Presentation2
1.Definition of truss structure.
Truss is a structure meets following criteria:
1. All components are “two-force members” connected as pins in the joints called “nodes”.
2. The members are rectilinear.
3.Loads are applied at the nodes as concentrated force.
4.Structure is stable.
2.Describe methods of calculation for truss structures (joint method, section method, beam analogy method, bar replacement method).
– Method of joints: The forces in truss members could be calculated by considering of equilibrium equation for each joint. ∑x=0, ∑y=0.
Calculation is possible only for two unknown force member joint.
–Section method: Allows to calculate the force in three cutted members using static equilibrium equations. The effect of running cut is represented as the forces in the cutted members acting on the free body. The care have to exercise to institude the cut through the three or les unknown components.
– Beam analogy method: Truss Works in similar way as beam. The vending is carried out by the chord members similar to flanges of I-beam, while shear by the web members similar to web of I-beam.
– Bar replacement method: This method is used when application of Method of joints and Method of section is very complicated or impossible.
Bar replacement method involves the removal of one member and replacement it by another in new location.
3.Beam analogy method in calculation of influence lines of forces in members of truss structure.
Forces in chord members of truss are the functions of moments in beam, so the influence lines of these forces are the functions of the influence lines of moments in the equivalent beam. Forces in web members of truss are the functions of shear in beam, so the influence lines of these forces are the functions of the influence lines of shear in the equivalent beam.
Formulae for the influence lines: IL (B) = 1/h IL (MB), IL (T) = – 1/h IL (MT), IL (D) = (+-) 1/sin α IL (Vα-α), IL (W) = (+-) IL (Vβ-β).
4. Supernumerary restraints (absolutely necessary and conditionally necessary restraints.
–Conditionally necessary restraints: Are supernumerary restraint could be removed under the condition the structure will stable.
–Absolutely necessary restraints: the restraints could be removed in any case deleting these restraints provide the instability of the structure and make the structure become a collapse mechanism.
5. Degree of static indeterminacy (formulas for beams, frames, trusses).
Beam: n=r-3-p
r- Number of reaction in structure.
3- Number of static equilibrium equation.
p- Number of pins (hinges).
Frames: n=r-3+3·c-∑n·pn
c- Number of closed frames.
n- Degree of hinge.
Trusses: n=(r+m)-2j
m- Number of truss member.
j- Number of truss joint.
6. Properties of structures with supernumerary restraints.
1. Just one supernumerary restraint is enough to make structure statically indeterminate. That means the structure cannot be solved by using only the equilibrium equations.
2.To determine forces in the absolutely necessary restraints the equilibrium equations are sufficient.
3.The comply of all conditions of equilibrium of all forces occurring in statically indeterminate structures is a necessary but not sufficient condition for an accurate solution.
4.All conditionally necessary restraints in the structure are cooperating each other.
5.The internal forces in the conditionally necessary restraints depend on the stiffness of each elements (EI, EA, GA).
6.The supernumerary restraints reduce extreme values of the internal forces and displacements.
7.Statically indeterminate structures are sensitive to the activities of non-static loads (displacement of nodes, temperature loads) in the areas of conditionally necessary restraints
8.The absolutely necessary restraints are not working when the structure is loaded by the system of self-equalized forces on direction of conditionally necessary restraints.
7. Types of semi-rigid (flexible) supports. Calculation of reactions in semi-rigid supports.
8. Symmetry of the structure. Reduction of entire symmetrical structure to half structure system.
When structure is symmetrical, any systems of external loads can be accepted as a sum of symmetric and anti-symmetric burden.
Symmetrical loads applied to the symmetrical system generate only symmetrical internal forces or reactions in cross-section located on the axis of symmetry, while anti-symmetrical internal forces are zero.
Reducing structure to half-structure system internal and external restraints located on the axis of symmetry are treated in half-structure system as external restraints (support reactions). It allows to create virtual support on the axis of symmetry.
9. Stability of structure. Degree of freedom of the structure.
DOF = 3B – R – 2H
Where:
B – Number of rigid body.
R – Number of restraints.
H – Number of hinges.
DOF > 0 the structure is unstable (is a mechanism).
DOF = 0 the structure is statically determinate structure.
DOF < 0 the structure is statically indeterminate structure.
10. Lost of stability (parallel forces, concurrent forces).
1. – Parallel forces: when all the reactions are parallel, it means the structure is unstable.
2. – Concurrent forces: when the action of three or more reaction intersect in one point, the structure is isostable.
11. Static loads in Force Method.
This method is based on the removal as much restraints as degree of indeterminacy is. After removal, the structure is statically determinate. At the removed restraints, the unit loads are applied and denoted as Xi. The formulas under statics loads are:
12. Kinematic loads in Force Method.
A kinematic load is the impact on a structure as the change of the node position or as the rotation of structural member relative to the support node or other part of construction.
The influence of translation of rotation of the nodes of structure on the virtual displacements of the structure is included in the part of Maxwell-Mohr formula:
ΔiP = –∑ Ri · Δ
Ri reaction (or internal force) on the direction of node translation or rotation under the unit force Xi.
Δ– Kinematic load (support movement, support or node rotation, elongation or reduction of structural element.
The reaction has positive value when its direction is consistent with direction of the support (node) displacement.
13. Thermal loads in Force Method.
Due to the change of the temperature into the element the displacement of the structure is described the following formula:
Where:
t – Is calculated: t = tC – t0.
tC – current temperature in the element at the centroid of element cross-section.
t0 – the beginning temperature in the element.
αt –thermal expansion ratio.
Ni – the values of axial forces in the elements under the unit force Xi.
Due to the temperature difference around the element the displacement of the structure is described the following formula:
Where:
Δt – is calculated: Δt = tu – td.
αt – thermal expansion ratio.
h – a height of element cross-section.
Mi – the values of moments in the elements under the unit force Xi.
14. Flexible supports in Force Method.
If in the node exists restrain with specified stiffness (k, к) restricted movement on the direction of the restraint is allowed and the node is called semi-rigid.
Subsidence of support (translationally flexible support):
f – Translation of the support.
R – Reaction on the direction of displacement.
k – Translational stiffness of the support.
(Value of force causes the unit displacement).
1/k – translational flexibility of support.
Node displacement: f =1/k * r
Elastic rotation of support (rotationally flexible support).
j – Rotation of the support.
M – Reaction on the direction of displacement.
k – Rotational stiffness of the support (value of moment causes the unit rotation).
1/k – rotational flexibility of support.
Node displacement: f =1/k * M
15. Simplifying assumptions of Direct Displacement Method.
The effect of elongation or reduction of component is omitted as negligible, except where it is decisive (thermal loads, shrinkage, component is tie-bar or tie-rod).
The difference between length of deflected component and the length of its chord is omitted, because of the relatively very small real movement of the component nodes.
The effect of lateral and axial forces (V, N) is usually omitted.
16. Degree of kinematic indeterminacy. Calculation of unknown rotations and translations. Formula.
Is a sum of possible rotation of rigid nodes and independent movements of any nodes.
nk =∑ϕ + ∑Δ
∑ϕ – number of rigid nodes intersected by minimum two statically indeterminate bars.
∑Δ – number of restraint necessary to stop all possible linear movements (in kinematic chain of the frame).
17. Diagrams of moments for basic beams of Direct Displacement moments.
18. Kinematic chain of structure. Independent movements. Formulas. Dependency between displacements of beam axis and deformation angles.
Kinematic chain of structure is a structure with all nodes replaced by the pins. The number of restraints necessary to provide the stability is the number of independent linear movements.
The number of independent movements can be calculated by the next formula:
∑Δ = 2n – (m + r) + o – f
Where:
n – Number of nodes in kinematic chain.
m – Number of members in kinematic chain.
r – Number of reaction (restraints) in kinematic chain.
o – Number of over-rigidity (two or more reaction acting in one axis connected by elements of kinematic chain).
f – Number of free ends (nodes at the end of the bars with possibility to movement perpendicular to member axis).
The dependency between displacements of beams axis Δi and deformation angles ψi is shown with the next relation: ψi = Δi / li
19. Loads applied in the nodes of structure in Direct Displacement Method.
In the nodes of a structure could be applied two types of loads: rotation and displacement.
20. Kinematic loads in Direct Displacement Method.
A kinematic load is the impact on a structure as the change of the node position or as the rotation of structural member relative to the support node or other part of construction. There are two types: rotation (in a node or in a support) and translation.
21. Thermal loads in Direct Displacement Method.
There are two types of thermal loads in Direct Displacement Method: difference of temperature around the element and change of temperature in the element, which in turn, can be in two ways: increase of the temperature (elongation of element’s length) and decrease of the temperature (reduction of element’s length).
22. Semi-rigid (flexible) supports in Direct Displacement Method.
In semi-rigid supports, movement (translation or rotation) is possible on the direction of flexible restraints. Translation is enabled in translationally flexible support and rotation is enabled in rotationally flexible support.
Because in Direct Displacement Method the unknowns are the displacements then translation or rotation of semi-rigid supports increase the number of unknowns.
Dependences between displacement of semi-rigid support and the reaction are presented by following formulae:
Because the displacement are unit, the reaction are equal to the support stiffness. These reactions are treated as additional force (on the direction of translation) or moment (on the direction of rotation).
23.DEFINITION OF INFLUENCE LINE
Influence line is the graphical representation of the response function of the structure as the downward unit load moves across the structure. The ordinate of the influence line show the magnitude and character of the function.
The most common response functions of our interest are support reaction, shear at a section, bending moment at a section, and force in truss member.
24.EXPLAIN HOW TO PUT INFLUENCE LINE OF FORCE, MOMENT OR REACTION BY DEVELOPING THE EQUATION IN STATICALLY DETERMINATE STRUCTURES
This is done by solving for the reaction, shear, or moment in a point A caused by a unit load placed at x feet along the structure instead of a specific distance. This method is similar to the tabulated values method, but rather than obtaining a numeric solution, the outcome is an equation in terms of a variable.
It is important to understanding where the slope of the influence line changes for this method because the influence-line equation will change for each linear section of the influence line. Therefore, the complete equation will be a piecewise linear function which has a separate influence-line equation for each linear section of the influence line
25. Explain how to plot influence line of Force, Moment or Reaction using Mueller- Breslau Principle in statically determinate structures.
To plot the influence line of axial force in selected member of truss the restraint generating appearance of expected force has to be removed.
Force in the truss member is constant ton entire length of component; therefore, the truss bar represents the restrain. (ejemplo de la barra quitada)
26. Calculation of internal forces using Influence Lines (uniform load, concentrated force, concentrated moment).
For calculate internal forces using influence lines we have to use the following formulas:
B = MB / h , T = -MT / h , D = (+-) 1/sinα * Vα-α , W = (+-) Vβ-β
Uniform load: is a force applied over an area, denoted by ‘q’, which is force per unit length.
Concentrated force: is a force considered to act along a single line in space.
Concentrated moment: can be applied at any point of the structural beam.