Airy

Examples to Solve
in the course TMHL03 Mechanics of Light Weight Structures

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Chapter

1

In-plane loaded plates in Cartesian coordinates
1.1
A rectangular plate is fixed at a rigid wall and it is carrying a triangularly distributed total load Q in accordance to the figure below. Define the boundary conditions along all four edges. Plane stress conditions can be assumed. The length ofthe plate is L, the height H and the thickness t.
y Q

H x

L

1.2
A rectangular plate of length L, height H and thickness t is loaded by a bending moment. Show that the stresses in the plate can be derived from an Airy’s stress function φ(y) = 2M 3 y tH 3 (1.1)

Check that both the boundary conditions and bi-harmonic differential equation are fulfilled.

3

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1. In-plane loadedplates in Cartesian coordinates
y

M H/2

M

x

H/2

L/2

L/2

1.3
A rectangular plate of length L, height H and thickness t is loaded in accordance to the figure below. Determine the unknown constants A2 to D3 used in the Airy’s stress function φ(x, y) below so all boundary conditions are fulfilled. φ(x, y) = φ2 (x, y) + φ3 (x, y) where φ2 (x, y) = φ3 (x, y) = A2 2 C2 2 x + B2 xy + y2 2 A3 3 B3 2 C3 2 D3 3 x + x y+ xy + y 6 2 2 6
y

(1.2)

(1.3) (1.4)

p

qy x

qy

p

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1.4
A rectangular plate is fixed at a rigid wall and it is carrying a shear stress τ0 along its upper boundary. Determine the unknown constant C used in the Airy’s stress function φ(x, y) below. φ(x, y) = C(axy − xy 2 − xy 3 /a + by 2 + by 3 /a) (1.5)

Hint: Focus on how to at leastfulfill the bending stress situation along the left vertical boundary from an equilibrium point of view.
y

τ0 a x a

b

1.5
A rectangular plate is loaded in accordance to the figure below. Determine the unknown constants C and D used in the Airy’s stress function φ(x, y) below so that the bending stress is approximated as good as possible. The thickness of the plate is t. φ(x, y) = Cy 2 /2 + Dy 3/6 (1.6)

y

P H

P

x
H

L

L

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1. In-plane loaded plates in Cartesian coordinates

1.7
A triangular plate is exposed to a linearly increasing pressure shown in the figure below. The pressure is q0 at x = a. A proper choice of an Airy’s stress function φ(x, y) is given below. The thickness of the plate is t. φ(x, y) = Ax3 + Bx2 y + Cxy 2 + Dy 3 (1.7)

Determine theconstants A to D and calculate the stresses along the fixed boundary. The results can be compared to a FE-analysis.

y

a

x q(x) a

Chapter

2

In-plane loaded plates in polar coordinates
1.12
A thin arc-shaped plate, shown in the figure below, is loaded by a bending moment M . Plane stress conditions can be assumed. Calculate the stresses! A polar coordinate system is most convenientlyused. The thickness of the plate is t. The stress function φ(r) to be used is φ(r) = a0 ln r + b0 r2 + c0 r2 ln r

b

r
M

a

M

φ

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2. In-plane loaded plates in polar coordinates

1.13
An infinite plate is exposed to a uniform tensile stress σ0 i the x-direction. In the plate there is a hole with a radius a. Calculate the stresses, especially close to the hole and compare itto the stress level σ0 ! A polar coordinate system is most conveniently used.
y

r

φ σ0 x σ0

2a

1.16
A semi-infinite plate is exposed to a concentrated force P acting perpendicular to the boundary as shown i the figure below. The force P is distributed as a uniform line load across the thickness t. Calculate the stresses.
P

φ r

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1.17
A semi-infinite plate is exposed to aconcentrated force P acting parallel to the boundary as shown i the figure below. The force P is distributed as a uniform line load across the thickness t. Calculate the stresses. Study especially the maximum shear stress.
P φ r

1.18
A sector of a infinite plate with an opening angle α is exposed to a concentrated force P acting perpendicular to symmetry line through the material as shown i...