The reaction RA at the upper beam is obtained from the free-body diagram of the bar (Figure 2.28). Expressing that the total deformation of the rod must be zero due to the imposed constraints, we write
87SAMPLE PROBLEM 2.4
SOLUTION
PROBLEMS
- A 250-mm bar of 150 3 30-mm rectangular cross section consists of two aluminum layers, 5 mm thick, brazed to a center brass layer
- Compressive centric forces of 40 kips are applied at both ends of the assembly shown by means of rigid end plates. Knowing that
- The rigid bar ABCD is suspended from four identical wires. Deter- mine the tension in each wire caused by the load P shown
- A rod consisting of two cylindrical portions AB and BC is restrained at both ends. Portion AB is made of steel (E s 5 29 3 10 6 psi,
- POISSON’S RATIO 93
- MULTIAXIAL LOADING; GENERALIZED HOOKE’S LAW
Determine (a) the maximum allowable change in temperature if the stress in the aluminum shell is not to exceed 6 ksi, (b) the corresponding change in length of the assembly. Determine (a) the final normal stress in the cable, (b) the final length of the cable.
2.13 DILATATION; BULK MODULUS
SHEARING STRAIN
Consider first a cubic element of side one (Fig. 2.42) subject to no stresses other than the shear stresses txy and tyx acting on faces of the element perpendicular to the x and y axes, respectively. As was the case for normal stresses and strains, the initial portion of the shear stress-strain diagram is a straight line.
1012.15 FURTHER DISCUSSION OF DEFORMATIONS
It follows from Hooke's law of shear stress and strain that the shear stress g9 associated with the element in Fig. But the angle formed by the inclined and horizontal faces of the element in fig.
2.16 STRESS-STRAIN RELATIONSHIPS FOR FIBER-REINFORCED COMPOSITE MATERIALS
These contractions depend on the placement of the fibers in the matrix and will generally be different. The changes in the cube dimensions are obtained by multiplying the corresponding deformations by the length L 5 0.060 m of the side of the cube:.
107SAMPLE PROBLEM 2.5
- The change in diameter of a large steel bolt is carefully measured as the nut is tightened. Knowing that E 5 29 3 10 6 psi and n 5
- A 30-mm square was scribed on the side of a large steel pressure vessel. After pressurization the biaxial stress condition at the square
- The aluminum rod AD is fitted with a jacket that is used to apply a hydrostatic pressure of 6000 psi to the 12-in. portion BC of the
- The plastic block shown is bonded to a fixed base and to a hori- zontal rigid plate to which a force P is applied. Knowing that for
- STRESS AND STRAIN DISTRIBUTION UNDER 113 AXIAL LOADING; SAINT-VENANT’S PRINCIPLE
- STRESS CONCENTRATIONS
- PLASTIC DEFORMATIONS 117
Knowing that sx 5 2180 MPa, determine (a) the magnitude of side for which the change in the height of the block will be zero, (b) the corresponding change in the area of the plane ABCD, (c) the corresponding change in the volume of the block. Draw the load-deflection diagram of the rod-tube assembly when a load P is applied to the plate as shown.
2.20 RESIDUAL STRESSES
The stress caused by the discharge is the same in the rod and in the tube. For purposes of discussion, the plug will be considered as a small rod AB that will be attached to a small hole in the plate (Fig. 2.67). As a result of the welding operation, a residual stress is created approximately equal to the yield strength of the steel used in the socket and in the weld.
The result is residual longitudinal tensile stresses in the inner core and residual compressive stresses in the outer layers.
SAMPLE PROBLEM 2.6
Determine the maximum value of the force P and the permanent set of the bar after the force is removed, knowing that (a) dm 5 4.5 mm, (b) dm 5 8 mm. Determine (a) the maximum value of P, (b) the maximum stress in the hardened steel bars, (c) the permanent set after the load is removed. 2.111, if P is gradually increased from zero to 98 kips and then decreased back to zero, determine (a) the maximum deformation of the bar, (b) the maximum stress in the hardened steel bars, (c) the permanent set after the load is removed.
Knowing that a 5 0.640 m, determine (a) the value of the normal stress in each link, (b) the maximum deflection of point B.
REVIEW AND SUMMARY
This relationship is known as Hooke's law and the coefficient E as the material's modulus of elasticity. The value of s for which no failure occurs even for an infinite number of cycles is known as the endurance limit of the material used in the test. This strain is referred to as the lateral strain, and the ratio of lateral strain over the axial strain is called Poisson's ratio and denoted by n (Greek letter now).
The change in volume per unit volume is called the expansion of the material and is denoted by e.
REVIEW PROBLEMS
Knowing that the coefficient of friction is 0.60 between the strip and the support at B, determine the temperature drop for which slippage will occur. Knowing that E psi and h 5 4 ft, determine the largest allowable stress in AC so that the deflections in members AB and CD do not exceed 0.04 in. Determine the largest allowable temperature rise if the stress in the steel core is not to exceed 8 ksi.
Knowing that the maximum allowable shear stress is 420 kPa, determine (a) the minimum allowable dimension b, (b) the minimum required thickness a.
COMPUTER PROBLEMS
Write a computer program to determine the allowable load P for given values of r, D, thickness t of the strip, and allowable material load. Replacing the cone with n circular cylinders of equal thickness, write a computer program that can be used to calculate the elongation of the truncated cone.
This chapter is devoted to the study of torsion and of the stresses and
Torsion 3.1 Introduction
- INTRODUCTION
- PRELIMINARY DISCUSSION OF THE STRESSES IN A SHAFT
- DEFORMATIONS IN A CIRCULAR SHAFT
- STRESSES IN THE ELASTIC RANGE
It also shows that g is proportional to the distance r from the axis of the axis to the point under consideration. Thus, the shear stress in a circular shaft varies linearly with distance from the shaft axis. Let us consider the resulting stresses and forces on the faces that are at 458 with the axis of the shaft.
We conclude that the stresses on the surfaces of element c at 458 relative to the axis of the shaft (Figure 3.19) are normal stresses equal to 6tmax.
SAMPLE PROBLEM 3.1
153SAMPLE PROBLEM 3.2
ANGLE OF TWIST IN THE ELASTIC RANGE 159
In each case the angle of rotation f of the shaft was therefore equal to the angle of rotation of its free end. However, when both ends of a shaft rotate, the angle of rotation of the shaft is equal to the angle through which one end of the shaft rotates with respect to the other. Since the end D of shaft AD is fixed, the angle of rotation fA of gear A is equal to the angle of rotation of the shaft and is obtained by writing
Now considering axis BE, we remember that the angle of rotation of the axis is equal to the angle fEyB through which end E rotates with respect to end B.
TLJG 5 5TL
- STATICALLY INDETERMINATE SHAFTS
By drawing the free-body diagram of the shaft and by TA and TB indicating the torques exerted by the supports (Fig. 3.26a), we obtain the equilibrium equation. Since this equation is not sufficient to determine the two unknown torques TA and TB, the shaft is statically indeterminate. From the free-body diagram of a small portion of the shaft including end A (Fig. 3.26b), we note that the internal torque T1 in AC is equal to TA; from the free-body diagram of a small portion of the shaft including end B (Fig. 3.26c), we note that the internal torque T2 in CB is equal to TB. 3.16) and observing that parts AC and CB of the shaft are twisted in opposite senses, we write.
Passing a section through the shaft between A and B and using the free body shown, we find.
SAMPLE PROBLEM 3.3
Since the shaft consists of three sections AB, BC and CD, each of uniform cross-section and each with a constant internal torque, Eq.
SAMPLE PROBLEM 3.4
By denoting T1 the torque exerted by the tube on the disk and by T2 the torque exerted by the shaft we find. Since both the tube and the shaft are connected to the rigid disk, we have. 2), we calculate the corresponding value T2 and then find the maximum shear stress in the steel shaft.
We note that the allowable steel stress of 120 MPa is exceeded; your assumption was wrong.
SAMPLE PROBLEM 3.5
- and 3.56 Two solid steel shafts are fitted with flanges that are then connected by bolts as shown. The bolts are slightly undersized
- The torque T is applied to flange B
- The torque T is applied to flange C
- DESIGN OF TRANSMISSION SHAFTS
- STRESS CONCENTRATIONS IN CIRCULAR SHAFTS 179
Knowing that section CD of the copper rod is hollow and has an inner diameter of 40 mm, determine the twist angle at A. Design specifications require that the diameter of the shank be uniform from A to D and that the twist angle between A and D be no higher than 1.58. Determine the required diameter of the shaft, knowing that the shaft is made of (a) steel with an allowable shear stress of 90 MPa and a stiffness modulus of 77 GPa, (b) bronze with an allowable shear stress of 35 MPa and a stiffness modulus of 42 GPa .
Knowing that a torque T is applied to end A of shaft AB and that end D of tube CD is fixed, (a) determine the magnitude and location of the maximum shear stress in the annular plate, (b) show that the angle through which end B of the shaft rotates relative to end C of the tube.
SAMPLE PROBLEM 3.6
Knowing that the maximum allowable shear stress is 8 ksi, determine the required diameter (a) of shaft AB, (b) of shaft CD. Knowing that the permissible shear stress in the shaft is 40 MPa and that the radius of the fillet is r 5 6 mm, determine the minimum permissible speed for the shaft. Knowing that the radius of the fillet is r 5 8 mm and the allowable shear stress is 45 MPa, determine the maximum power that can be transmitted.
Knowing that the shaft speed is 2400 rpm and the allowable shear stress is 7500 psi, determine the maximum power that can be transmitted by the shaft.
3.9 PLASTIC DEFORMATIONS IN CIRCULAR SHAFTS
Assuming that the maximum value tmax of the shear stress t is determined, we can obtain a graph of the dependence of t on r as follows. The important torque value is the ultimate torque TU which causes the shaft to fail. Assuming a fictitious linear stress distribution, Eq. 3.9) is then used to determine the corresponding maximum shear stress RT:.
Using the stress-strain diagram of the material, we can then obtain the corresponding shear stress value t and plot t against r.
3.10 CIRCULAR SHAFTS MADE OF AN ELASTOPLASTIC MATERIAL
This can be done by the expression used in Art. 3.3 for the shear stress g in terms of f, r and the length L of the shaft:. Equation (3.26) will be used to determine the value of the torque T corresponding to a given radius rY of the elastic core. If we in Eq. 3.32) the expression obtained for rYyc, we express the torque T as a function of the angle of rotation f,.
On the other hand, values of the shear stress g can be obtained from Eq. 3.2) and from the values of f and L measured in the tubular part of the specimen.
189EXAMPLE 3.08
3.11 RESIDUAL STRESSES IN CIRCULAR SHAFTS
As a result, the unloading of the shaft is represented by a straight line on the T-f diagram (Fig. 3.39). By adding the two groups of stresses, we obtain the distribution of the residual stresses in the shaft (Fig. 3.40c). We recall from example 3.08 that the yield strength tY 5 is 150 MPa and the radius of the elastic core corresponding to the given torque rY 5 is 15.8 mm.
The distribution of the stress in the loaded shaft is therefore as in Fig.
SAMPLE PROBLEM 3.7
When the plastic zone reaches the inner surface, the stresses are evenly distributed as shown.
193SOLUTION
SAMPLE PROBLEM 3.8
3.99, determine (a) the magnitude of the torque T required to turn the shaft through an angle of 158, (b) the radius of the corresponding elastic core. For a length of 1.2 m of the shaft, determine the maximum shear stress and the angle of twist caused by a 200 N. Determine the magnitude T of the torque and the corresponding angle of rotation (a) at the onset of yield change, (b) ) when the plastic zone is 10 mm deep.
Determine (a) the magnitude and location of the maximum residual shear stress, (b) the permanent rotation angle.
3.12 TORSION OF NONCIRCULAR MEMBERS
Determine the magnitude and location of the maximum residual shear stress in the bar. 3.118, determine the permanent angle of rotation of the rod. We conclude that there is no shear stress at the corners of the bar cross-section. The determination of the stresses in non-circular elements subjected to a torsional load is beyond the scope of this text.
Thus we verify that the maximum shear stress in a bar of rectangular cross-section will occur at the midpoint N of the larger side of that section.
3.13 THIN-WALLED HOLLOW SHAFTS
