1/8M.Chrzanowski: Strength of Materials

SM1-02: Statics 1: Internal forces in bars

INTERNAL FORCESIN BARS

2/8M.Chrzanowski: Strength of Materials

SM1-02: Statics 1: Internal forces in bars

DefinitionsL

H

B

•Bar – a body for which L»H,B

•Bar axis - locus of gravitational centres of bar sections cutting its surface

•Prismatic bar – when generator of bar surface is parallel to the bar axis

•Straight bar – when bar axis is a straight line

Bar axis

3/8M.Chrzanowski: Strength of Materials

SM1-02: Statics 1: Internal forces in bars

Assumptions

•Bar axis represents the whole body and loading is applied not to the bar surface but the bar axis

•Set of bar and loading will be considered as the plane one if forces acts in plane of the bar.

P

q

M

.M

4/8M.Chrzanowski: Strength of Materials

SM1-02: Statics 1: Internal forces in bars

Agreements

•Reduction centre O is located on the bar axis by vector r0

•Internal forces are determined on the planes perpendicular to the bar axis (vector n is parallel to the axis)

•Vector n is an outward normal vector

n

n

O

x

y

zr0

5/8M.Chrzanowski: Strength of Materials

SM1-02: Statics 1: Internal forces in bars

Components of internal forces resultants Swx , Swy , Swz and Mwx , Mwy , Mwz are called cross-sectional forces

In 3D vectors of internal forces resultants have three components each

Sw{ Swx , Swy , Swz } Mw{ Mwx , Mwy , Mwz }

x

y

z

Swz

Sny

Swx

SwMwz

MwxMwy

Mw

6/8M.Chrzanowski: Strength of Materials

SM1-02: Statics 1: Internal forces in bars

Sw = Sw(rO , n)

Mw = Mw(rO , n)

Sw = Sw(rO)

Mw = Mw(rO)

Vector n is known if we know the shape of bar axis

.n

. n

.

n

.n

. n

Thus, resultants of internal forces for known bar structure are function of only one vector r0

Resultants of internal forces are vector functions of two vectors ro and n

7/8M.Chrzanowski: Strength of Materials

SM1-02: Statics 1: Internal forces in bars

In 2D number of cross-sectional forces is reduced, because loading and bars axes are in the same plane (x, z):

Sw{ Sx , 0, Sz }

Mw{ 0, My , 0 }

x

y

z

P q

.M

M

Sx

Sz

My

We will use following notations and names for these components:

Sx=N - axial forces

Sz=Q - shear force

My = M - bending moment

8/8M.Chrzanowski: Strength of Materials

SM1-02: Statics 1: Internal forces in bars

Special cases of internal forces reductions are called:

TENSION – when internal forces reduce to the sum vector only, which is parallel to the bar axis

SHEAR – when internal forces reduce to the sum vector only, which is perpendicular to the bar axis

BENDING – when internal forces reduce to the moment vector only, which is perpendicular to the bar axis

TORSION – when internal forces reduce to the moment vector only, which is parallel to the bar axis

M

Ms

Q

N

9/8M.Chrzanowski: Strength of Materials

SM1-02: Statics 1: Internal forces in bars

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