Structural Analysis: Thermal Loads and Flexible Supports in Force Method

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 = t_C – t₀.

t_C - current temperature in the Element at the centroid of element cross-section.

t₀ - the beginning temperature in The element.

α_t –thermal expansion ratio.

N_i – the values of axial forces in The elements under the unit force X_i.

Due to the temperature Difference around the element the displacement of the structure is Described the following formula:

Where:

Δt – is Calculated: Δt = t_u – t_d.

α_t – thermal expansion ratio.

h – a height of element cross-section.

M_i – the values of moments in the Elements under the unit force X_i.

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.

n_k =∑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).