# 1-1 Locate the centroid of the plane area shown. 1-2 Determine the location of centroid of the composite area shown.

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1 Chapter 1 Review of Mechanics of Materials 1-1 Locate the centroid of the plane area shown 650 mm 1000 mm 650 x 1- Determine the location of centroid of the composite area shown mm radius 00 mm radius 00 mm 00 mm 600 mm 1- Verif that the radius of gration for a circle of diameter d with respect to a centroidal axis is d/ Determine the moment of inertia of the shaded area with respect to the x- axis. 400 mm 600 mm 00 mm 400 mm x 400 mm 400 mm 00

2 1-5 Determine the product moment of inertia of the triangle with respect to the x and axes. h G b x x 1-6 Determine the product moment of inertia of the triangle in the previous question with respect to the x and axes. The centroid of the triangle is at G. Answers: 1-1 x = 76mm, = 08 from the bottom left corner 1- x = 601mm, = 00mm from bottom left corner 1- Hint: find the area moment of inertia and the area m (b h )/ (b h )/7

3 Chapter Basic Elasticit -1 The two dimensional stress state at a point of an element of a material is given as shown. 0 MPa 40 MPa 75 MPa Calculate (a) the axial and shear stress on a plane whose normal is 40 0 clockwise to the x-direction (b) the magnitude and directions of the principal stresses and (c) the maximum shear stress. - A plane element is subjected to a constant axial stress of 50 MPa in the x- direction and an axial stress varing from -50 MPa to 50 MPa in the - direction. Plot the maximum shear stress acting in the plane element with respect to the axial stress in the -direction. What is the largest shear stress magnitude? - Determine the magnitude and directions of the principal strains and the maximum shear strain on an element with the following strains: ε x = 160 x 10-6 ; ε = -80 x 10-6 ; γ x = 10 x The principal strains have been found to be and respectivel. Determine (a) the maximum shear strain and (b) the maximum shear stress given that the shear modulus of elasticit is 6. GPa. -5 The element shown is subject to 50 MPa and 75 MPa compressive stresses in the x and directions respectivel and a shear stress of unknown magnitude but acting in the described sense. When this element is rotated clockwise at 5 o, the shear stress magnitude is equal but acts in the opposite sense; while the axial stress magnitudes are unchanged. Determine the value of the unknown shear stress. 75 MPa unknown 50 MPa

4 -6 The element shown is subject to an unknown axial stress in the x direction and zero axial stress in the direction. The shear stress is 0 MPa. When this element is rotated around, the maximum shear stress recorded is 50 MPa. Determine (a) the axial stress in the x direction, and (b) the principal stresses. 0 MPa 0 MPa unknown -7 A pair of strain gages gave the following readings: with 0 o gage = 500 microstrains, with 90 o gage = 100 microstrains. The strain gages register equal values after a 0 o anti-clockwise rotation. Determine (a) the maximum shear strain, and (b) the principal strains. -8 A beam of length l with a thin rectangular cross-section is clamped at the end x = 0 and loaded at the tip with vertical force P. Show that the stress distribution can be represented b φ = A + B x + Cx Determine the coefficients A, B, and C. -9 The cantilever beam shown is in a state of plane strain and is rigidl supported at x = L. Examine if the stress function given meets the biharmonic equation and boundar conditions. w 5 φ = (15h x 5x h + ) 0h

5 Answers: -1 (a) 71 MPa -6 MPa MPa (b) 88.5 MPa -4.5 MPa (c) 66 MPa - 50 MPa when the axial stress = -50 MPa x 10-6, -94 x 10-6, 7 x 10-6, 1 o -4 (a) (b) MPa MPa -6 (a) 80 MPa (b) 90 MPa, -10 MPa -7 (a) 680 microstrains (b) 540 microstrains, -140 microstrains -8 Pl / td, -P/td, P/td

6 Chapter Principles of Aircraft Construction -1 The Ford Trimotor, nicknamed The Tin Goose, was a three engine civil transport aircraft first produced in 195 b Henr Ford and continued until June 7, 19. The structure of the plane consists of a truss-work of U- shaped aluminum beams, with a thin skin of aluminum riveted on top, using skin corrugations instead of wing ribs and fuselage stringers. Briefl discuss the benefits and disadvantages with such a construction. - The Gossamer Albatross is a human-powered aircraft built b American aeronautical engineer Paul B. MacCread. Briefl discuss the merits of the external wire bracing construction used over truss-work or monocoque construction. - Briefl explain wh composite materials have led to huge advances in the monocoque construction of aircrafts.

7 -4 The double riveted joint shown connects two plates. If the failure strength of the rivets in shear is 70 N/mm, and the tensile strength of the plate is 465 N/mm, determine the rivet pitch if the joint is to be designed so that failure due to shear in the rivets and failure due to tension in the plate occur simultaneousl. Find also the joint efficienc. Answers: -4 1mm, 75%

8 Chapter 4 Airframe Loads 4-1 The aircraft shown weighs 15kN and has landed such that at the instant of impact the ground reaction on each main undercarriage wheel is 00kN and its vertical velocit is.5m/s. Find (i) the acceleration experienced. Each undercarriage wheel weighs.5kn and is attached to a strut. Calculate the (ii) axial load, and (iii) bending moment in the strut. At section AA the wing outboard of this section weighs 6.6kN and the center of gravit is.05m from AA. Calculate the (iv) shear force and (v) bending moment at section AA. 4- An aircraft makes a correctl banked turn at radius 610m at a speed of 168m/s. Find (i) the angle of bank, and (ii) load factor. Immediatel after making the turn and restoring to smmetric flight, the figure shows the relative positions of the center of gravit, aerodnamic center of the complete aircraft less the tailplane, and the tailplane center of pressure at zero lift incidence. The specifications are: Weight (W) = 1,500N; Wing area (S) = 46.5m ; Wing mean cord (c) = m; C D = C L ; C M,O = Find (iii) the lift coefficient, (iv) drag force, and (v) pitching moment. If the change in lift coefficient per wing incidence is 4.5/rad. Determine (vi) the tail load.

9 4- During pullout from a dive with zero thrust at 15m/s, an aircraft weighing 8,000N has the flight path at 40 o to the horizontal with radius of curvature 155m. The distance between the CG and tail is 1.m. The angular velocit of pitch is checked b appling an angular retardation of 0.5 rad/s. The moment of inertia of the aircraft for pitching is 04,000 kgm. Find (i) the additional tail load required to check the angular velocit in pitch. The aircraft has wings 88.5m in area, mean cord of 1m, and the pitching moment coefficient for all parts excluding the tailplane through the CG is given b C M.CG.c = 0.47C L Find (ii) the amount of lift, (iii) the lift coefficient, and (iv) pitching moment, and (v) tail load. (Hint: neglect the tail loads for the first approximation of lift, iterations is sufficient) Answers: m/s 19.kN 9kNm (clockwise) 0.m 19.5kN 59.6kNm o, 4.8, 0.80,,707N, -7,9Nm, 7,160N N, N, 0.59, 0880Nm, 1895N

10 Chapter 5 Torsion of Solid Sections 5-1 The stress function φ = k(r a ) is applicable to the solution of a solid circular section bar of radius a. Determine the stress distributions τ z, τ zx in the bar in terms of the applied torque, dw/dx, dw/d, and warping of the cross section. 5- A torque T is applied on the section comprising narrow rectangular strips shown. Determine (i) the torsional constant, (ii) the stress distributions τ z, τ zx, and (iii) the maximum shear stress. 5- The stress function φ = m(x a )( b ) is applicable to the solution of the rectangular section bar shown. Determine the stress distributions τ z, τ zx dw/dx, dw/d in the bar in terms of the applied torque. b x a Answers: 5-1 -Tx/πa 4, -T/πa 4, 0, 0, 0 (a + b) t d 5-, Gx θ T, 0, ± dz (a + b) t 9Tx( b ) 9T( x a ) dw 9T( x a + b ) 5- τ z =, τ zx =, =, 16a b 16a b dx a b dw 9Tx( x a + b ) = d a b

11 Chapter 6 Bending of Thin-Walled Beams 6-1 A bending moment of 000Nm is applied on the section shown at 0 o to the vertical axis. The sense of the bending moment is such that its components M x and M both produce tension in the positive x quadrant. Find the distances of C from edges BC and AB. Deduce the point where the flexural stress is maximum and calculate the amount. 6- A thin-walled cantilever beam of unsmmetrical cross-section supports the shear forces at the free end of the section shown. Calculate the flexural stress midwa along A on the beam. It can be assumed that no twisting of the beam occurs.

12 6- A thin walled beam has the cross-section shown. If the beam is subjected to a bending moment Mx in the plane of web, calculate the distribution of flexural stress in the beam cross section. Answers: mm, 8.4mm, C, 6.N/mm N/mm σ z,1 = Mx, σ Mx z, =, σ z, = Mx, σ z,4 = Mx h t h t h t h t

13 Chapter 7 Shear of Thin-Walled Beams 7-1 A beam has singl smmetrical thin-walled cross section shown. The thickness of the walls is constant throughout. Show that the distance of the shear centre from the web is given b ρ sin α cosα ξ s = d for ρ = d / h 1+ 6ρ + ρ sin α 7- A beam has singl smmetrical thin-walled cross section shown. Each wall of the section is flat and has the same length a and thickness t. Calculate the distance of the shear centre from point.

14 7- A uniform thin walled beam of thickness t has a cross-section in the shape of an isosceles triangle. It is loaded b a vertical shear force S applied at the apex. Calculate the shear flow over the cross section. Answers: 7-1 -Tx/πa 4, -T/πa 4, 0, 0, θ a a x x dz d G, + θ a x dz d G, dz d a x θ, dz d a a x θ +, dz d x a θ ) ( 1 7- ) ( ) / ( 1 1 d h h d h d s S q + =, ) ( ) 6 6 ( d h h h hs s S q + + =

15 Chapter 8 Virtual Work & Energ Methods 8-1 During a routine manufacturing operation, rod AB must acquire an elastic strain energ of 1 J. Determine the ield strength of the steel if the factor of safet = 5 and E = 00 GPa. B 18 mm diameter A P 1.5 m 8- Evaluate the strain energ of the prismatic beam for the loading shown. A P D B a L b 8- The element shown is taken from part of a bar subjected to axial stresses in x and axis. The shear stress is zero. Find the strain energ stored in the bar of volume.75 x 10-5 m. The modulus of elasticit is 00 GPa and the Poisson s ratio is 0.8. x 10 MPa 60 MPa 8-4 Determine the force in member AB in the truss shown in (a) using the principle of virtual work given the deformation described in (b).

16 8-5 Determine the slope A of the beam ABC at A using the principle of virtual work. 8-6 Calculate the vertical displacements of B and C in the simpl supported beam of length L and flexural rigidit EI using the energ method. 8-7 Calculate the loads in the members of the singl redundant pin-jointed framework using the energ method. The members AC and BD are 0mm in cross section and all other members are 0mm in cross section. The members AD, BC, and DC are 800mm long. E = 00,000N/mm.

17 Answers: MPa 8- Pab ( a+ b) 6EIL J kn 8-5 WL 16EI wL 5wL, 4576EI 84EI 8-7 R =.1 N

18 Chapter 9 Matrix Methods 9-1 The square smmetrical pin-jointed truss is pinned to rigid supports at and 4; whilst loaded at 1. The axial rigidit for all members is EA. Use the matrix method to (a) find the displacements in 1 and and (b) solve for all internal member forces and support reactions. 9- The displacement at node 4 of the pin-jointed frame is zero. Use the matrix method to find (a) the ratio H/P and the (b) displacements of nodes and. Answers: PL 9-1 v1 =, AE H 9- = , v P 0.9PL P v =, s 1 = s14 =, s = s4 = 0. 07P AE 4Pl 6Pl =, v = (9 + ) AE (9 + ) AE

19 Chapter 10 Stress/Strain Measurement 10-1 A cantilever bar is to be loaded as shown and the strain axial strain measured at midspan with strain gages. Briefl suggest a readout scheme wherein the highest voltage is obtained for the load applied. P L 10- In certain strain gage applications, it is necessar to record strains over a long period of time without having the opportunit to recheck the zero reading. The strain indicator will have an effect of the zero position drifting. Suggest how the measuring method can be done in order to eliminate the strain indicator drifting effect and how the instrumentation drift amount can be determined. 10- A birefringent disk of thickness of 5mm and material fringe value of 1.5 N/mm is viewed under a circular polariscope. Along a horizontal section in the middle, the outer ends have zero relative retardation. Find the principal stress difference at the middle of the disk. Answers: 10-15N/mm

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