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P are identical except for length. Before the load W was attached, the bar was horizontal and the rods were stress-free. P is pinned at B and connected to two vertical rods. P, a rigid beam with negligible weight is pinned at one end and attached to two vertical rods. Find the vertical movement of W.
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P, a rigid bar with negligible mass is pinned at O and attached to two vertical rods. Assuming that the rods were initially stress-free, what maximum load P can be applied without exceeding stresses of MPa in the steel rod and 70 MPa in the bronze rod. P is a section through a balcony. The total uniform load of kN is supported by three rods of the same area and material.
Compute the load in each rod. Assume the floor to be rigid, but note that it does not necessarily remain horizontal. Assuming that there was no slack or stress in the rods before the load was applied, find the stress in each rod. Initially, the assembly is stress free. Horizontal movement of the joint at A is prevented by a short horizontal strut AE. If temperature deformation is permitted to occur freely, no load or stress will be induced in the structure.
In some cases where temperature deformation is not permitted, an internal stress is created. The internal stress created is termed as thermal stress. For a homogeneous rod mounted between unyielding supports as shown, the thermal stress is computed as: If the wall yields a distance of x as shown, the following calculations will be made: Take note that as the temperature rises above the normal, the rod will be in compression, and if the temperature drops below the normal, the rod is in tension.
Solution to Problem Thermal Stress A steel rod with a cross-sectional area of 0. At what temperature will the stress be zero? At what temperature will the rails just touch? What stress would be induced in the rails at that temperature if there were no initial clearance? Solution a Without temperature change: See figure above. Find the temperature at which the compressive stress in the bar will be 35 MPa. Assume that the supports are unyielding and that the bar is suitably braced against buckling.
Neglect the deformation of the wheel caused by the pressure of the tire. P is pinned at B and attached to the two vertical rods. Initially, the bar is horizontal and the vertical rods are stress-free. Neglect the weight of bar ABC. Solution Contraction of steel rod, assuming complete freedom: In terms of aluminum, this movement is by ratio and proportion: P, there is a gap between the aluminum bar and the rigid slab that is supported by two copper bars. Solution Assuming complete freedom: Solution Before temperature change: Assume the coefficients of linear expansion are If the system is initially stress-free.
Calculate the temperature change that will cause a tensile stress of 90 MPa in the brass rod. Assume that both rods are subjected to the change in temperature. Each bar has a cross-sectional area of mm2. Torsion 2. Flanged Bolt Couplings 3. Torsion of Thin-Walled Tubes 4. Such a bar is said to be in torsion. For solid cylindrical shaft: Determine the maximum shearing stress and the angle of twist.
What maximum shearing stress is developed? What power can be transmitted by the shaft at 20 Hz? If the shearing stress is limited to 12 ksi, determine the maximum horsepower that can be transmitted. Solution Hollow circular shaft: Solution to Problem Torsion An aluminum shaft with a constant diameter of 50 mm is loaded by torques applied to gears attached to it as shown in Fig. Determine the maximum length of the shaft if the shearing stress is not to exceed 20 ksi. What will be the angular deformation of one end relative to the other end?
Solution Based on maximum allowable shearing stress: At the right end, 30 kW are removed and another 20 kW leaves the shaft at 1. Determine the maximum permissible value of T subject to the following conditions: The two shafts are then fastened rigidly together at their ends.
What torque can be applied to the composite shaft without exceeding a shearing stress of psi in the bronze or 12 ksi in the steel? Determine the maximum shearing stress in each segment and the angle of rotation of the free end.
P is attached to rigid supports. What torque T is required? Solution From Solution P, to a solid shaft with built-in ends. How would these values be changed if the shaft were hollow? Equations 1 and 2a: Therefore, the values of T1 and T2 are the same no change if the shaft were hollow. Solution to Problem Torsion A solid steel shaft is loaded as shown in Fig. Solution Based on maximum allowable shear: Determine the maximum shearing stress developed in each segment.
Solution Stress developed in each segment with respect to TA: The rotation of B relative to A is zero. For the bronze segment AB, the maximum shearing stress is limited to psi and for the steel segment BC, it is limited to 12 ksi. P, each with one end built into a rigid support have flanges rigidly attached to their free ends. The shafts are to be bolted together at their flanges.
Determine the maximum shearing stress in each shaft after the shafts are bolted together. See figure. For rigid flanges, the shear deformations in the bolts are proportional to their radial distances from the shaft axis. Determine the torque capacity of the coupling if the allowable shearing stress in the bolts is 40 MPa.
Determine the torque capacity of the coupling if the allowable shearing stress in the bolts is psi. What torque can be applied without exceeding a shearing stress of 60 MPa in the bolts? Solution For one bolt in the outer circle: Determine the shearing stress in the bolts. What torque can be applied without exceeding psi in the steel or psi in the aluminum?
Solution to Problem Flanged bolt couplings A plate is fastened to a fixed member by four mm-diameter rivets arranged as shown in Fig. Compute the maximum and minimum shearing stress developed. P to the fixed member. Using the results of Prob. What additional loads P can be applied before the shearing stress in any rivet exceeds psi?
Solution Without the loads P: P is fastened to the fixed member by five mm-diameter rivets. Compute the value of the loads P so that the average shearing stress in any rivet does not exceed 70 MPa. Solution Solving for location of centroid of rivets: Determine the wall thickness t so as not to exceed a shear stress of 80 MPa. What is the shear stress in the short sides?
Neglect stress concentration at the corners. Solution to Problem Torsion of thin-walled tube A tube 0.
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What torque will cause a shearing stress of psi? Determine the smallest permissible dimension a if the shearing stress is limited to psi. Assume that the shearing stress at any point is proportional to its radial distance. J J Helical Springs When close-coiled helical spring, composed of a wire of round rod of diameter d wound into a helix of mean radius R with n number of turns, is subjected to an axial load P produces the following stresses and elongation: For heavy springs and considering the curvature of the spring, a more precise formula is given by: Use Eq.
Compute the number of turns required to permit an elongation of 4 in. P supports a load P. The upper spring has 12 turns of mm-diameter wire on a mean radius of mm. The lower spring consists of 10 turns of mm diameter wire on a mean radius of 75 mm. If the maximum shearing stress in either spring must not exceed MPa, compute the maximum value of P and the total elongation of the assembly.
Compute the equivalent spring constant by dividing the load by the total elongation. Determine the maximum load W that may be supported if the shearing stress in the springs is limited to 20 ksi.
Each spring consists of 20 turns of mm wire having a mean diameter of mm. Compute the maximum shearing stress in the springs, using Eq. Neglect the mass of the rigid bar. P, a homogeneous kg rigid block is suspended by the three springs whose lower ends were originally at the same level.
Compute the maximum shearing stress in each spring using Eq. According to determinacy, a beam may be determinate or indeterminate. Statically Determinate Beams Statically determinate beams are those beams in which the reactions of the supports may be determined by the use of the equations of static equilibrium.
The beams shown below are examples of statically determinate beams. Statically Indeterminate Beams If the number of reactions exerted upon a beam exceeds the number of equations in static equilibrium, the beam is said to be statically indeterminate. In order to solve the reactions of the beam, the static equations must be supplemented by equations based upon the elastic deformations of the beam.
The degree of indeterminacy is taken as the difference between the umber of reactions to the number of equations in static equilibrium that can be applied. Types of Loading Loads applied to the beam may consist of a concentrated load load applied at a point , uniform load, uniformly varying load, or an applied couple or moment. These loads are shown in the following figures. Assume that the beam is cut at point C a distance of x from he left support and the portion of the beam to the right of C be removed.
The portion removed must then be replaced by vertical shearing force V together with a couple M to hold the left portion of the bar in equilibrium under the action of R1 and wx. The couple M is called the resisting moment or moment and the force V is called the resisting shear or shear. The sign of V and M are taken to be positive if they have the senses indicated above. Write shear and moment equations for the beams in the following problems.
In each problem, let x be the distance measured from left end of the beam. Also, draw shear and moment diagrams, specifying values at all change of loading positions and at points of zero shear. Neglect the mass of the beam in each problem. See the instruction. Solution From the load diagram: In segment AB, the shear is uniformly distributed over the segment at a magnitude of —30 kN.
In segment BC, the shear is uniformly distributed at a magnitude of 26 kN. In segment CD, the shear is uniformly distributed at a magnitude of —24 kN. To draw the Moment Diagram: At segment AB, the shear is uniformly distributed at lb.
A shear of — lb is uniformly distributed over segments BC and CD. Note that the maximum moment occurs at point of zero shear. For segment AB, the shear is uniformly distributed at 20 kN.
The shear for segment CD is uniformly distributed at —40 kN. Solution Segment AB: Solution to Problem Shear and Moment Diagrams Cantilever beam carrying a distributed load with intensity varying from wo at the free end to zero at the wall, as shown in Fig. To draw the Moment diagram: The shear is uniformly distributed at — lb along segments CD and DE. Shear is uniform along segment CD at —20 kN. To draw the Moment Diagram 1. Solution By symmetry: The other half of the diagram can be drawn by the concept of symmetry.
Solution to Problem Shear and Moment Diagrams A total distributed load of 30 kips supported by a uniformly distributed reaction as shown in Fig.
For the next half of the beam, the shear diagram can be accomplished by the concept of symmetry. P if a the load P is vertical as shown, and b the load is applied horizontally to the left at the top of the arch. Properties of Shear and Moment Diagrams The following are some important properties of shear and moment diagrams: The area of the shear diagram to the left or to the right of the section is equal to the moment at that section.
The slope of the moment diagram at a given point is the shear at that point. The slope of the shear diagram at a given point equals the load at that point. The maximum moment occurs at the point of zero shears. This is in reference to property number 2, that when the shear also the slope of the moment diagram is zero, the tangent drawn to the moment diagram is horizontal. When the shear diagram is increasing, the moment diagram is concave upward.
When the shear diagram is decreasing, the moment diagram is concave downward. Sign Convention The customary sign conventions for shearing force and bending moment are represented by the figures below. A force that tends to bend the beam downward is said to produce a positive bending moment.
An easier way of determining the sign of the bending moment at any section is that upward forces always cause positive bending moments regardless of whether they act to the left or to the right of the exploratory section. Without writing shear and moment equations, draw the shear and moment diagrams for the beams specified in the following problems. Give numerical values at all change of loading positions and at all points of zero shear.
Note to instructor: Problems to may also be assigned for solution by semi-graphical method describes in this article. Solution To draw the Shear Diagram 1.
Solving for x: For segment BC, the location of zero moment can be accomplished by symmetry and that is 5 ft from B. Solving for point of zero moment: This point is the appropriate location for construction joint of concrete structures.
Location of zero moment at segment BC: By squared property of parabola: Location of zero shear: P consists of two segments joined by a frictionless hinge at which the bending moment is zero. The location of zero moment in segment BH can easily be found by symmetry.
P consists of two segments joined by frictionless hinge at which the bending moment is zero. Draw shear and moment diagrams for each of the three parts of the frame. It is subjected to the loads shown in Fig. P, which act at the ends of the vertical members BE and CF.
These vertical members are rigidly attached to the beam at B and C. Draw shear and moment diagrams for the beam ABCD only.
Shear in segments AB and BC is zero. Moment in segment AB is zero 2. Location of zero shear C: The shear in AB is a parabola with vertex at A, the starting point of uniformly varying load. The load in AB is 0 at A to downward wo or —wo at B, thus the slope of shear diagram is decreasing. For decreasing slope, the parabola is open downward. The shear diagram is second degree curve, thus the moment diagram is a third degree curve.
The maximum moment highest point occurred at C, the location of zero shear. VBC is also parabolic since the load in BC is linear. MAC is third degree because the shear diagram in AC is second degree. The shear from A to C is decreasing, thus the slope of moment diagram from A to C is decreasing. The shear diagram in AB is second degree curve. The shear in AB is from —wo downward wo to zero or increasing, thus, the slope of shear at AB is increasing upward parabola.
The shear diagram in BC is second degree curve. The shear in BC is from zero to —wo downward wo or decreasing, thus, the slope of shear at BC is decreasing downward parabola To draw the Moment Diagram 1. The shear diagram from A to C is decreasing, thus, the moment diagram is a concave downward third degree curve. To draw the Shear Diagram 1. From the load diagram: The location of zero shear is obviously at the midspan or 2 m from B.
Load and moment diagrams for a given shear diagram Instruction: In the following problems, draw moment and load diagrams corresponding to the given shear diagrams. Specify values at all change of load positions and at all points of zero shear.
Solution To draw the Load Diagram 1. A lb upward force is acting at point A. No load in segment AB. No load in segment BC. No load in segment CD. No load in segment DE. A downward force of lb is concentrated at point E. To draw the Load Diagram 1. Downward lb force is concentrated at A and no load in segment AB. Upward force of lb is concentrated at end of span F. The locations of zero shear points G and H can be easily determined by ratio and proportion of triangle.
Another concentrated force is acting downward at D with a magnitude of lb. For segment DE, the diagram is downward parabola with vertex at G. G is the point where the extended shear in DE intersects the line of zero shear. The moment diagram in EF is a downward parabola with vertex at F.
Upward concentrated load at A is 10 kN. The shear in AB is a 2nd-degree curve, thus the load in AB is uniformly varying. The shear in DE is uniformly increasing, thus the load in DE is uniformly distributed and upward. To find the location of zero shear, F: For segment DE, the moment diagram is parabola open upward with vertex at E.
Moving Loads Moving Loads From the previous section, we see that the maximum moment occurs at a point of zero shears. For beams loaded with concentrated loads, the point of zero shears usually occurs under a concentrated load and so the maximum moment. The problem here is to determine the moment under each load when each load is in a position to cause a maximum moment.
The largest value of these moments governs the design of the beam.
Single Moving Load For a single moving load, the maximum moment occurs when the load is at the midspan and the maximum shear occurs when the load is very near the support usually assumed to lie over the support. Three or more moving loads In general, the bending moment under a particular load is a maximum when the center of the beam is midway between that load and the resultant of all the loads then on the span. With this rule, we compute the maximum moment under each load, and use the biggest of the moments for the design.
Usually, the biggest of these moments occurs under the biggest load.What minimum diameter pulleys can be used without exceeding a flexural stress of MPa? It is necessary to tighten the bolts initially to press the gasket to the flange, to avoid leakage of steam. Maximum longitudinal and circumferential stress Solution Longitudinal Stress: Draw shear and moment diagrams for the beam ABCD only.
Neglect the deformation of the wheel caused by the pressure of the tire. Classical mechanics: Using the results of Prob. It is subjected to the loads shown in Fig.
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