plates exam

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Tentamen 29 januari 2014, vragen

Technische Universiteit Delft | Plates and Slabs

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Delft University of Teclinology Write your name and student number Faculty of Civil Engineering and Geosciences on every page of your worl

Problem 2 (3.5 point)

A plate is loaded by a compressive distributed load S at its left and right edges, and by a bending moment M at its top and bottom edge. The loading condition of the plate is shown in the figure below. In the origin of the coordinate system, which lies in the middle of the plate, no rotations and translations are allowed. The Young's modulus of the plate is E, and it's Poisson's ratio is v and the moment of inertia is I.

M

M

a Formulate the boundary conditions in the origin of the coordinate system,

b From which of the following terms the displacement field is build up?

A) Constant and linear terms B) Cubic terms C) Constant and linear + quadratic terms D) Quadratic + cubic terms

Justify your answer.

0 Show how the most general trial solution for a displacement field, given by:

Ux = a^+ a2X+ a^^y + 34^^ + a^xy + agy^ + ajx^ + agx^y + agxy^ + a^o/^ Uy = b^ + + b^y + b4X^ + bgxy + bgy^ + byx'^ + bgx^ y + bgxy^ + b^Qy^

can be simplified by utilizing your answer to 2b.

d Use the chosen trial solution in 2c to compute the displacements Ux and Uy, taking into account the boundary conditions in 2a.

2


Problem 2 (continued)

e Consider a square plate of dimensions 2Lx2/ . under certain loading conditions tliat lead to a displacement field given by:

^y=\{y-L)

in the coordinate system as sketched below.

(-LrL) O

(OrL) - o -

(l-rL) O

(0,0) 0 o

O a (-U L)

Give a rough sketch of the final configuration (shape) of the square plate given the loading conditions provided. Is this deformation accompanied by rigid body translations and/or rotations? (this can be shown either graphically or numerically)

Hint: find the new configuration for some of the special points, as depicted in the image.

Problems (1.0 points)

A rectangular thin slab in the figure has a fixed edge at AO and simply supported edge at OC. The edges AB and BC are free. The corner B is supported. The bending moment Mo and line load qi are applied at edges AB and BC respectively. Express the boundary conditions of this slab in terms of the deflection w.

3


Problem 4 (4.0 points)

A rectangular thin slab ABCD consisting of linear-elastic material is shown in the figure below. The distributed bending load P=24a^C is applied on the slab. The slab stiffness is D (unit: Nm) and Poisson's ratio is zero. The bending load creates a displacement field of the form:

a Draw the displacement w along the lines x = 0 and x =-a, and also along the lines y = 0, y = -a and y = + a. What are the boundary conditions along the edges AB, AC, BD and CD?

b What is the dimension of C (in units N and m) ?

c Derive and plot the shear stress Vx along the two edges parallel to the y-axis, and Vy along the two edges parallel to the x-axis. (note: mind the +/- signs)

d Compute the total shear forces along the entire boundaries by taking the integrals of the shear stresses obtained from c. (note: mind the +/- signs)

e Determine the concentrated corner forces at corners A, B, C and D.

f Compute the boundary reaction forces along slab edges AC, DB, AB and CD.

g Verify the vertical force equilibrium condition.

where C is a constant.

Note: pay attention of the origin of the global coordinate system.

X

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