Deflections by Integration of the substitute the expression of M(x) into the deflection equation then integrating to satisfy d'y M(x) (i) boundary conditions dx² EI (ii) continuity conditions (iii) symmetry conditions to obtain the slope 0 = Bending-Moment Equation and the deflection y of the beam this method is called method of successive integration VA=0 VB=0

Problem 2 -A concentrated couple, = 600 Nm acts at the center of the simply Supported beam which is also asted upon by the distributed load shown in the figure. Assume that the beam itself is weightless. (a) Find the internal shear Vx) and moment M(x) as a functions of x. (b) Draw the shear and moment diagrams. Label the values at x = 0, x = 6, x = 12, and at any point where V(x) or M(x) is a local maximum or minimum. y 50 N/m Mo 6 m 6 m

a. Find the value of R1 for the beam loaded as shown in KN b. Find the value of R2 for the beam loaded as shown in KN c. Given the beam loaded as shown, find the value of C1 which is the first constant integration when you form the moment equation of the beam loaded as shown and integrate it once d. Determine the value of C2 when you use double integration method in determining the deflection of the beam shown

1) Find the transfer function from the force F (applied at x= L/3 the bar) to the motion Y of the location x=2L/3 a bar (with fixed-fixed) boundary conditions. Consider only the first mode of vibration with zero damping. The mass of the rod is M, mass-density of the rod is p, its length is L, Young's modulus is E, and cross-sectional area is A. 2) Find the response Y(t) for a unit step input with zero initial conditions. F(t) Y(t) →L/3 → 2L/3

A Mo ( C 1 Q: al Find Slope at Ends and deflection at c using the conjugate beam method 61 what if a → 0

Derive the equation of the resonance frequency of the uniform cantilever beam shown below by using method of conservation of energy. Solution procedure must be the principle of conservation of energy! Umax, maximum strain energy is equal to max. kinetic energy, Tmax of vibrating body. Tmax 1/2 S. y³²m dx U= 10. M(x)2 EI

Problem 2

Determine completely the deflection and rotation at point C of the beam shown using: 1.) Double Integration 2.)Area moment 3.) Conjugate method Just pick two method. Given: E=200 GPa |=250x10^6 mm^4 w=10KN/m L1=6m L2=4m w (kN/m) A В Li L2 RA RB

B.2. Elementary beam theory predicts that the axial bending stress ox in a prismatic beam is given by: σ, X (M.1, M,1,)y+ (M, I. - M.1,-) z (1,1.-1²-) where My and M₂ are bending moments applied to a cross-section, and where ly, Iz and lyz are second moments of area in the usual notation (Oxyz is a Cartesian coordinate system in which the x axis corresponds to the centroidal axis of the beam). (i) What assumptions have been made in the derivation of the above expression? (ii) Indicate by means of a sketch the directions in which positive values of the bending moments My and M₂ act on a cut plane facing in the positive x direction.

QUESTION 1 (a) Sketch a bar element, a plane truss element, and a beam element. Label the axes, nodes and nodal degrees of freedom (dof). Identify the physical quantity of each dof. (b) Figure Q1(a) below shows a loaded cantilevered beam. The beam is made of a material with E = 200 GPa while the cross section of the beam is shown in Figure Q1(b). کسیس 2 kN/m 1 6 m 10 KN (a) 2 4 m 8 kN 40 kN-m Figure Q1 200 mm (b) 400 mm