Published on December 6, 2013
Structural Integrity Analysis 4. Strength of Materials 4. STRENGTH OF MATERIALS Igor Kokcharov 4.1 TENSION AND COMPRESSION Tension and compression can be characterized by the following parameters: deformation, strain and stress. Stress Stress is the intensity of inner forces and is expressed in force per unit of area (megapascals, pound-force per square inch) and is negative for compression. External forces cause deformation of members. Deformation Deformation is the increase in the original size of a member. It is measured in units of length. Elongation Elongation is the increase in gauge length of a body subjected to a tensile force, referenced to a gauge length on the body. The parameter is considered as strain. Strain Strain is a deformation of the member divided by the original length of the member. According to Hooke's law for elastic deformation, stress is proportional to strain. The ratio of Copyrighted materials 1
Structural Integrity Analysis 4. Strength of Materials elastic stress/strain is a constant of the material. This constant is known as Young's modulus or modulus of elasticity. This parameter has units of stress (MPa, GPa). Polyethylene : Glass : Aluminum : Steel : E E E E = = = = 1 GPa = 60 GPa = 73 GPa = 207 GPa = 1000 MPa 60 000 MPa 73 000 MPa 207 000 MPa For the pressure vessel, the summed force in the bolts is equal to the sum of the force from inner pressures. This condition helps to find the necessary number of bolts n. Long cables are flexible and the primary source of deformation in the cables is the axial. There is a compressive stress in the base due to the weight of a heavy block and its own weight. The figure below shows the optimal shape of the base. For such a design, stresses in the bottom and at the top of the base are equal. Copyrighted materials 2
Structural Integrity Analysis 4. Strength of Materials 4.2 SHEAR AND TORSION Shear stress Shear stress is the stress component tangential to the plane on which the forces act. Shear stress is expressed in shear force per unit of area (megaPascals, pound-force per square inch). Shear stress is greatest in the most remote rivet from the center of the sum area. There is practically no shear stress in the central rivet. Copyrighted materials 3
Structural Integrity Analysis 4. Strength of Materials Torque transforms a square at the cylinder surface into a rhomb. Absolute values of shear stress at all side surfaces of the rhomb are equal. The stress state of pure shear is equivalent to bi-axial tension-compression state. Tensile stress can result in the brittle fracture of shafts under torsion. The figure shows the typical failure of a cylindrical shaft made from brittle material. There are two angles that help to describe torsion: shear angle γ and angle of twist ϕ. The shear angle does not depend on the length of a shaft with constant torque. The longer the shaft, the larger the angle of twist. Copyrighted materials 4
Structural Integrity Analysis 4. Strength of Materials A cross section without a sharp corner corresponds to uniform shear stress and effective use of the material. Rigidity depends on polar moment of inertia J, which is proportional to r4. The larger the radius r, the greater the rigidity. The rigidity of an open thin-walled shell is sufficiently smaller than the rigidity of a closed section. Torsional stress Torsional stress is the shear stress on a transverse cross section resulting from a twisting action, and is at a maximum at the surface. It is equal to zero in the center. The stress is proportional to the applied torque. For a rectangular cross section the maximum shear stress acts along the longer side, closest to the center point. Shear stress is at a maximum for the shaft with the highest torque. The moment is the highest for the shaft with lowest rotation speed - the last shaft in the kinematic chain. Copyrighted materials 5
Structural Integrity Analysis 4. Strength of Materials 4.3 STRESS-STRAIN STATE Normal stress in simple tension is given by s=Force/Area. If we cut the section at an angle ϕ, there are two stress components perpendicular σn and parallel τ to the incline plane. The maximum shear stress occurring at ϕ=45o is equal to half the maximum axial stress σ. For a bi-axial state of stress, normal stress σn and shear stress τ on an inclined plane depend on the two shown stress components. Copyrighted materials 6
Structural Integrity Analysis 4. Strength of Materials For common plane stress state there are two perpendicular planes (principle planes) where there are no shear stress and normal stresses are a minimum and maximum. The two components are known as the principle stresses. Maximum shear stress acts at the planes inclined 45o to principle planes. Hooke's law generally includes two constants of the material: Young's modulus E and Poisson's ratio µ. Copyrighted materials 7
Structural Integrity Analysis 4. Strength of Materials Generally, maximum shear stress depends on two principle stresses only - maximum and minimum. According to the first theory of strength, a fracture occurs if the maximum principle stress exceeds its critical value. According to the second theory of strength, a fracture occurs if the maximum tensile strain exceeds its critical value. This can be transformed into an equation with the equivalent stress depending on all three stress components and Poisson's ratio. Copyrighted materials 8
Structural Integrity Analysis 4. Strength of Materials 4.4 BENDING: FORCE AND MOMENT DIAGRAMS A cantilever beam is fully constrained at one end (for example attached to a wall) and has a load acting at the other end. Static equilibrium conditions Static equilibrium conditions are applied to determine the external support reactions, and the external moment acting on the beam in the embedded end: The sum of forces including internal transverse (shear) force Q in Y-direction is equal to zero. The sum of the moments relative to the chosen point, including internal bending moment Mx, is equal to zero. The resulting internal bending moment in the beam is proportional to the distance from the free end z. The situation is similar if the load is distributed q. The figures show different loading schemes and corresponding diagrams of shear force Q and bending moment M. For the same external load, the support structure can affect the maximum values of the bending moment M. Copyrighted materials 9
Structural Integrity Analysis Copyrighted materials 4. Strength of Materials 10
Structural Integrity Analysis Copyrighted materials 4. Strength of Materials 11
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Structural Integrity Analysis 4. Strength of Materials 4.5 GEOMETRICAL CHARACTERISTICS OF SECTIONS Bending tensile and shear stresses depend on the section geometry, not only the crosssectional area A. The section can be characterized by the following: moments of inertia Ixx, Iyy, polar moment of inertia J. The last characteristic is important for analysis of torsion deformation. The moments of inertia have units of [length4]. The polar moment of inertia is equal to the sum of the other two moments. It does not depend on the orientation of the axes. The moments of inertia depend on the placement and orientation of the axes. Practical engineers use values of the moments of inertia which pass through the center of the mass called the centroid (xc,yc). Copyrighted materials 13
Structural Integrity Analysis 4. Strength of Materials For simple geometry the moment of inertia can be easy calculated. For more complicated sections, the information can be found in reference books. The moment of inertia Ixx is higher if many elements of the area are located far from the neutral axis. The removed area of the circle decreases the moment of inertia. The decrease is small if the distance from the neutral axis to the center of the hole is small. For a symmetrical geometric shape, the principle axes pass through the centroid (center of area). An axis coincides with the line of symmetry. Copyrighted materials 14
Structural Integrity Analysis 4. Strength of Materials For a composite cross section, an axis tends towards the region with the most area and that is furthest from the neutral axis. 4.6 BENDING: STRESS AND DEFORMATION Tensile bending stress is at a maximum in the outer layers of the beam. The smaller the moment of inertia Ixx, the larger the maximum bending stress. Shear stress is equal to zero at the outer layers. It is at a maximum on the neutral axis of the beam. Tensile bending stress is equal to zero at the neutral axis. Copyrighted materials 15
Structural Integrity Analysis 4. Strength of Materials The deflection is proportional to force and length3. This means that a doubled length increases deflection by eight times. Copyrighted materials 16
Structural Integrity Analysis 4. Strength of Materials For a distributed load the order for length is higher. Deflection and rotation in the center can affect deflection of the right end of the cantilever beam. Heating of the upper flange of the cantilever beam causes thermal expansion of upper layers and changes the beam shape. The shape of the flexible beam depends on the external moments at the beam ends Mi = Fi hi. 4.7 MIXED MODE LOADING If an external force passes through the center of the cross section for the shown scheme, then there is no torque (Mz=0) and no projection of inner forces at axis z (Pz=0). Upper layers of the cantilever beam are under tension and lower layers are under Copyrighted materials 17
Structural Integrity Analysis 4. Strength of Materials compression. If forces do not lie in a plane with a principle axis, the neutral axis swings through some angle. The stress is greatest in points that are furthest from the shown neutral axis. The top left point of the section has the maximum tensile stress. The summed stress pattern can be obtained by considering the tension and bending of the beam. For long cantilever beams, bending stress is usually higher than tensile stress if both forces are equal. If a compressive force was applied outside the center of the cross section, a tensile bending stress may occur. There is an area in which the addition of a compressive force does not cause tensile stress in the column. The shape of the zone for cylindrical sections is a circle. Copyrighted materials 18
Structural Integrity Analysis 4. Strength of Materials If the line of force application and the center of the cross section do not coincide by a distance h, there is a bending moment F*h that causes compressive and tensile stresses. Additional bending moments cause a change in the shape. The shape depends on the sign of the bending moment, either positive or negative. The principle stress is at a maximum in the embedded end of the structure, where torque T and bending moment M are highest. Force and moment diagrams for a complex structure can be obtained by the static equilibrium conditions. The maximum number of inner forces and moments is six. There are geometries where less than six components of inner forces and moments act. Copyrighted materials 19
Structural Integrity Analysis 4. Strength of Materials 4.8 BUCKLING In 1757, Leonard Euler proposed a relationship for the critical load that would produce buckling in a pinned-pinned column. For other supporting schemes (fixed-pinned, fixed-fixed, fixed-free, etc.) engineers use an effective length Le = k*L, where k is the effective length constant. The numbers show the percent of critical buckling force for each column in comparison with the uniform column. The last case demonstrates the most efficient use of the material. Copyrighted materials 20
Structural Integrity Analysis 4. Strength of Materials There are two extreme directions with maximum and minimum values of the moment of inertia. The buckling occurs in the direction of the minimum moment of inertia. For a rectangular section, buckling occurs perpendicular to the longer side. The resulting deformed shape of the column depends on how it is supported. The calculation shows that the critical force is the highest for a structure with maximum shear rigidity. This condition corresponds to an optimal angle of 35.26 degrees. For smaller or larger angles the shear stiffness is smaller. Copyrighted materials 21
Structural Integrity Analysis 4. Strength of Materials The critical stress corresponding to buckling depends on beam length and yield properties of the material. In short columns the critical stress can exceed the yield strength (tension). A tensile force in the cable compresses the beam. The loading can result in the beam buckling. The closer to the fixed end, the smaller the bending moment, and the closer to the neutral axis, the smaller the torque under buckling. The first example of the cantilever beams demonstrates the most efficient use of the material. The last two structures are unstable due to possible buckling and torsion. Copyrighted materials 22
Structural Integrity Analysis 4. Strength of Materials 4.9 STATICALLY INDETERMINATE SYSTEMS Statically indeterminate systems can be solved from conditions of equal or zero displacement of chosen points. If both parts of the structure have the same length, the fraction of the external load accepted by each part of the structure is proportional to its rigidity - the product of modulus of elasticity by cross-sectional area. Stress is proportional to the corresponding fraction of the external load. At the same time, the stress is inversely proportional to the cross-sectional area. This means that absolute values of the stresses in both parts are equal. For an angle of 60 degrees, the length of cable 1 is twice as long as cable 2. Displacement d2 is twice as large as d1. The fraction of external force depends on rigidity, cable length and the displacement. Cables 1 and 3 accept only 20% of the external vertical load. The shaft parts have different torsional stiffness and a common part in the center. The common part is rotated at the same angle for both parts, the angle of twist. Absolute values of the angle of twist for the left and right parts are equal. The gap δ1 can be deleted by elastic deformation of the cables. The equilibrium equation for moments relative to point O shows that forces in the cables are inversely proportional to the distance from point O. Copyrighted materials 23
Structural Integrity Analysis 4. Strength of Materials Both beams have equal deflection that is proportional to L3. Cutting the length in half increases the fraction of external force by 23=8. Shapes of the moment diagram are similar, but for the second beam there are reaction moments in embedded ends. They shift the curve down and decrease the maximum moment. The second beam is more rigid. The bending moment is at a maximum in the support zones. The maximum value of the bending moment M=FL over the right support does not depend on the number of supports. Copyrighted materials 24
Structural Integrity Analysis 4. Strength of Materials To solve the system it is necessary to find the shown and unknown inner forces and reactions from conditions of equal displacements. There are no transverse forces for the symmetrical system. The number of equations n depends on the chosen equivalent scheme: the smaller the number the better. 4.10 THREE-DIMENSIONAL STRUCTURES For three-dimensional structures there are six components of moments and forces in a point. If we cut the curved beam perpendicular to the beam axis the torque is equal to the product of Force by Arm : F*r1 and the bending moment is equal to F*r2. Internal forces are equal to the sum of corresponding projections of external forces. The figure below shows examples of bending moment and torque diagrams for threedimensional structures. The diagrams are built by the above-mentioned rules. Copyrighted materials 25
Structural Integrity Analysis 4. Strength of Materials There are tensile as well as compressive stresses at the external surface of the arc. The bending moment changes its sign. For a thin-walled sphere, absolute value of radial stress at the external surface is equal to external pressure. It is smaller than meridian and tangential stress components. REFERENCES M. Gere, S.P.Timoshenko Mechanics of Materials, 4th edition, Pws Pub Co, 1997. F. B. Seely, Advanced Mechanics of Materials, John Wiley and Sons, Inc., New York, 1952. L. Spiegel, G. Limbrunner Applied Statics and Strength of Materials, 3rd edition, Prentice-Hall Career & Technology, 1999. S. Timoshenko Strength of Materials, Part I & II, Van Nostrand Company, Princeton, N. J., 1956. Copyrighted materials 26
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