the load is applied only along the axis of the rod, the normal stress is defined as 6

Lecture 11 MECN 4600 Inter - Bayamon Stress-Strain  Where A C is the cross-sectional area and F N is the tension force applied to the rod, normal to the area A C. The ratio of the change in length of the road (which results from applying the load) to the original length is the axial strain, defined as  where ε a is the average strain over the length L, is the change in length, and L is the original unloaded length. 7

Lecture 11 MECN 4600 Inter - Bayamon Stress-Strain  For most engineering materials, strain is small quantity; strain is usually reported in units of in./in. or m/m. These units are equivalent to a dimensionless unit called a microstrain (με).  Stress-strain diagrams are very important in understanding the behavior of a material under load. 8

Lecture 11 MECN 4600 Inter - Bayamon Stress-Strain A typical stress-strain curve for mild steel 9 Stress (psi) Stress (MN/m 2 ) Strain Elastic Plastic

Lecture 11 MECN 4600 Inter - Bayamon Hooke’s Law of Elasticity In mechanics, and physics, Hooke's law of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load applied to it. Many materials obey this law as long as the load does not exceed the material's elastic limit. Where: k is a constant called the rate or spring constant or stiffness (in SI units: N·m −1 or kg·s −2 ). 10

Lecture 11 MECN 4600 Inter - Bayamon Hooke’s Law of Elasticity Materials for which Hooke's law is a useful approximation are known as linear-elastic or "Hookean" materials. Hooke's law in simple terms says that strain is directly proportional to stress, hence, Where E is the modulus of elasticity, or Young’s modulus 11

Lecture 11 MECN 4600 Inter - Bayamon Beams  Members that are slender and support loads applied perpendicular to their longitudinal axis. Span, L Distributed Load, w(x) Concentrated Load, P Longitudinal Axis

Lecture 11 MECN 4600 Inter - Bayamon Types of Beams  Depends on the support configuration M FvFv FHFH Fixed FVFV FVFV FHFH Pin Roller Pin Roller FVFV FVFV FHFH

Lecture 11 MECN 4600 Inter - Bayamon Cantilever Beam A B c A B o’ xdx y From elementary calculus, simplified for beam parameters, Where: c is neutral axis “CURVATURE” of the beam

Lecture 11 MECN 4600 Inter - Bayamon Cantilever Beam  The strain (change in length of the element divided by the original length of element) is then:

Lecture 11 MECN 4600 Inter - Bayamon Moment – Curvature Equation A P L x B y

Lecture 11 MECN 4600 Inter - Bayamon Moment – Curvature Equation Where: is the “ area moment of inertia ” “Deflection Equation

Lecture 11 MECN 4600 Inter - Bayamon Laboratory: Find the value of Young’s Modulus 18  Using the Deflection Equation:  Integrating twice:

Lecture 11 MECN 4600 Inter - Bayamon Laboratory: Find the value of Young’s Modulus 19  Boundary conditions:  Evaluating the B.C.

Lecture 11 MECN 4600 Inter - Bayamon Laboratory: Find the value of Young’s Modulus 20  In our case: x=0:  Then: In this case b=0 intercept. Using Least Squared Method we calculate m. From m we will obtain the Young’s Modulus E. Inertia of a rectangle b h

Lecture 11 MECN 4600 Inter - Bayamon Laboratory: Find the value of Young’s Modulus  Data Collection 21 #Mass(kg)P-Load(N)Y - Deflection(m)

Lecture 11 MECN 4600 Inter - Bayamon Laboratory: Find the value of Young’s Modulus  Obtain the stress and strain: Stress: Stress:  Momentum: PL  Distance from neutral axis to top/ bottom: c=h/2  Inertia of a rectangle: Strain: Strain: 22