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Mechanics of solids pdf

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Saqib Imran
Saqib Imran

1. The document is notes on solid mechanics written by Saqib Imran, a civil engineering student, for other students and engineers. 2. It covers topics in solid mechanics including stress, strain, tensile strength, yield strength, ductile and brittle materials, Young's modulus, elastic and plastic regions, shear force, and bending moment. 3. Saqib provides contact information and invites readers to contact him if they have any questions about the notes.

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1 | P a g e SAQIB IMRAN 0341-7549889 1 Assala mu alykum My Name is saqib imran and I am the student of b.tech (civil) in sarhad univeristy of science and technology peshawer. I have written this notes by different websites and some by self and prepare it for the student and also for engineer who work on field to get some knowledge from it. I hope you all students may like it. Remember me in your pray, allah bless me and all of you friends. If u have any confusion in this notes contact me on my gmail id: Saqibimran43@gmail.com or text me on 0341-7549889. Saqib imran.
2 | P a g e SAQIB IMRAN 0341-7549889 2 Mechanics of Solids What is Solid Mechanics | Strength of Materials | Purpose of Studying SOM Solid mechanics or Mechanics of solids Solid mechanics or Mechanics of solids is the branch of Mechanics, Physics, Mathematics that concerns the behavior of solid matter under external actions (e.g., external forces, temperature changes, applied displacements, etc.). Or It is a branch of science which deals with the internal effects of the forces on the bodies when they are loaded, from initial point to rupture or break. Strength of Materials Strength of materials deals with the relations between externally applied loads and their internal effects on the bodies. Purpose of studying strength of materials To ensure that the structure used will be safe against the maximum internal effects that may be produced by any combination of external loading. Types of forces Forces that are acting on the body, may be of the following types. Effects of these forces are studied in solid mechanics or mechanics of solids. Normal forces The forces that are acting perpendicular to the body are known as normal forces. Normal forces are of two types: 1. Tensile 2. Compressive.
3 | P a g e SAQIB IMRAN 0341-7549889 3 Tangential forces The forces that are acting parallel to the body are known as tangential forces. Shear forces and frictional forces are example of tangential forces. Independent forces Active forces are independent forces. Dependent forces Reactive forces are dependent forces. Concentrated forces If forces are acting on an area of a body which is negligible as compared to the body, then these are known as concentrated forces. Distributed forces If forces are acting on an area of a body which is not negligible as compared to the body, then these are known as distributed forces. These may be of uniformly distributed load (UDL) or Uniformly varying load (UVL) or any kind of general loading. Effects of forces The effects of forces which are studied in solid mechanics or mechanics of solids are 1. Translation 2. Rotation (moment and torque) What is Stress | Yield stress | Compressive stress | Strain Stress is a feeling that’s created when we react to particular events. Stress may refer to: Mechanical, Biological, Music, other. Stress (physics): The average amount of force exerted per unit area. Yield stress: The stress at which a material begins to deform plastically.
4 | P a g e SAQIB IMRAN 0341-7549889 4 Compressive stress: The stress applied to materials resulting in their compaction. Strain: It is the deformation of materials caused by the action of stress. Strain is calculated by first assuming a change between two body states: the beginning state and the final state. Then the difference in placement of two points in this body in those two states expresses the numerical value of strain. Strain therefore expresses itself as a change in size and/or shape. If strain is equal over all parts of a body, it is referred to as homogeneous strain; otherwise, it is in homogeneous strain Stress Strain Curve: During testing of a material sample, the stress–strain curve is a graphical representation of the relationship between stress, and strain, i.e. elongation, compression, or distortion. The nature of the curve varies from material to material. Tensile Strength | Yield Strength | Ultimate Strength | Breaking Strength Tensile Strength Tensile strength is the stress at the maximum on the stress-strain curve. Or The greatest longitudinal or axial stress a material can bear without breaking. Tensile strength is the maximum amount of tensile stress that a material can take before failure. There are three definitions of tensile strength  Yield strength  Ultimate strength  Breaking strength Yield Strength The stress at which material strain changes from elastic deformation to plastic deformation, causing it to deform permanently. This point is not a well defined point. We
5 | P a g e SAQIB IMRAN 0341-7549889 5 can say that yield strength is a strength which a material can bear without permanent deformation. Yield strength of Structural steel A36 is 250 MPa. Ultimate Strength The maximum stress a material can withstand. Ultimate tensile strength of Structural steel A36 is 400 MPa. Breaking Strength The stress coordinate on the stress-strain curve at the point of rupture. Ductile and Brittle Materials | Difference Between Ductile and Brittle Material Ductile Materials:  Ductile materials will withstand large strains before the specimen ruptures.  Ductile materials often have relatively small Young’s moduli and ultimate stresses.  Ductile materials exhibit large strains and yielding before they fail.  Steel and aluminum usually fall in the class of Ductile Materials Brittle Materials:  Brittle materials fracture at much lower strains.  Brittle materials often have relatively large Young’s moduli and ultimate stresses.  Brittle materials fail suddenly and without much warning.  Glass and cast iron fall in the class of Brittle Materials. Youngs Modulus and its Units | Proportional Limit | Ultimate Stress | Rupture Stress Modulus of Elasticity: In solid mechanics, Young’s modulus (E) is a measure of the stiffness of a material.It is defined as the ratio of stress over strain in the region in which Hooke’s Law is obeyed for the material.
6 | P a g e SAQIB IMRAN 0341-7549889 6 Units Young’s modulus is the ratio of stress, which has units of pressure to strain, which is dimensionless; therefore Young’s modulus itself has units of pressure. Proportional Limit : The point up to which the stress and strain are linearly related is called the proportional limit. Ultimate Stress : The largest stress in the stress strain curve is called the ultimate stress. Rupture Stress : The stress at the point of rupture is called the fracture or rupture stress. Elastic Region | Plastic Region | Yield Point | Plastic Strain | Strain Hardening | Necking Elastic Region: The region of the stress-strain curve in which the material returns to the undeformed state when applied forces are removed is called the elastic region. Plastic Region: The region in which the material deforms permanently is called the plastic region. Yield Point: The point demarcating the elastic from the plastic region is called the yield point. The stress at yield point is called the yield stress. Plastic Strain: The permanent strain when stresses are zero is called the plastic strain. The off-set yield stress is a stress that would produce a plastic strain corresponding to the specified off-set strain.
7 | P a g e SAQIB IMRAN 0341-7549889 7 A material that can undergo large plastic deformation before fracture is called a ductile material. A material that exhibits little or no plastic deformation at failure is called a brittle material. Hardness is the resistance to indentation. The raising of the yield point with increasing strain is called strain hardening. Necking Phenomenon: The sudden decrease in the area of cross-section after ultimate stress is called necking. Modulus of Resilience | Modulus of Toughness | Poisson’s Ratio Modulus of Resilience The work done on a unit volume of material, as a simple tensile force is gradually increased from zero to such a value that the proportional limit of the material is reached, is defined as the modulus of resilience. This may be calculated as the area under the stress-strain curve from the origin up to the proportional limit and is represented as the shaded area in the figure below. Modulus of Toughness The work done on a unit volume of material as a simple tensile force is gradually increased from zero to the value causing rupture is defined as the modulus of toughness. Poisson’s Ratio When a bar is subject to a simple tensile loading there is an increase in length of the bar in the direction of the load, but a decrease in the lateral dimensions perpendicular to the load. The ratio of the strain in the lateral direction to that in the axial direction is defined as Poisson’s ratio.
8 | P a g e SAQIB IMRAN 0341-7549889 8 Classification of Materials | Isotropic Material | Homogeneous Material | Specific Strength and Its Units Classification of Materials An isotropic material: Has a stress-strain relationship that is independent of the orientation of the coordinate system at a point. An isotropic material is the one having the same elastic properties (E,µ) in all directions at any one point of the body. A Homogeneous Material: A material is said to be homogeneous if the material elastic properties (E,µ) are the same at all points in the body. Specific Strength: This quantity is defined as the ratio of the ultimate (or tensile) strength to specific weight, i.e., weight per unit volume. we have And, In the SI system, we have So that in either system specific strength has units of length. This parameter is useful for comparisons of material efficiencies. Definition of Shear Force and Bending Moment Shear Force: The Shear Force at the cross section of the beam may be defined as the unbalanced vertical force to the right or left of the section. Bending Moment:
9 | P a g e SAQIB IMRAN 0341-7549889 9 The Bending Moment at the cross section of the beam may be defined as the algebraic sum of the moments of the forces, to the right or left of the section. Important Note: While considering the shear force and bending moment at the section , the end reactions must also be considered along with other external loads. Direct Stresses | Shearing Stresses | Tensile Stresses | Compressive Stresses Direct Stresses : The Stresses which are acting normal to the plane of the body are called as normal or direct stresses. These are called normal because these are acting perpendicular to the plane of the body. Shearing Stresses : The Stresses which act parallel to the stressed surfaces are called as shearing stresses. Tensile Stresses : If a straight bar is subjected to a pair of collinear forces acting in opposite direction and coinciding with the ends of the bar and directed away from the bar, then the bar tends to increase in length and the stresses developed in the bar will be tensile. Compressive Stresses : If a straight bar is subjected to a pair of collinear forces acting in opposite direction and coinciding with the ends of the bar and directed towards the bar, then the bar tends to shorter in length and the stresses developed will be compressive stresses. Definition of Column | Difference between strut and column | Type of Failure in Columns Definition of Column A long slender bar subjected to axial compression is called a column.
10 | P a g e SAQIB IMRAN 0341-7549889 10 The term is frequently use to describe a vertical member. Sometimes direct stresses dominate and sometimes flexural stresses dominate. Axial Compression means the compressive forces act at the two ends of the member in the opposite direction are along the same axis . Difference between strut and column The difference between column and strut is that former is used to describe a vertical member whereas latter is used for the inclined members. Strut is any inclined member subjected to compression loads. Types of Columns Based upon mode of failure, there are three types. Short Column The failure initiates due to crushing of material and direct stresses are dominant. For short column, if L < 4d and KL/r min < 30 Where d = least lateral dimension. L = Unbraced length of the column. k = effective length factor depends upon the end conditions of the column. r min = least radius of gyration. Slender or long Column In these, failure initiates due to lateral buckling and flexural stresses are dominant. If L > 30d or
11 | P a g e SAQIB IMRAN 0341-7549889 11 KL/r min > critical slenderness ratio. Intermediate Column Failure initiates due to lateral buckling as well as crushing of the material. Direct and flexural stresses both are dominant. Column is intermediate when 4d < L < 30d and 30 < KL /r min < 100 or Critical slenderness ratio. There is no agreed equation for intermediate column. There are many approaches that are available for design and analysis of intermediate columns such as: 1. Rankine Gordon approach. 2. Tangent Modulus formula. 3. Straight line formula. 4. Parabolic formula. 5. AISC equations. Remember all these are empirical approaches. Type of Failure in Columns Long compression members fails due to buckling whereas short compression member may fails due to yielding of material. Buckling of a compression member may occur even the maximum stresses in the material are less than the yield point of the material . By Buckling of a compression member it means lateral deflection of the bar. Definition of critical load of a Column | Slenderness ratio of a Column Critical load :The critical load of a column is that value of axial force that is just sufficient to keep the bar in a slightly deflected shape. Slenderness ratio of a column : The ratio of effective length of the column (an axial compression member) to the minimum radius of gyration of the cross-sectional area is termed as slenderness ratio of the bar.
12 | P a g e SAQIB IMRAN 0341-7549889 12 As it is a ratio so it has no units and of course it is dimensionless. Note : Radius of gyration is always the property of cross-section that is why the word cross- section is used with radius of gyration. Definition of a Beam | Cantilever Beams | Simple Beams Definition of a Beam: A bar subjected to forces and couples that lie in a plane containing its longitudinal axis is called a beam. The Forces will act perpendicular to the longitudinal axis of the bar. Cantilever Beams : If a beam is rigidly supported at only one end in such a manner that the beam cannot rotate at that point, it is called as cantilever beam. The other end of the beam is free to deflect but from the end where it is fixed, it cannot rotate. The end which is fixed is said to be ” restrained.” The reaction of the beam at the fixed end consists of a vertical force and a couple acting in the plane of the applied loads. Simple Beams : A beam which is freely supported at both ends is known as simple beam. The term “freely supported” implies that the end supports of the beam are capable of exerting forces on the beam not any moment. So in simple beams no “restrained” is offer to the angular rotation as the beam deflects under the loads. Overhanging Beams | Statically determinate Beams | Statically Indeterminate Beams Overhanging Beam : A beam which is freely supported at two points and having one or both the ends extending beyond these supports is known as overhanging beam.
13 | P a g e SAQIB IMRAN 0341-7549889 13 Statically Determinate Beams : All the beams in which reactions of the supports may be determined by the use of equations of static equilibrium are known as statically determinate beams. The value of these reactions are independent of the deformation of the beam. Statically indeterminate Beams : All the beams in which the number of reactions exceeds than the total number of equations of static equilibrium are known as statically indeterminate beams. In these beams the static equations must be supplemented by the equation based upon the deformation of the beam. Deflection of a Beam | Slope of a Beam | Flexural Rigidity of Beam Deflection of a Beam : The deflection at any point on the axis of the beam is the distance between its position before and after loading. Slope of a Beam : Slope at any section in a deflected beam is defined as the angle in radians which the tangent at the section makes with the original axis of the beam. Flexural Rigidity of Beam : The Product ” EI” is called flexural rigidity of the beam and is usually constant along the beam. Theorem 1 of Moment Area Method Theorem 1 is stated as follows : ” The change in slope between tangents drawn to the elastic curve at any two points A and B is equal to the product of 1/EI and the area of the moment diagram between these two points.”
14 | P a g e SAQIB IMRAN 0341-7549889 14 Mathematically, it can be written as For simplification we can say first draw the elastic curve diagram of the beam, then draw tangents at those two points for which change in slope has to be determined ( in the above diagram these points are A and B ) . Then Θab is the change in slope between tangents drawn to the elastic curve at these two points A and B. Theorem 2 of Moment Area Method It is stated as follows : ” The deviation of any point B relative to a tangent drawn to the elastic curve at any other point A in a direction perpendicular to the original position of the beam, is equal to the product of 1/EI and the moment of area about b of that part of the moment diagram between points A and B.” Mathematically it can be written as Similarly Where Āb and Āa are the distances from center of gravity of the moment diagram to the point B and A respectively. For simplification we can say that Ā is the distance from center of gravity of the moment diagram to the point where we have to find deviation. For instance we are finding the deviation at point B then Ā will be the distance from center of gravity of the moment diagram to the point B, because we are finding deflection at B. Same is the case with any other point such as A.
15 | P a g e SAQIB IMRAN 0341-7549889 15 Definition of Thin-walled open sections Thin-walled open sections Thin-walled open sections may be defined as The sections in which the thickness of material is small as compared with the overall geometry of the material and there is only one or no axis of symmetry. These sections are widely used. Their main use is in air craft industry. Common examples are channel section, top hat, zed, angle and circular sections. Within the section, shear stress variation is almost constant and we can assume shear stress constant. Definition of shear Center Shear Center Shear center is defined as the point on the beam section where load is applied and no twisting is produced. At shear center, resultant of internal forces passes. On symmetrical sections, shear center is the center of gravity of that section. In unsymmetrical sections, if the external applied forces act through the centroid of the section, then in addition to bending, twisting is also produced. To avoid twisting, and cause only bending, it is necessary for the forces to act through the particular point, which may not coincide with the centroid. The position of the this point is a function only of the geometry of the beam section. It is termed as shear center. Such sections in which there is a sliding problem, we place loads at the shear center. Strain measuring instruments | Strain Gauges Strain measuring instruments : These are of the following types :
16 | P a g e SAQIB IMRAN 0341-7549889 16  Mechanical.  Electrical. (Strain gauges)  Photoelectrical.  Basic working principle of strain gauges : In 1856, Kelvin observed that resistance of the conductor is directly proportional to the length and inversely proportional to the area of the conductor. This principle was used in 1933 by defense department of US. They form gauges of alloys of  Copper + Nickle ( 55 : 45 ).  Nickle + Chromium + Iron. Advantages of strain gauges :  We can use strain gauges after cementing them to structural components. We can use them for a longer time.  Remote reading is possible.  The same strain meter can be used to measure the strains at 20 to 30 different places.  It can measure static and dynamic strains.  Strain can be measured in any required direction.  Micro strain measurement is possible. Disadvantages :  It can not be used immediately after cementing it as curing is needed. Curing :It is to put something in favourable conditions to gain strength.  It measures only surface strain.  We can not use them beyond the elastic limit of the material of the strain gauge.  It is very much sensitive to change in temperature.  Sensitive to change in humidity Construction of strain gauges :  Length of strain gauges varies from 2-25 mm.  Gauges for compression members varies from 2-4 mm.  Whereas gauges for tension members ranges from 10-25 mm. Diameter of strain gauge wire is 0.025 mm.  Range of strain gauge resistance is 60-10000 ohm.  Range of safe current is 25-50 mAmp.  Range of safe voltage is 35-50 volts
17 | P a g e SAQIB IMRAN 0341-7549889 17 Cementing procedure of strain gauges :  Clean oil, grease or dust from the surface usually from sand paper. The paint should be removed through sand paper and then paste.  Make the surface rough for good bond. Structural material and strain gauge should have proper bond. If there will be slippage then some of the strain will not be determined.  Clean oil, grease and dust from the backside of the strain gauge with tissue paper.  Apply paste/cementing material on the structural component e.g nitrocellulose and epoxy resins.  Paste strain gauge very carefully in proper direction and proper location by pressing strain gauge from one end to other end using finger. So that the surplus cementing material should be taken out.  Apply four coats of water proofing agent to avoid from temperature and humidity effects.  Give sufficient time for curing, normally 1-2 days.  Connect strain gauge with wheat stone bridge circuit. Definition of Principal stresses | Principal plane | Principal Axis  Principal stresses : Principal stresses may be defined as ” The extreme values of the normal stresses possible in the material.” These are the maximum normal stress and the minimum normal stress. Maximum normal stress is called major principal stress while minimum normal stress is called minor principal stress.  Principal plane : Principal plane may be defined as ” The plane on which normal stress attains its maximum and minimum value.” So these planes are also called as major principal plane and minor principal plane. The shear stress on principal plane is zero.  Principal Axis :
18 | P a g e SAQIB IMRAN 0341-7549889 18 Principal axis may be defined as ” The axis where moment of inertia is maximum or minimum and product of inertia is zero.” Note :  Principal plane is the property of stress whereas principal axis is the property of moment of inertia. There is no relation between principal axis and principal plane. Maximum Principal strain theory or Saint Venant theory with equation Maximum principal strain theory : This theory states that “ Yielding in a member subjected to complex/compound stress system commences when maximum principal strain of the system becomes equal to the strain corresponding to the yield point under simple tension test of the same material.” For a 3 – dimensional state of stress system
19 | P a g e SAQIB IMRAN 0341-7549889 19 For a 2 – dimensional state of stress system
20 | P a g e SAQIB IMRAN 0341-7549889 20 Where  є1 = maximum principal strain.  υ = Poisson’s ratio.  σ1= Major principal stress in x direction.  σ2 = Minor principal stress in y direction.  σ3 = Minor principal stress in z direction.  σyt= Maximum Tensile strength in yielding.  σyc =Maximum Compressive strength in yielding. Merits :  It computes the failure load based on all 3 values of principal stresses. De merits :  This theory gives unsatisfactory results for various materials.  This theory is not much used for design purposes.  This is not a very popular theory. Maximum principal stress theory of failure with equations February 5, 2012 by admin Leave a Comment  Maximum principal stress theory : This theory states that “A structural component will fail when maximum principal stress of the system will become equal to the yield strength of same material in a simple tension test.”  3-d equations for ductile materials: σ1=σyt
21 | P a g e SAQIB IMRAN 0341-7549889 21 σ3=σyc  2-d equations for ductile materials: σ1=σyt σ2=σyc  3-d Equation for brittle materials: σ1= σut σ3= σ uc  2-d equation for brittle materials: σ1= σ ut σ2= σ uc where  σ1 = Major principal stress in x direction.  σ2 = Minor principal stress in y direction.  σ3 = Minor principal stress in z direction.  σyt= Maximum Tensile strength in yielding.  σyc =Maximum Compressive strength in yielding.  σut= Ultimate tensile strength.  σut = Ultimate compressive strength. This theory gives satisfactory results for brittle materials because brittle materials fails in tension. It gives unsatisfactory results for ductile materials because ductile materials fail in maximum shear stress. This hypothesis, proposed by Rankine, which was also intended for use to predict yielding of a ductile material. Merits :
22 | P a g e SAQIB IMRAN 0341-7549889 22  Calculations are easy.  Gives satisfactory results for brittle materials. De-merits :  It computes failure load based only on values of principal stress.  Gives unsatisfactory results for ductile materials. Shear stress is zero at principal plane Proof of the statement “Shear stress is zero at principal plane.” At principal plane shear stress is zero. There is maximum direct or normal stress but no shear stress at principal plane.
23 | P a g e SAQIB IMRAN 0341-7549889 23 Derivation of the angle at which shear stress is maximum: Shear stress is maximum at 45 degree from the principal plane. Below is the derivation of the angle at which shear stress is maximum. Principal plane: At principal plane direct or normal stress is maximum and there is no shear stress on principal plane. Derivation:  Shear stress equation on an oblique section of body:
24 | P a g e SAQIB IMRAN 0341-7549889 24 Below is the derivation of the shear stress equation on an oblique section of a body in terms of co-ordinate stresses.
25 | P a g e SAQIB IMRAN 0341-7549889 25
26 | P a g e SAQIB IMRAN 0341-7549889 26 Derivation of maximum normal stress at principal plane To find the maximum value of normal stress at some angle, we have to differentiate the normal stress transformation equation, (which we derived in the previous article) with respect to dθ. Principal Plane : It is the plane at which normal stress attains the maximum value. Below is the derivation of maximum normal stress at principal plane.
27 | P a g e SAQIB IMRAN 0341-7549889 27
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30 | P a g e SAQIB IMRAN 0341-7549889 30 Normal Stress transformation equation: Below is the derivation of normal stress transformation equation when the axis is inclined at some angle. Sign conventions:  For normal stresses: We will show tension with positive sign and compression with negative sign.  For shear stresses: If vertical couple is Counter clockwise then (+) sign and if vertical couple is Clockwise (-) sign will use. If horizontal couple is Counter clockwise then (-) sign and if horizontal couple is clockwise then + sign.
31 | P a g e SAQIB IMRAN 0341-7549889 31
32 | P a g e SAQIB IMRAN 0341-7549889 32 Plane stress | Plain strain | Normal or Direct stress | Induced stress Plane stress :
33 | P a g e SAQIB IMRAN 0341-7549889 33 The plane stress is the case in which stress is zero in one direction. In this case stress is in 3 directions. Plane strain: In plane strain case, strain is in 3 directions and in one direction strain value is zero. Normal or Direct stress : The stress which is applied by us intentionally is known as direct or normal stress. Induced stress : This is the stress which we do not want to produce but is induced due to direct or normal stress. Direct & Shear Stress on a plane inclined to the direction of loading: If a piece of material is cut along a plane inclined at some angle θ to the direction of loading, then in order to maintain equilibrium, a system of forces would have to be applied to the plane. In figure 1 below, a bar is subjected to an axial tensile force P. The area of cross section normal to the axis of the bar is A. If the bar is cut along the plane BC, inclined at an angle θ, then the force P applied to this portion of the bar can be reacted by the component forces N normal and T tangential to the plane BC. Below is the derivation of the Direct and normal stress on a plane inclined to the direction of loading.
34 | P a g e SAQIB IMRAN 0341-7549889 34
35 | P a g e SAQIB IMRAN 0341-7549889 35 Conclusion:
36 | P a g e SAQIB IMRAN 0341-7549889 36 From the above we conclude that on the plane, where direct stress is maximum, shear stress is zero.
37 | P a g e SAQIB IMRAN 0341-7549889 37
38 | P a g e SAQIB IMRAN 0341-7549889 38 Strength of Materials What is Beam A structural member acted upon by a system of external forces at right angle to its axis is called beam. The beams may be classified :- 1. Cantilever : A beam whose one end is fixed and other end being free is called cantilever. 2. Simply supported beam : A beam whose both the ends are resting on supports,which may be walls or columns is called freely supported beam. 3. Encastre beam : A beam whose ends are built in into the walls is called encastre or built in beam. 4. Continuous beam : A beam which is continuous over intermediate supports is called continuous beam. Define Columns and struts A member subjected to compressive stress is termed as post, strut or pillar. Main supporting members of building are generally called columns o pillars. The load carried by short column is P = p X A where, p is the intensity of stress and A is the area of cross-section of the column. A column that fails primarily due to direct stress is called a short column and the one that fails primarily due to the bending stress is called the long column. Equivalent lengths for various end conditions
39 | P a g e SAQIB IMRAN 0341-7549889 39 Define Hookes law Under the elastic limit stress is directly proportional to strain. Stress ∝ Strain Stress = E strain E = Stress / Strain where, E is a constant and called modulus of elasticity. What is Modulus of rigidity
40 | P a g e SAQIB IMRAN 0341-7549889 40 It is defined as the ratio of tangential stress to shearing strain. Modulus of rigidity G or N = Stress / Strain = F/A/ θ Define Poissons ratio The ratio of lateral strain to longitudinal strain is called Poisson s ratio and denoted by 1/m. Define Rankines formula For long column with both ends hinged carrying a load W is given by where, fc is crushing stress k = Least value of radius of gyration a = Rankine's constant. What is Shear centre It is the point of intersection of bending axis and the plane of transverse section of a beam. The beams composed of thin sections generally are rolled I-sections, angles, channels etc. Therefore load must act through the shear centre to avoid twisting of the beam. Shear stress The shear stress is produced when the body is subjected to two equal and opposite forces P and tends to slide two adjacent layers. The intensity of shear stress q = P/A Explain Springs Spring is a device in which the material is arranged in such a way that it can undergo considerable change without being permanently distorted. Spring are of three types :- 1. Laminated spring 2. Closely coiled helical spring 3. Open coiled helical spring What is Stress When a body is loaded, the restoring force per unit area, comes into play inside the body is called stress. Its unit is Pascal.
41 | P a g e SAQIB IMRAN 0341-7549889 41 Stress = Force applied / Area of cross-section = P/A. Define Temperature stress When expansion or contraction of a body due to a change in temperature is prevented thermal stress and thermal strain is produced. Temperature stress = E.α.t where E = Modulus of elasticity α = Coefficient of thermal expansion t = Change in temperature Testing of Materials Different tests and methods used for testing materials are :- 1. Tensile test machine : It consists of a mean of applying load to specimen, a device to measure the applied load and accessories for griping the test piece, recording the load and power unit etc. 2. Tensile test of ductile metals 3. Percentage elongation 4. Tensile test of brittle metals 5. Compression test : This test is opposite to tensile test with respect to the direction of the applied stress. 6. Shear test : A force acting parallel to the plane produces shear stress which is of two types : (a) Direct shear (b) Torsional shear. 7. Hardness test : This test is carried out by pressing a loaded diamond against the surface under test. 8. Fatigue test : This test is carried out in parts subjected to continuously varying stresses either in magnitude or from one kind stress to the reversed kind. 9. Impact test : This load applied suddenly on a machine component is called impact blow. The impact load may be applied in flexure, tension, compression or torsion. Test enables to determination of toughness. Thin walled cylinder Thin walled cylindrical vessels are used to store fluids under pressure upto less than 2500 kg/cm2 . Beyond this pressure vessels used in a way. The thickness of thin walled cylinder is about d/20 cm. Due to internal pressure in the thin walled cylinder the three types of stresses may be observed 1. Hoop or circumferential stress acting across the longitudinal section. 2. Longitudinal stress acting across the section perpendicular to the longitudinal section. 3. Radial stress acting radially, being too small and can be ignored. These three types of stresses are mutually perpendicular to the area. Define Youngs modulus
42 | P a g e SAQIB IMRAN 0341-7549889 42 It is defined as the ratio of stress to longitudinal strain. Young's modulus (E) = Stress / Strain = F/A / δl/l Define Bulk modulus It is defined as the ratio of stress to volume strain. Bulk modulus = Stress / Strain = p / Δv / V Direct stress When the section is subjected to axial pull or push acting normally across the section the stress so produced is direct stress. Euler's Column Formulae What is Load A load on a material is the combination of external forces acting on it. When it is constant then called dead load and when it is variable then called as live load. Mohr's theorem (a) The angle in radians between the tangents to the elastic curve at two points on a straight member under bending is equal to the area between those two points. (b) The deflection of a point on a straight member under bending in the direction perpendicular to the original straight axis of the member measured from the
43 | P a g e SAQIB IMRAN 0341-7549889 43 tangent at another point on the member, is equal to the member between those two points, where the deflection occurs. Different Properties of Materials The following are the main properties of materials :- 1. Hardness : It is the property which enables the material to resist scratching, wear and abrasion, penetration by hard objects. 2. Malleability : It is the property which enables the material to undergo great changes in size without rupture under compressive stress. 3. Ductility : It is the property due to which material can be drawn into wires. 4. Plasticity : The property of the material by virtue of which it can not return to its original position when the load is removed, is known as plasticity. In this case there is a permanent deformation in the material. 5. Elasticity : It is the property of material by virtue of which material springs back to its original position with in certain limit (Elastic limit) with the removal of external load. 6. Brittleness : It is the property of the material due to which it breaks under load without permanent distortion. 7. Toughness : It is the property of the material enabling it to resist fracture due to high impact loads like hammer blows. Relationship between elastic constants What is Shear flow The shear flow at a point in the section is the force per unit length, and is expressed by  t = VQ / I
44 | P a g e SAQIB IMRAN 0341-7549889 44 where,  = Shear stress t = Thickness V = Total vertical load Q = a y analogous I M.O.I. of cross-section about N.A. Simple bending A beam or a cantilever is said to be subjected to simple bending when it bends under the action of uniform bending moment action of any load. Assumptions in theory of simple bending :- 1. Plane of transverse sections of the beam, remain plane even after bending. 2. The material of the beam always obeys Hooke's law and is assumed homogeneous. 3. The beam is initially straight of constant cross-section. 4. The moduli of elasticity for tension and compression are same. 5. The stresses in the beam do not exceed elastic limit. 6. The beam bends to a circular arc. 7. The radius of curvature is large compared to the section dimensions. 8. The loads act perpendicular to the beam axis. What is Strain It is defined as the ratio of change in length, volume, or shape to the original length, volume or shape. It is dimensionless. It is of three types :- 1. Longitudinal strain = Change in length / Original length 2. Volume strain = Change in volume / Original volume 3. Shearing strain = Change in shear Stress on inclined plane Stress on a plane inclined through O to the cross-section carrying a load P. Normal stress Pn = P cos2 θ Tangential stress Pi = P sin 2θ / 2 Thick walled cylinders Thick wall cylindrical vessels are used to store fluids under pressure more than 2500 kg/cm2 . A thick walled cylinder is subjected to the following three principal stresses 1. The radial pressure p 2. The hoop stress sfn 3. The longitudinal tensile stress P
45 | P a g e SAQIB IMRAN 0341-7549889 45 Ultimate stress The maximum stress which a material is subjected to before rupture is called ultimate stress.

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Mechanics of solids pdf

  • 1. 1 | P a g e SAQIB IMRAN 0341-7549889 1 Assala mu alykum My Name is saqib imran and I am the student of b.tech (civil) in sarhad univeristy of science and technology peshawer. I have written this notes by different websites and some by self and prepare it for the student and also for engineer who work on field to get some knowledge from it. I hope you all students may like it. Remember me in your pray, allah bless me and all of you friends. If u have any confusion in this notes contact me on my gmail id: Saqibimran43@gmail.com or text me on 0341-7549889. Saqib imran.
  • 2. 2 | P a g e SAQIB IMRAN 0341-7549889 2 Mechanics of Solids What is Solid Mechanics | Strength of Materials | Purpose of Studying SOM Solid mechanics or Mechanics of solids Solid mechanics or Mechanics of solids is the branch of Mechanics, Physics, Mathematics that concerns the behavior of solid matter under external actions (e.g., external forces, temperature changes, applied displacements, etc.). Or It is a branch of science which deals with the internal effects of the forces on the bodies when they are loaded, from initial point to rupture or break. Strength of Materials Strength of materials deals with the relations between externally applied loads and their internal effects on the bodies. Purpose of studying strength of materials To ensure that the structure used will be safe against the maximum internal effects that may be produced by any combination of external loading. Types of forces Forces that are acting on the body, may be of the following types. Effects of these forces are studied in solid mechanics or mechanics of solids. Normal forces The forces that are acting perpendicular to the body are known as normal forces. Normal forces are of two types: 1. Tensile 2. Compressive.
  • 3. 3 | P a g e SAQIB IMRAN 0341-7549889 3 Tangential forces The forces that are acting parallel to the body are known as tangential forces. Shear forces and frictional forces are example of tangential forces. Independent forces Active forces are independent forces. Dependent forces Reactive forces are dependent forces. Concentrated forces If forces are acting on an area of a body which is negligible as compared to the body, then these are known as concentrated forces. Distributed forces If forces are acting on an area of a body which is not negligible as compared to the body, then these are known as distributed forces. These may be of uniformly distributed load (UDL) or Uniformly varying load (UVL) or any kind of general loading. Effects of forces The effects of forces which are studied in solid mechanics or mechanics of solids are 1. Translation 2. Rotation (moment and torque) What is Stress | Yield stress | Compressive stress | Strain Stress is a feeling that’s created when we react to particular events. Stress may refer to: Mechanical, Biological, Music, other. Stress (physics): The average amount of force exerted per unit area. Yield stress: The stress at which a material begins to deform plastically.
  • 4. 4 | P a g e SAQIB IMRAN 0341-7549889 4 Compressive stress: The stress applied to materials resulting in their compaction. Strain: It is the deformation of materials caused by the action of stress. Strain is calculated by first assuming a change between two body states: the beginning state and the final state. Then the difference in placement of two points in this body in those two states expresses the numerical value of strain. Strain therefore expresses itself as a change in size and/or shape. If strain is equal over all parts of a body, it is referred to as homogeneous strain; otherwise, it is in homogeneous strain Stress Strain Curve: During testing of a material sample, the stress–strain curve is a graphical representation of the relationship between stress, and strain, i.e. elongation, compression, or distortion. The nature of the curve varies from material to material. Tensile Strength | Yield Strength | Ultimate Strength | Breaking Strength Tensile Strength Tensile strength is the stress at the maximum on the stress-strain curve. Or The greatest longitudinal or axial stress a material can bear without breaking. Tensile strength is the maximum amount of tensile stress that a material can take before failure. There are three definitions of tensile strength  Yield strength  Ultimate strength  Breaking strength Yield Strength The stress at which material strain changes from elastic deformation to plastic deformation, causing it to deform permanently. This point is not a well defined point. We
  • 5. 5 | P a g e SAQIB IMRAN 0341-7549889 5 can say that yield strength is a strength which a material can bear without permanent deformation. Yield strength of Structural steel A36 is 250 MPa. Ultimate Strength The maximum stress a material can withstand. Ultimate tensile strength of Structural steel A36 is 400 MPa. Breaking Strength The stress coordinate on the stress-strain curve at the point of rupture. Ductile and Brittle Materials | Difference Between Ductile and Brittle Material Ductile Materials:  Ductile materials will withstand large strains before the specimen ruptures.  Ductile materials often have relatively small Young’s moduli and ultimate stresses.  Ductile materials exhibit large strains and yielding before they fail.  Steel and aluminum usually fall in the class of Ductile Materials Brittle Materials:  Brittle materials fracture at much lower strains.  Brittle materials often have relatively large Young’s moduli and ultimate stresses.  Brittle materials fail suddenly and without much warning.  Glass and cast iron fall in the class of Brittle Materials. Youngs Modulus and its Units | Proportional Limit | Ultimate Stress | Rupture Stress Modulus of Elasticity: In solid mechanics, Young’s modulus (E) is a measure of the stiffness of a material.It is defined as the ratio of stress over strain in the region in which Hooke’s Law is obeyed for the material.
  • 6. 6 | P a g e SAQIB IMRAN 0341-7549889 6 Units Young’s modulus is the ratio of stress, which has units of pressure to strain, which is dimensionless; therefore Young’s modulus itself has units of pressure. Proportional Limit : The point up to which the stress and strain are linearly related is called the proportional limit. Ultimate Stress : The largest stress in the stress strain curve is called the ultimate stress. Rupture Stress : The stress at the point of rupture is called the fracture or rupture stress. Elastic Region | Plastic Region | Yield Point | Plastic Strain | Strain Hardening | Necking Elastic Region: The region of the stress-strain curve in which the material returns to the undeformed state when applied forces are removed is called the elastic region. Plastic Region: The region in which the material deforms permanently is called the plastic region. Yield Point: The point demarcating the elastic from the plastic region is called the yield point. The stress at yield point is called the yield stress. Plastic Strain: The permanent strain when stresses are zero is called the plastic strain. The off-set yield stress is a stress that would produce a plastic strain corresponding to the specified off-set strain.
  • 7. 7 | P a g e SAQIB IMRAN 0341-7549889 7 A material that can undergo large plastic deformation before fracture is called a ductile material. A material that exhibits little or no plastic deformation at failure is called a brittle material. Hardness is the resistance to indentation. The raising of the yield point with increasing strain is called strain hardening. Necking Phenomenon: The sudden decrease in the area of cross-section after ultimate stress is called necking. Modulus of Resilience | Modulus of Toughness | Poisson’s Ratio Modulus of Resilience The work done on a unit volume of material, as a simple tensile force is gradually increased from zero to such a value that the proportional limit of the material is reached, is defined as the modulus of resilience. This may be calculated as the area under the stress-strain curve from the origin up to the proportional limit and is represented as the shaded area in the figure below. Modulus of Toughness The work done on a unit volume of material as a simple tensile force is gradually increased from zero to the value causing rupture is defined as the modulus of toughness. Poisson’s Ratio When a bar is subject to a simple tensile loading there is an increase in length of the bar in the direction of the load, but a decrease in the lateral dimensions perpendicular to the load. The ratio of the strain in the lateral direction to that in the axial direction is defined as Poisson’s ratio.
  • 8. 8 | P a g e SAQIB IMRAN 0341-7549889 8 Classification of Materials | Isotropic Material | Homogeneous Material | Specific Strength and Its Units Classification of Materials An isotropic material: Has a stress-strain relationship that is independent of the orientation of the coordinate system at a point. An isotropic material is the one having the same elastic properties (E,µ) in all directions at any one point of the body. A Homogeneous Material: A material is said to be homogeneous if the material elastic properties (E,µ) are the same at all points in the body. Specific Strength: This quantity is defined as the ratio of the ultimate (or tensile) strength to specific weight, i.e., weight per unit volume. we have And, In the SI system, we have So that in either system specific strength has units of length. This parameter is useful for comparisons of material efficiencies. Definition of Shear Force and Bending Moment Shear Force: The Shear Force at the cross section of the beam may be defined as the unbalanced vertical force to the right or left of the section. Bending Moment:
  • 9. 9 | P a g e SAQIB IMRAN 0341-7549889 9 The Bending Moment at the cross section of the beam may be defined as the algebraic sum of the moments of the forces, to the right or left of the section. Important Note: While considering the shear force and bending moment at the section , the end reactions must also be considered along with other external loads. Direct Stresses | Shearing Stresses | Tensile Stresses | Compressive Stresses Direct Stresses : The Stresses which are acting normal to the plane of the body are called as normal or direct stresses. These are called normal because these are acting perpendicular to the plane of the body. Shearing Stresses : The Stresses which act parallel to the stressed surfaces are called as shearing stresses. Tensile Stresses : If a straight bar is subjected to a pair of collinear forces acting in opposite direction and coinciding with the ends of the bar and directed away from the bar, then the bar tends to increase in length and the stresses developed in the bar will be tensile. Compressive Stresses : If a straight bar is subjected to a pair of collinear forces acting in opposite direction and coinciding with the ends of the bar and directed towards the bar, then the bar tends to shorter in length and the stresses developed will be compressive stresses. Definition of Column | Difference between strut and column | Type of Failure in Columns Definition of Column A long slender bar subjected to axial compression is called a column.
  • 10. 10 | P a g e SAQIB IMRAN 0341-7549889 10 The term is frequently use to describe a vertical member. Sometimes direct stresses dominate and sometimes flexural stresses dominate. Axial Compression means the compressive forces act at the two ends of the member in the opposite direction are along the same axis . Difference between strut and column The difference between column and strut is that former is used to describe a vertical member whereas latter is used for the inclined members. Strut is any inclined member subjected to compression loads. Types of Columns Based upon mode of failure, there are three types. Short Column The failure initiates due to crushing of material and direct stresses are dominant. For short column, if L < 4d and KL/r min < 30 Where d = least lateral dimension. L = Unbraced length of the column. k = effective length factor depends upon the end conditions of the column. r min = least radius of gyration. Slender or long Column In these, failure initiates due to lateral buckling and flexural stresses are dominant. If L > 30d or
  • 11. 11 | P a g e SAQIB IMRAN 0341-7549889 11 KL/r min > critical slenderness ratio. Intermediate Column Failure initiates due to lateral buckling as well as crushing of the material. Direct and flexural stresses both are dominant. Column is intermediate when 4d < L < 30d and 30 < KL /r min < 100 or Critical slenderness ratio. There is no agreed equation for intermediate column. There are many approaches that are available for design and analysis of intermediate columns such as: 1. Rankine Gordon approach. 2. Tangent Modulus formula. 3. Straight line formula. 4. Parabolic formula. 5. AISC equations. Remember all these are empirical approaches. Type of Failure in Columns Long compression members fails due to buckling whereas short compression member may fails due to yielding of material. Buckling of a compression member may occur even the maximum stresses in the material are less than the yield point of the material . By Buckling of a compression member it means lateral deflection of the bar. Definition of critical load of a Column | Slenderness ratio of a Column Critical load :The critical load of a column is that value of axial force that is just sufficient to keep the bar in a slightly deflected shape. Slenderness ratio of a column : The ratio of effective length of the column (an axial compression member) to the minimum radius of gyration of the cross-sectional area is termed as slenderness ratio of the bar.
  • 12. 12 | P a g e SAQIB IMRAN 0341-7549889 12 As it is a ratio so it has no units and of course it is dimensionless. Note : Radius of gyration is always the property of cross-section that is why the word cross- section is used with radius of gyration. Definition of a Beam | Cantilever Beams | Simple Beams Definition of a Beam: A bar subjected to forces and couples that lie in a plane containing its longitudinal axis is called a beam. The Forces will act perpendicular to the longitudinal axis of the bar. Cantilever Beams : If a beam is rigidly supported at only one end in such a manner that the beam cannot rotate at that point, it is called as cantilever beam. The other end of the beam is free to deflect but from the end where it is fixed, it cannot rotate. The end which is fixed is said to be ” restrained.” The reaction of the beam at the fixed end consists of a vertical force and a couple acting in the plane of the applied loads. Simple Beams : A beam which is freely supported at both ends is known as simple beam. The term “freely supported” implies that the end supports of the beam are capable of exerting forces on the beam not any moment. So in simple beams no “restrained” is offer to the angular rotation as the beam deflects under the loads. Overhanging Beams | Statically determinate Beams | Statically Indeterminate Beams Overhanging Beam : A beam which is freely supported at two points and having one or both the ends extending beyond these supports is known as overhanging beam.
  • 13. 13 | P a g e SAQIB IMRAN 0341-7549889 13 Statically Determinate Beams : All the beams in which reactions of the supports may be determined by the use of equations of static equilibrium are known as statically determinate beams. The value of these reactions are independent of the deformation of the beam. Statically indeterminate Beams : All the beams in which the number of reactions exceeds than the total number of equations of static equilibrium are known as statically indeterminate beams. In these beams the static equations must be supplemented by the equation based upon the deformation of the beam. Deflection of a Beam | Slope of a Beam | Flexural Rigidity of Beam Deflection of a Beam : The deflection at any point on the axis of the beam is the distance between its position before and after loading. Slope of a Beam : Slope at any section in a deflected beam is defined as the angle in radians which the tangent at the section makes with the original axis of the beam. Flexural Rigidity of Beam : The Product ” EI” is called flexural rigidity of the beam and is usually constant along the beam. Theorem 1 of Moment Area Method Theorem 1 is stated as follows : ” The change in slope between tangents drawn to the elastic curve at any two points A and B is equal to the product of 1/EI and the area of the moment diagram between these two points.”
  • 14. 14 | P a g e SAQIB IMRAN 0341-7549889 14 Mathematically, it can be written as For simplification we can say first draw the elastic curve diagram of the beam, then draw tangents at those two points for which change in slope has to be determined ( in the above diagram these points are A and B ) . Then Θab is the change in slope between tangents drawn to the elastic curve at these two points A and B. Theorem 2 of Moment Area Method It is stated as follows : ” The deviation of any point B relative to a tangent drawn to the elastic curve at any other point A in a direction perpendicular to the original position of the beam, is equal to the product of 1/EI and the moment of area about b of that part of the moment diagram between points A and B.” Mathematically it can be written as Similarly Where Āb and Āa are the distances from center of gravity of the moment diagram to the point B and A respectively. For simplification we can say that Ā is the distance from center of gravity of the moment diagram to the point where we have to find deviation. For instance we are finding the deviation at point B then Ā will be the distance from center of gravity of the moment diagram to the point B, because we are finding deflection at B. Same is the case with any other point such as A.
  • 15. 15 | P a g e SAQIB IMRAN 0341-7549889 15 Definition of Thin-walled open sections Thin-walled open sections Thin-walled open sections may be defined as The sections in which the thickness of material is small as compared with the overall geometry of the material and there is only one or no axis of symmetry. These sections are widely used. Their main use is in air craft industry. Common examples are channel section, top hat, zed, angle and circular sections. Within the section, shear stress variation is almost constant and we can assume shear stress constant. Definition of shear Center Shear Center Shear center is defined as the point on the beam section where load is applied and no twisting is produced. At shear center, resultant of internal forces passes. On symmetrical sections, shear center is the center of gravity of that section. In unsymmetrical sections, if the external applied forces act through the centroid of the section, then in addition to bending, twisting is also produced. To avoid twisting, and cause only bending, it is necessary for the forces to act through the particular point, which may not coincide with the centroid. The position of the this point is a function only of the geometry of the beam section. It is termed as shear center. Such sections in which there is a sliding problem, we place loads at the shear center. Strain measuring instruments | Strain Gauges Strain measuring instruments : These are of the following types :
  • 16. 16 | P a g e SAQIB IMRAN 0341-7549889 16  Mechanical.  Electrical. (Strain gauges)  Photoelectrical.  Basic working principle of strain gauges : In 1856, Kelvin observed that resistance of the conductor is directly proportional to the length and inversely proportional to the area of the conductor. This principle was used in 1933 by defense department of US. They form gauges of alloys of  Copper + Nickle ( 55 : 45 ).  Nickle + Chromium + Iron. Advantages of strain gauges :  We can use strain gauges after cementing them to structural components. We can use them for a longer time.  Remote reading is possible.  The same strain meter can be used to measure the strains at 20 to 30 different places.  It can measure static and dynamic strains.  Strain can be measured in any required direction.  Micro strain measurement is possible. Disadvantages :  It can not be used immediately after cementing it as curing is needed. Curing :It is to put something in favourable conditions to gain strength.  It measures only surface strain.  We can not use them beyond the elastic limit of the material of the strain gauge.  It is very much sensitive to change in temperature.  Sensitive to change in humidity Construction of strain gauges :  Length of strain gauges varies from 2-25 mm.  Gauges for compression members varies from 2-4 mm.  Whereas gauges for tension members ranges from 10-25 mm. Diameter of strain gauge wire is 0.025 mm.  Range of strain gauge resistance is 60-10000 ohm.  Range of safe current is 25-50 mAmp.  Range of safe voltage is 35-50 volts
  • 17. 17 | P a g e SAQIB IMRAN 0341-7549889 17 Cementing procedure of strain gauges :  Clean oil, grease or dust from the surface usually from sand paper. The paint should be removed through sand paper and then paste.  Make the surface rough for good bond. Structural material and strain gauge should have proper bond. If there will be slippage then some of the strain will not be determined.  Clean oil, grease and dust from the backside of the strain gauge with tissue paper.  Apply paste/cementing material on the structural component e.g nitrocellulose and epoxy resins.  Paste strain gauge very carefully in proper direction and proper location by pressing strain gauge from one end to other end using finger. So that the surplus cementing material should be taken out.  Apply four coats of water proofing agent to avoid from temperature and humidity effects.  Give sufficient time for curing, normally 1-2 days.  Connect strain gauge with wheat stone bridge circuit. Definition of Principal stresses | Principal plane | Principal Axis  Principal stresses : Principal stresses may be defined as ” The extreme values of the normal stresses possible in the material.” These are the maximum normal stress and the minimum normal stress. Maximum normal stress is called major principal stress while minimum normal stress is called minor principal stress.  Principal plane : Principal plane may be defined as ” The plane on which normal stress attains its maximum and minimum value.” So these planes are also called as major principal plane and minor principal plane. The shear stress on principal plane is zero.  Principal Axis :
  • 18. 18 | P a g e SAQIB IMRAN 0341-7549889 18 Principal axis may be defined as ” The axis where moment of inertia is maximum or minimum and product of inertia is zero.” Note :  Principal plane is the property of stress whereas principal axis is the property of moment of inertia. There is no relation between principal axis and principal plane. Maximum Principal strain theory or Saint Venant theory with equation Maximum principal strain theory : This theory states that “ Yielding in a member subjected to complex/compound stress system commences when maximum principal strain of the system becomes equal to the strain corresponding to the yield point under simple tension test of the same material.” For a 3 – dimensional state of stress system
  • 19. 19 | P a g e SAQIB IMRAN 0341-7549889 19 For a 2 – dimensional state of stress system
  • 20. 20 | P a g e SAQIB IMRAN 0341-7549889 20 Where  є1 = maximum principal strain.  υ = Poisson’s ratio.  σ1= Major principal stress in x direction.  σ2 = Minor principal stress in y direction.  σ3 = Minor principal stress in z direction.  σyt= Maximum Tensile strength in yielding.  σyc =Maximum Compressive strength in yielding. Merits :  It computes the failure load based on all 3 values of principal stresses. De merits :  This theory gives unsatisfactory results for various materials.  This theory is not much used for design purposes.  This is not a very popular theory. Maximum principal stress theory of failure with equations February 5, 2012 by admin Leave a Comment  Maximum principal stress theory : This theory states that “A structural component will fail when maximum principal stress of the system will become equal to the yield strength of same material in a simple tension test.”  3-d equations for ductile materials: σ1=σyt
  • 21. 21 | P a g e SAQIB IMRAN 0341-7549889 21 σ3=σyc  2-d equations for ductile materials: σ1=σyt σ2=σyc  3-d Equation for brittle materials: σ1= σut σ3= σ uc  2-d equation for brittle materials: σ1= σ ut σ2= σ uc where  σ1 = Major principal stress in x direction.  σ2 = Minor principal stress in y direction.  σ3 = Minor principal stress in z direction.  σyt= Maximum Tensile strength in yielding.  σyc =Maximum Compressive strength in yielding.  σut= Ultimate tensile strength.  σut = Ultimate compressive strength. This theory gives satisfactory results for brittle materials because brittle materials fails in tension. It gives unsatisfactory results for ductile materials because ductile materials fail in maximum shear stress. This hypothesis, proposed by Rankine, which was also intended for use to predict yielding of a ductile material. Merits :
  • 22. 22 | P a g e SAQIB IMRAN 0341-7549889 22  Calculations are easy.  Gives satisfactory results for brittle materials. De-merits :  It computes failure load based only on values of principal stress.  Gives unsatisfactory results for ductile materials. Shear stress is zero at principal plane Proof of the statement “Shear stress is zero at principal plane.” At principal plane shear stress is zero. There is maximum direct or normal stress but no shear stress at principal plane.
  • 23. 23 | P a g e SAQIB IMRAN 0341-7549889 23 Derivation of the angle at which shear stress is maximum: Shear stress is maximum at 45 degree from the principal plane. Below is the derivation of the angle at which shear stress is maximum. Principal plane: At principal plane direct or normal stress is maximum and there is no shear stress on principal plane. Derivation:  Shear stress equation on an oblique section of body:
  • 24. 24 | P a g e SAQIB IMRAN 0341-7549889 24 Below is the derivation of the shear stress equation on an oblique section of a body in terms of co-ordinate stresses.
  • 25. 25 | P a g e SAQIB IMRAN 0341-7549889 25
  • 26. 26 | P a g e SAQIB IMRAN 0341-7549889 26 Derivation of maximum normal stress at principal plane To find the maximum value of normal stress at some angle, we have to differentiate the normal stress transformation equation, (which we derived in the previous article) with respect to dθ. Principal Plane : It is the plane at which normal stress attains the maximum value. Below is the derivation of maximum normal stress at principal plane.
  • 27. 27 | P a g e SAQIB IMRAN 0341-7549889 27
  • 28. 28 | P a g e SAQIB IMRAN 0341-7549889 28
  • 29. 29 | P a g e SAQIB IMRAN 0341-7549889 29
  • 30. 30 | P a g e SAQIB IMRAN 0341-7549889 30 Normal Stress transformation equation: Below is the derivation of normal stress transformation equation when the axis is inclined at some angle. Sign conventions:  For normal stresses: We will show tension with positive sign and compression with negative sign.  For shear stresses: If vertical couple is Counter clockwise then (+) sign and if vertical couple is Clockwise (-) sign will use. If horizontal couple is Counter clockwise then (-) sign and if horizontal couple is clockwise then + sign.
  • 31. 31 | P a g e SAQIB IMRAN 0341-7549889 31
  • 32. 32 | P a g e SAQIB IMRAN 0341-7549889 32 Plane stress | Plain strain | Normal or Direct stress | Induced stress Plane stress :
  • 33. 33 | P a g e SAQIB IMRAN 0341-7549889 33 The plane stress is the case in which stress is zero in one direction. In this case stress is in 3 directions. Plane strain: In plane strain case, strain is in 3 directions and in one direction strain value is zero. Normal or Direct stress : The stress which is applied by us intentionally is known as direct or normal stress. Induced stress : This is the stress which we do not want to produce but is induced due to direct or normal stress. Direct & Shear Stress on a plane inclined to the direction of loading: If a piece of material is cut along a plane inclined at some angle θ to the direction of loading, then in order to maintain equilibrium, a system of forces would have to be applied to the plane. In figure 1 below, a bar is subjected to an axial tensile force P. The area of cross section normal to the axis of the bar is A. If the bar is cut along the plane BC, inclined at an angle θ, then the force P applied to this portion of the bar can be reacted by the component forces N normal and T tangential to the plane BC. Below is the derivation of the Direct and normal stress on a plane inclined to the direction of loading.
  • 34. 34 | P a g e SAQIB IMRAN 0341-7549889 34
  • 35. 35 | P a g e SAQIB IMRAN 0341-7549889 35 Conclusion:
  • 36. 36 | P a g e SAQIB IMRAN 0341-7549889 36 From the above we conclude that on the plane, where direct stress is maximum, shear stress is zero.
  • 37. 37 | P a g e SAQIB IMRAN 0341-7549889 37
  • 38. 38 | P a g e SAQIB IMRAN 0341-7549889 38 Strength of Materials What is Beam A structural member acted upon by a system of external forces at right angle to its axis is called beam. The beams may be classified :- 1. Cantilever : A beam whose one end is fixed and other end being free is called cantilever. 2. Simply supported beam : A beam whose both the ends are resting on supports,which may be walls or columns is called freely supported beam. 3. Encastre beam : A beam whose ends are built in into the walls is called encastre or built in beam. 4. Continuous beam : A beam which is continuous over intermediate supports is called continuous beam. Define Columns and struts A member subjected to compressive stress is termed as post, strut or pillar. Main supporting members of building are generally called columns o pillars. The load carried by short column is P = p X A where, p is the intensity of stress and A is the area of cross-section of the column. A column that fails primarily due to direct stress is called a short column and the one that fails primarily due to the bending stress is called the long column. Equivalent lengths for various end conditions
  • 39. 39 | P a g e SAQIB IMRAN 0341-7549889 39 Define Hookes law Under the elastic limit stress is directly proportional to strain. Stress ∝ Strain Stress = E strain E = Stress / Strain where, E is a constant and called modulus of elasticity. What is Modulus of rigidity
  • 40. 40 | P a g e SAQIB IMRAN 0341-7549889 40 It is defined as the ratio of tangential stress to shearing strain. Modulus of rigidity G or N = Stress / Strain = F/A/ θ Define Poissons ratio The ratio of lateral strain to longitudinal strain is called Poisson s ratio and denoted by 1/m. Define Rankines formula For long column with both ends hinged carrying a load W is given by where, fc is crushing stress k = Least value of radius of gyration a = Rankine's constant. What is Shear centre It is the point of intersection of bending axis and the plane of transverse section of a beam. The beams composed of thin sections generally are rolled I-sections, angles, channels etc. Therefore load must act through the shear centre to avoid twisting of the beam. Shear stress The shear stress is produced when the body is subjected to two equal and opposite forces P and tends to slide two adjacent layers. The intensity of shear stress q = P/A Explain Springs Spring is a device in which the material is arranged in such a way that it can undergo considerable change without being permanently distorted. Spring are of three types :- 1. Laminated spring 2. Closely coiled helical spring 3. Open coiled helical spring What is Stress When a body is loaded, the restoring force per unit area, comes into play inside the body is called stress. Its unit is Pascal.
  • 41. 41 | P a g e SAQIB IMRAN 0341-7549889 41 Stress = Force applied / Area of cross-section = P/A. Define Temperature stress When expansion or contraction of a body due to a change in temperature is prevented thermal stress and thermal strain is produced. Temperature stress = E.α.t where E = Modulus of elasticity α = Coefficient of thermal expansion t = Change in temperature Testing of Materials Different tests and methods used for testing materials are :- 1. Tensile test machine : It consists of a mean of applying load to specimen, a device to measure the applied load and accessories for griping the test piece, recording the load and power unit etc. 2. Tensile test of ductile metals 3. Percentage elongation 4. Tensile test of brittle metals 5. Compression test : This test is opposite to tensile test with respect to the direction of the applied stress. 6. Shear test : A force acting parallel to the plane produces shear stress which is of two types : (a) Direct shear (b) Torsional shear. 7. Hardness test : This test is carried out by pressing a loaded diamond against the surface under test. 8. Fatigue test : This test is carried out in parts subjected to continuously varying stresses either in magnitude or from one kind stress to the reversed kind. 9. Impact test : This load applied suddenly on a machine component is called impact blow. The impact load may be applied in flexure, tension, compression or torsion. Test enables to determination of toughness. Thin walled cylinder Thin walled cylindrical vessels are used to store fluids under pressure upto less than 2500 kg/cm2 . Beyond this pressure vessels used in a way. The thickness of thin walled cylinder is about d/20 cm. Due to internal pressure in the thin walled cylinder the three types of stresses may be observed 1. Hoop or circumferential stress acting across the longitudinal section. 2. Longitudinal stress acting across the section perpendicular to the longitudinal section. 3. Radial stress acting radially, being too small and can be ignored. These three types of stresses are mutually perpendicular to the area. Define Youngs modulus
  • 42. 42 | P a g e SAQIB IMRAN 0341-7549889 42 It is defined as the ratio of stress to longitudinal strain. Young's modulus (E) = Stress / Strain = F/A / δl/l Define Bulk modulus It is defined as the ratio of stress to volume strain. Bulk modulus = Stress / Strain = p / Δv / V Direct stress When the section is subjected to axial pull or push acting normally across the section the stress so produced is direct stress. Euler's Column Formulae What is Load A load on a material is the combination of external forces acting on it. When it is constant then called dead load and when it is variable then called as live load. Mohr's theorem (a) The angle in radians between the tangents to the elastic curve at two points on a straight member under bending is equal to the area between those two points. (b) The deflection of a point on a straight member under bending in the direction perpendicular to the original straight axis of the member measured from the
  • 43. 43 | P a g e SAQIB IMRAN 0341-7549889 43 tangent at another point on the member, is equal to the member between those two points, where the deflection occurs. Different Properties of Materials The following are the main properties of materials :- 1. Hardness : It is the property which enables the material to resist scratching, wear and abrasion, penetration by hard objects. 2. Malleability : It is the property which enables the material to undergo great changes in size without rupture under compressive stress. 3. Ductility : It is the property due to which material can be drawn into wires. 4. Plasticity : The property of the material by virtue of which it can not return to its original position when the load is removed, is known as plasticity. In this case there is a permanent deformation in the material. 5. Elasticity : It is the property of material by virtue of which material springs back to its original position with in certain limit (Elastic limit) with the removal of external load. 6. Brittleness : It is the property of the material due to which it breaks under load without permanent distortion. 7. Toughness : It is the property of the material enabling it to resist fracture due to high impact loads like hammer blows. Relationship between elastic constants What is Shear flow The shear flow at a point in the section is the force per unit length, and is expressed by  t = VQ / I
  • 44. 44 | P a g e SAQIB IMRAN 0341-7549889 44 where,  = Shear stress t = Thickness V = Total vertical load Q = a y analogous I M.O.I. of cross-section about N.A. Simple bending A beam or a cantilever is said to be subjected to simple bending when it bends under the action of uniform bending moment action of any load. Assumptions in theory of simple bending :- 1. Plane of transverse sections of the beam, remain plane even after bending. 2. The material of the beam always obeys Hooke's law and is assumed homogeneous. 3. The beam is initially straight of constant cross-section. 4. The moduli of elasticity for tension and compression are same. 5. The stresses in the beam do not exceed elastic limit. 6. The beam bends to a circular arc. 7. The radius of curvature is large compared to the section dimensions. 8. The loads act perpendicular to the beam axis. What is Strain It is defined as the ratio of change in length, volume, or shape to the original length, volume or shape. It is dimensionless. It is of three types :- 1. Longitudinal strain = Change in length / Original length 2. Volume strain = Change in volume / Original volume 3. Shearing strain = Change in shear Stress on inclined plane Stress on a plane inclined through O to the cross-section carrying a load P. Normal stress Pn = P cos2 θ Tangential stress Pi = P sin 2θ / 2 Thick walled cylinders Thick wall cylindrical vessels are used to store fluids under pressure more than 2500 kg/cm2 . A thick walled cylinder is subjected to the following three principal stresses 1. The radial pressure p 2. The hoop stress sfn 3. The longitudinal tensile stress P
  • 45. 45 | P a g e SAQIB IMRAN 0341-7549889 45 Ultimate stress The maximum stress which a material is subjected to before rupture is called ultimate stress.