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Consider a rectangular plate with sides a and b and thickness t, as shown in the figure at left b...
62 (1%) Consider a rectangular plate with sides a and b and thickness t, as shown...
62 (1%) Consider a rectangular plate with sides a and b and thickness t, as shown in the figure at left below. a. Biharmonic equation. If we assume an Airy's stress function: Ф(x,y)-p(x'-y'), where p is a constant, will the biharmonic equation be satisfied? Ignore the body forces. b. Stresses. Determine the stresses (σ.σ..). c. Resultant forces & moment. Apply the Saint-Venant's Principle to find the resultant normal and shearing boundary forces as well as resuitant bending moment along all...
2. A rectangular plate with sides 6 m and 4 m of thickness l 12 mm (Fig. 1) is subjected to the s...
2. A rectangular plate with sides 6 m and 4 m of thickness l 12 mm (Fig. 1) is subjected to the stresses σο Ơy and Fry. An Airy stress function is proposed to be φ = 12x3y. (a) Is φ a possible valid Airy stress function? Prove your answer by a calculation. (b) If φ is valid, determine the values of stresses (Ox, σ, and τ, at point (xy,z) which satisfy the stress function. (c) Draw a sketch showing...
The rectangular plate shown in the figure below has the given loads uniformly distributed over the...
The rectangular plate shown in the figure below has the given loads uniformly distributed over the edges. The plate is 50 mm thick, AB is 500 mm and BC is 400 mm (a) (b) (c) (d) (e) Determine the shear forces which must operate on the edges BC and DA to maintain the equilibrium of the plate Relative to the x,y reference axes, determine the state of stress at any point P in the interior of the plate For the...
1. Consider a rectangular plate with sides a and b of thickness t, as showen in...
1. Consider a rectangular plate with sides a and b of thickness t, as showen in Fig. P1, for the stress function 0 = px?y, where p is a constant, what will be the resultant normal bounday force Py allong the edge of the plate y = b? (a) 3pabt, (b) pat, (c) 2pabt, (d) 0, (e) none of above YA 3 4 6 4 25 6 51 MPa X a Fig. P1 Fig. P2
1. Consider a rectangular plate with sides a and b of thickness t, as showen in...
1. Consider a rectangular plate with sides a and b of thickness t, as showen in Fig. P1, for the stress function Ø = px3y, where p is a constant, what will be the resultant normal bounday force Py allong the edge of the plate y = b? (a) 3pabt, (b) pa’t, (c) 2pa’bt, (d) 0, (e) none of above YA b 3 4 6 4 2 5 6 5 1 MPa X k a Fig. P1 Fig. P2
7.7 A thin plate is rigidly fixed at its edges (see Figure Ex7.7). The plate has...
7.7 A thin plate is rigidly fixed at its edges (see Figure Ex7.7). The plate has a height L and thickness t (normal to the plane of the figure). A crack moves from left to right through the plate. Every time the crack moves a distance Ax, two things happen: 1. Two new surfaces (with specific surface energy) are created. 2. The stress falls to zero behind the advancing crack front in a certain volume of the material. Obtain an...
The cantilever beam shown in the figure is subjected to a distributed shear stress oon the upper face. The following Airy stress function is proposed for this problem Determine the constants c and fi...
The cantilever beam shown in the figure is subjected to a distributed shear stress oon the upper face. The following Airy stress function is proposed for this problem Determine the constants c and find the stress distribution in the beam. Use resultant force boundary conditions at the ends. (Answer: C1-ToC/12/, С2-Tp/201.c3- -to/24cl,...) TOXI Suggested process is as follows: (a) Write out the boundary conditions using St. Venant's principle for the semi-inverse (b) Write the stress field corresponding to the Airy...
1. Consider a thin rectangular plate in the ry-plane, the figure. The PDE describing the temperature...
1. Consider a thin rectangular plate in the ry-plane, the figure. The PDE describing the temperature of the plate is the heat equation shown in as 0 xa, 0< y < b, t>0. D + at where u(x, y, t) is the temperature at point (x, y) diffusivity at time t andD> 0 is the thermal (a) Suppose that the solution to the PDE (once we impose initial and boundary con ditions) reaches equilibrium when t o, that is there...
Problem 1 A parallelepiped with edges a, b, c was deformed as shown on the figure. Find the strai...
Problem 1 A parallelepiped with edges a, b, c was deformed as shown on the figure. Find the strain tensor in the linear approximation assuming that 0 is small. Remember, the answer must be a 3x3 tensor Problem 2. For the deformation from the previous problem assuming that the sheer modulus is u find: 1. The stress tensor 2. The force F which has to be applied to the top surface 3. Total torque of all external forces. 4. Are...
A uniform load Po acts on a long and narrow rectangular plate with edges at y 0 and y b both clamped as shown in figure...
A uniform load Po acts on a long and narrow rectangular plate with edges at y 0 and y b both clamped as shown in figure Q 4b below. For Po 20 kN/m, b 0.6 m, t 12 mm, E 200 GPa, and v 0.333. Determine the maximum -deflection of the plate (6.5 Marks) b
A uniform load Po acts on a long and narrow rectangular plate with edges at y 0 and y b both clamped as shown in...
62 (1%) Consider a rectangular plate with sides a and b and thickness t, as shown in the figure at left below. a. Biharmonic equation. If we assume an Airy's stress function: Ф(x,y)-p(x'-y'), where p is a constant, will the biharmonic equation be satisfied? Ignore the body forces. b. Stresses. Determine the stresses (σ.σ..). c. Resultant forces & moment. Apply the Saint-Venant's Principle to find the resultant normal and shearing boundary forces as well as resuitant bending moment along all...
2. A rectangular plate with sides 6 m and 4 m of thickness l 12 mm (Fig. 1) is subjected to the stresses σο Ơy and Fry. An Airy stress function is proposed to be φ = 12x3y. (a) Is φ a possible valid Airy stress function? Prove your answer by a calculation. (b) If φ is valid, determine the values of stresses (Ox, σ, and τ, at point (xy,z) which satisfy the stress function. (c) Draw a sketch showing...
The rectangular plate shown in the figure below has the given loads uniformly distributed over the edges. The plate is 50 mm thick, AB is 500 mm and BC is 400 mm (a) (b) (c) (d) (e) Determine the shear forces which must operate on the edges BC and DA to maintain the equilibrium of the plate Relative to the x,y reference axes, determine the state of stress at any point P in the interior of the plate For the...
1. Consider a rectangular plate with sides a and b of thickness t, as showen in Fig. P1, for the stress function 0 = px?y, where p is a constant, what will be the resultant normal bounday force Py allong the edge of the plate y = b? (a) 3pabt, (b) pat, (c) 2pabt, (d) 0, (e) none of above YA 3 4 6 4 25 6 51 MPa X a Fig. P1 Fig. P2
1. Consider a rectangular plate with sides a and b of thickness t, as showen in Fig. P1, for the stress function Ø = px3y, where p is a constant, what will be the resultant normal bounday force Py allong the edge of the plate y = b? (a) 3pabt, (b) pa’t, (c) 2pa’bt, (d) 0, (e) none of above YA b 3 4 6 4 2 5 6 5 1 MPa X k a Fig. P1 Fig. P2
7.7 A thin plate is rigidly fixed at its edges (see Figure Ex7.7). The plate has a height L and thickness t (normal to the plane of the figure). A crack moves from left to right through the plate. Every time the crack moves a distance Ax, two things happen: 1. Two new surfaces (with specific surface energy) are created. 2. The stress falls to zero behind the advancing crack front in a certain volume of the material. Obtain an...
The cantilever beam shown in the figure is subjected to a distributed shear stress oon the upper face. The following Airy stress function is proposed for this problem Determine the constants c and find the stress distribution in the beam. Use resultant force boundary conditions at the ends. (Answer: C1-ToC/12/, С2-Tp/201.c3- -to/24cl,...) TOXI Suggested process is as follows: (a) Write out the boundary conditions using St. Venant's principle for the semi-inverse (b) Write the stress field corresponding to the Airy...
1. Consider a thin rectangular plate in the ry-plane, the figure. The PDE describing the temperature of the plate is the heat equation shown in as 0 xa, 0< y < b, t>0. D + at where u(x, y, t) is the temperature at point (x, y) diffusivity at time t andD> 0 is the thermal (a) Suppose that the solution to the PDE (once we impose initial and boundary con ditions) reaches equilibrium when t o, that is there...
Problem 1 A parallelepiped with edges a, b, c was deformed as shown on the figure. Find the strain tensor in the linear approximation assuming that 0 is small. Remember, the answer must be a 3x3 tensor Problem 2. For the deformation from the previous problem assuming that the sheer modulus is u find: 1. The stress tensor 2. The force F which has to be applied to the top surface 3. Total torque of all external forces. 4. Are...
A uniform load Po acts on a long and narrow rectangular plate with edges at y 0 and y b both clamped as shown in figure Q 4b below. For Po 20 kN/m, b 0.6 m, t 12 mm, E 200 GPa, and v 0.333. Determine the maximum -deflection of the plate (6.5 Marks) b
A uniform load Po acts on a long and narrow rectangular plate with edges at y 0 and y b both clamped as shown in...
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