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Bài giảng Strength of materials - Chương 2: Internal forces - Nguyễn Sỹ Lâm

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CHAPTER 2: INTERNAL FORCES  
2.1 Free Body Diagram  
2.2 Internal Forces & Method of Section  
2.3 Sign Conventions  
2.4 Procedure  
2.5 Diagrams of internal forces  
2.6 Relationships between loads, shear force and bending  
moment  
2.1 FREE BODY DIAGRAM  
Useful definition  
Free body diagram: A sketch of the outlines shape of the  
body isolated from its surrounding. On this sketch, all forces  
and couple moments that the surrounding exert on the body  
together with any support reactions must be shown correctly.  
Only then applying equilibrium equations will be useful.  
2.2 INTERNAL FORCES & METHOD OF SECTIONS  
Internal loadings: These internal loading acting on a specific  
region within the body can be attained by the Method of  
Section.  
Method of Section:  
Imaginary cut is made through the body in the region where  
the internal loading is to be determined.  
The two parts are separated and a free body diagram of one of  
the parts is drawn. Only then applying equilibrium would  
enable us to relate the resultant internal force and moment to  
the external forces.  
2.2 INTERNAL FORCES & METHOD OF SECTIONS  
CROSS SECTION METHOD  
2.2 INTERNAL FORCES & METHOD OF SECTIONS  
Point O is often chosen as the  
centroid of the sectioned area  
CROSS SECTION METHOD  
2.2 INTERNAL FORCES & METHOD OF SECTIONS  
CROSS SECTION METHOD  
2.2 INTERNAL FORCES & METHOD OF SECTIONS  
Four types of internal forces can be defined:  
Normal force, N. This force acts perpendicular to the area.  
Shear Force, V. This force lies in the plane of the area (parallel)  
Torsional Moment, T. This torque is developed when the external loads tend to  
twist one segment of the body with respect to the other  
Bending Moment, M. This moment is developed when the external loads tend  
to bend the body  
CROSS SECTION METHOD  
2.2 INTERNAL FORCES & METHOD OF SECTIONS  
• If the body is subjected to a co-planar force system then only normal  
force N, 01 shear force V, and 01 bending moment M component exist  
on the section.  
CROSS SECTION METHOD  
2.2 INTERNAL FORCES & METHOD OF SECTIONS  
Consider a bar at “balance” state (ie. free body diagram)  
6
P1  
P4  
F 0  
iy  
i1  
6
Fi : P1, P2…, P6  
P5  
P2  
F 0  
iz  
i1  
6
m
x i 0  
F
P3  
P6  
i1  
An imaginary cross section  
P4  
Qy > 0  
P1  
Mx > 0  
P5  
Nz > 0  
Nz > 0  
O
P2  
O
z
B
A
z
Mx > 0  
P6  
Qy > 0  
P3  
y
y
Fi : P4, P5, P6, Nz, Qy, Mx  
Fi : P1, P2, P3, Nz, Qy, Mx  
CROSS SECTION METHOD  
2.3 SIGN CONVENTION  
CROSS SECTION METHOD  
2.3 SIGN CONVENTION  
CROSS SECTION METHOD  
2.3 SIGN CONVENTION  
CROSS SECTION METHOD  
2.3 SIGN CONVENTION  
Shear force: clockwise  
Bending moment: compresses the upper part of the bar or  
elongates the lower part  
Normal force: elongates  
CROSS SECTION METHOD  
2.3 SIGN CONVENTION  
• If the internal shear force causes a  
clockwise rotation of the beam segment.  
Then it is considered to positive.  
• If the internal moment causes compression  
in the top fibers then it is considered to be  
positive  
CROSS SECTION METHOD  
2.3 SIGN CONVENTION  
Stress Under General Loadings  
• A member subjected to a general  
combination of loads is cut into  
two segments by a plane passing  
through Q  
• The distribution of internal stress  
components may be defined as,  
x
F  
  
lim  
x
A  
A0  
x
x
V  
V  
y
z
  
  
xz  
lim  
lim  
xy  
A  
A  
A0  
A0  
• For equilibrium, an equal and  
opposite internal force and stress  
distribution must be exerted on  
the other segment of the member.  
CROSS SECTION METHOD  
2.4 PROCEDURE  
Determination of support reactions by studying the  
equilibrium of the whole structure  
(Xác định phản lực liên kết bằng cách xét cân  
bằng tòan hệ)  
Imagine a section passing through the body  
(Tưởng tượng mặt cắt qua vật thể)  
Equilibrium of one divided part  
(Xét cân bằng một phần bị chia)  
CROSS SECTION METHOD  
2.4 PROCEDURE  
1. After sectioning, decide which segment of the body will be studied. If this  
segment has a support or connection than a free body diagram for the  
entire body must be done first to calculate the reactions of these supports.  
2. Pass an imaginary section through the body at the point where the  
resultant internal loadings are to be determined and put the three  
unknowns (V, Mo, N) at the cut section. Then apply equilibrium.  
Suggestion: take the summation of moment around a point on the cut section  
(V and N will not appear in this equation) and solve directly for Mo)  
CROSS SECTION METHOD  
2.4 PROCEDURE  
Determine the reactions  
using the equilibrium  
conditions of the overall  
structure  
Cut the beam at the cross section  
at which shear force and bending  
moment are to be determined.  
Draw a free-body diagram  
Set up equilibrium equations of the F.B.D. to  
determine shear force and bending moment at the  
cross section  
CROSS SECTION METHOD  
2.4 PROCEDURE  
Example: Determine internal forces on the cross section at C  
(Xác định nội lực tại C)  
  
F 0 N 0  
x
C
F 0 V  58,8N  
  
y
C
  
M 0 M  5,69N.m  
C
C
CROSS SECTION METHOD  
2.5 DIAGRAMS OF INTERNAL FORCES  
Beams are long straight bars having constant cross section  
area and support loads that are applied perpendicular to its  
longitudinal axis  
CROSS SECTION METHOD  
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