CONTENTS
A Definitions 8
A1 Important Properties of Engineering Materials 8
A2 Simple Stresses and Strains 9
A3 General Definitions 10
B Conversion Tables 26
B1 Introduction 26
B2 List of Units and Abbreviations 26
B3 Conversion factors Grouped by Category 28
B4 Miscellaneous Data 33
C Section Shear Center 34
C1 Location of Shear Center – Open Section 35
C2 Shear Center – Curved Web 36
C3 Shear Center Beam with Constant Shear Flow Between Booms 38 C4 Shear Center – Single Cell Closed Section 39 C5 Shear Center – Single Cell Closed Section with Booms 40 C6 Shear Center for Standard Section (Tables) 41
D Warping Constant 46
D1 Warping Constant – Typical Sections (Tables) 48
E Mechanical Properties 53
E1 DATA Basis 53
E2 Stress Strain Data 54
E3 Definitions of Terms 54
E4 Temperature Effects 57
2
1 Prismatic Bar In Tension 63
1.1 Limit Stress Analysis 63
1.2 Ultimate Stress Analysis 63
1.3 Problem 64
2. Prismatic Bar In Compression 65
2.1 Compression Stress Analysis 65
2.1.2 Problem 65
2.2 Euler Buckling 67
2.3 For Other Buckling Cases 67
2.3.1 Buckling For Eccentric Loading 67
2.3.2 Beam-Column 68
2.3.2.1 Notations And Conventions 68
2.3.2.2 Terms Used 69
2.3.2.3 Calculation Of Bending Moment 69
2.3.2.4 Allowable Stress 74
2.3.3 Local Buckling (Thin Walled Structures) 75
2.3.3.1 Flanges 75
2.3.3.2 Thin Webs 75
2.3.3.3 Plates And Shells 76
2.3.3.4 Analysis Of Thin Walled Structures 80
2.4 Problems 80
2.4.1 Problem For Euler Buckling 80
2.4.2 Problem For Beam – Column 83
2.4.3 Problem For Local Buckling 86
3 Lug Analysis 88
3.1 Failure In Tension 88
3.2 Failure By Shear Tear Out 89
3.3 Failure By Bearing 90
3.4 Lug Strength Analysis Under Transverse Loading 90 3.5 Lug Strength Analysis Under Oblique Loading 91
3.6 Lug Analysis (Avro Method) 92
3.7 Problems 93
4. Member In Bending 97
4.1 Introduction To Bending 98
4.2 Stress Analysis Of Beam 100
4.3 Bending Stresses 101
4.3.1 Limit Stress Analysis 101
4.3.2 Ultimate Stress Analysis 101
4.3.3 Rectangular Moment Of Inertia and Product of Inertia 102
4.3.4 Parallel Axis Theorem 103
4.4 Elastic Bending Of Homogeneous Beams 114
4.5 Elastic Bending Of Non Homogeneous Beams 128
4.6 Inelastic Bending of Homogeneous Beams 133
4.7 Elastic Bending Of Curved Beams 146
4.8 Miscellaneous Problems 148
4.8.1 Bending Stress In Symmetrical Section 151 4.8.2 Bending Stress In Unsymmetrical Section 153
4.8.3 Beams Of Uniform Strength 159
4.8.4 Beams Of Composite Section 160
4.8.5 Bending Of Unsymmetrical Section About Principal Axis 163 4.8.6 Bending Of Unsymmetrical Section About Arbitrary Axis 164
4
5 Member In Torsion 172
5.1 Torsion of Circular Bars 173
5.1.1 Elastic and Homogeneous 176
5.1.2 Elastic and Non-homogeneous 178
5.1.3 Inelastic and Homogeneous 180
5.1.4 Inelastic and Non-homogeneous 182
5.1.5 Residual Stress Distribution 184
5.1.6 Power Transmission 185
5.2 Torsion of Non-Circular Bars 189
5.3 Elastic Membrane Analogy 190
5.4 Torsion of Thin-Wall Open Sections 194
5.5 Torsion of Solid Non-Circular Shapes 198
5.6 Torsion of Thin-Wall Closed Sections 202
5.7 Torsion-Shear Flow Relations in Multiple-Cell Closed Sections 208 5.8 Shear Stress Distribution and Angle of Twist for Two-Cell Thin-Wall Closed Section 209
5.9 Shear Stress Distribution and θ for Multiple-Cell Thin-Wall Closed Section 214 5.10 Torsion of Stiffened Thin-Wall Closed Sections 215
5.11 Effect of End Restraint 217
5.12 Miscellaneous problems 222
5.12.1 Problem For Torsion On Circular Section 222 5.12.2 Problem For Torsion In Non Circular Section 225 5.12.3 Problem For Torsion In Open Section Composed Of Thin Plates 226 5.12.4 Problem For Torsion On Thin Walled Closed Section 228 5.12.5 Problem For Torsion In Thin Walled Unsymmetrical Section 233
6 Transverse Shear Loading Of Beams With Solid Or Open Sections 236
6.1 Introduction 237
6.2 Shear Center 238
6.3 Flexural Shear Stress And Shear Flow 239
6.5 Shear Flow Analysis For Unsymmetric Beams 286 6.6 Analysis Of Beams With Constant Shear-Flow Webs (I.E., Skin-Stringer Type Sections) 298
7. Transverse Shear loading Of Beams With Closed C.S 310 7.1 Single-Cell Unstiffened Box Beam: Symmetrical About One Axis 311 7.2 Statically Determinate Box Beams With Constant-Shear-Flow Webs 325 7.3 Single-Cell Multiple-Flange Box Beams. Symmetric And Unsymmetric Cross Sections 331 7.4 Multiple-Cell Multiple-Flange Box Beams. Symmetric And Unsymmetric Cross Sections 338 7.5 The Determination Of The Flexural Shear Flow Distribution By
Considering The Changes In Flange Loads (The Delta P Method) 346
7.6 Shear Flow In Tapered Sheet Panels 357
8. Combined Transverse Shear, Bending, And Torsion Loading 360
9 Internal Pressure 360
9.1 Membrane Equations Of Equilibrium 360
9.2 Special Problems In Pressurized Cabin Stress Analysis 368
10 Analysis Of Joints 373
10.1 Analysis Of Riveted Joints 373
10.1.1 Shear Failure Of Rivets 375
10.1.2 Tensile Failure Of Plate Along Rivet Line 375 10.1.2.1 Single Row And Rivets Having Uniform Diameter And Equal Pitch375 10.1.2.2 More Than One Row And Rivets With Equal Spacing 376 10.1.2.3 Rivet Line With Varying Number Of Rivets 377 10.1.2.4 Rivets Of Varying Diameter, Sheet Thickness And Pitch Distance 378
10.1.3 Failure By Double Shear Of Plate 378
10.1.4 Rivets Subjected To Tensile Loads 379 10.1.5 Rivets Subjected To Eccentric Loads 380
6
10.1.8 Problems 382
10.2 Analysis Of Bolted Joints 397
10.2.1 Failure By Bolt Shear 397
10.2.2 Failure By Bolt Bending 398
10.3 Analysis Of Welded Joints 399
10.3.1 Shear Failure In Weld 399
10.3.2 Tensile Failure Of Weld 400
11 Combined Stress Theories For Yield And Ultimate Failure 401 11.1 Determination Of Yield Strength Of Structural Member 401
11.1.1 Maximum Shearing Stress Theory 401
11.1.2 Maximum Energy Of Distortion Theory 402
11.1.3 Maximum Strain Energy Theory 402
11.1.4 Octahedral Shear Stress Theory 403
11.2 Determination Of Ultimate Strength Of A Structural Member 404
11.2.1 Maximum Principal Stress Theory 404
11.3 Problem 404
12 Cutouts In Plane Panels 409
12.1 Framing Members Around The Cutouts 409
12.2 Framing Cutouts With Doublers And Bends 412 12.3 Ring Or Donut Doublers For Round Holes 414 12.4 Webs With Lightning Holes Having Flanges 416 12.5 Structures With Non Circular And Non Rectangular Cutouts 417
12.6 Cutouts In Fuselage 418
12.6.1 Load Distribution Due To Fuselage Skin Shear 419 12.6.2 Load Distribution Due To Fuselage Bending 421 12.6.3 Load Distribution Due To Fuselage Cabin Pressurization 422
12.6.3.1 Panels Above And Below The Cutout 423
12.6.3.2 Panels At Sides Of The Cutout 424
12.7 Problem 425
13 Tension Clips 430
13.1 Single Angle 433
13.2 Double Angle 433
13.3 Problem 435
14 Pre - Tensioned Bolts 438
14.1 Optimum Preload 439
14.2 Total Bolt Load 440
14.2.1 Mating Surfaces Not In Initial Contact 440 14.2.2 Mating Surfaces In Initial Contact 440
14.3 Preload For A Given Wrench Torque 441
14.4 Bolt Strength Requirements 441
8 A. DEFINITIONS:
A1. Import Properties of Engineering Material:
Its necessary to know from engineering point of view the following import properties of solids.
a. Elasticity: It is property by virtue of which the body regains its original shape and size after the removal of the external force acting on the body.
b. Isotropic Material: It is the material, which is equally elastic in all directions. c. Ductility: It is the property of the material by virtue of which it can be drawn out
into wires of smaller dimension e.g., copper, Aluminium.
d. Plasticity: It is the property of the material wherein the strain does not disappear even after the load or stress is not acting on the material. The material becomes plastic in nature and behaves like a viscous liquid under the influence of large forces.
e. Malleability: It is the property possessed by the material enabling the material to extend uniformly in a direction without rupture. As such the material can be hot rolled, forged or drop stamped. A malleable material is highly plastic.
f. Brittleness: A material is said to be brittle when it cannot be drawn out into thon wires. This due to lack of ductility and the material breaks into pieces under loading.
g. Toughness: It is property of the material due to which it is capable of absorbing energy without fracture.
h. Hardness: It is the property of the material by virtue of which the is capable of resisting abrasion or indentation.
A2. Simple Stresses and Strains. Allowable Stress (Working Stress):
If a member is so designed that the maximum stress as calculated for the expected condition of service is less than some certain value, the member will have a proper margin of security against damage or failure. This certain value is the allowable stress of the kind and for the material and condition of service in question. The allowable stress is less than the damaging stress because of uncertainty as to the conditions of service, non uniformity of material, and inaccuracy of stress analysis. The margin between the allowable stress and the damaging stress may be reduced in proportion to the certainty with which the condition of service are known, the intrinsic reliability of material, the accuracy with which the stress produced by the loading can be calculated and the degree to which failure is unaffected by danger or loss. (Compare Damaging stress, Factor of Safety; Factor of utilization; Margin of safety)
Apparent Elastic Limit:
The stress at which the rate of change of strain with respect to stress is 50 percent greater than at zero stress. It is more definitely determine from the stress strain diagram than is the proportional limit, and is useful for comparing material s of the same general class(Compare Elastic limit, Proportional limit, Yield point; Yield strength).
Apparent Stress:
The stress corresponding to a given unit strain on the assumption of uniaxial elastic stress is called apparent stress. It is calculated by multiplying the unit strain by the modulus of elasticity and may differ from the true stress because the effect of transverse stresses is not taken into account.
10 A3. General Definitions:
Bending Moment:
The bending moment at any section of the beam is the moment of all forces that act on the beam to the left of that section, taken about the horizontal axis of the section. The bending moment is positive when clockwise and negative when counter clock wise; a positive bending moment therefore bends it so that it is concave downward. The moment equation is an expression for the bending moment at any section in terms of x, the distance to that section measured from a chosen origin, usually taken at the left end of the beam.
Boundary Conditions:
As used in strength of materials, the term usually refers to the condition of stress, displacement, or slope at the ends or edges of a member, where these conditions are apparent from the circumstances of the problem. Thus for a beam with fixed ends the zero slope at each end is a boundary condition are apparent from the circumstances of the problem. Thus for beam with fixed ends, the zero slope at each is a boundary condition: for a pierced circular plate with freely supported edges, the zero radial stress at each edge is a boundary condition.
Brittle Fracture:
The tensile failure with negligible plastic deformation of an ordinarily ductile metal is called brittle fracture.
Bulk Modulus of Elasticity:
The ratio of a tensile or compressive stress, triaxial and equal in all direction(e.g., hydrostatic pressure), to the relative change it produces in volume.
Central Axis (Centroidal Axis):
A central axis of an area is one that passes through the centroid; it is understood to lie in the plane of the area unless contrary is stated. When taken normal to the plane of the area, it is called the central polar axis.
Centroid of an Area:
That point in the plane of the area about any axis through which the moment of the area is zero; it coincides with the center of gravity of the area materialized as an infinitely thin homogenous and uniform plate.
Corrosion Fatigue:
Fatigue aggravated by corrosion , as in parts repeatedly stressed while exposed to salt water.
Creep:
Continuous increase in deformation under constant or decreasing stress. The term is ordinarily used with reference to the behaviour of metals under tension at elevated temperatures. The similar yielding of a material under compressive stress is usually called plastic flow, or flow. Creep at atmospheric temperature due to sustained elastic stress is sometimes called drift, or elastic drift. Another manifestation creep, the diminution in stress when deformation is maintained constant, is called relaxation.
Damaging Stress:
The least unit stress of a given kind and for a given material and condition of service that will render a member unfit for service before the end of its normal life. It may do this by producing excessive set, by causing creep to occur at an excessive rate, or by causing fatigue cracking, excessive strain hardening, or rupture..
Damping Capacity:
The amount of work dissipated into heat per unit of strain energy present at maximum strain for a complete cycle.
12 Deformation (Strain):
Change in the form or dimensions of a body produced by stress. Elongation is often used for tensile strain, compression or shortening for compressive strain, and detrusion for shear strain. Elastic deformation is such deformation as disappears on removal of stress; permanent deformation is such deformation as disappears on remains on removal of stress.
Eccentricity:
A load or component of a load normal to a given cross section of a member is eccentric with respect to that section if it does not act through the centroid The perpendicular distance from the line of action of the load to either principal central axis is the eccentricity with respect to that axis.
Elastic:
Capable of sustaining stress without permanent deformation; the term is also used to denote conformity to the law of stress strain proportionality. An elastic stress or strain within the elastic limit.
Elastic Axis:
The elastic axis of a beam is the line, lengthwise of the beam, along which transverse loads must be applied in order to produce bending only, with no torsion of the beam at any section. Strictly speaking, no such line exists except for a few conditions of loading. Usually the elastic axis is assumed to be the line that passes through the elastic center of every section. The term is most often used with reference to an airplane wing of either the shell or multiple-spar type (Compare Torsional center; Flexural center; Elastic center).
Elastic Center:
The elastic center of given section of a beam is that point in the plane of the section lying midway between the flexural center and center of twist of that section. The three points may be identical and are usually assumed to be so. (Compare Flexural center; Torsional center; Elastic axis).
Elastic Curve:
The curve assumed by the axis of a normally straight beam or column when bent by loads that do not stress it beyond the proportional limit.
Elastic Limit:
The least stress that will cause permanent set (Compare Proportional limit; Apparent elastic limit; Yield point; Yield strength).
Elastic Ratio:
The ratio of the elastic limit to the ultimate strength
Ellipsoid of Strain:
An ellipsoid that represent the state of strain at any given point in a body; it has the form assumed under stress by a sphere centered at that point.
Ellipsoid of Stress:.
An ellipsoid that represents the state of stress at a given point in a body; its semi axes are vectors representing the principal stresses at that point, and any radius vector represents the resultant stress on a particular plane through the point. For a condition of plane stress (one principal stress zero) the ellipsoid becomes the ellipse becomes the ellipse of stress.
Endurance Limit(Fatigue strength):
The maximum stress that can be reversed an indefinitely large number of times without producing fracture of a material.
Endurance Ratio:
Ratio of the endurance limit to the ultimate static tensile strength is called endurance ratio.
14 Endurance Strength:
The highest stress that a material can withstand with repeated application or reversal without rupture for a given number of cycles is the endurance strength of that material for that number of cycles. Unless otherwise specified, reversed stressing is usually implied (Compare Endurance Limit)
Energy of Rupture:
The work done per unit volume in producing fracture is the energy of rupture. It is not practical to establish a definite energy of rupture value for a given material because the result obtained depends upon the form and proportion of the test specimen and manner of loading. As determined by similar tests specimen, the energy of rupture affords a criterion for comparing the toughness of different materials.
Equivalent Bending Moment:
A bending moment that, acting alone, would produce in a circular shaft a normal (tensile or compressive) stress of the same magnitude as a maximum normal stress produced by a given bending moment and a given twisting moment acting simultaneously.
Equivalent Twisting Moment:
A twisting moment that, acting alone would produce in a circular shaft a shear stress of the same magnitude as the shear produced by a given twisting and a given bending moment acting simultaneously.
Factory of Safety:
The ratio of the load that would cause failure of a member or structure to the load that is imposed upon it in service. The term usually has this meaning; it may also be used to represent the ratio of breaking to service value of speed, deflection, temperature variation, or other stress producing factor against possible increase in which the factor of safety is provide as a safeguard (Compare Allowable stress; Margin of safety).
Factor of Strain:
In the presence of stress raisers, localized peak strains are developed. The factor of strain concentration is the ratio of the localized maximum strain at a given cross section to the nominal average strain on that cross section. The nominal average strain behaviour of the material. In a situation where all stresses and strains are elastic, the factors of stress concentration and strain concentration are equal (Compare Factor of stress concentration).
Factor of Stress Concentration:
Irregularities of form such as holes, screw threads notches, and sharp shoulders, when present in a beam, shaft, or other member subject to loading may produce high localized stresses. This phenomenon is called stress concentration, and the form irregularities that causes it are called stress raisers. The ratio of the true maximum stress to the stress calculated by the ordinary formulas of mechanics (flexural formula, torsion formula etc.), using the net section but ignoring the changed distribution of stresses is the factor of stress concentration for the particulars type of stress raiser.
Factor of Stress Concentration in Fatigue:
At a specified number of loading cycles, the fatigue strength of a given geometry depends upon the stress concentration factor and upon material properties. The factor of stress concentration in fatigue is the ratio of the fatigue strength without a stress concentration. It may vary with the specified number of cycles as well as with material
Factor of Utilization:
The ratio of the allowable stress to the ultimate strength. For cases in which stress is proportional to load, the factor of utilization is the reciprocal to the factor of safety.
Fatigue:
Tendency of material to fracture under many repetitions of a stress considerably less than the ultimate static strength.
16 Fibre Stress:
A term used for convenience to denote the longitudinal tensile or compressive stress in a beam or other member subject to bending. It is sometimes used to denote this stress at the point or points most remote from the neutral axis, but the term stress in extreme fibre is preferable for this purpose. Also, for convenience, the longitudinal elements of filaments of which a beam may be imagined as composed are called fibres.
Fixed (Clamped, built-in, encastre):
A conditioned of support at the ends of a beam or column or at the edges of a plate or shell that prevents rotation and transverse displacements of the edge of the neutral surface but permits longitudinal displacements.
Flexural Center (Shear Center):
With reference to a beam, the flexural center of any section is that point in the plane of the section through which a transverse load, applied at that section, must act if bending deflection only is to be produced with no twist of the section (Compare Torsional center; Elastic center; Elastic axis).
Form Factor:
The terms pertains to a beam section of a given shape and means the ratio of the modulus of rupture of a beam having a section adopted as standard. The standard section is usually taken as rectangular or square; for wood it is a 2 by 2 in square, with edges horizontal and vertical. The term is also used to mean, for a given maximum fiber stress within the elastic limit, the ratio of the actual resisting moment of a wide-flanged beam to the resisting moment the beam would develop if the fiber stress were uniformly distributed across the entire width of the flanges. So used, the term expresses the strength –reducing effect of shear lag.
Fretting Fatigue:
Guided:
A condition of support at the ends of a beam or column or at the edge of a plate or shell that prevents rotation of the edge of the neutral surface but permits longitudinal and transverse displacement (Compare Fixed; Guided; Supported).
Held:
A condition of support at the ends of a beam or columns or at the edge of a plate or shell that prevents longitudinal and transverse displacement of the edge of the neutral surface but permits rotation in the plane of bending (Compared Fixed; Guided Supported).
Influence Line:
Usually pertaining to a particular section of a beam, influence line is a curve drawn; so that its ordinate at any point represent the value of the reaction; vertical shear, bending moment, or deflection produced at the particular section by a unit load applied at the point where the ordinate is measured. An influence line may be used to show the effect of load position on any quantity dependent thereon, such as the stress in a given truss member, the deflection of a truss, or the twisting moment in a shaft.
Isoclinic:
A line (in a stressed body) at all points on which the corresponding principal stresses have the same directions.
Isotropic:
Having the same properties in all properties in all direction. In discussions pertaining to strengths of materials, isotropic usually means having the same strength and elastic properties (modulus of elasticity, modulus of rigidity, and poisson’s ratio) in all directions.
Korn (Kornal):
18 Margin of safety:
As used in aeronautical design, margin of safety is the percentage by which the ultimate strength of a member exceeds the design load. The design load is the applied load, or maximum probable load, multiplied by a specified factor of safety. [The use of the term margin of safety and design load in this sense is practically restricted to aeronautical engineering]
Mechanical Hysteresis:
The dissipation of energy as heat during a stress cycle, which is revealed graphically by failure of the descending and ascending branches of the stress-strain diagram to coincide.
Modulus of Elasticity (Young’s Modulus):
The rate of change of unit tensile or compressive stress with respect to unit tensile or compressive strain for the condition of uniaxial stress within the proportional limit. For most, but not all, materials, the modulus of elasticity is the same for tension and compression. For non-isotropic material such as wood, it is necessary distinguish between the moduli of elasticity in different directions.
Modulus of Resilience:
The strain energy per unit volume absorbed up to the elastic limit under the condition of uniform uniaxial stress.
Modulus of Rigidity (Modulus of Elasticity in Shear):
The rates of change of unit shear stress with respect to unit shear strain for the condition of pure shear within the proportional limit. For non-isotropic materials such as wood, it is necessary to distinguish between the moduli of rigidity in different directions.
Modulus of Rupture in Bending (Computed ultimate Twisting Strength):
The fictitious tensile or compressive stress in the extreme fiber of a beam computed by the flexure equation
I Mc
=
Modulus of Rupture in Torsion (Computed Ultimate twisting moment):
The fictitious shear stress at the surface of circular shaft computed by the tension formula
J Tr
=
τ , where T is the Twisting moment that causes rupture.
Moment of Area (First Moment of an Area, Statical Moment of an Area):
With respect to an axis, the sum of the products obtained by multiplying each element of the area dA by its distance from the axis y; it is therefore the quantity
∫
dAy . An axisin the plane of the area is implied.
Moment of Inertia of an area (Second Moment of an Area):
The moment of inertia of an area with respect to axis is the sum of the products obtained by multiplying each element of the area dA by the square of its distance from the axis y; it is therefore the quantity
∫
dAy . An axis in the plane of the area is implied, if the axis is 2normal to that plane, the term polar moment of inertia.
Neutral Axis:
The line of zero fiber stress in any given section of a member subject to bending; it contains the neutral axis of every section.
Notched – Sensitivity Ratio:
Used to compare stress concentration factor kt and fatigue-strength reduction
factor kf, the notch sensitivity ratio is commonly defined as the ratio
(
kf −1)
/(
kt −1)
.It variesfrom 0, for some ductile materials, to 1, for some hard brittle materials.
Neutral Surface:
The longitudinal surface of zero fibre stress in a member subject to bending; it contains the neutral axis of every section.
20 Plasticity:
The property of sustaining appreciable (visible to the eye) permanent deformation without rupture. The term is also used to denote the property of yielding or flowing under steady load.
Poisson’s Ratio:
The ratio of lateral unit strain to longitudinal unit strain under the condition of uniform and uniaxial longitudinal unit within the proportional limit.
Principal Axes:
The principal axes of an area for a given point in its plane are the two mutually perpendicular axes, passing through the point and lying in the plane of the area, for one of which the moment of inertia is greater and for the other less than for any other coplanar axis passing through that point. If the point in question is the centroid of the area, these axes are called principal central axes.
Principal Planes; Principal Stresses:
Through any point in stressed body there pass three mutually perpendicular planes, the stress on each of which is purely normal, tension, or compression; these are the principal planes for that point. The stresses on these planes are the principal stresses; one of them is the maximum stress at that point, and one of them is the minimum stress at that point. When one of the principal stresses is zero, the condition is one of uniaxial stress.
Product of Inertia of an Area:
With respect to a pair of rectangular axes in its plane, the sum of the products obtained b multiplying each element of the area dA by its coordinates with respect to those axes x and y; it is therefore the quantity
∫
dAxy.Proof Stress:
Pertaining to acceptance tests of metals, a specified tensile stress that must be sustained without deformation in excess of a specified amount.
Proportional Limit:
The greatest stress that a material can sustain without deviating from the law of stress-strain proportionality. (Compare Elastic limit; apparent elastic limit; Yield point; Yield strength)
Radius of Gyration:
The radius of gyration of area with respect to a given axis is the square root of the quantity obtained by dividing the moment of inertia of the area with respect to that axis by the area.
Reduction of Area:
The difference between the cross sectional area of a tension specimen at the section of rupture before loading and after rupture.
Rupture Factor:
Used in reference to brittle materials, i.e. materials in which failure occurs through tensile rupture rather than through excessive deformation. For a member of given form, size, and material, loaded and supported in a given manner, the rupture factor is the ratio of the fictitious maximum tensile stress at failure, as calculated by the appropriate formula for elastic stress, to the ultimate tensile strength of the material, as determined by a conventional tension test.
Section Modulus (Section Factor):
Pertaining to the cross section of a beam, the section modulus with respect to either principal central axis is the moment of inertia with respect to that axis divided by the distance from that axis to the most remote point of the section. The section modulus largely determines the ability of a beam to carry load with no plastic deformation
22 Set (Permanent set, Permanent deformation, Plastic strain, Plastic deformation):
Strain remaining after removal of stress. Shakedown Load (Stabilizing Load):
The maximum load that can be applied to a beam or rigid frame and on removal leave such residual moments that subsequent application of the same or a smaller load will cause only elastic stresses.
Shear Lag:
Because of shear strain, the longitudinal tensile or compressive bending stress in wide beam flanges diminishes with the distance from the web or webs, and this stress diminution is called shear lag.
Singularity Functions:
A class of function that, when used with some caution, permit expressing in one equation what would normally be expressed in several separate equations, with boundary conditions being matched at the ends of the intervals over which the several separate expressions are valid.
Ultimate Strength:
The ultimate strength of a material in tension, compression, or shear, respectively, is the maximum tensile, compressive, or shear stress that the material can sustain calculated on the basis of the ultimate load and the original or unstrained dimensions. It is implied that the condition of stress represents uniaxial tension, uniaxial compression, or pure shear, as the case may be.
Unit Stress:
The amount of stress per unit of area. The unit stress (tensile, compressive, or shear) at any point on a plane is the limit, as Δ approaches 0, A Δ /P ΔA where Δ is the total P
Vertical Stress:
Refers to a beam, assumed for convenience to be horizontal and loaded and supported by forces that all lies in a vertical plane. The vertical shear at any section of the beam is the vertical component of all the forces that act on the beam to the left of the section. The shear equation is an expression for the vertical shear at any section in terms of x, the distance to that section measured from a chosen origin, usually taken at the left end of the beam.
Yield Point:
The lowest stress at which strain increase in stress. For some purposes it is important to distinguish between the upper yield point, which is the stress at which is the stress-strain diagram first becomes horizontal, and lower yield point, which is the somewhat lower and almost constant stress under which the metal continues to deform. Only a few materials exhibit a true yield point; for other materials the term is sometimes used synonymously with yield strength (Compare yield strength; Elastic limit; Apparent elastic; proportional limit)
Yield Strength:
The stress at which a material exhibits a specified permanent deformation or set. The set is usually determined by measuring the departure of the actual stress-strain diagram from an extension of the initial straight portion. The specified value is often taken as a unit strain of 0.002.
Slenderness Ratio:
The ratio of length of a uniform column to the least radius of gyration of the cross section.
Slip Lines:
24 Strain:
Any forced change in the dimensions of a body. A stretch is tensile strain; a shortening is a compressive strain; and angular distortion is a shear strain. The word strain is commonly used to connote unit strain.
Strain energy:
Other wise called as elastic energy, potential energy of deformation. Mechanical energy stored up in the stressed material. Stress with in the elastic limit is implied; therefore, the strain energy is equal to the work done by the external forces in producing the stress and is recoverable.
Strain rosette:
At any point on the surface of a stressed body, strains measured on each of the intersecting gauge line make possible the calculation of the principal stresses. Such gauge lines and the corresponding strains are called strain rosettes.
Stress:
Internal force exerted by either of two adjustant parts of a body upon the other across an imagined plane of separation. When the forces are parallel to the plane, stress is called shear stress; when the forces are normal to the plane, the stress is called normal stress; when the normal stress is directed towards the part on which it acts, it is called compressive stress; and when it is directed away from the part on which it acts, it is called tensile stress. Shear, compressive, and tensile stresses, respectively, resists the tendency of the parts to mutually slide, approach, or separate under the action of applied forces.
Stress solid:
The solid figure formed by the surfaces bounding vectors drawn at all points of the cross section of a member and representing the unit normal stress at each such point. The solid stress gives a picture of the stress distribution on a section.
Stress trajectory (isostatic):
A line in a stressed body tangent to the direction of one of the principal stresses at every point through which it passes id called stress trajectory.
Torsional center:
Otherwise called as center of twist or center of torsion or center of shear. If a twisting couple is applied at a given section of a straight member, that section rotates about some point in its plane. This point which does not move when the member twists, is the torsional center of that section.
True stress:
For an axially loaded bar, the load divided by corresponding actual cross section area. It differs from the stress as ordinarily defined because of the change in area due to loading.
Twisting moment (torque):
At any section of the member, the moment of all forces that act on the member to the left of that section, taken about a polar axis through the flexure center of that section. For the sections that are symmetrical about each principal central axis, the flexural center coincides with the centroid.
Ultimate elongation:
The percentage of permanent deformation remaining after tensile rupture measured over an arbitrary length including the section of rupture.
26 B Conversion Tables:
B1. Introduction:
The conversion factors in this section include the System International (S.I) Units as these are now widely accepted as the standard units of measurement.
There are six basics units in the S.I. system. These are follows
The main advantage of the S.I. System is the ease with which mechanical units expressed in mass length and time can be related to electrical units.
Decimal Prefixes:.
The use of double prefixes should be avoided when single prefixes are available. Basic Units in Stress Analysis
28 B3. Conversion Factors Grouped by Category:
The following pages give a list of various conversion factors used grouped by category
Length:
Area:
Volume:
Mass:
Density:
30 Force Length Combination:
Moment of Inertia:
Energy:
Acceleration:
Electrical:
32 Power:
Stress or Pressure:
Time:
Viscosity:
34 C. Section Shear Centre:
The shear centre can be defined as that point in the plane of a cross-section through which an applied transverse load must pass for bending to occur unaccompanied by twisting. To obtain the shear centre a unit transverse load is applied to the section at that point. The shear stress distribution, as thus shear flow distribution, thus shear flow distribution is obtained for each element of the section. Equating the torque given by the shear distribution to the torque given by the transverse load will produce position of shear centre.
C1. Location of Shear Centre-Open sections: Section Constants 2 xy yy xx xx x I I I I Z − = 2 xy yy xx yy y I I I I Z − = 2 xy yy xx xy xy I I I I Z − =
Shear flow in an element
(
)
[
]
∫
+ − − + = t i y i x xy i x x i y y o f t xV yV Z xV Z yV Z ds q q Where q0is value of q at s = 0 fxi = horizontal distance from the element centroid to the section centroid
yi = vertical distance from the element centroid to the section centroid
Shear force in an element
ds q Ff = i
Torque/Moment given by element
i i
i FR
36 Horizontal Location of shear Centre:
To Locate the horizontal position of the shear centre, we apply a vertical unit force through the unknown position of the shear centre x0,y0 ie Vy = 1, and Vx = 0.
(
xZ y Z)
t ds q0qi = i xy − i y t +
∴
∫
Shear forces and total torque ∑ are calculated as defined previously. Ti
Vertical Location Of Shear Centre:
To locate the vertical position of the shear centre, we apply a horizontal unit through the unknown position of the shear centre x0,y0 ie Vy = 0, and Vx = 1
(
)
00 yZ xZ t ds q
qi = i xy − i x t +
∴
∫
C2. Shear Centre-curved Web:
Horizontal Location:
∫
⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − = θ θ 0 1 tds I y q xx θ cos R y= ds= Rdθ∫
⋅ = θ θ θ R Rd I t q xx cos θ θ R2sin I t q xx =R qds× = R qRd × = θ θ θd I t R x sin 4 Total Moment
∫
= π θ θ 0 4 sin d I t R xx[
θ]
π 0 4 cos − xx I t RVertical component of force of element
θ
sin
qds
∴ Total Vertical Force
∫
= θdθ I t R xx 2 3 sin xx I t R 2 3 π =Taking Moment about 0
xx xx I t R e I t R3 2 4 2 = π Horizontal Location, π R e= 4 Alternatively for a solid section:
38 with q = web shear flow
A = Enclosed area h = web depth Vertical Equilibrium Vy qh= Moment Equilibrium e Vy Aq= × 2 . h A e=2 /
C3. Shear centre-Beams with constant shear flow between boom members:
xy yy xx I I
I , , evaluated using boom areas only(skin shear carrying only) and Zx,Zy,Zxy using above values of Ixx,Iyy,Ixy
Shear flow in each element
(
)
[
xV yV Z xV Z yV Z]
AHorizontal location:
Firstly set Vy = 1 and Vx = 0
Giving qi =
∑
[
( )
x Zxy −( )
y Zy]
δA+q( )
i−1Where x and y are locations to element area δA
Vertical location:
Firstly set Vy = 0 and Vx = 1
Giving qi =
∑
[
( )
y Zxy −( )
x Zx]
δA+q( )
i−1Where x and y are locations to element area δA
C4. Shear Centre – Single Cell Closed Section:
For Horizontal Location:
Cut element at 0 and assume qo =0
Evaluate shear flow distribution around section.
Take moment about 0to give the out of balance moment Mo.
Balance Mo with balancing shear flow qb.
Where A M qb 2 0 =
40 For Vertical Location:
Cut element at a and assume qa =0
Evaluate shear flow distribution around section.
Take moment about 0to give the out of balance moment Mo.
Balance Mo with balancing shear flow qb.
Where A M qb 2 0 =
With resultant shear flow distribution; take moments about convenient point to give moment MR.
y R
v M e=
C5. Shear centre – single cell closed section with booms:
Initially the member is dealt with as a beam i.e. a cut is made at appropriate skin element thus reducing it from a single cell to an open beam section. Having established the open beam element shear floes, moments are taken about a convenient point and out of balance moment Mo established.
Mo is balanced by shear flow
A M q o b 2 =
With resultant shear flow distribution; take moments about a convenient point to give MR.
Shear centre horizontal location and vertical location:
y R V M e= , x R V M e=
46 D. Warping Constant:
When a torque, is applied to a non-circular beam, the beam will twist with each section of the beam rotating about torsional centre. The sections of the beam will not remain plane and will wrap out of plane. Depending on how the beam is constrained the effects of the applied torque, T will vary.
Consider a torque, T, applied to the section to the right.
The diagrams below illustrate the effects of the torque with different constraints.
The warping constant Γ is calculated by adding Γ1 and Γ2. The constant Γ1 relates to the strain in the middle plane of the walls arising from variations of warping of the cross-section along length due to twist. The constant Γ2 relates to the strain across the thickness, t, of the walls. Therefore the warping constant is defined by
(
Γ1 +Γ2)
= Γ
Consider the following section
We can now show equations for the warping constants based on the generalised section shown above.
( )
2 0 0 2 1 1∫
− ⎢⎣⎡∫
⎥⎦⎤ = Γ s AWdA A tds W Where ds r W =∫
s t 0The secondary warping constant Γ2 is obtained from the formula
( )
∫
= Γ A rnts dA 0 2 2 12 1 in6where A = Total cross-section area
s = distance from any free end along middle of wall.
rt = the perpendicular from the shear centre to the tangent of the point defined by s.
rn = distance from the point of rotation to the normal of the tangent line of the
middle of the wall. A = Total cross-section area
rt = distance from the point of rotation to the tangent line of the centre line of
the section wall at s
s = distance from any free end along the middle of the wall.
Constraint Warping about a Point
It is sometimes necessary to know the warping constant
( )
Γ may be obtained by 1 0 means of the formula[ ]
Γ1 0 =Γ1+x02Ixx + y02Iyy −2x0y0Ixy48 D1 Warping Constant – Tables for Typical Sections:
E. Mechanical properties: E1. DATA Basis:
A-values:
When applied loads are eventually distributed through a single member within an assembly, the failure of which would results in the loss of the structural integrity of the component involved, the guaranteed minimum design mechanical properties (“A” - values) must be met.
At least 99% of the populations of values are expected to equal or exceed the A-value mechanical property allowable, with a confidence of 95%
Examples of such items are:
Single members such as drag struts, push-pull rods, etc.
B-values:
Redundant structures, in which the failure of individual elements would result in applied loads being safely distributed to other load-carrying members, may be designed on the basis of 90% probability.
“B”-values are above which at least 90% of the population of values is expected to with a confidence of 95%.
Examples of such items are: Sheet – stiffener combinations.
Multi – rivet or multiple bolt connections. Thick skin type wing construction.
S-values:
The S-value is the minimum value specified by the governing industry specification of federal or military standards for the material. For certain products heat treated by the user the S- values may reflect a specified quality control requirement. Statistical assurance associated with this value is not known.
54 E2. Stress-Strain Data:
The significant properties of material for structural design purposes are determined experimentally. Tension and Compression coupon tests provide the simplest fundamental information on material mechanical properties. Experimental data is recorded on stress – strain curves that record the measures units strain as a function of the applied force per unit area. Data of importance for structural analysis is derived from the stress –strain as shown above.
E3. Definition of terms: Modulus of Elasticity:
Slope of initial straight-line part of the stress-strain curve, E (force per unit area)
Secant Modulus:
Slope of secant line from the origin to any point (stress) on the stress-strain curve a function of the stress level applied, Es
Tangent Modulus:
Slope of the line tangent to strain curve at any point (stress) on the stress-strain curve, Et. The tangent modulus is the same as the modulus of elasticity in the range.
Proportional Limit:
Stress at which the stress-strain curve deviates from linearity, indication of plastic or permanent set is called proportionality limit. Sometimes defined arbitrarily as stress at 0.01% offset, Fpl (psi
Yield Strength or Stress:
At 0.2% offset, Fcy, Ftu (psi)
Ultimate Strength or Stress:
Maximum force per unit area, Ftu (psi)
Poisson’s Ratio:
Ratio of the lateral strain measured normal to the loading direction, to the normal strain measured in the direction of the load. Poisson ratio varies from the elastic value of around 0.3 to 0.5 in the plastic range.
Secant stress:
Stress at intercept of any given secant modulus line with the stress – strain curve is called secant stress. F0.7, F0.85 are secant lines of 0.7 and 0.85 % of young’s modulus. Elongation:
Permanent strain at fracture (tension) in the direction of loading e(%). Sometimes this property is considered a measure of ductility/brittleness.
Fracture Strain:
56 Elastic Strain:
Strains equal to or less than the strain a t the proportional limit stress,εeor the strain to the projected young’s modulus line of any stress level,
Plastic Strain:
Increment of strain beyond the elastic strain at stresses above the proportional limit.
Shear Properties:
The results of torsion tests on round tubes or round solid section s are sometimes plotted as torsion stress-strain diagrams. The modulus of elasticity in shear as determined from such a diagram is a basic shear property. Other properties, such as proportional limit and ultimate shearing stress, cannot be treated as basic properties because of the form factor.
Bearing Properties:
Bearing strength are of value in the design of joints and lugs. Only yield and ultimate values are obtained from bearing tests. The bearing stress is obtained by dividing the load on a pin, which bears against the edge of the hole, by the bearing area, where the area is the product of the pin diameter and sheet thickness.
The bearing test requires the use of special cleaning procedures as specified in ASTM E 238. In the “various Room Temperature Property Tables. When the indicated values are based on tests with clean pins, the values are footnoted as “dry pin” values. Designers should consider the use of a reduction factor in applying these values to structural analysis.
In the definition of bearing values, t is sheet thickness, D is the hole diameter and e is the edge distance measured from the hole centre to the edge of the material in the direction of applied stress.
E4. Temperature Effects:
Temperature below room temperature generally causes an increase in all strength properties of metals. However ductility usually decreases below room temperature. Temperatures above room temperature have the opposite effect which is dependent on many factors such as time of exposure and the characteristic of the material.
E5. Fatigue Properties:
Practically all materials will break under numerous repetitions of a stress is not as great the stress required to produce immediate rupture. This phenomena is known as fatigue. Fatigue crack initiation is normally associated with the endurance limit of a material below which fatigue does not occur is usually presented in the form of S-N curves, where the cycles stress amplitude is plotted versus the number of cycles to failure. Another major factor affecting the fatigue behaviour is the stress concentration caused by detail design of the structure.
58 manifested by carbide precipitation, spheroidization, sigma-phase formation, temper
embrittlement and internal or structural transformation, depending upon the material and the condition.
Biaxial and Multiaxial Properties:
Many structural geometries, or load applications, are such that induced stresses are not uniaxial but are bi-or triaxial. Because of the difficulty in testing, few triaxial test data exist. However considerable biaxial testing has been conducted. If stresses are referred to as mutually perpendicular x, y, and z direction of the usual rectangular co-ordinates, a biaxial stress is a condition such that there are either a positive or negative stress in the x, y directions and the stresses in the z direction is usually zero.
Fracture Strength:
The occurrence of flaws in a structural component is an unavoidable circumstance of material processing, fabrication, or service. These flaws may be cracks, metallurgical
inclusion or voids, weld defects, design discontinuities, or combination thereof. If severe enough, these flaws can include structural failure at loads below those of normal design. The strength of a component containing a flaw is dependent on the flaw size, the component geometry, and a material property termed “fracture toughness”. The fracture toughness of a material is literally a measure of resistance to fracture. It also is considered a measure of its tolerance or lack of sensitivity to flaws. As with many other material properties, fracture toughness is dependent on processing variables, product form geometry, temperature, loading rate, and other environmental factors.
Fatigue-Crack Propagation Behaviour:
Between the crack initiating phenomenon of fatigue and the critical instability of cracked structural elements as identified by fracture toughness and residual strength lies an important fact of material behaviour known as fatigue-crack propagation.
In small size laboratory fatigue specimens, crack initiation and specimen failure may be nearly synonymous. However in larger structural components the existence of a crack does not
necessarily indicate imminent failure of the component. The structural life during the cycle crack extension phase is the basis for damage tolerance certification of aircraft structure.
Material Failures:
Fracture of a metal can be very complex, it can occur in either ductile or brittle state, in the same material depending on the state of stress and the environment. Fracture can occur after elongation of the metal over a relatively large uniform length, or after a concentrated elongation in a short length. Shear deformation will also vary dependent on metal and stress state, because of these variations in magnitude and mode of deformation the ductility of a metal can have a profound effect on the ability of a fabricated part to withstand applied loads.
E.6 Generalised Stress-Strain Equations: Introduction:
This section describes a generalised method for representing the stress-strain curves of materials which exhibits a smooth curves. The method uses a reference stress and a characteristic index, in addition to the modulus of elasticity, and this section indicates how these two quantities may be determined
Generalised Equations for Stress-Strain curves:
A smooth continuous stress- strain curve can be represented by the equation
m n n n f f m f f f E ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + = 1 ε
in which f is the stress, fn is taken to be the stress at which the tangent modulus is one half the
modulus of elasticity and the index m has been found to characterize the shape of the stress-strain curve and is therefore referred to as the material characteristic. The equation above only applies if f/fn is positive and this needs to be taken into account in cases where the chosen sign
60 in which f is the stress, fn is taken to be the stress at which the tangent modulus is one half the
modulus of elasticity and the index m has been found to characterize the shape of the stress-strain curve and is therefore referred to as the material characteristic. The equation above only applies if f/fn is positive and this needs to be taken into account in cases where the chosen sign
convention leads to either for ε being of opposite sign to fn.
This equation can then be rearranged and the value of Tangent Modulus obtained as
(
df dε)
Et = / 1 1 1 − − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + = m n t f f E EThis can be rearranged to give
m n n t n f f f f E f f E ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + =
The secant modulus Es, is the ratio of stress to total strain and may be determined directly from
stress – strain curve equation.
1 1 1 1 − − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + = m n s f f m E E
From the above set of equations the simple relationship between Es and Et follows
⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ −1 1 1 t s E E m E E
The value of Poisson’s ratio is according to the equation
(
)
(
)
1 1 1 1 1 / − − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + = − − = m n m n p e e p s p f f m f f m v v v v E E v v ev = Fully elastic value of Poisson’s ratio
p
v = Fully plastic value of Poisson’s ratio
as the material yields Poisson’s ration will increase from its elastic value up to its fully plastic value
The plastic value of Poisson’s ratio v can be assumed to be 0.5 for all aluminium alloys. The p
elastic value for aluminium alloys can be assumed to be 0.3.
Determination of m and fn
The value of m, the material characteristic, can be found from standard fitting procedure if the full stress-strain curve is known. However, it may be estimated, provided that two points on the non-proportional region of the curve are known, from the equation.
(
)
(
')
' / log / log R R R R f f m= ε εwhere fR and f’R are known stress values on the stress-strain curve of the actual material and εR
and '
R
ε are the strain in excess of the elastic strain corresponding to these stresses. For example , if fR and fR’ are the 0.1% and 0.2% tensile proof stresses of the material then εR and εR' are
0.001 and 0.002 respectively.
For a known value of m the reference stress, fn, may be evaluated from the equation
( 1) / 1 − − ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = m R R R n f E m f f ε E.6.1. Problem:
Calculate the strain when a stress of 30000 lb.in2 is applied to sample 2024 T3 aluminium alloy sheet using B-basis in compression.(0.010<t<0.128 in.thick)
For 2024 T3 Aluminium alloy steel sheet we can obtain the following values from MIL-HDBK-5 E = 10.5×106 lb/in2 f= 30000 ld/in2 002 . 0 = R ε fR = 3.9×104lb/in2
m = 15 (material characteristic as defined in MIL-HDBK-5) We can calculate fn from
62 Therefore fn = 33593.97 lbin
To calculate the strain,ε , we must simplify the equation
m n n n f f m f f f E ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + = 1 ε To give E f f f m f f n m n n ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ + = 1 ε Which gives us a strain of 0.003.
1. PRISMATIC BAR IN TENSION:
For Ductile material yield stress is to be considered during analysis. If material is brittle or exhibits brittle fracture upon loading, ultimate stress is to be considered for analysis. Load considered for ultimate analysis is factor of safety times limit load. t2 values are the
maximum permissible tensile stresses for the proof loading condition. ft values are given for the
ultimate loading conditions. In case of a few materials, which have high proof to ultimate strength ratio, allowance may have to be made for the effects of stress concentration.
1.1 Limit Stress Analysis:
For limit stress analysis t2 is considered. It is then compared with limit stress or actual stress.
Limit load = P Working stress A P = σ
Allowable Proof stress (Or) 0.2% Proof stress = t2
Reserve factor =
σ2
t
If reserve factor value is greater than 1 the design is considered to be safe. 1.2 Ultimate Stress Analysis:
For ultimate stress analysis ultimate load has to be considered. It has to be compared with UTS value of the material.
64 1.3 Problem: Material BSL 168T6511 Mpa t2 =370 Mpa ft =415 Limit load P = 1500kg A = 10mm×10mm Limit stress analysis:
Applied proof stress PA 150Mpa
100 15000 = =
=
σ
Allowable proof stress t2 = 370Mpa
Reserve factor = 370/150 = 2.47
Ultimate stress analysis:
Actual ultimate stress = fos ×actual proof stress = 1.5×150=225
Allowable ultimate stress = 415 Reserve factor = 415/225 = 1.84
2. PRISMATIC BAR IN COMPRESSION:
For the stable members in pure compression an ultimate compressive stress equal to the 0.2% proof stress (c2) is recommended. Where the c2 value is not given tensile values t2
may be used. For thin sections or flanges, consideration should be given to the possibility of local or Torsional instability. For slender columns buckling load is to be considered. Where magnesium alloys are used in compression, it is recommended that special test made to be establish the compressive proof stress values, since they may be as low as 0.5 of the corresponding proof stresses.
2.1 Compression Stress Analysis: (Block Compression) For Stress analysis c2 is considered.
Actual Load = P
Working Stress σ = P /A Allowable stress = c2.
Reserve Factor = c2/ σ
If the reserve factor value is greater than 1 design is safe.
2.1.2 Problem:
66 Actual ultimate stress = 225 / 2
100 15000 5 . 1 mm N = ×
Allowable ultimate stress = 370 Reserve factor = 370/225 = 1.64
2.2 Euler Buckling: Buckling stress A CL EI cr 2 2 π σ =
Pcr value depends upon end fixing condition C:
Hinged - Hinged C = 1 Fixed - Free C = 4 Fixed - Fixed C = (1/4) Fixed - Hinged C = (1/2) 2.3 For other Buckling cases: Section Modulus = Z
2.3.1 Buckling for Eccentric Loading:
In an actual structure it is not possible for a column to be perfectly straight or to be loaded exactly at the centroid of the area. Therefore apart from normal load there will be additional load due to moment caused by eccentricity.
Z EI P L Pe A P cr ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ + = 2 sec σ
This is a Transcendental in P. The solution will be taken as Pall.
Eccentricity of the load = e in mm. Moment of inertia = I in mm4.
68 2.3.2 Beam - Columns:
A beam is called Beam-Column when it is submitted to both compression and bending loads. When a structural element similar to a beam is subjected simultaneously to a normal load and bending moment, superposition theorem cannot be used to determine the stresses. The lateral deflections appreciably change the moment arms of the compression forces and deflections are not proportional to the loads. The verification of Beam – Column will include two steps as following
1 .The study of column.
2. The study of Beam – column if axial load is lower than the critical load calculated in the previous step.
2.3.2.1 Notations and Conventions:
1. The compressive loads are taken to be positive.
2. A positive bending moment compresses the upper fibres of the beam. 3. A distributed lateral loads leads to the positive moment.
4. A positive lateral load compresses the upper fibre of the beam. 5. The compression stresses are positive.
2.3.2.2 Terms used: Compression load =P in N.
Distributed lateral load = q in N/mm. Applied isolated load = F in N.
Critical buckling stress = σcrin N/mm2.
Modulus of Elasticity in compression = Ec in N/mm2.
Moment of inertia = I in mm4. Area of cross section = A in mm2. Clamping factor = K.
Column effective buckling length = L in mm. Real column length = S in mm.
2.3.2.3 Calculation of Bending Moment:
Let us consider a beam where the ends are on single supports with compression loads and bending loads.
70
Final bending moment is ⎟⎟
⎠ ⎞ ⎜⎜ ⎝ ⎛ + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − = j x M j x j L j L M M M A A B Z Z Z Z sin cos sin cos in N-mm.
Maximum bending moment is at
⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − = j L j L M M jArc x A B Z Z sin cos tan in N-mm.
Common expression for Beam – Column bending moment
