Struct-notes-influence Lines-beams.pdf

Notes in Structural Analysis – Andres W.C. Oreta / De La Salle University

INFLUENCE LINES When one is designing a specific part of a structure, it is necessary to proportion the part under consideration so that it has sufficient strength to withstand the greatest stress to which it may be subjected during the life of the structure. In order to design such a part, the greatest contribution of the live load to the total design stress is one of the items that must be determined. The stress produced in a given part by the live load varies with the position of the load on the structure. There is always one position of the live loads on a structure that will cause the maximum live stresses in any particular part of a structure. It should be therefore be clear that it is essential for the structural engineer to understand clearly the methods by which the position of live load which causes the maximum stress at any point can be determined. Definition: An influence line is a curve the ordinate to which at any point equals the value of some particular function (e.g. reaction, bending moment, axial force, or shear) due to a unit load acting at that point. Influence lines can be used for two very important purposes: (1) To determine what position of live loads that will lead to a maximum value of the particular function for which the influence line has been constructed; and (2) To compute the value of that function due to the live loads. Theorems: (1) To obtain the maximum value of a function due to a single concentrated live load, the load should be placed at the point where the ordinate to the influence line for that function is a maximum. (2) The value of a function due to the action of a single concentrated live load equals the product of the magnitude of the load and the ordinate to the influence line for that function, measured at the point of application of the load. (3) To obtain the maximum value of the function due to a uniformly distributed live load, the load should be placed over all those portions of the structure for which the ordinates to the influence line for that function have the sign of the character of the function desired. (4) The value of a function due to a uniformly distributed live load is equal to the product of the intensity of the loading and the net area under that portion of the influence line, for the function under consideration, which corresponds to the portion of the structure loaded.

Notes in Structural Analysis – Andres W.C. Oreta / De La Salle University

INFLUENCE LINES FOR A SIMPLY SUPPORTED BEAM: To draw an influence line, simply apply a unit load from the left end to the right end of the beam and determine the corresponding value of the function (e.g. reaction, shear, moment) at that position of the unit load. I.

Draw the influence line for the reactions at A and B.

X

Unit Load = 1 N

C 5m

A

B 20 m

X

Unit Load = 1 N

C

∑M

B

= 0 : 20 R A −(1)(20 − x) = 0

20 − x x = 1− 20 20 ∑ Fy = 0 :R A + RB − 1 = 0 RA =

5m

RA

20 m

RB

1.0

Fig. 1. Influence Line for Reaction at A

1.0

Fig. 2. Influence Line for Reaction at B

RB =

x 20

Notes in Structural Analysis – Andres W.C. Oreta / De La Salle University

II. Draw the influence line for the bending moment and shear at C. Apply the unit load on the beam and then draw the FBD of either AC or CB. Shown below are FBDs of AC when the unit load is between A and C and between C and B. Unit Load between A and C 0≤X<5

Unit Load between C and B 5 ≤ X < 20

1

MC

MC

X C

A

C

A 5m

5m

VC

VC RA = 1-X/20

RA = 1-X/20

M C = RA (5) − 1(5 − X )

Note: The unit load will not appear in the FBD of AC.

M C = RA (5)

= (1 − X / 20)(5) − 5 + X = X − X /4 VC = RA − 1

= (1 − X / 20)(5) = 5− X /4 VC = RA = (1 − X / 20)

= (1 − X / 20) − 1

C A

5m

B 20 m 3.75

Fig. 3. Influence Line for Bending Moment at C (N-m)

0.75

0.25 Fig. 4. Influence Line for Shear at C (N)

Notes in Structural Analysis – Andres W.C. Oreta / De La Salle University

III. Apply theorems on influence lines for the moment and shear at C: (a) Compute the maximum positive moment at C due to a live load P = 20 kN (b) Compute the maximum positive moment at C due to uniform live load w = 10 kN/m (c) Compute the maximum positive shear at C due to a live load P = 20 kN (d) Compute the maximum positive shear at C due to uniform live load w = 10 kN/m

C A

5m

B 20 m 3.75

Influence Line for Bending Moment at C (N-m)

0.75

0.25 Influence Line for Shear at C (N)

Notes in Structural Analysis – Andres W.C. Oreta / De La Salle University

LBYCVT2 – Influence Lines in Beams

1. Draw the influence line for (a) vertical reaction at A, (b) bending moment at A, (c) the shear at B, and (d) the bending moment at B of the cantilever beam. A 4m

B

8m

(e) What is the maximum shear at B due to a concentrated live load P = 25 kN? Show with a sketch where the load will be applied to obtain the maximum value. (f) What is the maximum negative moment at B due to a uniform live load with intensity w = 15 kN/m? Show with a sketch where the load will be applied to obtain the maximum value.

2. Draw the influence line for (a) the vertical reaction at A, (b) the shear at C, (c) the moment at C, (d) the shear D of the simple beam with overhang.

6m 2m D

4m

A

6m C

B

(e) What is the maximum shear at C due to a concentrated live load P = 25 kN? Show with a sketch where the load will be applied to obtain the maximum value. (f) What is the maximum negative moment at C due to a uniform live load with intensity w = 15 kN/m? Show with a sketch where the load will be applied to obtain the maximum value.