Solved Problems Review of Mechanics of Materials 1-1 Locate the centroid of the complex plate shown.

1-2 A beam is constructed by gluing a 20mm x 40mm wooden plank to a second wooden plank 20mm x 80mm to form a cross section as shown. Determine the second moment of the cross-sectional area with respect to (a) the x-axis (b) the y-axis (c) the xc axis, which passes through the centroid of the area and is parallel to the x-axis, Determine also the radius of gyration with respect to the xc axis

x

y

80

20 40

201

2

1-3 Determine the product moment of inertia of the beam about the x and y centroidal axis.

x

600 mm

400 mm

100 mm

100 mm

A

B

C

O

y

100 mm

300 mm

400 mm

Basic Elasticity 2-1 For the state of stress shown, determine the stress components exerted on the

element by rotating counterclockwise through 300.

2-2 For the state of plane stress in the element shown, determine (a) the principal stresses and (b) the maximum shearing stress. Indicate the rotation of the element necessary to bring them about.

2-3 Solve Problem 4-1 using Mohr's circle for stress diagram.

2-4 Solve Problem 4-2 using Mohr's circle for stress diagram. 2-5 For a material in a state of plane strain, it is known that a horizontal side of

10mm x 10mm square elongates by 4 microns while its vertical side remains unchanged. The angle of at the lower left corner increases by 0.4 x 10-3 rad. Determine (a) the principal strains and (b) the maximum shear strain.

2-6 A strain gage rosette on a plate under stress gave the following readings: with

0o gage = 592 microstrains, with 45o gage = 308 microstrains, with 90o gage = –432 microstrains. Determine the principal strains acting on the plate.

x

y

100 MPa

48 MPa

60 MPa

x

y

50 MPa

40 MPa10 MPa

y

x

y

x10 mm

10 mm

mmm μ+ 410

rad3104.02

−×+π

2-7 A cylindrical pressure vessel is fabricated by welding a rolled 5mm thick sheet of mild steel along a spiral which is 650 to the pipe axis. The inner diameter of the pipe is 280mm. In order for the weld not to fail under an operating internal pressure of 2.5 MPa, calculate the maximum axial and shear stress normal to the weld line.

2-8 A cylindrical vessel of thickness t and radius r were both subjected to an

internal pressure P. Derive expressions for the circumferential and meriodinal strain if the Poisson's ratio is μ and the modulus of elasticity is E.

2-9 Find the required loading condition on the rectangular sheet shown using a

complex polynomial for the stress function:

6226

3223 DyCxyyBxAx+++=φ

Determine the form of loading on the plate.

weld-line

65o

Principles of Aircraft Construction 3-1 Briefly explain how the monocoque and semi-monocoque constructions are

used in relation to the types of load acting on the aircraft.

3-2 The fuselage skin in an aircraft is constructed by riveting the skins between two

straps as shown. If the tensile stress in the skin must not exceed 125 N/mm2, and shear stress in the rivet is limited to 120 N/mm2, determine the maximum allowable rivet spacing such that the joint is equally strong in shear and tension.

Airframe Loads 4-1 An aircraft having a total weight of 45kN lands on the deck of an aircraft

carrier and is brought to rest by means of a cable engaged by an arrester hook. If the deceleration induced by the cable is 3g, determine the tension T in the cable on an undercarriage strut and the shear and axial loads in the fuselage at section AA; the weight of the aircraft aft of AA is 4.5kN. Calculate also the length of deck covered by the aircraft before it is brought to rest if the touchdown speed is 25m/s.

4-2 An aircraft having weight 250kN and a tricycle undercarriage lands at a vertical

velocity of 3.7m/s such that the vertical and horizontal reactions on the main wheel are 1200kN and 400kN respectively. At this instant the nose wheel is 1m from the ground. If the moment of inertia of the aircraft about its C.G. is 5.65 x 108 Ns2mm, determine the inertia forces on the aircraft. Find also the time taken for its vertical velocity to become zero and its angular velocity.

4-3 The curves CD, α and CM,CG for a light aircraft is as shown. The aircraft weight

is 8000N, its wing area 14.5m2 and its mean chord 1.35m. Determine the lift, drag, tail load, and forward inertia force for a symmetric maneuver corresponding to n = 4.5 and a speed of 60m/s. Assume that engine-off conditions apply and the air density is 1.223kg/m3.

4-4 A supersonic airliner shown flies 610m/s at an altitude where ρ = 0.116kg/m3

and has the following relevant data: Wing area (S) = 280m2; tail area (ST) = 28m2; weight (W) = 1,600,000N; dCL/dα = 1.5; dCL,T/dα = 2.0; CM,O = -0.01, mean chord (c) = 22.8m Find the (i) lift, (ii) wing incidence angle, and (iii) tail load. It can be assumed that CL= (α)dCL/dα. The aircraft encounters a sharp vertical-up gust of 18m/s. Calculate the changes in (iv) lift, and (v) tail load.

Torsion of Solid Sections 5-1 An elliptical bar shown is subjected to a torsion T. Find expressions for the (i)

torsional constant, shear stress distributions (ii) τzx, (iii) τzy, and (iv) warping.

5-2 Show that the stress function

⎥⎦⎤

⎢⎣⎡ −−−+

θ−=φ 22322

272)3(

21)(

21 axyx

ayx

dzdG

is correct for a bar having a cross-section in the form of an equilateral triangle shown. Find the expressions for the torque T, the stress distributions τyz, τzx, the angle of twists dw/dx, dw/dy, and warping of the cross section.

5-3 A torque T is applied on narrow rectangular strip shown. Determine (i) the

torsional constant, (ii) the stress distributions τyz, τzx, and (iii) the maximum shear stress.

x

y

a

b

a

t x

y

Bending of Thin Walled Beams 6-1 The cross section of a beam has dimensions shown. If the beam is subjected to

a negative bending moment of 100kNm applied in the vertical plane, determine the stress distribution through the depth of the section.

6-2 A beam having the cross-section shown is subjected to a bending moment of

1500Nm in a vertical plane. Calculate the maximum flexural stress due to bending starting at the point in which it acts.

6-3 Determine the horizontal and vertical components of the tip deflection of the cantilever shown.

6-4 Determine the flexural stress distribution in the thin-walled section shown that

is produced by a positive bending moment Mx.

6-5 A thin-walled cantilever with walls of constant thickness t is loaded by a

vertical force W at the tip and a horizontal force 2W at the mid-section. Both these forces act through the shear center. Determine the distribution of flexural stress along the length of the beam for points 1 and 2 of the cross section.

Shear of Thin-Walled Beams 7-1 Determine the shear flow distribution in the thin-walled Z section shown due to

a shear load Sy applied through the shear center of the section.

7-2 A thin-walled closed section has a single symmetric cross section shown. Each

wall of the section is flat, has the same thickness t and shear modulus G. A shear load Sy is applied in the vertical axis through S. Find the shear flow distribution, and the distance of shear center from point 4.

7-3 The cross-section of a thin singly symmetrical I is as shown. Show that the distance of the shear center from the vertical web is given by

ρ+β−ρ

=ξ121

)1(3s

Virtual Work and Energy Methods 8-1 Determine the bending moment at point B in the simply supported beam using

virtual work.

8-2 Determine the vertical deflection of the free end of the cantilever beam.

8-3 Confirm the strain energy equations for members under the action of (a) tensile force and (b) torsion.

8-4 Determine the deflection of the mid-span point of the linearly elastic, simply

supported beam shown. The lower extremity of a compound shaft is subject to a torque of 5 kNm. At the junction of the shaft, a torque of 8 KNm in opposite direction is applied. Determine the total energy stored in the shaft. G = 80 GNm-2.

8-5 Find the deflection at the free end of a cantilever subjected to a concentrated load at the free end.

1 m

1 m

A

B

C

100 mm

75 mm

8 kNm

5 kNm

8-6 Find the reaction R in member BD of the truss shown when a horizontal force P is applied at C.

Matrix Methods 9-1 Determine the horizontal and vertical components of the deflection of node 2

and the forces in the members of the pin-jointed framework shown. Assume that the cross-sectional area A and modulus of elasticity E to be constant for all members.

Stress/Strain Measurement 10-1 List some of the characteristics that a well performing strain gage should

exhibit. 10-2 The ruler was observed under a plane polariscope. Explain why fringe patterns

are present even when no load is applied.