Question 1 A pin-jointed plane frame, shown in Figure Q1, is fixed to rigid supports at nodes and 4 to prevent their nodal displacements. The frame is loaded at nodes 1 and by a horizontal and a vertical downward force of 40 N and 100 N respectively, as shown. Using the finite element method of structural analysis, determine displacements at nodes 1 and. All members of the frame have the same value of the axial rigidity, equal to 400 N. [5 mars] 100 N 4 1 m 1 40 N 1 m 1 m Figure Q1 Page 1 of 9 AE0 Aircraft Struct Paper1_ay.doc
Question A rigidly jointed plane frame is shown in Figure Q with rigid supports at points 1 and. The structure is loaded at node as shown. The cross-sectional area and the second moment of area for both the structural members of the frame are 7500 mm and 5 10 6 mm 4 respectively The oung s modulus E for the material is 00 GPa. Using the finite element method of structural analysis, calculate the displacements at node. [5 mars] 150 N 4.5 m 150 Nm 1 6 m 6 m Figure Q Page of 9 AE0 Aircraft Struct Paper1_ay.doc
Question (a) n the usual notation, derive the governing differential equation for Euler (elastic) critical bucling analysis of a column. [9 mars] (b) Obtain the general solution of the equation derived in (a) and then derive the expressions for slope, bending moment and shear force. [6 mars] (c) Determine the critical bucling load of a column whose one end is built-in and the other end is free. [10 mars] Page of 9 AE0 Aircraft Struct Paper1_ay.doc
Question 4 A uniform straight strut has a thin-walled hollow cross-section in the shape of a rectangle 100 mm by 50 mm along the wall median lines. The wall thicness is the same throughout at.5mm. The strut is 1. m long and is fixed to foundations at each end which allow only axial movement at the ends. The strut material has the stress-strain relationship of the form. ε = E + B R n where ε = direct strain, = direct stress B and n are constants to be found. E = oung's odulus = 168 Gpa 1 = 0.1% proof stress = 56 pa = 0.% proof stress = 61 pa Estimate the minimum flexural bucling load for this strut and compare it with the elastic bucling load, maing a brief appropriate comment from a design perspective. [5 mars] Page 4 of 9 AE0 Aircraft Struct Paper1_ay.doc
Question 5 A uniform thin-walled beam of length m, which is built-in at point A, is subjected to a concentrated load of 100 N at the free-end position B, as shown in Figure Q5(a). The load at the free-end position B is acting vertically downward in the cross section of the beam and the beam has an asymmetrical cross section of wall thicness 1.5 mm throughout and dimensions as shown in Figure Q5(b). The wall thicness may be considered to be small in comparison with other dimensions when calculating the cross-sectional properties of the beam. (a) Determine the location of the centroid G and sectional properties xy of the beam s cross section., and xx yy [ 7 mars] (b) Calculate the maximum stresses in the cross section at the built-in end of the beam. [1 mars] (c) Determine the neutral axis and setch the stress distribution in the cross section. [6 mars] 100N 100 N A B 1 y m G x 50 mm 1.5 mm Figure Q5(a) 4 5 0 mm 0 mm Figure Q5(b) (Not to scale) Page 5 of 9 AE0 Aircraft Struct Paper1_ay.doc
Question 6 n the two-dimensional stress system, shown in Figure Q6, the shear stress on an inclined plane AC at 5 o to the reference plane AB is 70 N/m. Determine the following: (a) The normal stress y on plane BC and the normal stress AC on the inclined plane AC. [8 mars] (b) The magnitudes and directions of the principal stresses relative to AB. [8 mars] (c) The magnitudes and directions of the maximum shear stress plane relative to AB. [4 mars] (d) The magnitudes of the principal strains that would occur in this two-dimensional stress field. [5 mars] Tae E = 07 10 N/m and ν = 0.. y A 0 N/m AC 10 N/m 10 N/m 5 o 70 N/m B C 0 N/m y Figure Q6 Page 6 of 9 AE0 Aircraft Struct Paper1_ay.doc
Question 7 Using a 60 strain gauge rosette in a uniform strain field, the strains recorded to be 700, 100, and 00 µε, respectively, as shown in Figure Q7. (a) Determine the magnitudes and directions of the principal strains. ε a, ε b and ε c are [1 mars] (b) What are the principal stresses associated with these strains and how do they act? [5 mars] (c) f gauge [b] is set incorrectly with an error of 10, i.e. at 50 o instead of 60 o from gauge [a], what strain would be recorded in gauge [b]? What would be the consequent percentage error for the principal strains if gauge [b] setting error were not detected? [8 mars] Tae E = 07 x 10 N/m and ν = 0. [a] 700 µε 60 0 [c] 00 µε [b] 100 µε Figure Q7 Examiners: Prof. J.R. Banerjee Dr. C.W. Cheung External Examiner: Prof. D..A. Poll Page 7 of 9 AE0 Aircraft Struct Paper1_ay.doc
nformation Sheet 1 (A) The stiffness matrix of a bar element in global co-ordinates is given by 11 11 1 = where 1 [ ] EA l lm = = 1 = 1 = L lm m where EA and L are respectively the axial rigidity and the length of the bar; l and m are the direction cosines of the local co-ordinate axes with respect to the global coordinate axes. (B) The stiffness matrix of a beam element in global co-ordinates is given by 11 1 = where 1 [ ] F H [ ] = G P Q [ ] H G 11, Q R F = G H G P Q H Q R F H [ ] = G P Q 1 and [ ] H G Q B 1 F = G H G P Q H Q B where EA 1E F + L L EA 1E 6E =, H = m L L L = l m, G lm lm EA 1E P + L L 6E =, L = m l, Q l R = 4E, B = E. L L EA and E are respectively the axial and bending rigidity of the beam; l and m are the direction cosines of the local co-ordinate axes with respect to the global co-ordinate axes and L is the length of the beam. Page 8 of 9 AE0 Aircraft Struct Paper1_ay.doc
nformation Sheet The following relationships are given in the usual notation. (i) Shear flow in a singly symmetric closed section q S = q o S S 0 t y ds (ii) Rate of twist and torque are related as d θ = d z T GJ (iii) Torsion constant (J) for closed and open sections are respectively given by (iv) Bredt-Batho formula 4A J C = d s t t, J O = d s T = A q L O (v) Bending stress due to asymmetric bending Z = y + x or Z y = + x (vi) Differential equations governing vertical and horizontal deflections due to asymmetric bending are respectively given by d d z v = E, d d z u = E where / =, 1 / / =. 1 / Page 9 of 9 AE0 Aircraft Struct Paper1_ay.doc