ENGR310 1 07

1. Stress and Strain: 

1. Stress and Strain ENGR 310 “Mechanics of Materials” Fall, 2007 Tomasz Arciszewski

Hierarchy of Sciences: 

2 Hierarchy of Sciences Physics - focus on the relationships between the properties of matter and energy Mechanics - a sub-domain of physics, focus on the action of forces on bodies or fluid that are at rest or in motion Applied/Engineering Mechanics - a sub-domain of mechanics, focus on engineering applications Mechanics of Materials - a sub-domain of applied mechanics, focus on the relationships between the external loads applied to a deformable body and the intensity of internal forces acting within the body

Mechanics of Materials: 

3 Mechanics of Materials An engineering science dealing with the modeling of behavior (analysis) of structural members considering: External loads Internal forces and stresses Deformations and strains Stability

Statics versus Mechanics of Materials : 

4 Statics versus Mechanics of Materials Statics - focus on a rigid body, determination of forces applied to this body Mechanics of Materials - focus on a deformable body, determination of its behavior

Slide5: 

5 Rigid Versus Deformable Body Rigid body: AB distance constant Deformable body: AB distance changes when external loads are applied

Statics versus Mechanics of Materials: 

6 Statics versus Mechanics of Materials Statics - an outside look, a global view A system of forces and couples applied to a given rigid body A system of forces of interactions in a given configuration of rigid bodies Mechanics - an inside look, a local view Internal forces (inside a given member) Deformations at a given point

External Loads: 

7 External Loads A system of forces, couples of forces, surface forces, temperature field, etc. applied to a given deformable body

Surface Forces: 

8 Surface Forces ... are forces of interaction between two bodies which are distributed over a contact surface

Slide9: 

9 External Forces

Idealized Surface Forces: 

10 Idealized Surface Forces Concentrated force: a force applied at a point Linear distributed load: a system of forces distributed along a line described by the distributed loading function (curve) w(s) with a resultant (resultant force) FR equivalent to the area under w(s) applied at the centroid of the loading curve

Body Forces (Mass Forces): 

11 Body Forces (Mass Forces) Forces acting on a given body without any direct contact with another body and distributed through the body: Gravity forces (weights) caused by the field of gravity Earthquake forces caused by the movements of the entire structural system Forces caused by electromagnetic field Applied at the centroids of the individual bodies (structural members)

Support Reactions: 

12 Support Reactions Forces of interaction between a given body and its supports In general, they are surface forces Usually, they are idealized as concentrated forces and couples of forces Support translations are prevented by forces (reactant forces, reactions) Support rotations are prevented by couples of forces (reactant couples of forces, reactions)

Slide13: 

13 Supports & Connections: their Idealization

Slide14: 

14 Hinges …are connections between structural members which do not prevent the relative rotation of connected members and transfer only forces of interactions

Conditions of Equilibrium: 

15 Conditions of Equilibrium A body is in equilibrium when both the resultant force and the resultant couple are equal to zero, in vector terms: ∑F = 0 and ∑Mo=0 A balance of forces and a balance of moments occurs, in scalar terms: ∑Fx = 0, ∑Fy = 0, ∑Fz = 0 ∑Mx=0, ∑My=0, ∑Mz=0 or ∑Fx = 0, ∑Fy = 0, ∑Mz=0 for a planar system

Free Body Diagram: 

16 Free Body Diagram A graphical representation (visualization) of all necessary and sufficient information about a given member or a structural system to use conditions of equilibrium for various analytical purposes. It contains: A representation of a given member External loads and their locations Reactions and their locations

Method of Sections: 

17 Method of Sections Construction of imaginary sections, or cuts, through the various parts of a given solid body Opening by cuts a given body to reveal the distribution of forces of interaction between two parts of a body, which balance external loads Conditions of equilibrium allow the determination of resultants of forces of interaction in the form of resultant force FR and resultant moment MRo at any specific point O

Forces of Interaction & Conditions of Equilibrium: 

18 Forces of Interaction & Conditions of Equilibrium Conditions of equilibrium are necessary and sufficient to determine resultants of forces of interaction Conditions of equilibrium are insufficient to determine the distribution of forces of interaction

Slide19: 

19 Section, Internal Loading & Internal Forces at a Point

Cross Section: 

20 Cross Section A concept related to the analysis of structural members A section perpendicular to the longitudinal axis of a given member It is usually a vertical section for horizontally positioned beams Usually, the point O is located at the centroid of a given cross section

Components of Resultant Force & Moment: 

21 Components of Resultant Force & Moment Both vectors can be resolved into components normal and tangent to the section Vector MRO is resolved into: M - called “bending moment” and tangent to the plane T - called “torsional moment” and normal to the plane Vector FR is resolved into: N - called “normal force” and normal to the plane V - called “shear force” and tangent to the plane

Slide22: 

22 Normal and Tangent Components

Right-Hand Rule: 

23 Right-Hand Rule Use your right-hand curled hand The thumb gives the arrowhead sense of the vector The fingers show the tendency to rotate

Coplanar Loading: 

24 Coplanar Loading A body is subjected to a coplanar system of forces (loaded in a single, usually vertical plane) Only normal forces, shear forces and bending moments exist in all cross sections It is the main focus of our course (sorry)

Slide25: 

25 Coplanar Loading: Internal Forces

Limitation of Conditions of Equilibrium: 

26 Limitation of Conditions of Equilibrium Conditions of equilibrium are necessary and sufficient to determine resultants of forces of interaction Conditions of equilibrium are insufficient to determine the distribution of forces of interaction

Mechanics of Materials: Main Focus: 

27 Mechanics of Materials: Main Focus Determination of distribution of forces of interaction over a section of a deformable body

Material Assumptions: 

28 Material Assumptions Continuous - consists of continuum, or uniform distribution of matter with no voids Cohesive - all portions connected together, behaves as a single piece of matter (body) Deformable - distance between two points changes when loading applied

Slide29: 

29 Resultant Force and Moment

Distribution of Forces of Interaction: 

30 Distribution of Forces of Interaction Section divided into a very large number of very small but finite A area A finite yet very small force F acts on A area F is resolved into Fz (normal) Fy (tangent) Fz (tangent) components

Stress at a Point: 

31 Stress at a Point When both F and A approach zero, their ratio approaches a finite limit called STRESS AT A POINT  = lim F/A when A  0 The stress vector acts along the line of action of F Sometimes called “traction vector”

Stress at a Point: 

32 Stress at a Point Stress at a point is a measure of intensity of the internal forces (forces of interaction) on a specific plane passing through this point Fundamental concept of mechanic of materials, of structural engineering, and of life in general

Stress Resolution: 

33 Stress Resolution Stress at a point is a vector It can be resolved into three perpendicular components acting along x, y, and z axes Z axis is normal to the section X and y are in the plane of the section

Slide34: 

34

Normal Stress at a Point: 

35 Normal Stress at a Point z = lim Fz/A when A  0

Shear Stresses: 

36 Shear Stresses Two components in the plane normal to z axis: zx = lim Fx/A when A  0 Vector parallel to x zy = lim Fy/A when A  0 Vector parallel to y

General State of Stress: 

37 General State of Stress A specific point is selected A cubic element is cut out around the point Its faces are perpendicular to x, y, and z axes

Slide38: 

38 General State of Stress

Face Considered: 

39 Face Considered

Sign Convention: 

40 Sign Convention Normal stress - subscript represents the axis normal to a given face Shear stress: First subscript represents the axis to which a given face is normal Second subscript represents the axis to which the stress vector is parallel Two second subscripts represent together the face (xy - face parallel to axes x and y)

Units: 

41 Units Stress: a ratio of force to the area acted upon SI Units: Newton per square meter, N/m2, Pascal, Pa (very small) MN/m2, mega Pascal, MPa 1 MPa = 106 Pa 1 GPa = 109 Pa US Customary Units: Pound per square inch, psi 1 ksi = 103 psi

Average Normal Stress: Assumptions: 

42 Average Normal Stress: Assumptions Prismatic member Straight member both before and after load is applied Axial tensile force applied Saint-Venant’s Principle Distribution of stresses at both member ends NOT considered Homogeneous material Isotropic material

Prismatic Member: 

43 Prismatic Member Straight (longitudinal) centroidal axis connecting centroids of all x-sections All x-sections identical in terms of: Shape X-sectional area

Saint-Venant’s Principle (Assumption of Flat Sections): 

44 Saint-Venant’s Principle (Assumption of Flat Sections) Two cases of a deformable prismatic member under axial loading applied at both ends: Rigid plates at both ends, distributed loading, no shape change, identical deformations of all parts of a member, uniform distribution of normal stresses for all x-sections No plates, concentrated forces, significant shape changes at both ends of member, uniform distribution of normal stresses only in the central part

Saint-Venant’s Principle: 

45 Saint-Venant’s Principle At a distance equal to, or greater that the width of a member, the distribution of normal stresses at all x-sections is the same, whether the member is loaded by uniformly distributed forces or by concentrated forces. Also, the stress distribution is independent of the actual mode of application of the loads.

Saint-Venant’s Principle Illustration: 

46 Saint-Venant’s Principle Illustration

Saint-Venant’s Principle Limitations: 

47 Saint-Venant’s Principle Limitations The actual applied load and that used in the analysis must be statically equivalent (conditions of equilibrium are satisfied) The principle is incorrect for the vicinity of the load application points

Slide48: 

48 Axial Tensile Force …is a tensile force applied along the longitudinal centroidal axis of a member

Slide49: 

49 Homogeneous Material …is a material which has the same physical and mechanical properties throughout its volume, for any point within the body

Isotropic and Anisotropic Material: 

50 Isotropic and Anisotropic Material Isotropic material has the same mechanical properties in all directions for any point within a given body (examples: steel) Anisotropic material has different mechanical properties for different directions for any point within a given body (examples: concrete, wood)

Average Normal Stress Distribution: 

51 Average Normal Stress Distribution Assumptions: Constant uniform deformation Constant normal stress

Average Normal Stress: 

52 Average Normal Stress z = P/A where: z - average normal stress at any point on the x-section P - internal axial (centroidal) force (internal resultant normal force) A - x-sectional area of the member Units - psi, ksi, Pa, KPa, MPa

Uniaxial Stress: 

53 Uniaxial Stress Conditions of equilibrium must be satisfied for all cubes Resultant forces acting on the parallel faces of the cube (top& bottom) must be equal

Applications of Average Normal Stress : 

54 Applications of Average Normal Stress Both tension and compression When compression is considered, only short members (no buckling) can be properly analyzed

Maximum Average Normal Stress: 

55 Maximum Average Normal Stress Uniform distribution of normal stresses assumed for a given x-section Maximum average normal stress is equal to average normal stress Important from pragmatic point of view (dimensioning of members under axial tensile load)

Average Normal Stress: Example: 

56 Average Normal Stress: Example A prismatic steel member under two axial tensile forces F F = 50 KN Circular x-section, 20 mm diameters Calculate normal average stress

Average Shear Stress: 

57 Average Shear Stress Shear stress (shear stress component) is tangent to the cutting section Planes AB and CD

Average Shear Stress: 

58 Average Shear Stress External force F is balanced by two internal resultant shear forces V (resultants of shear stresses) V force is equivalent of a stream of shear stresses acting on a given x-section (resultant) Uniform shear stress distribution is assumed Pure shear, simple or direct shear, occurs only in simple connections

Average Shear Stress: 

59 Average Shear Stress avg = V/A where: avg - average shear stress at the section V - internal resultant shear force at the section ( A - area at the section Units - psi, ksi, Pa, KPa, MPa

Single Shear: 

60 Single Shear It occurs in simple connections of two members Single plane of shearing Shear force V is equal to external force F

Double Shear : 

61 Double Shear It occurs in simple connections of three members Two planes of shearing Shear force V is equal to half of external force F

Complementary Stresses: 

62 Complementary Stresses A stress cube is considered Only shear stresses in the vertical plane parallel to zy are show … are a pair of equal in magnitude shear stresses in 2 normal planes, which are both directed to or from the line of intersection of their planes

Stresses on Inclined Plane: 

63 Stresses on Inclined Plane Prismatic member Axial tensile (centroidal) loading Cross section is considered first carrying average normal stress uniformly distributed Inclined plane is considered next carrying uniformly distributed s stresses S stresses are resolved into normal and shear (tangential) stresses

Stresses on Inclined Plane: 

64 Stresses on Inclined Plane n = x cos2() n = - (1/2)xsin(2)

Structural Design Process (Designing): 

65 Structural Design Process (Designing) …is a process which starts when needs for a given structural system, or for a modification of a given system, are realized and it ends when the final design, a description of a new or a modified system is produced. It has two major stages, including: conceptual designing and detailed designing

Conceptual Designing: 

66 Conceptual Designing It is the 1st stage in the structural design process in which a design concept is developed. A design concept is an abstract description of a future structural system in terms of symbolic attributes (For example: type of members, type of joints, type of loading) A design concept, an example: a truss - a system of straight members connected by hinges and loaded at joints

Detailed Designing: 

67 Detailed Designing It is the 2nd stage in the structural design process in which a design concept is converted into a detailed design A detailed design is a description of a future structural system in terms of numerical attributes (dimensions, weights, etc.)

Detailed Designing, Major Activities: 

68 Detailed Designing, Major Activities Stress analysis Dimensioning, determination of x-sections of the individual members Optimization, determination of optimal x-section of the individual members (minimum weight, cost, security, etc.)

Stress Analysis: 

69 Stress Analysis Determination of stresses in the individual structural members Determination if the occurring stresses are safe

Allowable Load and F.S.: 

70 Allowable Load and F.S. Allowable load is the magnitude of load which can be safely applied to a given structural member Failure load is the magnitude of load which causes the structural failure of a given structural member (buckling, excessive deformations, collapse, fracture, etc. Factor of Safety, F.S. = Ffail/Fallow

Allowable Stress and F.S.: 

71 Allowable Stress and F.S. Stresses are assumed as linearly related to loads Allowable stress is the magnitude of stress which can safely occur in a given structural member Failure stress is the magnitude of stress which causes the structural failure of a given structural member Factor of Safety: F.S. = fail/ allow or F.S. = fail/ allow

Simple Connections Design: Assumptions: 

72 Simple Connections Design: Assumptions Isotropic and homogeneous material Perfectly linear elastic behavior Small deformations

Tension Member: 

73 Tension Member A prismatic member Axial tensile (centroidal) force Arequired = P/ allow

Connector subjected to Shear: 

74 Connector subjected to Shear Pinned or bolted connections Transfer of loading through the pinn Friction is neglected Uniform distribution of shear stresses A = P/ allow, A - x-section of the bolt

Contact (Bearing) Stress: 

75 Contact (Bearing) Stress Neglected in the book b = P/(bxd) (b times d) is “projected contact area” Required contact area: (bxd) = P/(b)allow

Area to Resist Bearing : 

76 Area to Resist Bearing Direct contact of two surfaces Uniform distribution of normal stresses Arequired = P/(b)allow

Area to Resist Shear caused by Axial Load: 

77 Area to Resist Shear caused by Axial Load Shear stresses act on the shearing surface Contact length l to be determined Lrequired = P/d allow

Deformation : 

78 Deformation External loading is applied to a body Shape and size of a body are changed Deformation occurred

Deformation: 

79 Deformation A rubber membrane subjected to tension Changes in white lines: Vertical line elongates Horizontal line shortens Inclined line changes length and rotates

Slide80: 

80

Slide81: 

81

Strain : 

82 Strain Strain is a formal measure of deformations Two deformation types: Linear deformations (elongation, contraction) Angular deformations (change in angle between two line segments originally normal)

Normal Strain: 

83 Normal Strain Consider line AD Initial length s Final length s’ Average normal strain avg = (s -s’)/s Normal strain at a point  = lim (s -s’)/s s  0 Normal strain is dimensionless quantity

Slide84: 

84 Shear Strain Consider angle CAB between lines n and t Initial angle /2, final angle ’ Shear strain at a point (A) nt = /2 - ’, when B  A and C  A

Slide85: 

85 Shear Strain Interpretation It measures the change in an angle It is measured in radians When ’ less /2, positive shear strain

Slide86: 

86 Cartesian Strain Components A deformable body is considered A rectangular elements is assumed around a given point Very small initial dimensions x, y, z Initial angles /2

Slide87: 

87 Cartesian Strain Components Deformed shape: a parallelepiped In general, for each pair of edges s’  (1 + ) s Final lengths for three sides: x’  (1 + x) x y’  (1 + y) y z’  (1 + z) z Final angles: /2 - xy /2 - yz /2 - xz

Slide88: 

88 Strain: Physical Interpretation Normal strains cause a volume change of a rectangular element Shear strains cause a change in shape

Slide89: 

89 Small (Engineering) Strain Analysis Strains assumed very small (deformations very small)  << 1  is very small sin  =  cos  = 1