Lec5

Jan 19, 20110 likes295 views

This document summarizes a lecture on forces and moments transmitted by slender members. It defines slender members as long, skinny structural elements like skis, golf clubs, and I-beams. It discusses axial forces that act along the member's axis, shear forces that act in the plane of the member's face, and bending moments. Sign conventions for these internal loads are also defined. An example is provided to demonstrate calculating the internal forces and moments in a beam by setting up free body diagrams at different points.

2.001 - MECHANICS AND MATERIALS I
Lecture #5
9/20/2006
Prof. Carol Livermore

TOPIC: FORCES AND MOMENTS TRANSMITTED BY SLENDER MEM-
BERS?
When will a structure fail?

Recall:
1. What can the structural elements tolerate?
2. What forces and moments are experienced in the various parts of the
structure?

EXAMPLE: Airplane

EXAMPLE: Truss

SLENDER MEMBERS
Long, skinny structural elements

1

L >> b, L >> h

EXAMPLE: Skis, Golf Clubs, I-Beams

Local axes follow the beam.

AXIAL FORCES:

Forces that act along the axis of the slender member.

FBD

Fij (x) = N (x), i: face, j: direction, x: location

SHEAR FORCES:
Forces that act in the plane of a face.

Summarize Internal Loads:

Moments:

Mxz (x2 ) > 0x: face, z: axis, x2 : location
Mxz (x) = Mz (x)
Bending Moment

Sign Convention

Summarize sign conventions

3

EXAMPLE: Calculating internal forces and moments

Step 1: Find reactions at supports.

Fx = 0
R Ax = 0

Fy = 0
R Ay ? P = 0
R Ay = P

MA = 0
MA ? P a = 0
MA = P a

5

Step 2: If you have a multi-component structure, you will need to ?nd joint
forces too.

Step 3: Find internal forces and moments.
1. Set up local coordinates.

2. Identify interesting points.

3. Cut beam and do FBDs.

6

For 0 < x < a

Fx = 0
N =0

Fy = 0
P ? Vy = 0
P = Vy

MA = 0
P a + Mz ? P x = 0

Mz = P x ? P a = ?P (a ? x)
OR (From the other end)

FBD

Fx = 0
?N = 0
N =0

Fy = 0
?P + Vy = 0
Vy = P

M? = 0
?Mz ? P (a ? x) = 0

7

Mz = ?P (a ? x)
Note: This is the same as before.
For a < x < L

Fx = 0
?N = 0
N =0

Fy = 0
Vy = 0

M? = 0
?Mz = 0
Mz = 0
Step 4: Plot result and sanity check.

8

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1. 2.001 - MECHANICS AND MATERIALS I Lecture #5 9/20/2006 Prof. Carol Livermore TOPIC: FORCES AND MOMENTS TRANSMITTED BY SLENDER MEM- BERS? When will a structure fail? Recall: 1. What can the structural elements tolerate? 2. What forces and moments are experienced in the various parts of the structure? EXAMPLE: Airplane EXAMPLE: Truss SLENDER MEMBERS Long, skinny structural elements 1

2. L >> b, L >> h EXAMPLE: Skis, Golf Clubs, I-Beams Local axes follow the beam. AXIAL FORCES: Forces that act along the axis of the slender member. FBD Fij (x) = N (x), i: face, j: direction, x: location SHEAR FORCES: Forces that act in the plane of a face.

3. Summarize Internal Loads: Moments: Mxz (x2 ) > 0x: face, z: axis, x2 : location Mxz (x) = Mz (x) Bending Moment Sign Convention Summarize sign conventions 3

4. N >0 N <0 V >0 V <0 M >0 M <0 4

5. EXAMPLE: Calculating internal forces and moments Step 1: Find reactions at supports. Fx = 0 R Ax = 0 Fy = 0 R Ay ? P = 0 R Ay = P MA = 0 MA ? P a = 0 MA = P a 5

6. Step 2: If you have a multi-component structure, you will need to ?nd joint forces too. Step 3: Find internal forces and moments. 1. Set up local coordinates. 2. Identify interesting points. 3. Cut beam and do FBDs. 6

7. For 0 < x < a Fx = 0 N =0 Fy = 0 P ? Vy = 0 P = Vy MA = 0 P a + Mz ? P x = 0 Mz = P x ? P a = ?P (a ? x) OR (From the other end) FBD Fx = 0 ?N = 0 N =0 Fy = 0 ?P + Vy = 0 Vy = P M? = 0 ?Mz ? P (a ? x) = 0 7

8. Mz = ?P (a ? x) Note: This is the same as before. For a < x < L Fx = 0 ?N = 0 N =0 Fy = 0 Vy = 0 M? = 0 ?Mz = 0 Mz = 0 Step 4: Plot result and sanity check. 8

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