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```					Problems for Matrix Structural Mechanics. Second Year Mechanical Engineering Students.

(1) Prove that for a 1-D bar that the summation of the coefficients of each column of the
structure stiffness matrix is equal to zero. Hint ∑ Fx = 0.
(2) Calculate the nodal displacements, reaction forces and the elemental forces. In addition,
calculate the normal stresses for Figs. 2a-c.

(3-a) Prove that for a plane truss that ∑ K2i-1,j = 0 and ∑ K2i,j = 0, where i = 1 to number-of-
nodes.
(3-b) Show that for a plane truss δ = (u2 - u1) cos θ + (v2 - v1) sin θ.
(4-a) Prove that the reduced equations are

 Fx1  k  4          3   u1 
                          . If v1 = 1 and u1 = 0, get the forces in each element. Take EA/L
 Fy1  = 2  3       4   v1 
                         
= k for all members.

Mechanics of Structures - Problems - Second Year - Mechanical Engineering - Cairo University       1/4
(4-b) Calculate the nodal displacements, element forces, and reaction forces.
(4-c) Calculate the nodal displacements, reaction forces and element stresses. The table
shows the relevant data.
Element number                       Material                              Area mm2
1                                    Aluminium, E = 70 GPa                 900
2                                    Steel,    E = 200 GPa                 600
3                                    Aluminium, E = 70 GPa                 1250

Mechanics of Structures - Problems - Second Year - Mechanical Engineering - Cairo University   2/4
(4-d) Calculate the nodal displacements and elemental forces. A(1) = A(2) = 30 mm2 and A(3) =
10 mm2. Take E = 200 GPa.
(5) Show that when the CCW moments and slopes are assumed positive, the stiffness matrix
for 2-D beam elements takes the following form:

 12    6 L − 12 6 L 
 6 L 4 L2 − 6 L 2 L2 
EI                        
L3  − 12 − 6 L 12 − 6 L 
                       
 6L 2L
2
− 6 L 4 L2 
Hint: put a minus sign before m1, m2, θ1, and θ2 in the element matrix equation given in the
notes.
(6) Get the linear and angular nodal displacements. Calculate the forces and moments
transmitted to each element.

Computer Problems
(7) The derrick structure has a load capacity of 90 kN. Choose suitable steel and use a
factor of safety of 4, determine the cross section for all members. Recommend a cross
section that will prevent any member from buckling. Check the forces in element 1-2, 2-5,
and 4-5 by the method of sections.
(8) The rod 2-4 has a diameter of 25 mm. Get the displacement, nodal forces and stresses
in the beam with and without the existence of the rod, (both ends of the rod are hinged).
Comment on the results.

Mechanics of Structures - Problems - Second Year - Mechanical Engineering - Cairo University   3/4
(9-a) Determine the displacement of node 3 and the maximum stress for the wooden roof
truss by two approaches. (i) assume it to be a truss, and (ii) assume all joints connecting the
wooden members to be rigid (i.e. capable of transmitting moments). Take E = 14 GPa.
Comment on the results.
(9-b) Solve again assuming the structure to be a symmetric truss.

(9-c) Solve for the member forces by the method of joints assuming the structure to be a
truss. Hint: the force in member 1-3 is equal to zero, because u1 = u3 = 0.

Mechanics of Structures - Problems - Second Year - Mechanical Engineering - Cairo University   4/4

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