Strength of Materials - Lecture Notes / Mehmet Zor

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10. ENERGY METHODS

DEFORMATION WORK

AND STRAIN ENERGY

10.1

(tvid- 10.1)

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Energy Methods/ Deformation Work and Strain Energy

An object under the influence of external loads and in static balance does not move but changes shape. In this case, acting external loads (forces, moments) do deformation work.

10.1.1.b) If there is more than one force, how is the deformation work calculated?

We will try to understand the answers to these questions through 2 examples..>>

m: The final position of point k under the influence of all forces.

k'': projection of point m in the Fk direction

(a)

Figure 10.1.1

(b)

Figure 10.1.2

k: Application point of constant force F_k ,

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Energy Methods/ Deformation Work and Strain Energy

Ayşe comes first and sits at point c1 on an elastic beam.

Work done by weight W_A :

Later, while Ayşe is sitting, Berrin comes and sits at point c2.

Work done by weight W_B :

(Because Berrin hasn't settled down yet. There is no W_B force at c₂ at this moment.)

(Because when Berrin sat down, Ayşe was already sitting. W_A force was present in C₁ .)

From the instant Ayşe starts to sit, the force and deflection at point c₁ increase linearly, and the force and deflection reach their maximum values ​​at the last position of the sitting. For this reason, the F-ΔL diagram appears as follows. (This is a general rule. Try to notice this situation while sitting on a flexible floor or chair.)

Total Deformation Work:

Example-1: Deformation work when there are 2 forces in the same direction:

If Berrin had sat first,then Ayşe, or if they had both sat at the same time, the total U_d would not have changed.(Superposition method). This rule will be used in the future.

Figure 10.1.2

Figure 10.1.3

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Example -2 : Deformation when more than 2 forces are present in different directions

k

F

• Let's consider an object placed at a fixed corner. Let's first apply horizontal force F to this object.

If first F,

then F₁, and last F₂

are applied to the object:

: The amount of deformation of k in the F direction due to the force F

: Additional deformation amount of k in the F direction due to other forces.

The deformation work of force F is:

The deformation work of force F₁ is:

The deformation work of force F₂ is:

(Contribution of F₁ and F₂)

(Contribution of F₂)

If there were n forces:

Total Deformation Work:

+

(Contribution of F and F₂)

(10.1.1.a)

(10.1.1.b)

(10.1.1.c)

(10.1.1.d)

• Force F increases linearly from zero, causing a deformation of ∆_𝑘𝐹 on the object, and brings the application point k to 𝑘′.

• Then, according to the superposition method, while the F force is on the object, we apply other forces such as F₁ and F₂ to the object respectively, and in this case, point k moves from 𝑘′ to a point such as m. (However, as explained in Example-1, F has no effect anymore in this second deformation.)

10.1.2 Deformation Work of external moments:

(10.1.2.a)

Similar to equation 10.1.1.a, we first apply the M_k bending moment and then other loads. In this case, the deformation work of M_k is:

Deformation Work of External Torsional Moment for Circular Sections:

(10.1.2.b)

Energy Methods/ Deformation Work and Strain Energy

Figure 10.1.5

Figure 10.1.4

Figure 10.1.6

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1- Using strain energy density (a_i) per unit volume or

2- Using internal force and internal moments.

10.1.3 Strain Energy (U_i) :

10.1.3.a- Calculation of Strain Energy Density (ai) and Total Deformation Energy (U_i)

The strain energy stored (absorbed) in unit volume is called Strain Energy Density and is denoted by ai.

We consider a cube element with edges of 1 unit (1 unit volume) and fixed at its back surface.

Internal normal force in x direction:

Strain in x direction :

From Hooke's equation for uniaxial loading:

or

(10.1.4.b)

(10.1.4.a)

Strain Energy (U_i) can be calculated by the following 2 methods:

(from equ. 10.1.1.a)

Tip 10.1.a: Total Strain Work (U_d) is stored in the object as strain energy (U_i).

(10.1.3)

If we squeeze an elastic ball, the total deformation work (U_d) of the squeezing forces is stored in the ball as strain energy (U_i), and this energy continues to remain on the ball as long as it remains in the squeezing position.

Now we will examine these 2 methods one by one:

Energy Methods/ Deformation Work and Strain Energy

Figure 10.1.7

Figure 10.1.8

(a)

(b)

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Hooke relation:

or

Or the same result can be obtained from the deformation work of a constant shear force S_xy.

Shear Force:

Elongation:

(from equ. 10.1.1.a)

or

or

By similar calculations

By similar calculations

or

or

(10.1.5.a)

(10.1.5.b)

(10.1.6.a)

(10.1.6.b)

(10.1.7.b)

(10.1.7.a)

(10.1.8.a)

(10.1.9.a)

(10.1.8.b)

(10.1.9.b)

25.03.2023

Energy Methods/ Deformation Work and Strain Energy

Figure 10.1.9

(a)

(b)

(c)

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(10.1.10)

(from equations 10.1.4.b,..10.1.9.b)

(10.1.11)

If we substitute the general Hooke relations into equation 10.1.11, :

(10.1.12.a)

Strain energy density in terms of Principal stresses

If we write equation 10.1.12.a in 1-2-3 principal axes:

Strain energy density in terms of Cartesian stress components

Strain energy density in terms of Principal strains

If we write equation 10.1.12.b in 1-2-3 principal axes:

(10.1.12.b)

(10.1.13.a)

Strain energy density in terms of Cartesian stress and strain components:

(10.1.13.b)

(shear strains are zero in the principal axes)

(shear stresses are zero on the principal axes)

If we obtain the stresses in terms of strains from the general Hooke relations and substitute them in equation 10.1.12.a:

Strain energy density in terms of Cartesian strains components

Question: The superposition method cannot be applied to equations (10.1.4.a-10.1.9.a) and they cannot be collected on top of each other. What do you think is the reason for this?

Total Strain Energy:

(10.1.14)

Energy Methods/ Deformation Work and Strain Energy

Reminder: General Hooke’s Laws (stress-strain relations in the elastic region)

Figure 10.1.10

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10.1.5 Calculation of the total strain energy (Ui) resulting from Section Effects (Internal forces and internal moments):

or from the strain energy density

We found the same result.

10.1.5.a-) Strain Energy due to Internal Normal Force (N):(U_N)

Equ.10.1.1.a:

(equ. 3.3a )

(Since there is no other force)

From equ. 10.1.4.b :

;

;

(10.1.15.a)

(10.1.15.b)

For L length piece in stepped bars :

For single rod of length L :

Now we will obtain the strain energy from the internal force and internal moment types (section effects) occurring inside the object.

A normal internal force N occurs in the bar section subjected to the tensile force P.

For this loading type:

Total Volume:

Energy Methods/ Deformation Work and Strain Energy

Figure 10.1.11

Figure 10.1.12

Figure 10.1.14

(a)

(b)

(c)

Figure 10.1.13

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Internal moments Mz arising on the surfaces of the differential part of length 𝑑𝑠 removed from a beam subject to simple bending cause a convexity of 𝑑𝜃.

10.1.5.b-) Strain Energy (U_M) due to Simple Internal Bending Moment (M_z)

Equ. (9.2.1) :

Strain energy in the beam due to internal bending moment M_z :>>

From geometry, for small deformations:

(10.1.16)

or we find from the energy density as follows:

From equa. 10.1.4.b:

Normal stress:

We got the same result

( equ. 5.1.1 )..>>

Energy Methods/ Deformation Work and Strain Energy

Figure 10.1.15

(a)

(b)

Figure 10.1.16

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10.1.5.c-) Strain Energy due to Internal Shear Force (V) in Simple Bending with Shear: Uv

We want to find the deformation work and therefore the deformation energy caused by the internal shear force of length dx of a beam subjected to simple bending with shear. (Actually, there are bending moments and normal stresses in the section, but we did not show them in the figure because we only examined the effect of shear force.)

Strain energy density due to 𝜏 stress is from equation (10.1.7.b):

Strain energy density due to shear force V:

Total strain energy due to shear force V:

Shear stress at all points on the 𝑎−𝑐 line:

(Static moment of area 𝐴′ )

Symmetrical beam section

(10.1.17)

Energy Methods/ Deformation Work and Strain Energy

Figure 10.1.17

(a)

(b)

(c)

(d)

,

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10.1.5.d-) - Strain Energy due to Internal Torsional Moment (T) for Circular Sections: U_T

or

It is found from the energy density as follows:

Total Strain Energy:

Angle of twist formed by the effect of the torsional moment T:

(Equation: 4.1.5)

(Area under the curve)

Differential deformation work and strain energy resulting from T_int moment in the section of length dx:

Angle of twist for the section of length dx:

(10.1.18b)

Strain energy density from equation 10.1.7.b:

Shear stress distribution occurring in torsion:

(equa. 4.1.6)

Total Deformation Work:

We found the same result.

(Polar moment of inertia)

Valid for circular sections.

As a result of the torsional moment T applied to a built-in shaft, a torsional internal moment T_int = T occurs at any cross-section..

Energy Methods/ Deformation Work and Strain Energy

(10.1.18a)

Figure 10.1.18

Figure 10.1.19

Figure 10.1.20

Figure 10.1.21

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10.1.6 When all cross-sectional effects are present simultaneously, Total Strain Energy: (𝑈_𝑖 )

For torsional moment T in circular cross-section members

for V shear force in simple bending with shear

For bending moment M in simple bending

For N normal forces

(tension or compression)

Equations (10.1.20) are valid for isotropic materials and in the elastic loading region.

Energy Methods/ Deformation Work and Strain Energy

a-) In Rods with Linear Axis

b-) In Rods with Curvilinear Axis

Since the position of any cross-section is expressed relative to the orbital coordinate s, equation 10.1.20.a will become:

Figure 10.1.22

Figure 10.1.23

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Example 10.1.1

Solution:

First of all, we must determine the section effects (internal forces and internal moments) in any section.

3

The + direction of the internal influences in the left part of cut I-I is selected in this way. (See: topic 5.2.3)

Strain energy due to tensile force N:

Strain energy due to shear force V:

Energy Methods/ Deformation Work and Strain Energy

From equation (10.1.15.b) :

From equation (10.1.17) :

Figure 10.1.24

Figure 10.1.25

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Strain energy due to bending moment

Total Strain Energy:

When all values ​​are substituted in the above equation

Numeric values :

From equ.10.1.19 :

Energy Methods/ Deformation Work and Strain Energy

From equation (10.1.16)

found.

, ν =0.25