MABY4400 Structural analysis and design
Oslo Metropolitan University
Spring 2025
Department of Built Environment
Aase Reyes
Exercise 1.2 Continuum mechanics – Stresses & Equilibrium
Problem 1.2.1 Traction
In a point P the stress matrix is given as [units in MPa]
10 20 30
[𝝈] = [20 40 60]
30 60 10
a) What is the traction vector {𝚽} in point P on a plane normal to the x-axis?
b) The vector {𝒏} = ⌊𝑙 𝑚 𝑛⌋𝑇 is normal to the plane 𝑙𝑥 + 𝑚𝑦 + 𝑛𝑧 = 0. What is the traction {𝚽} in point P
on a plane that is parallel to the plane 2𝑥 − 2𝑦 − 𝑧 = 0. What is the normal stress and the shear stress
on this plane?
Problem 1.2.2 Principal stress
In a point P the stress matrix is given as [units in MPa]
180 80 0
[𝝈] = [ 80 60 0 ]
0 0 −30
a) Calculate the stress invariants 𝐼1 , 𝐼2 and 𝐼3 . Find the principal stresses 𝜎1 , 𝜎2 and 𝜎3 and the corresponding
principal directions {𝒏}1, {𝒏}2 and {𝒏}3. Check that the stress invariants keep their numerical value when
calculated from the principal stresses.
b) Draw the stresses 𝜎𝑥 , 𝜎𝑦 and 𝜏𝑥𝑦 on an infinitesimal element in an xy coordinate system. Also draw the
principal stresses 𝜎1 , 𝜎2 and 𝜎3 on an element that is correctly rotated according to the principal stress
directions.
c) Calculate the hydrostatic and deviatoric stress.
d) Compare the invariant 𝐽2 with the equivalent von Mises stress 𝜎𝑉𝑀 .
MABY4400 Structural analysis and design
Oslo Metropolitan University
Spring 2025
Department of Built Environment
Aase Reyes
Problem 1.2.3 Equilibrium
y
y + dy xy
y xy + dy
y
Fy xy
x xy + dx
x
dy
Fx x
xy x x + dx
y dx x
xy
y
z x
The equilibrium equations can be deducted by considering an infinitesimal element in the body. (The figure above
shows only stresses and volume forces in the xy-plane.) Force equilibrium in the x-, y- and z-direction will give the
same result as we have obtained in class with “global” equilibrium and use of the divergence theorem:
𝜕𝜎𝑥 𝜕𝜏𝑥𝑦 𝜕𝜏𝑧𝑥
+ + + 𝐹𝑥 = 0
𝜕𝑥 𝜕𝑦 𝜕𝑧
𝜕𝜏𝑥𝑦 𝜕𝜎𝑦 𝜕𝜏𝑦𝑧
+ + + 𝐹𝑦 = 0
𝜕𝑥 𝜕𝑦 𝜕𝑧
𝜕𝜏𝑧𝑥 𝜕𝜏𝑦𝑧 𝜕𝜎𝑧
+ + + 𝐹𝑧 = 0
𝜕𝑥 𝜕𝑦 𝜕𝑧
Show how this can be done with the figure above as a starting point.
MABY4400 Structural analysis and design
Oslo Metropolitan University
Spring 2025
Department of Built Environment
Aase Reyes
Problem 1.2.4 Equilibrium
The figure shows an idealization of a support rib in a dam structure. The hydrostatic pressure acts on a plate in the
yz plane. This plate supported by equally spaced triangular ribs in the xy plane. Assume that the ribs are loaded by
a linearly increasing hydrostatic pressure along the vertical side x = 0, and the gravitational force in the y-direction.
The structure has density .
y g
p(y) = p0
h
n h
b
y
Assume the following stresses in the xy plane:
𝑝0 𝜌𝑔ℎ 2𝑝0 ℎ2 𝑝0 ℎ 𝑝0 ℎ
𝜎𝑥 = − 𝑦 𝜎𝑦 = ( − 3
) 𝑥 + ( 2 − 𝜌𝑔) 𝑦 𝜏𝑥𝑦 = − 𝑥
ℎ 𝑏 𝑏 𝑏 𝑏2
a) Show that the proposed stress field satisfies the equilibrium equations in the plane stress state.
b) Show that the unloaded inclined plane is stress free.
