Vertical Stress Increases Chapter 8 Point Load 1 3/25/2015 Point - PDF document

3/25/2015 Vertical Stress Increases Chapter 8 Point Load 1

3/25/2015 Point Load Point Load � � � 3� �� � � � � � � � � � � 2� ⁄ 2

3/25/2015 Point Load 1 � � ⁄ � 3� ⁄ � � � � � � ⁄ � � � 1 � � 2� �� � Point Load I 1 ‐ 4 ‐ 3 ‐ 2 ‐ 1 0 1 2 3 4 r/z 3

3/25/2015 Example A vertical point load of 25 kN is applied at the ground surface. What is the stress increase in the ground 2 m below the ground surface directly beneath the load? ∆� � � � � � � � 25 �� 2 � � 0.4775 � 2.98 �� � � ⁄ r = 0, z = 2 Example A vertical point load of 25 kN is applied at the ground surface. What is the stress increase in the ground 2 m below the ground surface and 2 m to the side? ∆� � � � � � � � 25 �� 2 � � 0.0844 � 0.5275 �� � � ⁄ r = 2, z = 2 4

3/25/2015 Line Load NOTE: q has units of kN/m Line Load � 2� ⁄ � � � 1 � � � 1 ⁄ � � � 1 � � � 2 � ⁄ � � �� �� � � �� � ⁄ ∆� 2 � � � � ⁄ � � � 1 � �� �� ⁄ �� � 5

3/25/2015 Line Load 2 � ⁄ � � ⁄ � � � 1 � �� � I ‐ 4 ‐ 3 ‐ 2 ‐ 1 0 1 2 3 4 x/z Example A vertical line load of 8 kN/m is applied at the ground surface. What is the stress increase in the ground 4 m below the ground surface directly beneath the load? ∆� � � � � � 8 ��/� 0.637 � 1.274 �� � � ⁄ 4 � x = 0, z = 4 6

3/25/2015 Example A vertical line load of 8 kN/m is applied at the ground surface. What is the stress increase in the ground 4 m below the ground surface and 2 m off to the side? ∆� � � � � � 8 ��/� 0.407 � 0.814 �� � � ⁄ 4 � x = 2, z = 4 Circular Loaded Area � �� NOTE: q has units of kN/m 2 or kPa 7

3/25/2015 Circular Loaded Area � �� � � ∆� 1 � � 1 � ⁄ � � � 1 �/� �� � Circular Loaded Area  /q 0 0.2 0.4 0.6 0.8 1 1.2 0 0.5 1 1.5 2 z/R 2.5 3 3.5 4 4.5 5 8

3/25/2015 Example A uniform load of 80 kPa is applied at the ground surface over a circular area with a diameter of 8 m. What is the stress increase directly beneath the center of the loaded area at a depth of 8 m? ∆� � �� � 80 ��� 0.2845 � 22.76 ��� z = 8, R = 4 Circular Loaded Area 9

3/25/2015 Example A uniform load of 80 kPa is applied at the ground surface over a circular area with a diameter of 8 m. What is the stress increase 2 m below the ground surface at a radial distance of 4 m from the load centerline? ∆� � �� � � 80 ��� 0.417 � 33.36 ��� r = 4 = R z = 8, R = 4 Rectangular Loaded Area ∆� � �� � ����2�� �� � � �� � � �� � �� � 1 , 2� � � � � �� � � �� � 1� 10

3/25/2015 Example A uniform load of 80 kPa is applied at the ground surface over a rectangular area with a length of 5 m and a width of 2 m. What is the stress increase beneath the corner of the loaded area at a depth of 10 m? � � � � � � 2 10 � 0.2 � � � � � � 5 10 � 0.5 ∆� � �� � � 80 ��� 0.04 � 3.2 ��� 11

3/25/2015 Example A uniform load of 80 kPa is applied at the ground surface over a rectangular area with a length of 5 m and a width of 2 m. What is the stress increase beneath the corner of the loaded area at a depth of 10 m? � � � � � � 5 10 � 0.5 � � � � � � 2 10 � 0.2 ∆� � �� � � 80 ��� 0.04 � 3.2 ��� Rectangular Loaded Area B A' L 12

3/25/2015 Rectangular Loaded Area L/2 L/2 B/2 B A' B/2 L ∆� � �� � = L/B z/(B/2) = 13

3/25/2015 Example Example � � � � � � 4 4 � 1.0 � � � � � � 2 4 � 0.5 ∆� � �� � � 150 ��� 0.12 � 18 ��� 14

3/25/2015 Example � � � � � �1��0.5� � 0.5 �′ � � 0.5 � � 0.25 �′ � � 1 � � 1 �′ � �′ � � �1��0.25� � 0.25 � � � 1 2�1��0.5� 1 � 0.25 � 1 1 � 0.25 � 2 � tan �� 2�1��0.5� 1 � 0.25 � 1 � 0.120175 4� 1 � 0.25 � 0.25 � 1 1 � 0.25 � 1 1 � 0.25 � 0.25 � 1 Example � � � � � � 1 4 � 0.25 � � � � � � 2 4 � 0.5 ∆� � �� � � �150 ��� 0.048 � �7.2 ��� 15

3/25/2015 Example ∆� � 18.0 � 7.2 � 10.8 ��� 16