Determine the equation of the elastic curve for the beam

An L2 steel strap having a thickness of 0.125 in. and a width of 2 in. is bent into a circular arc of radius 600 in. Determine the maximum bending stress in the strap.

The L2 steel blade of the band saw wraps around the pulley having a radius of 12 in. Determine the maximum normal stress in the blade. The blade has a width of 0.75 in. and a thickness of 0.0625 in.

A picture is taken of a man performing a pole vault, and the minimum radius of curvature of the pole is estimated by measurement to be 4.5 m. If the pole is 40 mm in diameter and it is made of a glass-reinforced plastic for which \(E_{\mathrm{g}}=131 \mathrm{GPa}\), determine the maximum bending stress in the pole.

Determine the equation of the elastic curve for the beam using the x coordinate that is valid for \(0 \leq x<L / 2\). Specify the slope at A and the beam’s maximum deflection. EI is constant.

Determine the deflection of end C of the 100-mm-diameter solid circular shaft. Take E = 200 GPa.

Determine the elastic curve for the cantilevered beam, which is subjected to the couple moment \(\mathbf{M}_{0}\). Also calculate the maximum slope and maximum deflection of the beam. EI is constant.

The A-36 steel beam has a depth of 10 in. and is subjected to a constant moment \(\mathbf{M}_{0}\), which causes the stress at the outer fibers to become \(\sigma_{Y}=36 \mathrm{ksi}\). Determine the radius of curvature of the beam and the beam’s maximum slope and deflection.

Determine the equations of the elastic curve using the coordinates \(x_{1}\) and \(x_{2}\). EI is constant.

Determine the equations of the elastic curve for the beam using the \(x_{1}\) and \(x_{2}\) coordinates. EI is constant.

Determine the equations of the elastic curve using the coordinates \(x_{1}\) and \(x_{2}\). What is the slope at C and displacement at B? EI is constant.

Determine the equations of the elastic curve using the coordinates \(x_{1}\) and \(x_{2}\). What is the slope at B and deflection at C? EI is constant.

Draw the bending-moment diagram for the shaft and then, from this diagram, sketch the deflection or elastic curve for the shaft’s centerline. Determine the equations of the elastic curve using the coordinates \(x_{1}\) and \(x_{2}\). EI is constant.

Determine the maximum deflection of the beam and the slope at A. EI is constant.

The simply supported shaft has a moment of inertia of 2I for region BC and a moment of inertia I for regions AB and CD. Determine the maximum deflection of the shaft due to the load P.

A torque wrench is used to tighten the nut on a bolt. If the dial indicates that a torque of \(60 \mathrm{lb} \cdot \mathrm{ft}\) is applied when the bolt is fully tightened, determine the force P acting at the handle and the distance s the needle moves along the scale. Assume only the portion AB of the beam distorts. The cross section is square having dimensions of 0.5 in. by 0.5 in. \(E=29\left(10^{3}\right) \mathrm{ksi}\).

The pipe can be assumed roller supported at its ends and by a rigid saddle C at its center. The saddle rests on a cable that is connected to the supports. Determine the force that should be developed in the cable if the saddle keeps the pipe from sagging or deflecting at its center. The pipe and fluid within it have a combined weight of 12.5 lb/ft. EI is constant.

Determine the equations of the elastic curve for the beam using the x1 and x2 coordinates. Specify the beam’s maximum deflection. EI is constant.

The bar is supported by a roller constraint at B, which allows vertical displacement but resists axial load and moment. If the bar is subjected to the loading shown, determine the slope at A and the deflection at C. EI is constant.

Determine the deflection at B of the bar in Prob. 12–18.

Determine the equations of the elastic curve using the \(x_{1}\) and \(x_{2}\) coordinates. What is the slope at A and the deflection at C? EI is constant.

Determine the maximum deflection of the solid circular shaft. The shaft is made of steel having E = 200 GPa. It has a diameter of 100 mm.

Determine the elastic curve for the cantilevered W14 X 30 beam using the x coordinate. Specify the maximum slope and maximum deflection. \(E=29\left(10^{3}\right) \mathrm{ksi}\).

Determine the equations of the elastic curve using the coordinates \(x_{1}\) and \(x_{2}\). What is the deflection and slope at C? EI is constant.

Determine the equations of the elastic curve using the coordinates \(x_{1}\) and \(x_{1}\). What is the slope at A? EI is constant.

The floor beam of the airplane is subjected to the loading shown. Assuming that the fuselage exerts only vertical reactions on the ends of the beam, determine the maximum deflection of the beam. EI is constant.

Determine the maximum deflection of the simply supported beam. The beam is made of wood having a modulus of elasticity of \(E=1.5\left(10^{3}\right) \mathrm{ksi}\).

The beam is made of a material having a specific weight \(\gamma\). Determine the displacement and slope at its end A due to its weight. The modulus of elasticity for the material is E.

Determine the slope at end B and the maximum deflection of the cantilever triangular plate of constant thickness t. The plate is made of material having a modulus of elasticity of E.

Determine the equation of the elastic curve using the coordinates \(x_{1}\) and \(x_{2}\). What is the slope and deflection at B? EI is constant.

Determine the equations of the elastic curve using the coordinates \(x_{1}\) and \(x_{3}\). What is the slope and deflection at point B? EI is constant.

The shaft is supported at A by a journal bearing and at C by a thrust bearing. Determine the equation of the elastic curve. EI is constant.

The shaft supports the two pulley loads shown. Determine the equation of the elastic curve. EI is constant.

The beam is made of a ceramic material. If it is subjected to the elastic loading shown, and the moment of inertia is I and the beam has a measured maximum deflection \(\Delta\) at its center, determine the modulus of elasticity, E. The supports at A and D exert only vertical reactions on the beam.

Determine the equation of the elastic curve, the maximum deflection in region AB, and the deflection of end C. EI is constant.

The beam is subjected to the load shown. Determine the equation of the elastic curve. EI is constant.

Determine the equation of the elastic curve, the slope at A, and the deflection at B. EI is constant.

Determine the equation of the elastic curve and the maximum deflection of the simply supported beam. EI is constant.

The shaft supports the two pulley loads. Determine the equation of the elastic curve. EI is constant.

Determine the maximum deflection of the cantilevered beam. Take E = 200 GPa and \(I=65.0\left(10^{6}\right) \mathrm{mm}^{4}\).

Determine the slope at A and the deflection of end C of the overhang beam. Take \(E=29\left(10^{3}\right) \mathrm{ksi}\) and \(I=204 \mathrm{in}^{4}\).

Determine the maximum deflection in region AB of the overhang beam. Take \(E=29\left(10^{3}\right) \mathrm{ksi}\) and \(I=204 \mathrm{in}^{4}\).

The shaft supports the two pulley loads shown. Determine the slope of the shaft at A and B. The bearings exert only vertical reactions on the shaft. EI is constant.

Determine the equation of the elastic curve. EI is constant.

Determine the equation of the elastic curve. EI is constant.

Determine the deflection at each of the pulleys C, D, and E. The shaft is made of steel and has a diameter of 30 mm. \(E_{\mathrm{st}}=200 \mathrm{GPa}\).

Determine the slope of the shaft at A and B. The shaft is made of steel and has a diameter of 30 mm. The bearings only exert vertical reactions on the shaft. \(E_{\mathrm{st}}=200 \mathrm{GPa}\).

Determine the equation of the elastic curve. Specify the slopes at A and B. EI is constant.

Determine the value of a so that the displacement at C is equal to zero. EI is constant.

Determine the displacement at C and the slope at A. EI is constant.

Determine the equations of the slope and elastic curve. EI is constant.

Determine the slope and deflection at C. EI is constant.

Determine the slope and deflection at C. EI is constant.

Determine the deflection of end B of the cantilever beam. EI is constant.

Determine the slope at B and the deflection at C. EI is constant.

The composite simply supported steel shaft is subjected to a force of 10 kN at its center. Determine its maximum deflection. \(E_{\mathrm{st}}=200 \mathrm{GPa}\) .

Determine the slope of the shaft at A and the displacement at D. EI is constant.

The simply supported shaft has a moment of inertia of 2I for region BC and a moment of inertia I for regions AB and CD. Determine the maximum deflection of the shaft due to the load P. The modulus of elasticity is E.

Determine the deflection at C and the slope of the beam at A, B, and C. EI is constant.

Determine the maximum deflection of the 50-mm-diameter A-36 steel shaft.

Determine the slope of the 50-mm-diameter A-36 steel shaft at the journal bearings at A and B. The bearings exert only vertical reactions on the shaft.

Determine the position a of the roller support B in terms of L so that the deflection at end C is the same as the maximum deflection of region AB of the overhang beam. EI is constant.

Determine the slope of the 20-mm-diameter A-36 steel shaft at the journal bearings A and B.

Determine the slope and the deflection of end B of the cantilever beam. EI is constant.

The two A-36 steel bars have a thickness of 1 in. and a width of 4 in. They are designed to act as a spring for the machine which exerts a force of 4 kip on them at A and B. If the supports exert only vertical forces on the bars, determine the maximum deflection of the bottom bar.

Determine the slope at A and the displacement at C. Assume the support at A is a pin and B is a roller. EI is constant.

Determine the deflection at C and the slopes at the bearings A and B. EI is constant.

Determine the maximum deflection within region AB. EI is constant.

Determine the slope at A and the maximum deflection of the simply supported beam. EI is constant.

Determine the slope at C and the deflection at B. EI is constant.

Determine the slope at A and the maximum deflection of the beam. EI is constant.

Determine the displacement of the 20-mm-diameter A-36 steel shaft at D.

The two force components act on the tire of the automobile. The tire is fixed to the axle, which is supported by bearings at A and B. Determine the maximum deflection of the axle. Assume that the bearings resist only vertical loads. The thrust on the axle is resisted at C. The axle has a diameter of 1.25 in. and is made of A-36 steel. Neglect the effect of axial load on deflection.

At what distance a should the journal bearing supports at A and B be placed so that the deflection at the center of the shaft is equal to the deflection at its ends? EI is constant.

The rod is constructed from two shafts for which the moment of inertia of AB is I and for BC it is 2I. Determine the maximum slope and deflection of the rod due to the loading. The modulus of elasticity is E.

Determine the slope at B and the deflection at C of the beam. E = 200 GPa and \(I=65.0\left(10^{6}\right) \mathrm{mm}^{4}\).

Determine the slope at point A and the maximum deflection of the simply supported beam. The beam is made of material having a modulus of elasticity E. The moment of inertia of segments AB and CD of the beam is I, and the moment of inertia of segment BC is 2I.

Determine the position a of roller support B in terms of L so that deflection at end C is the same as the maximum deflection in region AB of the overhang beam. EI is constant.

Determine the slope at B and deflection at C. EI is constant.

Determine the slope and displacement at C. EI is constant.

Determine the slope at C and deflection at B. EI is constant.

The two bars are pin connected at D. Determine the slope at A and the deflection at D. EI is constant.

Determine the maximum deflection of the beam. EI is constant.

The W10 X 15 cantilevered beam is made of A-36 steel and is subjected to the loading shown. Determine the slope and displacement at its end B.

The W10 X 15 cantilevered beam is made of A-36 steel and is subjected to the loading shown. Determine the displacement at B and the slope at B.

The W14 X 43 simply supported beam is made of A992 steel and is subjected to the loading shown. Determine the deflection at its center C.

The W14 X 43 simply supported beam is made of A992 steel and is subjected to the loading shown. Determine the slope at A and B.

The W14 X 43 simply supported beam is made of A-36 steel and is subjected to the loading shown. Determine the deflection at its center C.

The W14 X 43 simply supported beam is made of A-36 steel and is subjected to the loading shown. Determine the slope at A and B.

The W8 X 48 cantilevered beam is made of A-36 steel and is subjected to the loading shown. Determine the displacement at C and the slope at A.

The beam supports the loading shown. Code restrictions, due to a plaster ceiling, require the maximum deflection not to exceed 1/360 of the span length. Select the lightest-weight A-36 steel wide-flange beam from Appendix B that will satisfy this requirement and safely support the load. The allowable bending stress is \(\sigma_{\text {allow }}=24 \mathrm{ksi}\) and the allowable shear stress is \(\tau_{\text {allow }}=14 \mathrm{ksi}\). Assume A is a roller and B is a pin.

The W24 X 104 A-36 steel beam is used to support the uniform distributed load and a concentrated force which is applied at its end. If the force acts at an angle with the vertical as shown, determine the horizontal and vertical displacement at A.

The W8 X 48 cantilevered beam is made of A-36 steel and is subjected to the loading shown. Determine the displacement at its end A.

The rod is pinned at its end A and attached to a torsional spring having a stiffness k, which measures the torque per radian of rotation of the spring. If a force P is always applied perpendicular to the end of the rod, determine the displacement of the force. EI is constant.

Determine the vertical deflection and the change in angle at the end A of the bracket. Assume that the bracket is fixed supported at its base, and neglect the axial deformation of segment AB. EI is constant.

The pipe assembly consists of three equal-sized pipes with flexibility stiffness EI and torsional stiffness GJ. Determine the vertical deflection at A.

The assembly consists of a cantilevered beam CB and a simply supported beam AB. If each beam is made of A-36 steel and has a moment of inertia about its principal axis of \(I_{x}=118 \mathrm{in}^{4}\), determine the displacement at the center D of beam BA.

The relay switch consists of a thin metal strip or armature AB that is made of red brass C83400 and is attracted to the solenoid S by a magnetic field. Determine the smallest force F required to attract the armature at C in order that contact is made at the free end B. Also, what should the distance a be for this to occur? The armature is fixed at A and has a moment of inertia of \(I=0.18\left(10^{-12}\right) \mathrm{m}^{4}\).

Determine the moment \(M_{0}\) in terms of the load P and dimension a so that the deflection at the center of the shaft is zero. EI is constant.

Determine the reactions at the supports A and B, then draw the shear and moment diagram. EI is constant. Neglect the effect of axial load.

Determine the reactions at the supports, then draw the shear and moment diagram. EI is constant.

Determine the reactions at the supports A, B, and C, then draw the shear and moment diagrams. EI is constant.

Determine the reactions at the supports A and B, then draw the shear and moment diagrams. EI is constant.

Determine the reactions at the supports A and B, then draw the shear and moment diagrams. EI is constant.

Determine the moment reactions at the supports A and B. EI is constant.

Determine the reactions at the supports A and B, then draw the moment diagram. EI is constant.

Determine the reactions at the support A and B. EI is constant.

Determine the reactions at roller support A and fixed support B.

Determine the moment reactions at the supports A and B, then draw the shear and moment diagrams. Solve by expressing the internal moment in the beam in terms of \(A_{y}\) and \(M_{A}\). EI is constant.

The beam has a constant \(E_{1} I_{1}\) and is supported by the fixed wall at B and the rod AC. If the rod has a cross-sectional area \(A_{2}\) and the material has a modulus of elasticity \(E_{2}\), determine the force in the rod.

The beam is supported by a pin at A, a roller at B, and a post having a diameter of 50 mm at C. Determine the support reactions at A, B, and C. The post and the beam are made of the same material having a modulus of elasticity E = 200 GPa, and the beam has a constant moment of inertia \(I=255\left(10^{6}\right) \mathrm{mm}^{4}\).

Determine the moment reactions at the supports A and B. EI is constant.

Determine the reaction at the supports, then draw the shear and moment diagrams. EI is constant.

Determine the vertical reaction at the journal bearing support A and at the fixed support B, then draw the shear and moment diagrams for the beam. EI is constant.

Determine the reactions at the supports A and B, then draw the shear and moment diagrams. EI is constant.

Determine the reactions at the supports. EI is constant.

Determine the vertical reaction at the journal bearing support A and at the fixed support B, then draw the shear and moment diagrams for the shaft. EI is constant.

Determine the reactions at the journal bearing supports A, B, and C of the shaft, then draw the shear and moment diagrams. EI is constant.

Determine the reactions at the supports, then draw the shear and moment diagrams. EI is constant.

Determine the reactions at the supports, then draw the shear and moment diagrams. EI is constant.

Determine the reactions at the supports A and B. EI is constant.

The beam is used to support the 20-kip load. Determine the reactions at the supports. Assume A is fixed and B is a roller.

Determine the reactions at the supports A and B. EI is constant.

Determine the reactions at the supports A and B. EI is constant.

Before the uniform distributed load is applied to the beam, there is a small gap of 0.2 mm between the beam and the post at B. Determine the support reactions at A, B, and C. The post at B has a diameter of 40 mm, and the moment of inertia of the beam is \(I=875\left(10^{6}\right) \mathrm{mm}^{4}\). The post and the beam are made of material having a modulus of elasticity of E = 200 GPa.

The fixed supported beam AB is strengthened using the simply supported beam CD and the roller at F which is set in place just before application of the load P. Determine the reactions at the supports if EI is constant.

The beam has a constant \(E_{1} I_{1}\) and is supported by the fixed wall at B and the rod AC. If the rod has a cross-sectional area \(A_{2}\) and the material has a modulus of elasticity \(E_{2}\), determine the force in the rod.

The beam is supported by the bolted supports at its ends. When loaded these supports initially do not provide an actual fixed connection, but instead allow a slight rotation \(\alpha\) before becoming fixed after the load is fully applied. Determine the moment at the supports and the maximum deflection of the beam.

Each of the two members is made from 6061-T6 aluminum and has a square cross section 1 in. X 1 in. They are pin connected at their ends and a jack is placed between them and opened until the force it exerts on each member is 50 lb. Determine the greatest force P that can be applied to the center of the top member without causing either of the two members to yield. For the analysis neglect the axial force in each member. Assume the jack is rigid.

The beam is made from a soft linear elastic material having a constant EI. If it is originally a distance \(\Delta\) from the surface of its end support, determine the length a that rests on this support when it is subjected to the uniform load \(w_{0}\), which is great enough to cause this to happen.

The beam AB has a moment of inertia \(I=475 \mathrm{in}^{4}\) and rests on the smooth supports at its ends. A 0.75-in.- diameter rod CD is welded to the center of the beam and to the fixed support at D. If the temperature of the rod is decreased by 150°F, determine the force developed in the rod. The beam and rod are both made of A-36 steel.

The rim on the flywheel has a thickness t, width b, and specific weight \(\gamma\). If the flywheel is rotating at a constant rate of \(\omega\), determine the maximum moment developed in the rim. Assume that the spokes do not deform. Hint: Due to symmetry of the loading, the slope of the rim at each spoke is zero. Consider the radius to be sufficiently large so that the segment AB can be considered as a straight beam fixed at both ends and loaded with a uniform centrifugal force per unit length. Show that this force is \(w=b t \gamma \omega^{2} r / g\).

The box frame is subjected to a uniform distributed loading w along each of its sides. Determine the moment developed in each corner. Neglect the deflection due to axial load. EI is constant.