Download JNTUK M-Tech 2020 R19 Civil 1104 Analytical& Numerical Methods For Structural Engineering Model Question Paper

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I M. Tech I Semester (R19) Regular Examinations

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ANALYTICAL& NUMERICAL METHODS FOR STRUCTURAL ENGINEERING

(STRUCTURAL ENGINEERING)

MODEL QUESTION PAPER

TIME: 3 Hrs. Max. Marks: 75

Answer ONE Question from EACH UNIT

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All questions carry equal marks

UNIT -1

1. a). Using the Laplace transform method solve the Initial Boundary Value Problem (IBVP) described as PDE ?²u/?t² = c² ?²u/?x² - cos(ax) ; 0 < x < 8 , 0 < t < 8. Also given boundary conditions are u(x,0)=u_t(x, 0)=0.
   b). Write the Laplace transform of {1/vt}.

OR

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2. a). A string is stretched as fixed between two points (0, 0) & (/, 0). Motion is initiated by displacing the string in the form of u = 4sin(px/l) and released from rest at time t=0. Find the displacement of any point on the string at any time t.
   b). State the heat conduction problem in semi - infinite rod.

UNIT - II

3. a). Using the Fourier transform method solve the solution of 2D Laplace equation ?²u/?x² + ?²u/?y² = 0 , is valid in the half -plane , y > 0, is subjected to the condition U(x,0)=0 if x<0, u(x,0)=1 if x>0 and lim_y?8 u(x,y)=0 in the half plane.
   b). Write the change of scale property of Fourier transforms

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OR

4. a). Find the curves on which the functional ?₀¹ (y'² +12xy)dx with y(0)=0 and y(1)=1 can be extremised.
   b). Show that the curve which extremizes the functional I = ?_x1^x2 [y'² +x²]dx under the conditions

UNIT - III

5. a). Verify that u(x) = x² is a solution of the Volterra Integral equation u(x) = sinx + 2?₀^x cos(x - t)u(t)dt
   

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   b). Convert d²y/dx² +xy=1, y(0)=0, y(1) =1 into an integral equation

OR

6. a). Find the Eigen values and Eigen functions of the Integral Equation u(x) = ? ?₀¹ e^x u(t)dt
   b). Solve the homogeneous Fredholm Integral equation of second kind u(x)=? ?₀^2p sin(x + t)u(t)dt

UNIT - IV

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7. a). From the following table, estimate the number of students who obtain marks between 40 and 45.

   Marks	30-40	40-50	50-60	60-70	70-80
   No. of Students	31	42	51	35	31

   
   b). Find by Taylor’s series method the value of y at x =0.1 and x =0.2 to five places of decimals from dy/dx =x²y-1, y(0)=1

OR

8. a). A beam of length /, supported at n points carries a uniform load w per unit length. The bending moments M_i-1, M_i, M_i+1 at the supports satisfy the Clapeyron’s equation: M_i-1 + 4M_i + M_i+1 = (3/8)wl². If a beam weighing 30 kg is supported at its ends and at two other supports dividing the beam into three equal parts of 1 meter length, show that the bending moments at each of the two middle supports is 1 kg meter.
   b). The deflection of Beam is given by the equation d⁴y/dx⁴ +81y = f(x) , where f(x) is:

   X	1/3	2/3	1
   f(x)	81	162	243

   And boundary condition y(0)=y '(0) =y''(1) = y'''(1) = 0. Evaluate the deflection at the pivotal points of the beam using three sub intervals.

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UNIT -V

9. a). Given Values

   X	5	7	11	13	17
   f(x)	150	398	1492	2366	5202

   Evaluate f(9) using Lagrange Formula
   b). Use the Composite Trapezoidal Rule with m =n = 2 to evaluate the double integral ?₀¹?₀¹ e^x+ydxdy

OR

10. a). Evaluate the double integral ?₀¹?_x¹ (x² +y² )dydx
    

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    b). Apply Newmark Method with suitable example

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