1. SOLID MECHANICS II (BDA 3033) CHAPTER 2:
DEFLECTION OF BEAMS
WEEK 2-3
FAZIMAH BT MAT NOOR (UTHM)
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2. FAZIMAH BT MAT NOOR (UTHM)
THE ELASTIC CURVE
The deflection diagram of the longitudinal axis that passes through the centroid of each cross sectional area of the beam is called the elastic curve.
Deflection diagram = Elastic curve
Pin resist force  restrict displacement
Fixed wall  resist moment  restrict displacement and rotation
FAZIMAH BT MAT NOOR (UTHM)
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3. THE ELASTIC CURVE (CONT)
A positive internal moment tends to bend the beam concave upward.
Positive moment – smiley face 
A negative moment tends to bend the beam concave downward.
Negative moment – Sad face 
If the moment diagram is known, it will be easy to construct the elastic curve.
FAZIMAH BT MAT NOOR (UTHM)
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4. THE ELASTIC CURVE (CONT)
Force on beam in fig. (a) & moment diagram in fig. (b).
displacement B & D = 0.
Negative moment AC = sad  curve
Positive moment CD = smiley  curve
C = inflection point, moment change sign (sad  smile) = zero moment
Change moment slope = zero deflection slope, the beam’s deflection may be a maximum
Whether A < E depends on the relative magnitudes of P1 and P2 and the location of the roller at B.
FAZIMAH BT MAT NOOR (UTHM)
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5. Moment-Curvature Relationship
given
FAZIMAH BT MAT NOOR (UTHM)
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6. SLOPE AND DISPLACEMENT BY INTEGRATION
FAZIMAH BT MAT NOOR (UTHM)
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7. SLOPE AND DISPLACEMENT BY INTEGRATION (cont.)
The elastic curve for beam can be expressed as y=f(x)
To obtained this equation, the curvature (1/) must be expressed in terms of y and x.
In most calculus book, this relationship is shown as:
FAZIMAH BT MAT NOOR (UTHM)
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8. SLOPE AND DISPLACEMENT BY INTEGRATION (cont.)
FAZIMAH BT MAT NOOR (UTHM)
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9. SLOPE AND DISPLACEMENT BY INTEGRATION (cont.)
w = distributed load
V = shear load
M = moment
dy/dx = slope = θ
FAZIMAH BT MAT NOOR (UTHM)
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10. SLOPE AND DISPLACEMENT BY INTEGRATION (cont.)
Positive sign convention -w +w
FAZIMAH BT MAT NOOR (UTHM)
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11. Boundary and Continuity Conditions
Free end no bend, no shear
Unless force applied at free end of the beam (V=applied force)
Pin joint wont bend
Fixed end no slope
FAZIMAH BT MAT NOOR (UTHM)
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12. FAZIMAH BT MAT NOOR (UTHM)
Distributed load
Constant distributed load apply at 1/2 length
Linear distributed load apply at 2/3 length
FAZIMAH BT MAT NOOR (UTHM)
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13. FAZIMAH BT MAT NOOR (UTHM)
EXAMPLES
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35. FAZIMAH BT MAT NOOR (UTHM)
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36. Discontinuity Functions: Macaulay
If several different loadings act on a beam
integration become more tedious
separate loading or moment functions must be written for each region of the beam
The beam shown requires four moment functions to be written.
They describe the moment in regions AB, BC, CD and DE.
When applying the moment-curvature relationship EI d2y/dx2=M, and integrating each moment twice, we must evaluate eight constants of integration.
FAZIMAH BT MAT NOOR (UTHM)
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37. Discontinuity Functions: Macaulay (cont.)
If there is discontinuity use
Discontinuity functions
x = start point
a = start of discontinuity
Why bracket < >
B’cos = function valid for x≥a
FAZIMAH BT MAT NOOR (UTHM)
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38. Discontinuity Functions: Macaulay (cont.)
NOT GIVEN
Sign convention
positive:
Forces upward
Moment clockwise
Load towards end..
FAZIMAH BT MAT NOOR (UTHM)
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39. Discontinuity Functions: Macaulay (cont.)
FAZIMAH BT MAT NOOR (UTHM)
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40. Discontinuity Functions: Macaulay (cont.)
FAZIMAH BT MAT NOOR (UTHM)
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41. FAZIMAH BT MAT NOOR (UTHM)
EXAMPLES:
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52. Statically Indeterminate Beams and Shafts
In all of the problems discussed so far, it was possible to determine the forces and stresses in beams by utilizing the equations of equilibrium, that is Fx=0, Fy=0 and MA=0.
When the equations of equilibrium are sufficient to determine the forces and stresses in a structural beam, we say that this beam is statically determinate.
When the equilibrium equations alone are not sufficient to determine the loads or stresses in a beam, then such beam is referred to as statically indeterminate beam. (the number of unknown reactions exceeds the available number of equilibrium equations.
The additional support reactions on the beam or shaft that are not needed to keep it in stable equilibrium are called redundants.
The number of these redundants is referred to as the degree of indeterminacy.
FAZIMAH BT MAT NOOR (UTHM)
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53. Statically Indeterminate Beams and Shafts (cont.)
Determinacy of Beams
- for a coplanar (2D) beam, there are at most three equilibrium equations for each part, so that if there is a total of n parts and r reactions, we have
FAZIMAH BT MAT NOOR (UTHM)
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54. Statically Indeterminate Beams and Shafts (cont.)
Example:
Classify each of the beams shown as statically determinate or statically indeterminate. If statically indeterminate, report the degrees of determinacy. The beams are subjected to external loadings that are assumed to be known and can act anywhere on the beams.
FAZIMAH BT MAT NOOR (UTHM)
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55. Statically Indeterminate Beams and Shafts (cont.)
 Statically indeterminate to second degree (5-3=2)
FAZIMAH BT MAT NOOR (UTHM)
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56. Statically Indeterminate Beams and Shafts (cont.)
r3 = r6
r4 = r5
r = 6, n = 2;
3n = 3(2) = 6
r = 3n
 Statically determinate
FAZIMAH BT MAT NOOR (UTHM)
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57. FAZIMAH BT MAT NOOR (UTHM)
EXAMPLE
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58. FAZIMAH BT MAT NOOR (UTHM)
SOLUTION
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59. FAZIMAH BT MAT NOOR (UTHM)
SOLUTION (cont)
FAZIMAH BT MAT NOOR (UTHM)
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60. FAZIMAH BT MAT NOOR (UTHM)
SOLUTION (cont)
FAZIMAH BT MAT NOOR (UTHM)
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61. FAZIMAH BT MAT NOOR (UTHM)
EXAMPLE
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62. FAZIMAH BT MAT NOOR (UTHM)
SOLUTION:
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63. FAZIMAH BT MAT NOOR (UTHM)
SOLUTION (cont.)
FAZIMAH BT MAT NOOR (UTHM)
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64. FAZIMAH BT MAT NOOR (UTHM)
SOLUTION (cont)
FAZIMAH BT MAT NOOR (UTHM)
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