Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 1 ENT 450 MECHANICS OF MATERIALS (MoM) RC. Hibbler Lecture: DR. HAFTIRMAN Teaching Engineers: Mohd Shahril Bin Shariff School of Mechatronics Engineering UniMAP

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 2 Lecture 1. Stress 16 July 2007 Dr. HAFTIRMAN

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 3 STRESS  Introduction  Equilibrium of a deformable body  Stress  Average normal stress in an axially loaded bar  Average shear stress  Allowable stress

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 4 Objectives Describe the important principles of static. Determine the internal resultant loadings in a body. Describe the concepts of normal and shear stress. Specific applications of the analysis and design of members subjected to an axial load or direct shear.

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 5 Why Mechanics of Materials? (Why is Statics not enough?) Real world structures are typically designed for strength, stiffness, and stability considerations. None of these can be accomplished with what we have learned in Statics Strength: The ability of the structures to carry or transmit loads. Stiffness: The ability of the structures to resist changes in shape. Stability: The ability of the structure to resist buckling under compressive loads.

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 6 INTRODUCTION Mechanics of materials is a branch of mechanics that studies the relationships between the external loads applied to a deformable body and the intensity of internal force acting within the body. The subject also involves computing the deformations of the body, and it provides a study of the body’s stability when the body is subjected to external forces.

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 7 External Loads  Surface Forces  are caused by the direct contact of one body with the surface of another.  Concentrated Force idealization  this area is small in comparison with the total surface force can be idealized as a single concentrated force, which is applied to a point on the body.  Linear Distributed load idealization  If the surface loading is applied along a narrow area.  Body Force  Is developed when one boy exerts a force on another body without direct physical contact between the bodies.  Gravitation, this force called the weight of the body.

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 8 External Loads

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 9 Equilibrium of a Deformable Body  Equations of Equilibrium

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 10 Equilibrium of a Deformable Body

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 11 Internal Forces Fig 1-2a shows in equilibrium by the four external forces. Fig 1-2b shows the two parts of the body are then separated, and a free-body diagram of a distribution on internal forces acting on the exposed area of the section. Fig 1-2c shows resultant force and moment, F R and M Ro at any specific point O on the section area.

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 12 Internal Forces  Define resultant force (F R ) and moment (M Ro ) in 3D:

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 13 Internal Forces Coplanar system of forces Shear forces,V=> ΣF y =0 Bending moment, M=> ΣM o =0 Normal force, N=>ΣF x =0

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 14 Internal Forces  Normal force, N=> the force acts perpendicular to the area. It is developed whenever the external loads tend to push or pull on the two segments of the body.  Shear force, V=> the shear force lies in the plane of the area and is developed when external loads tend to cause the two segments of the body to slide over one another.

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 15 Internal Forces  Torsional moment or torque, T=> this effect is developed when the external loads tend to twist one segment of the body within respect to the other.  Bending moment, M=> the bending moment is caused by the external loads that tend to bend the body about an axis lying within the plane of the area.

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 16 Support Reactions The surface forces that develop at the supports or point of contact between bodies are called reactions. A support produces a force in a particular direction on its attached member if it prevents translation of the member in that direction, and it produces a couple moment on the member if it prevents rotation.

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 17 Support Reactions

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 18 Support reactions for 2D problems

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 19 Procedure for analysis Procedure for analysis to determine the resultant internal loadings at a point located section of a body. Method of sections  Choose segment to analyze.  Determine Support Reactions.  Draw free-body diagram for whole body.  Apply equations of equilibrium

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 20 Procedure for analysis Free-body diagram  Keep all external loadings in exact locations before “sectioning”.  Indicate unknown resultants, N, V, M, and Tat the section, normally at centroid Co f sectioned area.  Coplanar system of forces only include N, V, and M.  Establish x, y, zcoordinate axes with origin at centroid

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 21 Procedure for analysis Equations of equilibrium  Sum moments at section, about each coordinate axes where resultants act.  This will eliminate unknown forces Nand V, with direct solution for M(and T).  Resultant force with negative value implies that assumed direction is opposite to that shown on free-body diagram.

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 22 STRESS Concept of stress  To obtain distribution of force acting over a sectioned area.  Assumptions of material:  It is continuous (uniform distribution of matter).  It is cohesive (all portions are connected together).

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 23 STRESS Concept of stress  Consider ΔAi n figure below.  Small finite force, ΔFacts on ΔA.  As ΔA →0, ΔF →0.  But stress (ΔF / ΔA) →finite limit (∞)

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 24 STRESS

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 25 STRESS

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 26 STRESS

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 27 STRESS  Tensile stress: normal force “pulls”or “stretches”the area element ΔA.  Compressive stress: normal force “pushes”or “compresses”area element ΔA

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 28 STRESS

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 29 STRESS General state of stress  Figure shows the state of stress acting around a chosen point in a body Units (SI system)  Newtons per square meter (N/m 2 ) or a pascal(1 Pa = 1 N/m 2 ) kPa= 10 3 N/m 2 (kilo-pascal) MPa= 10 6 N/m 2 (mega-pascal) GPa= 10 9 N/m 2 (giga-pascal)

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 30 AVERAGE NORMAL STRESS IN AXIALLY LOADED BAR

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 31 AVERAGE NORMAL STRESS IN AXIALLY LOADED BAR  Uniform deformation: Bar remains straight before and after load is applied, and cross section remains flat or plane during deformation.  In order for uniform deformation, force P be applied along centroidal axis of cross section.

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 32 AVERAGE NORMAL STRESS IN AXIALLY LOADED BAR

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 33 AVERAGE NORMAL STRESS IN AXIALLY LOADED BAR

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 34 AVERAGE NORMAL STRESS IN AXIALLY LOADED BAR

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 35 AVERAGE NORMAL STRESS IN AXIALLY LOADED BAR

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 36 Maximum average normal stress  For problems where internal force P and x- sectional A were constant along the longitudinal axis of the bar, normal stress σ= P/A is also constant.  If the bar is subjected to several external loads along its axis, change in x-sectional area may occur.  Thus, it is important to find the maximum average normal stress.  To determine that, we need to find the location where ratio P/A is a maximum

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 37 Maximum average normal stress  Draw an axial or normal force diagram (plot of P vs. its position x along bar’s length) Sign convention:  P is positive (+) if it causes tension in the member  P is negative (−) if it causes compression  Identify the maximum average normal stress from the plot

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 38 Procedure for Analysis Average normal stress  Use equation of σ= P/A for x-sectional area of a member when section subjected to internal resultant force P

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 39 Procedure for Analysis Axially loaded members Internal Loading:  Section member perpendicularto its longitudinal axis at pt where normal stress is to be determined.  Draw free-body diagram.  Use equation of force equilibrium to obtain internal axial force Pat the section

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 40 Procedure for Analysis Axially loaded members  Average Normal Stress:  Determine member’s x-sectional area at the section.  Compute average normal stress σ= P/A

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 41 AVERAGE SHEAR STRESS  Shear stress is the stress component that act in the plane of the sectioned area.  Consider a force F acting to the bar.  For rigid supports, and F is large enough, bar will deform and fail along the planes identified by AB and CD.  Free-body diagram indicates that shear force, V= F/2 be applied at both sections to ensure equilibrium

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 42 AVERAGE SHEAR STRESS

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 43 AVERAGE SHEAR STRESS Average shear stress over each section is:  Τ avg =average shear stress at section, assumed to be same at each pt on the section.  V =internal resultant shear force at section determined from equations of equilibrium.  A= area of section

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 44 AVERAGE SHEAR STRESS

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 45 AVERAGE SHEAR STRESS  Case discussed above is example of simple or direct shear.  Caused by the direct action of applied load F.  Occurs in various types of simple connections, e.g., bolts, pins, welded material.

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 46 Single shear

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 47 Single shear  For equilibrium, x-sectional area of bolt and bonding surface between the two members are subjected to single shear force, V= F.  The average shear stress equation can be applied to determine average shear stress acting on colored section in (d).

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 48 Double shear

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 49 Double shear  The joints shown below are examples of double-shear connections, often called double lap joints.  For equilibrium, x-sectional area of bolt and bonding surface between two members subjected to double shear force, V= F/2.  Apply average shear stress equation to determine average shear stress acting on colored section in (d).

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 50 Double shear

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 51 Shear Stress The shear stress acting on the top face of the element.

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 52 Shear Stress Internal shear  Section member at the pt where the τ avg is to be determined.  Draw free-body diagram.  Calculate the internal shear force V Average shear stress  Determine sectioned area A.  Compute average shear stress τ avg = V/A

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 53 Equilibrium Consider a volume element of material taken at a point located on the surface of any sectioned area on which the average shear stress acts. Equilibrium in the y direction;

Lecture 1 Stress 16 July 2007 ENT 450 Mechanics of Materials Dr. Haftirman 54