Short Notes: Influence Line Diagram | Short Notes for Civil Engineering - Civil Engineering (CE) PDF Download

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INFLUENCE LINE DIAGRAM 
1. MULLER BRESLAU PRINCIPLE:  
As per this principle, “If an internal stress component or reaction component is considered to 
act through and tends to deflect a structure than the deflected shape of the structure will be 
the influence line for the stress or reaction component to some scale.”  
Note: This Principle gives for quantitative and qualitative deflected shape for determinate 
structure and qualitative deflected shape for indeterminate structures. 
2. MAXIMUM SHEAR FORCE AND BENDING MOMENT FOR A BEAM SUBJECTED TO 
MOVING LOADS 
2.1. Due to Single Point Load 
The influence line diagram for a single point load for Shear force and Bending moment is 
 
So, bending moment will be maximum if the load is at the section. For maximum 
negative shear force the load should be just to the left of section and for maximum 
positive shear force the load should be just right to the section. 
Absolute Maximum shear force and Bending Moment:  
Absolute maximum shear force:  
For absolute maximum negative shear force, x/L should be maximum. Thus, for absolute 
maximum negative shear force the value of x would be L i.e. at support B. For absolute 
maximum positive shear force, (1 -
?? ?? ) should be maximum. Thus, for absolute maximum 
positive shear force the value of x would be zero i.e. at support A. 
For absolute maximum bending moment: 
?? ?? =
???? (?? - ?? )
?? 
?? ?? ?? ????
=
?? (?? - 2?? )
?? = 0 
Thus, absolute maximum bending moment will occur at mid span in case of point load. 
 
 
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INFLUENCE LINE DIAGRAM 
1. MULLER BRESLAU PRINCIPLE:  
As per this principle, “If an internal stress component or reaction component is considered to 
act through and tends to deflect a structure than the deflected shape of the structure will be 
the influence line for the stress or reaction component to some scale.”  
Note: This Principle gives for quantitative and qualitative deflected shape for determinate 
structure and qualitative deflected shape for indeterminate structures. 
2. MAXIMUM SHEAR FORCE AND BENDING MOMENT FOR A BEAM SUBJECTED TO 
MOVING LOADS 
2.1. Due to Single Point Load 
The influence line diagram for a single point load for Shear force and Bending moment is 
 
So, bending moment will be maximum if the load is at the section. For maximum 
negative shear force the load should be just to the left of section and for maximum 
positive shear force the load should be just right to the section. 
Absolute Maximum shear force and Bending Moment:  
Absolute maximum shear force:  
For absolute maximum negative shear force, x/L should be maximum. Thus, for absolute 
maximum negative shear force the value of x would be L i.e. at support B. For absolute 
maximum positive shear force, (1 -
?? ?? ) should be maximum. Thus, for absolute maximum 
positive shear force the value of x would be zero i.e. at support A. 
For absolute maximum bending moment: 
?? ?? =
???? (?? - ?? )
?? 
?? ?? ?? ????
=
?? (?? - 2?? )
?? = 0 
Thus, absolute maximum bending moment will occur at mid span in case of point load. 
 
 
 
 
2.2. Due to Uniformly Distributed Load Longer than span 
Maximum negative shear force occurs when the load covers portion AC only and 
maximum positive shear force occurs when the load covers the portion CB only. 
Maximum bending moment at any section will be due to UDL covering the entire span. 
Absolute Maximum Value of SF and BM anywhere in the span: 
For Absolute maximum Shear Force: 
On observing the influence line diagram for Shear force, it is clear that maximum 
negative shear force will occurs at support B, when the UDL covers the entire span and 
maximum positive shear force will occurs at support A, when the UDL covers the entire 
span. 
For Absolute maximum bending moment: 
Maximum bending moment at any section, 
?? ?? =
1
2
× ?? ×
?? (?? - ?? )
?? × ?? 
?? ?? ?? ????
=
?? (?? - 2?? )
2
= 0 
? ?? =
?? 2
 
Thus, absolute maximum bending moment will occur at mid span in case of UDL larger 
than the span.   
2.3. UDL shorter than the span 
For UDL shorter than the span, maximum negative shear force will take place when 
entire UDL is just left of the section and maximum positive shear force will take place 
when entire UDL is just right to the section.  
For maximum bending moment at C 
 
Load should be placed such that 
?? ?? =
?? ?? - ?? 
2.4. Due to Train of Concentrated Loads 
Maximum Bending Moment at a section: Due to train of concentrated loads 
maximum bending moment will occur at the section if the loads are placed such that the 
average loading to the left of the section is equal to average loading to the right of the 
section.  
Maximum Bending Moment under a wheel load: Maximum bending moment under a 
wheel load occurs if the loads are placed such that the load and the resultant of the 
loading is equidistant from the centre of span.  
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INFLUENCE LINE DIAGRAM 
1. MULLER BRESLAU PRINCIPLE:  
As per this principle, “If an internal stress component or reaction component is considered to 
act through and tends to deflect a structure than the deflected shape of the structure will be 
the influence line for the stress or reaction component to some scale.”  
Note: This Principle gives for quantitative and qualitative deflected shape for determinate 
structure and qualitative deflected shape for indeterminate structures. 
2. MAXIMUM SHEAR FORCE AND BENDING MOMENT FOR A BEAM SUBJECTED TO 
MOVING LOADS 
2.1. Due to Single Point Load 
The influence line diagram for a single point load for Shear force and Bending moment is 
 
So, bending moment will be maximum if the load is at the section. For maximum 
negative shear force the load should be just to the left of section and for maximum 
positive shear force the load should be just right to the section. 
Absolute Maximum shear force and Bending Moment:  
Absolute maximum shear force:  
For absolute maximum negative shear force, x/L should be maximum. Thus, for absolute 
maximum negative shear force the value of x would be L i.e. at support B. For absolute 
maximum positive shear force, (1 -
?? ?? ) should be maximum. Thus, for absolute maximum 
positive shear force the value of x would be zero i.e. at support A. 
For absolute maximum bending moment: 
?? ?? =
???? (?? - ?? )
?? 
?? ?? ?? ????
=
?? (?? - 2?? )
?? = 0 
Thus, absolute maximum bending moment will occur at mid span in case of point load. 
 
 
 
 
2.2. Due to Uniformly Distributed Load Longer than span 
Maximum negative shear force occurs when the load covers portion AC only and 
maximum positive shear force occurs when the load covers the portion CB only. 
Maximum bending moment at any section will be due to UDL covering the entire span. 
Absolute Maximum Value of SF and BM anywhere in the span: 
For Absolute maximum Shear Force: 
On observing the influence line diagram for Shear force, it is clear that maximum 
negative shear force will occurs at support B, when the UDL covers the entire span and 
maximum positive shear force will occurs at support A, when the UDL covers the entire 
span. 
For Absolute maximum bending moment: 
Maximum bending moment at any section, 
?? ?? =
1
2
× ?? ×
?? (?? - ?? )
?? × ?? 
?? ?? ?? ????
=
?? (?? - 2?? )
2
= 0 
? ?? =
?? 2
 
Thus, absolute maximum bending moment will occur at mid span in case of UDL larger 
than the span.   
2.3. UDL shorter than the span 
For UDL shorter than the span, maximum negative shear force will take place when 
entire UDL is just left of the section and maximum positive shear force will take place 
when entire UDL is just right to the section.  
For maximum bending moment at C 
 
Load should be placed such that 
?? ?? =
?? ?? - ?? 
2.4. Due to Train of Concentrated Loads 
Maximum Bending Moment at a section: Due to train of concentrated loads 
maximum bending moment will occur at the section if the loads are placed such that the 
average loading to the left of the section is equal to average loading to the right of the 
section.  
Maximum Bending Moment under a wheel load: Maximum bending moment under a 
wheel load occurs if the loads are placed such that the load and the resultant of the 
loading is equidistant from the centre of span.  
 
 
Maximum bending moment will occur under W3 if the loads are placed as shown below.