Derive an expression of the maximum statical moment of area, Q and its location from the given x-axis.
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- Problem II. Refer to the cross section shown below. b h 30 a) Derive an expression of the maximum statical moment of area, Q and its location from the given x-axis. b) Derive an expression for the maximum bending stress. c) Derive an expression for the maximum horizontal shear stress.Problem 4. Determine the value of the constant C in the stress function 0 = C{r?(x -0) + p?sin 0 cos 0 - r? cos? 0 tan a required to satisfy on the upper and lower edges of a triangular plate shown below. Evaluate the stress components o, Ty for a vertical section mn. Draw curves for the case a = 20° and draw also for comparison the curves given by elementary beam theory. m a yThe T-beam is subjected to a bending moment of M = 150 kip in. directed as shown. Determine the maximum bending stress in the beam and the orientation of the neutral axis. Make sure you check all possible points (corners) for max stress and/or state other reasoning for the point that you pick. Answer: [σmax = 3.33 ksi, β = -63.1°]
- SOLVE THE PROBLEM USING DOUBLE INTEGRATION METHOD Problem II. The midpoint B of the steel cantilever beam ABC is supported by a vertical aluminum rod. Find the maximum allowable value of the applied force P if the stress in the rod is not to exceed 120 MPa. Use Est = 200 GPa, I = 50 x 10 mm4 for the beam, Eal = 70 GPa, and A = 42 mm² for the rod. 5 m 2 m Aluminum -Steel 2 m C30 mm C A B 150 mm M 150 mm 300 mm 30 mm - determine the bending stress developed at points A, B, and C. Sketch the bending stress distribution on the cross section. - Assume that the cross-section has a moment of inertia about the neutral axis of 1.907 x 10-4 m². - Assume that the neutral axis is located 97.5 mm from the bottom of the cross-section - Assume the internal bending moment M = 61.4 kN.mThe cantilever beam is subjected to a concentrated load P. The cross-sectional dimensions of the rectangular tube shape are shown in the second figure. Assume b = 133 mm, d = 207 mm, y = 62 mm, yk = 37 mm and t = 6 mm. (a) Compute the value of Q that is associated with point H, which is located 62 mm above the centroid of the rectangular tube shape. (b) If the allowable shear stress for the rectangular tube shape is 117 MPa, determine the maximum concentrated load P that can be applied to the cantilever beam. Answers: (a) Q = (b) P = i i P Z mm³. kN. ун Ук H K b t (typ.)
- From the previous problem, determine the shear stress 30 mmabove the neutral axis.(c) Beam AB is made of three planks glued together and is subjected, in its plane of symmetry, to the loading as shown. Knowing that the width of each glued joint is 20 mm, determine the average shearing stress in each joint at section n-n of the beam. The location of the centroid of the section is given in the sketch and the centroidal moment of inertia is known to be l = 8.63 x 10 m². 1.5 KN 1.5 KN n n - 0.4 m 20 mm 80 mm 20 mm 0.2 m 100 mm C -Joint a -20 mm 0.4 m- 60 mm- 68.3 mm Joint b BFor a rectangular curved beam in pure bending, polar coordinates are used. Neglecting body forces, the following Airy stress function has been suggested: Pre = A+ Blnr + Cr² + Dr²Inr, (r,0) where A, B, C, and D are constants. Can re) be an Airy stress function? If so, establish: re [0,]=| r
- The T-shaped beam shown above is supporting a concentrated load P at its free end. The beam has an allowable bending stress of ?????? = 250 MPa and an allowable shear stress of ?????? = 100 MPa. a) Determine the distance to the neutral axis (?̅), second moment of area (?), and the section modulus (?) of the cross-section.b) Draw the shear force diagram (SFD) and bending moment diagram (BMD) of the beam. On your diagrams, express the values of shear and moment in terms of the applied load P.c) Determine the maximum value of P such that bending failure will not occur.d) Determine the maximum value of P such that shear failure will not occur.e) Based on your answers to (c) and (d), what is the maximum load P that can be applied to the beam? Is this beam bending or shear governed?If σsafety=140MPa (safety stress) in the beam whose loading condition and cross-section are visible, find the maximum value of the P force, Based on the assumption of Maximum shear stress (Tresca) and deformation energy (Von Mises) theorems. Draw the shear stress diagram for the force P found.P 3 m 3 m 120 mm 50 mm 50 mm 20 mm 20 mm 150 mm N/A y The T-shaped beam shown above is supporting a concentrated load P at its free end. The beam has an allowable bending stress of allow = 250 MPa and an allowable shear stress of Tallow= 100 MPa. a) Determine the distance to the neutral axis (y), second moment of area (I), and the section modulus (S) of the cross-section. b) Draw the shear force diagram (SFD) and bending moment diagram (BMD) of the beam. On your diagrams, express the values of shear and moment in terms of the applied load P. c) Determine the maximum value of P such that bending failure will not occur. d) Determine the maximum value of P such that shear failure will not occur. e) Based on your answers to (c) and (d), what is the maximum load P that can be applied to the beam? Is this beam bending or shear governed?