Exercises

Consider the typical floor of a multistorey building, with the centre of mass located in position X=9.0, Y=9.0 and the seismic force applied at the ground floor is equal to 2000 kN. Calculate the displacements and stress resultants of columns C₁, C₃, C₄, C₆, C₁₁, C₁₃, C₁₆ under seismic action in direction X. The ground floor height is 3.0 m. Concrete class C30/37.

 

	δx	δy	Vx	Vy	Mx,1	Mx,2	My,1	My,2
	(mm)	(mm)	(kN)	(kN)	(kNm)	(kNm)	(kNm)	(kNm)
C1	3,67	0,97	47,5	12,6	71,3	-71,3	18,8	-18,8
C3	3,67	-0,32	116,0	-10,2	174,0	-174,0	-15,3	15,3
C4	3,67	-0,97	47,5	-12,6	71,3	-71,3	-18,8	18,8
C6	4,31	0,32	136,4	10,2	204,6	-204,6	15,3	-15,3
C11	4,96	-0,32	156,9	-10,2	235,3	-235,3	-15,3	15,3
C13	5,60	0,97	72,6	12,6	108,9	-108,9	18,8	-18,8
C16	5,60	-0,97	72,6	-12,6	108,9	-108,9	-18,8	18,8

The sum of shear forces in each direction shall be equal to the corresponding seismic force, regardless of the stiffness distribution or the deviation of the C_M from the C_T, i.e. in this particular case Σ(V_x)=2000 kN and Σ(V_y)=0.

In the structure of the previous exercise, assuming that the number of storeys is 10 and the interstorey height is 3.0 m, calculate the displacement the centre of stiffness C_T of each floor and the maximum displacement of the higher floor of the building. Consider two cases: (a) orthogonal and (b) triangular distribution of seismic forces.

Base seismic shear = 2000 kN, orthogonal distribution of seismic forces

Conclusions:

• ·    The maximum absolute displacement δ₁₀=55a occurs at the 10^th level, and the minimum δ₁=10a at the 1^st level. On the contrary, the maximum relative displacement δ_z1=10a occurs at the 1^st level, and the minimum δ_z10=a at the 10^th level.
• ·    The maximum stress is developed at the 1^st level due to the maximum relative displacement, while the minimum at the 10^th level.
• ·    Due to non-coincident center of mass C_Mand center of stiffness C_T, the diaphragm moments are proportional to the shear forces and the corresponding displacements are proportional to the C_T displacements.  Therefore the maximum displacement occurs at the 10^thlevel in column c13 and is 55/10=5.5 times greater than the one of the ground floor, i.e. δ_xx,10,13=5.5·5.60=30.8 mm and δ_xy,10,13=5.5·0.97=5.3 mm.

Base seismic shear = 2000 kN, triangular distribution of seismic forces

Conclusions:

• ·     At the 1^st level, all types of seismic force distribution, including the triangular and the orthogonal, give the same displacement and stress.
• ·     Given the same seismic shear, the displacement in the last floor δ₁₀=70a according to the triangular distribution is greater than the respective value δ₁₀=55a of the orthogonal one.
• ·     The area of the shear forces diagram and heights represents the total moment of the floors and is larger in the triangular distribution than in the rectangular one.
• ·     The maximum displacement, developed at the 10^th level in column c13, is 70/10=7.0 times greater than the one of the ground floor, i.e. δ_xx,10,13=7.0×5.60=39.2 mm and δ_xy,10,13=7.0×0.97=6.8 mm

The actual structure of the building with the wireframe model

Seismic action with base shear 2000 kN and orthogonal distribution of seismic actions

In the floor of exercise 5.4.7-1 above, calculate the corresponding results if the columns C3, C5, C12, C14 are replaced by walls with cross-section 2000/300 parallel to each perimeter side.

	δx	δy	Vx	Vy	Mx,1	Mx,2	My,1	My,2
	(mm)	(mm)	(kN)	(kN)	(kNm)	(kNm)	(kNm)	(kNm)
C1	1,76	0,26	22,4	3,3	33,5	-33,5	4,9	-4,9
C3	1,76	-0,09	564,5	-2,3	846,8	-846,8	-3,5	3,5
C4	1,76	-0,26	22,4	-3,3	33,5	-33,5	-4,9	4,9
C6	1,93	0,09	59,3	2,6	88,9	-88,9	3,9	-3,9
C11	2,10	-0,09	64,5	-2,6	96,8	-96,8	-3,9	3,9
C13	2,27	0,26	28,9	3,3	43,3	-43,3	4,9	-4,9
C16	2,27	-0,26	28,9	-3,3	43,3	-43,3	-4,9	4,9

 In the presence of walls, the shear effect should be taken into account, because they affect significantly both the displacements magnitude and the stress resultants distribution.

 The magnitude of the maximum displacement is equal to 2.27 mm compared to 5.60 mm of the frame type structure of exercise 5.4.7-1 thus requiring smaller seismic joints, when necessary. The resulting wall displacements are smaller leading to less important or nil damages in the event of an earthquake. Moreover smaller stresses are developed in the columns and less reinforcement is required.

In the storey of exercise 5.4.7-1 calculate the results for the following cases:

(a) replacement of columns C₅, C₁₂, C₁₄ with walls of cross-section 2000/300 parallel to the respective sides of the perimeter.

 

	δx	δy	Vx	Vy	Mx,1	Mx,2	My,1	My,2
	(mm)	(mm)	(kN)	(kN)	(kNm)	(kNm)	(kNm)	(kNm)
C1	3,365	-0,370	42,8	-4,7	64,3	-64,3	-7,1	7,1
C3	3,365	0,123	103,6	3,8	155,4	-155,4	5,7	-5,7
C4	3,365	0,369	42,8	4,7	64,3	-64,3	7,1	-7,1
C6	3,118	-0,124	96,0	-3,8	144,0	-144,0	-5,7	5,7
C11	2,872	0,123	88,4	3,8	132,6	-132,6	5,7	-5,7
C13	2,625	-0,370	33,4	-4,7	50,1	-50,1	-7,1	7,1
C16	2,625	0,369	33,4	4,7	50,1	-50,1	7,1	-7,1

(b) replacement of the columns C₁₂, C₁₄,  with walls of cross-section 2000/300 parallel to the respective sides of the perimeter.

 

	δx	δy	Vx	Vy	Mx,1	Mx,2	My,1	My,2
	(mm)	(mm)	(kN)	(kN)	(kNm)	(kNm)	(kNm)	(kNm)
C1	3,620	-0,795	46,1	-10,1	69,1	-69,1	-15,2	15,2
C3	3,620	-0,045	111,4	-1,4	167,1	-167,1	-2,1	2,1
C4	3,620	0,330	46,1	4,2	69,1	-69,1	6,3	-6,3
C6	3,245	-0,420	99,9	-12,9	149,8	-149,8	-19,4	19,4
C11	2,870	-0,045	88,3	-1,4	132,5	-132,5	-2,1	2,1
C13	2,495	-0,795	31,8	-10,1	47,7	-47,7	-15,2	15,2
C16	2,495	0,330	31,8	4,2	47,7	-47,7	6,3	-6,3

(c) replacement of the columns C₅, C₁₂, with walls of cross-section 2000/300 parallel to the corresponding sides of the perimeter.

 

	δx	δy	Vx	Vy	Mx,1	Mx,2	My,1	My,2
	(mm)	(mm)	(kN)	(kN)	(kNm)	(kNm)	(kNm)	(kNm)
C1	4,438	0,407	56,5	5,2	84,7	-84,7	7,8	-7,8
C3	4,438	-0,136	136,6	-4,2	204,9	-204,9	-6,3	6,3
C4	4,438	-0,407	56,5	-5,2	84,7	-84,7	-7,8	7,8
C6	4,709	0,136	144,9	4,2	217,4	-217,4	6,3	-6,3
C11	4,980	-0,136	153,3	-4,2	229,9	-229,9	-6,3	6,3
C13	5,251	0,407	66,9	5,2	100,3	-100,3	7,8	-7,8
C16	5,251	-0,407	66,9	-5,2	100,3	-100,3	-7,8	7,8

(d) replacement of the column C5 with wall of cross-section 2000/300 in parallel to the respec-tive side of the perimeter.

	δx	δy	Vx	Vy	Mx,1	Mx,2	My,1	My,2
	(mm)	(mm)	(kN)	(kN)	(kNm)	(kNm)	(kNm)	(kNm)
C1	4,113	0,406	52,4	5,2	78,5	-78,5	7,7	-7,7
C3	4,113	-0,509	126,6	-15,7	189,9	-189,9	-23,5	23,5
C4	4,113	-0,967	52,4	-12,3	78,5	-78,5	-18,5	18,5
C6	4,570	-0,052	140,7	-1,6	211,0	-211,0	-2,4	2,4
C11	5,028	-0,509	154,8	-15,7	232,1	-232,1	-23,5	23,5
C13	5,485	0,406	69,8	5,2	104,8	-104,8	7,7	-7,7
C16	5,485	-0,967	69,8	-12,3	104,8	-104,8	-18,5	18,5